Topics in operator theory. Systems and mathematical physics

Operator Theory: Advances and Applications Vol. 203 Founded in 1979 by Israel Gohberg Editors: Harry Dym (Rehovot, Isr...

Author: Ball J. | et al. (eds.)

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Operator Theory: Advances and Applications Vol. 203 Founded in 1979 by Israel Gohberg

Editors: Harry Dym (Rehovot, Israel) Joseph A. Ball (Blacksburg, VA, USA) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland)

Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Topics in Operator Theory Volume 2: Systems and Mathematical Physics Proceedings of the XIXth International Workshop on Operator Theory and its Applications, College of William and Mary, 2008

A tribute to Israel Gohberg on the occasion of his 80 th birthday

Joseph A. Ball Vladimir Bolotnikov J. William Helton Leiba Rodman Ilya M. Spitkovsky Editors Birkhäuser

Editors: Joseph A. Ball Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA e-mail: [email protected] Vladimir Bolotnikov Department of Mathematics College of William and Mary P. O. Box 8795 Williamsburg, VA 23187-8795 USA e-mail: [email protected]

Leiba Rodman Department of Mathematics College of William and Mary P. O. Box 8795 Williamsburg, VA 23187-8795 USA e-mail: [email protected] Ilya M. Spitkovsky Department of Mathematics College of William & Mary Williamsburg, VA 23187-8795 USA e-mail: [email protected]

J. William Helton Department of Mathematics University of California San Diego 9500 Gilman Drive La Jolla, CA 92093-0112 e-mail: [email protected]

2010 Mathematics Subject Classiﬁcation: 15, 45, 46, 47, 93 Library of Congress Control Number: 2010920057

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-0160-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Birkhäuser, Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany

ISBN 978-3-0346-0157-3 Vol. 1 ISBN 978-3-0346-0160-3 Vol. 2 ISBN 978-3-0346-0163-4 Set

e-ISBN 978-3-0346-0158-0 e-ISBN 978-3-0346-0161-0

987654321

www.birkhauser.ch

Contents J.A. Ball, V. Bolotnikov, J.W. Helton, L. Rodman and I.M. Spitkovsky The XIXth International Workshop on Operator Theory and its Applications. II . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

T. Aktosun, T. Busse, F. Demontis and C. van der Mee Exact Solutions to the Nonlinear Schr¨ odinger Equation . . . . . . . . . . . . . .

1

J.A. Ball and S. ter Horst Robust Control, Multidimensional Systems and Multivariable Nevanlinna-Pick Interpolation . . . . . . . . . . . . . . . . . . . . . . . . .

13

P. Binding and I.M. Karabash Absence of Existence and Uniqueness for Forward-backward Parabolic Equations on a Half-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

P.A. Binding and H. Volkmer Bounds for Eigenvalues of the p-Laplacian with Weight Function of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

A. Boumenir The Gelfand-Levitan Theory for Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili On the Uniqueness of a Solution to Anisotropic Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 C. Bu¸se and A. Zada Dichotomy and Boundedness of Solutions for some Discrete Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 B. Cichy, K. Galkowski and E. Rogers Control Laws for Discrete Linear Repetitive Processes with Smoothed Previous Pass Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

P. Djakov and B. Mityagin Fourier Method for One-dimensional Schr¨ odinger Operators with Singular Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

vi

Contents

S. Friedland Additive Invariants on Quantum Channels and Regularized Minimum Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 I.M. Karabash A Functional Model, Eigenvalues, and Finite Singular Critical Points for Indeﬁnite Sturm-Liouville Operators . . . . . . . . . . . . . . . . . . . . . .

247

M. Klaus On the Eigenvalues of the Lax Operator for the Matrix-valued AKNS System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 S.A.M. Marcantognini and M.D. Mor´ an An Extension Theorem for Bounded Forms Deﬁned in Relaxed Discrete Algebraic Scattering Systems and the Relaxed Commutant Lifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 M. Martin Deconstructing Dirac Operators. III: Dirac and Semi-Dirac Pairs . . . . . 347 I. Mitrea Mapping Properties of Layer Potentials Associated with Higher-order Elliptic Operators in Lipschitz Domains . . . . . . . . . . . . . . . . 363 G.H. Rawitscher Applications of a Numerical Spectral Expansion Method to Problems in Physics; a Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A. Rybkin Regularized Perturbation Determinants and KdV Conservation Laws for Irregular Initial Proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

427

The XIXth International Workshop on Operator Theory and its Applications. II Joseph A. Ball, Vladimir Bolotnikov, J. William Helton, Leiba Rodman and Ilya M. Spitkovsky Abstract. Information about the workshop and comments about the second volume of proceedings is provided. Mathematics Subject Classiﬁcation (2000). 35-06, 37-06, 45-06, 93-06, 47-06. Keywords. Operator theory, diﬀerential and diﬀerence equations, system theory, mathematical physics.

The Nineteenth International Workshop on Operator Theory and its Applications – IWOTA 2008 – took place in Williamsburg, Virginia, on the campus of the College of William and Mary, from July 22 till July 26, 2008. It was held in conjunction with the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS) in Blacksburg, Virginia (Virginia Tech, July 28–August 1, 2008) and the 9th Workshop on Numerical Ranges and Numerical Radii (July 19–July 21, 2008) at the College of William and Mary. The organizing committee of IWOTA 2008 (Ball, Bolotnikov, Helton, Rodman, Spitkovsky) served also as editors of the proceedings. IWOTA 2008 celebrated the work and career of Israel Gohberg on the occasion of his 80th birthday, which actually fell on August 23, 2008. We are pleased to present this volume as a tribute to Israel Gohberg. IWOTA 2008 was a comprehensive, inclusive conference covering many aspects of theoretical and applied operator theory. More information about the workshop can be found on its web site http://www.math.wm.edu/~vladi/IWOTA/IWOTA2008.htm There were 241 participants at IWOTA 2008, representing 30 countries, including 29 students (almost exclusively graduate students), and 20 young researchers (those who received their doctoral degrees in the year 2003 or later). The scientiﬁc program included 17 plenary speakers and 7 invited speakers who gave overview of many topics related to operator theory. The special sessions covered

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J.A. Ball et al.

Israel Gohberg at IWOTA 2008, Williamsburg, Virginia

a broad range of topics: Matrix and operator inequalities; hypercomplex operator theory; the Kadison–Singer extension problem; interpolation problems; matrix completions; moment problems; factorizations; Wiener–Hopf and Fredholm operators; structured matrices; Bezoutians, resultants, inertia theorems and spectrum localization; applications of indeﬁnite inner product spaces; linear operators and linear systems; multivariable operator theory; composition operators; matrix polynomials; indeﬁnite linear algebra; direct and inverse scattering transforms for integrable systems; theory, computations, and applications of spectra of operators. We gratefully acknowledge support of IWOTA 2008 by the National Science Foundation Grant 0757364, as well as by the individual grants of some organizers, and by various entities within the College of William and Mary: Department of Mathematics, the Oﬃce of the Dean of the Faculty of Arts and Sciences, the Oﬃce of the Vice Provost for Research, and the Reves Center for International Studies.

IWOTA 2008 II

ix

One plenary speaker has been sponsored by the International Linear Algebra Society. The organization and running of IWOTA 2008 was helped tremendously by the Conference Services of the College of William and Mary. The present volume is the second of two volumes of proceedings of IWOTA 2008. Here, papers on systems, diﬀerential and diﬀerence equations, and mathematical physics are collected. All papers are refereed. The ﬁrst volume contains papers on operator theory, linear algebra, and analytic functions, as well as a commemorative article dedicated to Israel Gohberg. August 2009 Added on December 14, 2009: With deep sadness the editors’ ﬁnal act in preparing this volume is to record that Israel Gohberg passed away on October 12, 2009, aged 81. Gohberg was a great research mathematician, educator, and expositor. His visionary ideas inspired many, including the editors and quite a few contributors to the present volume. Israel Gohberg was the driving force of iwota. He was the ﬁrst and the only President of the Steering Committee. In iwota, just as in his other endeavors, Gohberg’s charisma, warmth, judgement and stature lead to the lively community we have today. He will be dearly missed. The Editors:

Joseph A. Ball, Vladimir Bolotnikov, J. William Helton, Leiba Rodman, Ilya M. Spitkovsky.

Joseph A. Ball Department of Mathematics Virginia Tech Blacksburg, VA 24061, USA e-mail: [email protected] Vladimir Bolotnikov, Leiba Rodman and Ilya M. Spitkovsky Department of Mathematics College of William and Mary Williamsburg, VA 23187-8795, USA e-mail: [email protected] [email protected] [email protected] J. William Helton Department of Mathematics University of California San Diego La Jolla, CA 92093-0112, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 203, 1–12 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Exact Solutions to the Nonlinear Schr¨ odinger Equation Tuncay Aktosun, Theresa Busse, Francesco Demontis and Cornelis van der Mee Dedicated to Israel Gohberg on the occasion of his eightieth birthday

Abstract. A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schr¨ odinger equation on the line with a cubic nonlinearity. With motivation by the inverse scattering transform and help from the state-space method, an explicit formula is obtained to express such exact solutions in a compact form in terms of a matrix triplet and by using matrix exponentials. Such solutions consist of multisolitons with any multiplicities, are analytic on the entire xt-plane, decay exponentially as x → ±∞ at each ﬁxed t, and can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates x and t. Various equivalent forms of the matrix triplet are presented yielding the same exact solution. Mathematics Subject Classiﬁcation (2000). Primary: 37K15; Secondary: 35Q51, 35Q55. Keywords. Nonlinear Schr¨ odinger equation, exact solutions, explicit solutions, focusing NLS equation, NLS equation with cubic nonlinearity, inverse scattering transform.

1. Introduction Our goal in this paper is to review and further elaborate on a recent method [3, 4] to construct certain exact solutions to the focusing nonlinear Schr¨ odinger (NLS) equation (1.1) iut + uxx + 2|u|2 u = 0, Communicated by J.A. Ball.

2

T. Aktosun, T. Busse, F. Demontis and C. van der Mee

with a cubic nonlinearity, where the subscripts denote the corresponding partial derivatives. The NLS equation has important applications in various areas such as wave propagation in nonlinear media [15], surface waves on deep waters [14], and signal propagation in optical ﬁbers [9–11]. It was the second nonlinear partial diﬀerential equation (PDE) whose initial value problem was discovered [15] to be solvable via the inverse scattering transform (IST) method. Recall that the IST method associates (1.1) with the Zakharov-Shabat system −iλ dϕ(λ, x, t) = dx −u(x, t)∗

u(x, t) iλ

ϕ(λ, x, t),

(1.2)

where u(x, t) appears as a potential and an asterisk is used for complex conjugation. By exploiting the one-to-one correspondence between the potential u(x, t) and the corresponding scattering data for (1.2), that method amounts to determining the time evolution u(x, 0) → u(x, t) in (1.1) with the help of solutions to the direct and inverse scattering problems for (1.2). We note that the direct scattering problem for (1.2) consists of determining the scattering coeﬃcients (related to the asymptotics of scattering solutions to (1.2) as x → ±∞) when u(x, t) is known for all x. On the other hand, the inverse scattering problem for (1.2) is to construct u(x, t) when the scattering data is known for all λ. Even though we are motivated by the IST method, our goal is not to solve the initial value problem for (1.1). Our aim is rather to construct certain exact solutions to (1.1) with the help of a matrix triplet and by using matrix exponentials. Such exact solutions turn out to be multisolitons with any multiplicities. Dealing with even a single soliton with multiplicities has not been an easy task in other methods; for example, the exact solution example presented in [15] for a onesoliton solution with a double pole, which is obtained by coalescing two distinct poles into one, contains a typographical error, as pointed out in [13]. In constructing our solutions we make use of the state-space method [6] from control theory. Our solutions are uniquely constructed via the explicit formula (2.6), which uses as input three (complex) constant matrices A, B, C, where A has size p × p, B has size p × 1, and C has size 1 × p, with p as any positive integer. We will refer to (A, B, C) as a triplet of size p. There is no loss of generality in using a triplet yielding a minimal representation [3, 4, 6], and we will only consider such triplets. As seen from the explicit formula (2.6), our solutions are well deﬁned as long as the matrix F (x, t) deﬁned in (2.5) is invertible. It turns out that F (x, t) is invertible if and only if two conditions are met on the eigenvalues of the constant matrix A; namely, none of the eigenvalues of A are purely imaginary and that no two eigenvalues of A are symmetrically located with respect to the imaginary axis. Our solutions given by (2.6) are globally analytic on the entire xt-plane and decay exponentially as x → ±∞ for each ﬁxed t ∈ R as long as those two conditions on the eigenvalues of A are satisﬁed.

Exact Solutions to the NLS Equation

3

In our method [3, 4] we are motivated by using the IST with rational scattering data. For this purpose we exploit the state-space method [6]; namely, we use a matrix triplet (A, B, C) of an appropriate size in order to represent a rational function vanishing at inﬁnity in the complex plane. Recall that any rational function R(λ) in the complex plane that vanishes at inﬁnity has a matrix realization in terms of a matrix triplet (A, B, C) as R(λ) = −iC(λI − iA)B,

(1.3)

where I denotes the identity matrix. The smallest integer p in the size of the triplet yields a minimal realization for R(λ) in (1.3). A minimal realization is unique up to a similarity transformation. The poles of R(λ) coincide with the eigenvalues of (iA). The use of a matrix realization in the IST method allows us to establish the separability of the kernel of a related Marchenko integral equation [1, 2, 4, 12] by expressing that kernel in terms of a matrix exponential. We then solve that Marchenko integral equation algebraically and observe that our procedure leads to exact solutions to the NLS equation even when the input to the Marchenko equation does not necessarily come from any scattering data. We refer the reader to [3, 4] for details. The explicit formula (2.6) provides a compact and concise way to express our exact solutions. If such solutions are desired to be expressed in terms of exponential, trigonometric (sine and cosine), and polynomial functions of x and t, this can also be done explicitly and easily by “unpacking” matrix exponentials in (2.6). If the size p in the matrices A, B, C is larger than 3, such expressions become long; however, we can still explicitly evaluate them for any matrix size p either by hand or by using a symbolic software package such as Mathematica. The power of our method is that we can produce exact solutions via (2.6) for any positive integer p. In some other available methods, exact solutions are usually tried to be produced directly in terms of elementary functions without using matrix exponentials, and hence any concrete examples that can be produced by such other methods will be relatively simple and we cannot expect those other methods to produce our exact solutions when p is large. Our method is generalizable to obtain similar explicit formulas for exact solutions to other integrable nonlinear PDEs where the IST involves the use of a Marchenko integral equation [1, 2, 4, 12]. For example, a similar method has been used [5] for the half-line Korteweg-de Vries equation, and it can be applied to other equations such as the modiﬁed Korteweg-de Vries equation and the sineGordon equation. Our method is also generalizable to the matrix versions of such integrable nonlinear PDEs. For instance, a similar method has been applied in the third author’s Ph.D. thesis [8] to the matrix NLS equation in the focusing case with a cubic nonlinearity. Our method also easily handles nonsimple bound-state poles and the time evolution of the corresponding bound-state norming constants. In the literature,

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T. Aktosun, T. Busse, F. Demontis and C. van der Mee

nonsimple bound-state poles are usually avoided due to mathematical complications. We refer the reader to [13], where nonsimple bound-state poles were investigated and complications were encountered. A systematic treatment of nonsimple bound states has recently been given in the second author’s Ph.D. thesis [7]. The organization of our paper is as follows. Our main results are summarized in Section 2 and some explicit examples are provided in Section 3. For the proofs, further results, details, and a summary of other methods to solve the NLS equation exactly, we refer the reader to [3, 4].

2. Main results In this section we summarize our method to construct certain exact solutions to the NLS equation in terms of a given triplet (A, B, C) of size p. For the details of our method we refer the reader to [3, 4]. Without any loss of generality, we assume that our starting triplet (A, B, C) corresponds to a minimal realization in (1.3). Let us use a dagger to denote the matrix adjoint (complex conjugate and matrix transpose), and let the set {aj }m j=1 consist of the distinct eigenvalues of A, where the algebraic multiplicity of each eigenvalue may be greater than one and we use nj to denote that multiplicity. We only impose the restrictions that no aj is purely imaginary and that no two distinct aj values are located symmetrically with respect to the imaginary axis on the complex plane. Let us set λj := iaj so that we can equivalently state our restrictions as that no λj will be real and no two distinct λj values will be complex conjugates of each other. Our method uses the following steps: (i) First construct the constant p × p matrices Q and N that are the unique solutions, respectively, to the Lyapunov equations Q A + A† Q = C † C,

(2.1)

A N + N A† = BB † .

(2.2)

In fact, Q and N can be written explicitly in terms of the triplet (A, B, C) as 1 dλ (λI + iA† )−1 C † C(λI − iA)−1 , (2.3) Q= 2π γ 1 dλ (λI − iA)−1 BB † (λI + iA† )−1 , (2.4) N= 2π γ where γ is any positively oriented simple closed contour enclosing all λj in such a way that all λ∗j lie outside γ. The existence and uniqueness of the solutions to (2.1) and (2.2) are assured by the fact that λj = λ∗j for all j = 1, 2, . . . , m and λj = λ∗k for k = j. (ii) Construct the p × p matrix-valued function F (x, t) as F (x, t) := e2A

†

x−4i(A† )2 t

2

+ Q e−2Ax−4iA t N.

(2.5)

Exact Solutions to the NLS Equation

5

(iii) Construct the scalar function u(x, t) via u(x, t) := −2B† F (x, t)−1 C † .

(2.6)

Note that u(x, t) is uniquely constructed from the triplet (A, B, C). As seen from (2.6), the quantity u(x, t) exists at any point on the xt-plane as long as the matrix F (x, t) is invertible. It turns out that F (x, t) is invertible on the entire xt-plane as long as λj = λ∗j for all j = 1, 2, . . . , m and λj = λ∗k for k = j. Let us note that the matrices Q and N given in (2.3) and (2.4) are known in control theory as the observability Gramian and the controllability Gramian, respectively, and that it is well known in control theory that (2.3) and (2.4) satisfy (2.1) and (2.2), respectively. In the context of system theory, the invertibility of Q and N is described as the observability and the controllability, respectively. In our case, both Q and N are invertible due to the appropriate restrictions imposed on the triplet (A, B, C), which we will see in Theorem 1 below. Our main results are summarized in the following theorems. For the proofs we refer the reader to [3, 4]. Although the results presented in Theorem 1 follow from the results in the subsequent theorems, we state Theorem 1 independently to clearly illustrate the validity of our exact solutions to the NLS equation. Theorem 1. Consider any triplet (A, B, C) of size p corresponding to a minimal representation in (1.3), and assume that none of the eigenvalues of A are purely imaginary and that no two eigenvalues of A are symmetrically located with respect to the imaginary axis. Then: (i) The Lyapunov equations (2.1) and (2.2) are uniquely solvable, and their solutions are given by (2.3) and (2.4), respectively. (ii) The constant matrices Q and N given in (2.3) and (2.4), respectively, are selfadjoint; i.e., Q† = Q and N † = N. Furthermore, both Q and N are invertible. (iii) The matrix F (x, t) deﬁned in (2.5) is invertible on the entire xt-plane, and the function u(x, t) deﬁned in (2.6) is a solution to the NLS equation everywhere on the xt-plane. Moreover, u(x, t) is analytic on the entire xt-plane and it decays exponentially as x → ±∞ at each ﬁxed t ∈ R. ˜ B, ˜ C) ˜ are equivalent if they We will say that two triplets (A, B, C) and (A, yield the same potential u(x, t) through (2.6). The following result shows that, as far as constructing solutions via (2.6) is concerned, there is no loss of generality is choosing our starting triplet (A, B, C) of size p so that it corresponds to a minimal representation in (1.3) and that all eigenvalues aj of the matrix A have positive real parts. ˜ B, ˜ C) ˜ of size p corresponding to a minimal Theorem 2. Consider any triplet (A, representation in (1.3), and assume that none of the eigenvalues of A˜ are purely imaginary and that no two eigenvalues of A˜ are symmetrically located with respect to the imaginary axis. Then, there exists an equivalent triplet (A, B, C) of

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T. Aktosun, T. Busse, F. Demontis and C. van der Mee

size p corresponding to a minimal representation in (1.3) in such a way that all eigenvalues of A have positive real parts. The next two results given in Theorems 3 and 4 show some of the advantages of using a triplet (A, B, C) where all eigenvalues of A have positive real parts. Concerning Theorem 2, we remark that the triplet (A, B, C) can be obtained from ˜ B, ˜ C) ˜ and vice versa with the help of Theorem 5 or Theorem 6 given below. (A, Theorem 3. Consider any triplet (A, B, C) of size p corresponding to a minimal representation in (1.3). Assume that all eigenvalues of A have positive real parts. Then: (i) The solutions Q and N to (2.1) and (2.2), respectively, can be expressed in terms of the triplet (A, B, C) as ∞ ∞ −As † −As ds [Ce ] [Ce ], N= ds [e−As B][e−As B]† . (2.7) Q= 0

0

(ii) Q and N are invertible, selfadjoint matrices. (iii) Any square submatrix of Q containing the (1, 1)-entry or (p, p)-entry of Q is invertible. Similarly, any square submatrix of N containing the (1, 1)-entry or (p, p)-entry of N is invertible. ˜ B, ˜ C) ˜ of size p corresponding to a minimal Theorem 4. Consider a triplet (A, representation in (1.3) and that all eigenvalues aj of the matrix A˜ have positive real parts and that the multiplicity of aj is nj for j = 1, 2, . . . , m. Then, there exists an equivalent triplet (A, B, C) of size p corresponding to a minimal representation in (1.3) in such a way that A is in a Jordan canonical form with each Jordan block containing a distinct eigenvalue aj and having −1 in the superdiagonal entries, and the entries of B consist of zeros and ones. More speciﬁcally, we have ⎡ ⎤ ⎤ ⎡ B1 0 A1 0 . . . ⎢ B2 ⎥ ⎢ 0 A2 . . . 0 ⎥ ⎢ ⎥ ⎥ ⎢ , B = ⎢ . ⎥ , C = C1 C2 . . . Cm , (2.8) A=⎢ . ⎥ . . . . . . . . ⎣ . ⎦ ⎣ . . . . ⎦ Bm 0 0 . . . Am ⎡

aj ⎢0 ⎢ ⎢ Aj := ⎢ 0 ⎢ .. ⎣. 0

−1 0 . . . aj −1 . . . 0 aj . . . .. .. .. . . . 0 0 ...

⎡ ⎤ ⎤ 0 0 ⎢0⎥ 0⎥ ⎢ ⎥ ⎥ ⎢.⎥ 0⎥ ⎥ , Bj := ⎢ .. ⎥ , Cj := cj(nj −1) ⎢ ⎥ .. ⎥ ⎣0⎦ .⎦ 1 aj

...

cj1

cj0 ,

where Aj has size nj × nj , Bj has size nj × 1, Cj has size 1 × nj , and the (complex) constant cj(nj −1) is nonzero. We will refer to the speciﬁc form of the triplet (A, B, C) given in (2.8) as a standard form.

Exact Solutions to the NLS Equation

7

The transformation between two equivalent triplets can be obtained with the help of the following two theorems. First, in Theorem 5 below we consider the transformation where all eigenvalues of A are reﬂected with respect to the imaginary axis. Then, in Theorem 6 we consider transformations where only some of the eigenvalues of A are reﬂected with respect to the imaginary axis. Theorem 5. Assume that the triplet (A, B, C) of size p corresponds to a minimal realization in (1.3) and that all eigenvalues of A have positive real parts. Consider the transformation ˜ B, ˜ C, ˜ Q, ˜ N ˜ , F˜ ), (A, B, C, Q, N, F ) → (A, (2.9) where (Q, N ) corresponds to the unique solution to the Lyapunov system in (2.1) and (2.2), the quantity F is as in (2.5), ˜ = −N −1 B, C˜ = −CQ−1 , Q ˜ = −Q−1, N ˜ = −N −1 , A˜ = −A† , B ˜ B, ˜ C, ˜ Q, ˜ N ˜) and F˜ and u ˜ are as in (2.5) and (2.6), respectively, but by using (A, instead of (A, B, C, Q, N ) on the right-hand sides. We then have the following: ˜ and N ˜ are selfadjoint and invertible. They satisfy the respec(i) The matrices Q tive Lyapunov equations

˜ A˜ + A˜† Q ˜ = C˜ † C, ˜ Q (2.10) ˜B ˜†. ˜ +N ˜ A˜† = B A˜N (ii) The quantity F is transformed as F˜ = Q−1 F N −1 . The matrix F˜ is invertible at every point on the xt-plane. To consider the case where only some of eigenvalues of A are reﬂected with respect to the imaginary axis, let us again start with a triplet (A, B, C) of size p and corresponding to a minimal realization in (1.3), where the eigenvalues of A all have positive real parts. Without loss of any generality, let us assume that we partition the matrices A, B, C as B1 A1 0 , B= , C = C1 C2 , (2.11) A= 0 A2 B2 so that the q×q block diagonal matrix A1 contains the eigenvalues that will remain unchanged and A2 contains the eigenvalues that will be reﬂected with respect to the imaginary axis on the complex plane, the submatrices B1 and C1 have sizes q × 1 and 1 × q, respectively, and hence A2 , B2 , C2 have sizes (p − q) × (p − q), (p − q) × 1, 1 × (p − q), respectively, for some integer q not exceeding p. Let us clarify our notational choice in (2.11) and emphasize that the partitioning in (2.11) is not the same partitioning used in (2.8). Using the partitioning in (2.11), let us write the corresponding respective solutions to (2.1) and (2.2) as N1 N2 Q1 Q2 , N= , (2.12) Q= Q3 Q 4 N3 N4

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T. Aktosun, T. Busse, F. Demontis and C. van der Mee

where Q1 and N1 have sizes q × q, Q4 and N4 have sizes (p − q) × (p − q), etc. Note that because of the selfadjointness of Q and N stated in Theorem 1, we have Q†1 = Q1 ,

Q†2 = Q3 ,

Q†4 = Q4 ,

N1† = N1 ,

N2† = N3 ,

N4† = N4 .

Furthermore, from Theorem 3 it follows that Q1 , Q4 , N1 , and N4 are all invertible. Theorem 6. Assume that the triplet (A, B, C) partitioned as in (2.11) corresponds to a minimal realization in (1.3) and that all eigenvalues of A have positive real ˜ B, ˜ C) ˜ having similar block repparts. Consider the transformation (2.9) with (A, resentations as in (2.11), (Q, N ) as in (2.12) corresponding to the unique solution to the Lyapunov system in (2.1) and (2.2), A˜1 = A1 ,

A˜2 = −A†2 ,

˜1 = B1 − N2 N −1 B2 , B 4

C˜1 = C1 − C2 Q−1 4 Q3 ,

˜2 = −N −1 B2 , B 4

C˜2 = −C2 Q−1 4 ,

˜ and N ˜ partitioned in a similar way as in (2.12) and given as and Q ˜ 1 = Q1 − Q2 Q−1 Q3 , Q 4

˜ 2 = −Q2 Q−1 , Q 4

˜ 3 = −Q−1 Q3 , Q 4

˜ 4 = −Q−1 , Q 4

˜1 = N1 − N2 N −1 N3 , N ˜2 = −N2 N −1 , N ˜3 = −N −1N3 , N ˜4 = −N −1 , N 4 4 4 4 ˜ ˜ ˜ ˜ ˜ ˜ and F and u ˜ as in (2.5) and (2.6), respectively, but by using (A, B, C, Q, N ) instead of (A, B, C, Q, N ) on the right-hand sides. We then have the following: ˜ and N ˜ are selfadjoint and invertible. They satisfy the respec(i) The matrices Q tive Lyapunov equations given in (2.10). (ii) The quantity F is transformed according to I I −Q2 Q−1 4 F˜ = F −1 −1 −N4 N3 0 −Q4

0 −N4−1

,

and the matrix F˜ is invertible at every point on the xt-plane. ˜ B, ˜ C) ˜ are equivalent; i.e., u (iii) The triplets (A, B, C) and (A, ˜(x, t) = u(x, t).

3. Examples In this section we illustrate our method of constructing exact solutions to the NLS equation with some concrete examples. Example 1. The well-known “n-soliton” solution to the NLS equation is obtained by choosing the triplet (A, B, C) as A = diag{a1 , a2 , . . . , an }, B † = 1 1 . . . 1 , C = c1 c2 . . . cn , where aj are distinct (complex) nonzero constants with positive real parts, B contains n entries, and the quantities cj are complex constants. Note that diag is

Exact Solutions to the NLS Equation

9

used to denote the diagonal matrix. In this case, using (2.5) and (2.7) we evaluate the (j, k)-entries of the n × n matrix-valued functions Q, N, and F (x, t) as Njk

2 n c∗j ck c∗j cs e−2as x−4ias t 1 2a∗ x−4i(a∗ )2 t j j = , Qjk = ∗ , Fjk = δjk e + , aj + a∗k aj + ak (a∗j + as )(as + a∗k ) s=1

where δjk denotes the Kronecker delta. Having obtained Q, N, and F (x, t), we construct the solution u(x, t) to the NLS equation via (2.6) or equivalently as the ratio of two determinants as 0 B † 2 u(x, t) = (3.1) . det F (x, t) C † F (x, t) For example, when n = 1, from (3.1) we obtain the single soliton solution ∗

∗ 2

−8c∗1 (Re[a1 ])2 e−2a1 x+4i(a1 ) t , u(x, t) = 2 4(Re[a1 ])2 + |c1 |2 e−4x(Re[a1 ])+8t(Im[a1 ]) where Re and Im denote the real and imaginary parts, respectively. From (1.1) we see that if u(x, t) is a solution to (1.1), so is eiθ u(x, t) for any real constant θ. Hence, the constant phase factor eiθ can always be omitted from the solution to (1.1) without any loss of generality. As a result, we can write the single soliton solution also in the form |c1 | u(x, t) = 2 Re[a1 ] eiβ(x,t) sech 2 Re[a1 ](x − 4t Im[a1 ]) − log , 2Re[a1 ] where it is seen that u(x, t) has amplitude 2 Re[a1 ] and moves with velocity 4 Im[a1 ] and we have β(x, t) := 2xIm[a1 ] + 4t Re[a21 ]. Example 2. For the triplet (A, B, C) given by 2 0 1 A= , B= , 0 −1 1

C = 1 −1 ,

(3.2)

we evaluate Q and N explicitly by solving (2.1) and (2.2), respectively, as 1/4 1 1/4 −1 N= , Q= , 1 −1/2 −1 −1/2 and obtain F (x, t) by using (2.5) as ⎤ ⎡ 1 −4x−16it 1 −4x−16it 1 2x−4it e e + e e4x−16it − e2x−4it + ⎥ ⎢ 16 4 2 F (x, t) = ⎣ ⎦. 1 −4x−16it 1 2x−4it 1 − e − e e−2x−4it − e−4x−16it + e2x−4it 4 2 4 Finally, using (2.6), we obtain the corresponding solution to the NLS equation as u(x, t) =

8e4it (9e−4x + 16e4x ) − 32e16it (4e−2x + 9e2x ) . −128 cos(12t) + 4e−6x + 16e6x + 81e−2x + 64e2x

(3.3)

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T. Aktosun, T. Busse, F. Demontis and C. van der Mee

It can independently be veriﬁed that u(x, t) given in (3.3) satisﬁes the NLS equation on the entire xt-plane. With the help of the results stated in Section 2, we can determine triplets ˜ B, ˜ C) ˜ that are equivalent to the triplet in (3.2). (A, The following triplets all yield the same u(x, t) given in (3.3): 9/α1 2 0 ˜ ˜ , B= , C˜ = α1 α2 , (i) A = −4/α2 0 1 where α1 and α2 are arbitrary (complex) nonzero parameters. Note that both eigenvalues of A˜ are positive, whereas only one of the eigenvalues of A in (3.2) is positive. 16/(9α3) −2 0 ˜ ˜ , B= (ii) A = , C˜ = α3 α4 , 0 1 −4/(9α4 ) where α3 and α4 are arbitrary (complex) nonzero parameters. Note that the eigenvalues of A˜ in this triplet are negatives of the eigenvalues of A given in (3.2). 1/α5 2 0 ˜ ˜ , B= (iii) A = , C˜ = α5 α6 , −1/α6 0 −1 where α5 and α6 are arbitrary (complex) nonzero parameters. Note that A˜ here agrees with A in (3.2). 16/α7 −2 0 ˜ ˜ , B= (iv) A = , C˜ = α7 α8 , 0 −1 −9/α8 where α7 and α8 are arbitrary (complex) nonzero parameters. Note that both eigenvalues of A˜ are negative. ˜ B, ˜ C) ˜ given by (v) Equivalent to (3.2) we also have the triplet (A, ⎡ ⎤ α9 α10 ⎦, A˜ = ⎣ (1 − α9 )(α9 − 2) 3 − α9 α10 5α210 α11 + α10 α12 − 5α9 α10 α12 ˜= B

14α10 α11 − 5α9 α10 α11 + 10α12 − 15α9 α12 + 5α29 α12

α210 α211 + 3α10 α11 α12 − 2α9 α10 α11 α12 + 2α212 − 3α9 α212 + α29 α212 C˜ = α11 α12 ,

,

where α9 , . . . , α12 are arbitrary parameters with the restriction that ˜ is nonzero; α10 α11 α12 = 0, which guarantees that the denominator of B when α10 = 0 we must have α11 α12 = 0 and choose α9 as 2 or 1. In fact, the

Exact Solutions to the NLS Equation

11

˜ B, ˜ C) ˜ guarantees that B ˜ is well deﬁned. For exminimality of the triplet (A, ample, the triplet is not minimal if α11 α12 = 0. We note that the eigenvalues of A˜ are 2 and 1 and that A˜ here is similar to the matrix A˜ in the equivalent triplet given in (i). Other triplets equivalent to (3.2) can be found as in (v) above, by exploiting the similarity for the matrix A˜ given in (ii), (iii), and (iv), respectively, and by ˜ and C˜ in the triplet. using (1.3) to determine the corresponding B Acknowledgment The research leading to this article was supported in part by the U.S. National Science Foundation under grant DMS-0610494, the Italian Ministry of Education and Research (MIUR) under PRIN grant no. 2006017542-003, and INdAM-GNCS.

References [1] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge Univ. Press, Cambridge, 1991. [2] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981. [3] T. Aktosun, T. Busse, F. Demontis, and C. van der Mee, Symmetries for exact solutions to the nonlinear Schr¨ odinger equation, preprint, arXiv: 0905.4231. [4] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the focusing nonlinear Schr¨ odinger equation, Inverse Problems 23, 2171–2195 (2007). [5] T. Aktosun and C. van der Mee, Explicit solutions to the Korteweg-de Vries equation on the half-line, Inverse Problems 22, 2165–2174 (2006). [6] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions. The state space method, Birkh¨ auser, Basel, 2007. [7] T. Busse, Ph.D. thesis, University of Texas at Arlington, 2008. [8] F. Demontis, Direct and inverse scattering of the matrix Zakharov-Shabat system, Ph.D. thesis, University of Cagliari, Italy, 2007. [9] A. Hasegawa and M. Matsumoto, Optical solitons in ﬁbers, 3rd ed., Springer, Berlin, 2002. [10] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric ﬁbers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142–144 (1973). [11] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric ﬁbers. II. Normal dispersion, Appl. Phys. Lett. 23, 171–172 (1973). [12] S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons, Consultants Bureau, New York, 1984. [13] E. Olmedilla, Multiple pole solutions of the nonlinear Schr¨ odinger equation, Phys. D 25, 330–346 (1987). [14] V.E. Zakharov, Stability of periodic waves of ﬁnite amplitude on the surface of a deep ﬂuid, J. Appl. Mech. Tech. Phys. 4, 190–194 (1968).

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[15] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34, 62–69 (1972). Tuncay Aktosun Department of Mathematics University of Texas at Arlington Arlington, TX 76019, USA e-mail: [email protected] Theresa Busse Department of Mathematics and Computer Science Northeast Lakeview College Universal City, TX 78148, USA e-mail: [email protected] Francesco Demontis and Cornelis van der Mee Dipartimento di Matematica e Informatica Universit` a di Cagliari Viale Merello 92 I-09123 Cagliari, Italy e-mail: [email protected] [email protected] Received: March 1, 2009 Accepted: August 20, 2009

Operator Theory: Advances and Applications, Vol. 203, 13–88 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Robust Control, Multidimensional Systems and Multivariable Nevanlinna-Pick Interpolation Joseph A. Ball and Sanne ter Horst Dedicated to Israel Gohberg on the occasion of his 80th birthday

Abstract. The connection between the standard H ∞ -problem in control theory and Nevanlinna-Pick interpolation in operator theory was established in the 1980s, and has led to a fruitful cross-pollination between the two ﬁelds since. In the meantime, research in H ∞ -control theory has moved on to the study of robust control for systems with structured uncertainties and to various types of multidimensional systems, while Nevanlinna-Pick interpolation theory has moved on independently to a variety of multivariable settings. Here we review these developments and indicate the precise connections which survive in the more general multidimensional/multivariable incarnations of the two theories. Mathematics Subject Classiﬁcation (2000). Primary: 47A57, 93D09; Secondary: 13F25, 47A56, 47A63, 93B52, 93D15. Keywords. Model-matching problem, Youla-Kuˇcera parametrization of stabilizing controllers, H ∞ -control problem, structured singular value, structured uncertainty, Linear-Fractional-Transformation model, stabilizable, detectable, robust stabilization, robust performance, frequency domain, state space, Givone-Roesser commutative/noncommutative multidimensional linear system, gain-scheduling, Finsler’s lemma.

1. Introduction Starting in the early 1980s with the seminal paper [139] of George Zames, there occurred an active interaction between operator theorists and control engineers in the development of the early stages of the emerging theory of H ∞ -control. The cornerstone for this interaction was the early recognition by Francis-Helton-Zames [65] that the simplest case of the central problem of H ∞ -control (the sensitivity Communicated by L. Rodman.

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J.A. Ball and S. ter Horst

minimization problem) is one and the same as a Nevanlinna-Pick interpolation problem which had already been solved in the early part of the twentieth century (see [110, 105]). For the standard problem of H ∞ -control it was known early on that it could be brought to the so-called Model-Matching form (see [53, 64]). In the simplest cases, the Model-Matching problem converts easily to a NevanlinnaPick interpolation problem of classical type. Handling the more general problems of H ∞ -control required extensions of the theory of Nevanlinna-Pick interpolation to tangential (or directional) interpolation conditions for matrix-valued functions; such extensions of the interpolation theory were pursued by both engineers and mathematicians (see, e.g., [26, 58, 90, 86, 87]). Alternatively, the Model-Matching problem can be viewed as a Sarason problem which is suitable for application of Commutant Lifting theory (see [125, 62]). The approach of [64] used an additional conversion to a Nehari problem where existing results on the solution of the Nehari problem in state-space coordinates were applicable (see [69, 33]). The book of Francis [64] was the ﬁrst book on H ∞ -control and provides a good summary of the state of the subject in 1987. While there was a lot of work emphasizing the connection of the H ∞ -problem with interpolation and the related approach through J-spectral factorization ([26, 90, 91, 86, 87, 33, 24]), we should point out that the ﬁnal form of the H ∞ -theory parted ways with the connection with Nevanlinna-Pick interpolation. When calculations were carried out in state-space coordinates, the reduction to ModelMatching form via the Youla-Kuˇcera parametrization of stabilizing controllers led to inﬂation of state-space dimension; elimination of non-minimal state-space nodes by ﬁnding pole-zero cancellations demanded tedious brute-force calculations (see [90, 91]). A direct solution in state-space coordinates (without reduction to Model-Matching form and any explicit connection with Nevanlinna-Pick interpolation) was ﬁnally obtained by Ball-Cohen [24] (via a J-spectral factorization approach) and in the more deﬁnitive coupled-Riccati-equation form of Doyle-Glover-Khargonekar-Francis [54]. This latter paper emphasizes the parallels with older control paradigms (e.g., the Linear-Quadratic-Gaussian and LinearQuadratic-Regulator problems) and obtained parallel formulas for the related H 2 problem. The J-spectral factorization approach was further developed in the work of Kimura, Green, Glover, Limebeer, and Doyle [87, 70, 71]. A good review of the state of the theory to this point can be found in the books of Zhou-Doyle-Glover [141] and Green-Limebeer [72]. The coupled-Riccati-equation solution however has now been superseded by the Linear-Matrix-Inequality (LMI) solution which came shortly thereafter; we mention speciﬁcally the papers of Iwasaki-Skelton [78] and Gahinet-Apkarian [66]. This solution does not require any boundary rank conditions entailed in all the earlier approaches and generalizes in a straightforward way to more general settings (to be discussed in more detail below). The LMI form of the solution is particularly appealing from a computational point of view due to the recent advances in semideﬁnite programming (see [68]). The book of Dullerud-Paganini [57] gives an up-to-date account of these latest developments.

Control and Interpolation

15

Research in H ∞ -control has moved on in a number of diﬀerent new directions, e.g., extensions of the H ∞ -paradigm to sampled-data systems [47], nonlinear systems [126], hybrid systems [23], stochastic systems [76], quantum stochastic systems [79], linear repetitive processes [123], as well as behavioral frameworks [134]. Our focus here will be on the extensions to robust control for systems with structured uncertainties and related H ∞ -control problems for multidimensional (N D) systems – both frequency-domain and state-space settings. In the meantime, Nevanlinna-Pick interpolation theory has moved on to a variety of multivariable settings (polydisk, ball, noncommutative polydisk/ball); we mention in particular the papers [1, 49, 113, 3, 35, 19, 20, 21, 22, 30]. As the transfer function for a multidimensional system is a function of several variables, one would expect that the same connections familiar from the 1D/single-variable case should also occur in these more general settings; however, while there had been some interaction between control theory and several-variable complex function theory in the older area of systems over rings (see [83, 85, 46]), to this point, with a few exceptions [73, 74, 32], there has not been such an interaction in connection with H ∞ -control for N -D systems and related such topics. With this paper we wish to make precise the interconnections which do exist between the H ∞ -theory and the interpolation theory in these more general settings. As we shall see, some aspects which are taken for granted in the 1-D/single-variable case become much more subtle in the N -D/multivariable case. Along the way we shall encounter a variety of topics that have gained attention recently, and sometimes less recently, in the engineering literature. Besides the present Introduction, the paper consists of ﬁve sections which we now describe: (1) In Section 2 we lay out four speciﬁc results for the classical 1-D case; these serve as models for the type of results which we wish to generalize to the N -D/multivariable settings. (2) In Section 3 we survey the recent results of Quadrat [117, 118, 119, 120, 121, 122] on internal stabilization and parametrization of stabilizing controllers in an abstract ring setting. The main point here is that it is possible to parametrize the set of all stabilizing controllers in terms of a given stabilizing controller even in settings where the given plant may not have a double coprime factorization – resolving some issues left open in the book of Vidyasagar [136]. In the case where a double-coprime factorization is available, the parametrization formula is more eﬃcient. Our modest new contribution here is to extend the ideas to the setting of the standard problem of H ∞ -control (in the sense of the book of Francis [64]) where the given plant is assumed to have distinct disturbance and control inputs and distinct error and measurement outputs. (3) In Section 4 we look at the internal-stabilization/H ∞-control problem for multidimensional systems. These problems have been studied in a purely frequencydomain framework (see [92, 93]) as well as in a state-space framework (see [81, 55, 56]). In Subsection 4.1, we give the frequency-domain formulation of the problem.

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When one takes the stable plants to consist of the ring of structurally stable rational matrix functions, the general results of Quadrat apply. In particular, for this setting stabilizability of a given plant implies the existence of a double coprime factorization (see [119]). Application of the Youla-Kuˇcera parametrization then leads to a Model-Matching form and, in the presence of some boundary rank conditions, the H ∞ -problem converts to a polydisk version of the Nevanlinna-Pick interpolation problem. Unlike the situation in the classical single-variable case, this interpolation problem has no practical necessary-and-suﬃcient solution criterion and in practice one is satisﬁed with necessary and suﬃcient conditions for the existence of a solution in the more restrictive Schur-Agler class (see [1, 3, 35]). In Subsection 4.2 we formulate the internal-stabilization/H ∞-control problem in Givone-Roesser state-space coordinates. We indicate the various subtleties involved in implementing the state-space version [104, 85] of the double-coprime factorization and associated Youla-Kuˇcera parametrization of the set of stabilizing controllers. With regard to the H ∞ -control problem, unlike the situation in the classical 1-D case, there is no useable necessary and suﬃcient analysis for solution of the problem; instead what is done (see, e.g., [55, 56]) is the use of an LMI/Bounded-Real-Lemma analysis which provides a convenient set of suﬃcient conditions for solution of the problem. This suﬃciency analysis in turn amounts to an N -D extension of the LMI solution [78, 66] of the 1-D H ∞ -control problem and can be viewed as a necessary and suﬃcient analysis of a compromise problem (the “scaled” H ∞ -problem). While stabilization and H ∞ -control problems have been studied in the statespace setting [81, 55, 56] and in the frequency-domain setting [92, 93] separately, there does not seem to have been much work on the precise connections between these two settings. The main point of Subsection 4.3 is to study this relationship; while solving the state-space problem implies a solution of the frequency-domain problem, the reverse direction is more subtle and it seems that only partial results are known. Here we introduce a notion of modal stabilizability and modal detectability (a modiﬁcation of the notions of modal controllability and modal observability introduced by Kung-Levy-Morf-Kailath [88]) to obtain a partial result on relating a solution of the frequency-domain problem to a solution of the associated state-space problem. This result suﬀers from the same weakness as a corresponding result in [88]: just as the authors in [88] were unable to prove that minimal (i.e., simultaneously modally controllable and modally observable) realizations for a given transfer matrix exist, so also we are unable to prove that a simultaneously modally stabilizable and modally detectable realization exists. A basic diﬃculty in translating from frequency-domain to state-space coordinates is the failure of the State-Space-Similarity theorem and related Kalman state-space reduction for N -D systems. Nevertheless, the result is a natural analogue of the corresponding 1-D result. There is a parallel between the control-theory side and the interpolationtheory side in that in both cases one is forced to be satisﬁed with a compromise solution: the scaled-H ∞ problem on the control-theory side, and the Schur-Agler

Control and Interpolation

17

class (rather than the Schur class) on the interpolation-theory side. We include some discussion on the extent to which these compromises are equivalent. (4) In Section 5 we discuss several 1-D variations on the internal-stabilization and H ∞ -control problem which lead to versions of the N -D/multivariable problems discussed in Section 4. It was observed early on that an H ∞ -controller has good robustness properties, i.e., an H ∞ -controller not only provides stability of the closed-loop system associated with the given (or nominal) plant for which the control was designed, but also for a whole neighborhood of plants around the nominal plant. This idea was reﬁned in a number of directions, e.g., robustness with respect to additive or multiplicative plant uncertainty, or with respect to uncertainty in a normalized coprime factorization of the plant (see [100]). Another model for an uncertainty structure is the Linear-Fractional-Transformation (LFT) model used by Doyle and coworkers (see [97, 98]). Here a key concept is the notion of structured singular value μ(A) for a ﬁnite square matrix A introduced by Doyle and Safonov [52, 124] which simultaneously generalizes the norm and the spectral radius depending on the choice of uncertainty structure (a C ∗ -algebra of matrices with a prescribed block-diagonal structure); we refer to [107] for a comprehensive survey. If one assumes that the controller has on-line access to the uncertainty parameters one is led to a gain-scheduling problem which can be identiﬁed as the type of multidimensional control problem discussed in Section 4.2 – see [106, 18]; we survey this material in Subsection 5.1. In Subsection 5.2 we review the purely frequencydomain approach of Helton [73, 74] toward gain-scheduling which leads to the frequency-domain internal-stabilization/H ∞-control problem discussed in Section 4.1. Finally, in Section 5.3 we discuss a hybrid frequency-domain/state-space model for structured uncertainty which leads to a generalization of Nevanlinna-Pick interpolation for single-variable functions where the constraint that the norm be uniformly bounded by 1 is replaced by the constraint that the μ-singular value be uniformly bounded by 1; this approach has only been analyzed for very special cases of the control problem but does lead to interesting new results for operator theory and complex geometry in the work of Bercovici-Foias-Tannenbaum [38, 39, 40, 41], Agler-Young [5, 6, 7, 8, 9, 10, 11, 12, 13], Huang-MarcantogniniYoung [77], and Popescu [114]. (5) The ﬁnal Section 6 discusses an enhancement of the LFT-model for structured uncertainty to allow dynamic time-varying uncertainties. If the controller is allowed to have on-line access to these more general uncertainties, then the solution of the internal-stabilization/H ∞-control problem has a form completely analogous to the classical 1-D case. Roughly, this result corresponds to the fact that, with this noncommutative enhanced uncertainty structure, the a priori upper bound μ (A) for the structured singular value μ(A) is actually equal to μ(A), despite the fact that for non-enhanced structures, the gap between μ and μ can be arbitrarily large (see [133]). In this precise form, the result appears for the ﬁrst time in the thesis of Paganini [108] but various versions of this type of result have also appeared elsewhere (see [37, 42, 60, 99, 129]). We discuss this enhanced

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noncommutative LFT-model in Subsection 6.1. In Subsection 6.2 we introduce a noncommutative frequency-domain control problem in the spirit of Chapter 4 of the thesis of Lu [96], where the underlying polydisk occurring in Section 4.1 is now replaced by the noncommutative polydisk consisting of all d-tuples of contraction operators on a ﬁxed separable inﬁnite-dimensional Hilbert space K and the space of H ∞ -functions is replaced by the space of scalar multiples of the noncommutative Schur-Agler class introduced in [28]. Via an adaptation of the Youla-Kuˇcera parametrization of stabilizing controllers, the internal-stabilization/H ∞-control problem can be reduced to a Model-Matching form which has the interpretation as a noncommutative Sarason interpolation problem. In the ﬁnal Subsection 6.3, we show how the noncommutative state-space problem is exactly equivalent to the noncommutative frequency-domain problem and thereby obtain an analogue of the classical case which is much more complete than for the commutative-variable case given in Section 4.3. In particular, if the problem data are given in terms of state-space coordinates, the noncommutative Sarason problem can be solved as an application of the LMI solution of the H ∞ -problem. While there has been quite a bit of recent activity on this kind of noncommutative function theory (see, e.g., [14, 22, 75, 82, 115, 116]), the noncommutative Sarason problem has to this point escaped attention; in particular, it is not clear how the noncommutative Nevanlinna-Pick interpolation problem studied in [22] is connected with the noncommutative Sarason problem. Finally we mention that each section ends with a “Notes” subsection which discusses more specialized points and makes some additional connections with existing literature.

2. The 1-D systems/single-variable case Let C[z] be the space of polynomials with complex coeﬃcients and C(z) the quotient ﬁeld consisting of rational functions in the variable z. Let RH ∞ be the subring of stable elements of C(z) consisting of those rational functions which are analytic and bounded on the unit disk D, i.e., 11with no poles in the closed unit G12 disk D. We assume to be given a plant G = G G21 G22 : W ⊕ U → Z ⊕ Y which is given as a block matrix of appropriate size with entries from C(z). Here the spaces U, W, Z and Y have the interpretation of control-signal space, disturbance-signal space, error-signal space and measurement-signal space, respectively, and consist of column vectors of given sizes nU , nW , nZ and nY , respectively, with entries from C(z). For this plant G we seek to design a controller K : Y → U, also given as a matrix over C(z), that stabilizes the feedback system Σ(G, K) obtained from the signal-ﬂow diagram in Figure 1 in a sense to be deﬁned precisely below. Note that the various matrix entries Gij of G are themselves matrices with entries from C(z) of compatible sizes (e.g., G11 has size nZ × nW ) and K is a matrix over C(z) of size nU × nY .

Control and Interpolation wvu1

z G

19

-

y v2

K

Figure 1. Feedback with tap signals The system equations associated with the signal-ﬂow diagram of Figure 1 can be written as ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ G11 0 0 z w 0 I −G12 ⎣0 I −K ⎦ ⎣u⎦ = ⎣ 0 I 0⎦ ⎣v1 ⎦ . (2.1) y 0 −G22 G21 0 I I v2 Here v1 and v2 are tap signals used to detect stability properties of the internal signals u and y. We say that the system Σ(G, K) is well posed if there is a wellw

z

deﬁned map from vv1 to uy . It follows from a standard Schur complement 2 computation that the system is well posed if and only if det(I − G22 K) = 0, and w

z

that in that case the map from vv1 to uy is given by 2 ⎡ ⎤ ⎡ ⎤ w z ⎣u⎦ = Θ(G, K) ⎣v1 ⎦ y v2 where

⎤−1 ⎡ ⎤ I −G12 0 G11 0 0 I −K ⎦ ⎣ 0 I 0⎦ Θ(G, K) := ⎣ 0 0 −G22 I G21 0 I ⎡ ⎤ G11 + G12 K(I − G22 K)−1 G21 G12 [I + K(I − G22 K)−1 G22 ] G12 K(I − G22 K)−1 −1 −1 −1 ⎦ K(I − G22 K) G21 I + K(I − G22 K) G22 K(I − G22 K) =⎣ (I − G22 K)−1 G21 (I − G22 K)−1 G22 (I − G22 K)−1 ⎤ ⎡ G12 (I − KG22 )−1 K G11 + G12 (I − KG22 )−1 KG21 G12 (I − KG22 )−1 −1 −1 −1 ⎦. (I − KG22 ) KG21 (I − KG22 ) (I − KG22 ) K =⎣ [I + G22 (I − KG22 )−1 K]G21 G22 (I − KG22 )−1 I + G22 (I − KG22 )−1 K (2.2) ⎡

We say that the system Σ(G, K) is internally stable if Σ(G, K) is well posed and, ∞ in addition, if the map Θ(G, K) maps RHW ⊕ RHU∞ ⊕ RHY∞ into RHZ∞ ⊕ RHU∞ ⊕ ∞ RHY , i.e., stable inputs w, v1 , v2 are mapped to stable outputs z, u, y. Note that this is the same as the condition that the entries of Σ(G, K) be in RH ∞ . We say that the system Σ(G, K) has performance if Σ(G, K) is internally stable and in addition the transfer function Tzw from w to z has supremum-norm over the unit disk bounded by some tolerance which we normalize to be equal to 1: Tzw ∞ := sup{Tzw (λ) : λ ∈ D} ≤ 1.

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J.A. Ball and S. ter Horst

Here Tzw (λ) refers to the induced operator norm, i.e., the largest singular value for the matrix Tzw (λ). We say that the system Σ(G, K) has strict performance if in addition Tzw ∞ < 1. The stabilization problem then is to describe all (if any exist) internally stabilizing controllers K for the given plant G, i.e., all K ∈ C(z)nU ×nY so that the associated closed-loop system Σ(G, K) is internally stable. The standard H ∞ -problem is to ﬁnd all internally stabilizing controllers which in addition achieve performance Tzw ∞ ≤ 1. The strictly suboptimal H ∞ -problem is to describe all internally stabilizing controllers which also achieve strict performance Tzw ∞ < 1. 2.1. The model-matching problem Let us now consider the special case where G22 = 0, so that G has the form 11 G12 . In this case well-posedness is automatic and Θ(G, K) simpliﬁes to G= G G21 0 ⎡

G11 + G12 KG21 KG21 Θ(G, K) = ⎣ G21

G12 I 0

⎤ G12 K K ⎦. I

Thus internal stability for the closed-loop system Σ(G, K) is equivalent to stability of the four transfer matrices G11 , G12 , G21 and K. Hence internal stabilizability of G is equivalent to stability of G11 , G12 and G21 ; when the latter holds a given K internally stabilizes G if and only if K itself is stable. Now assume that G11 , G12 and G21 are stable. Then the H ∞ -performance problem for G consists of ﬁnding stable K so that G11 + G12 KG21 ∞ ≤ 1. Following the terminology of [64], the problem is called the Model-Matching Problem. Due to the inﬂuence of the paper [125], this problem is usually referred to as the Sarason problem in the operator theory community; in [125] it is shown explicitly how the problem can be reduced to an interpolation problem. In general control problems the assumption that G22 = 0 is an unnatural assumption. However, after making a change of coordinates using the Youla-Kuˇcera parametrization or the Quadrat parametrization, discussed below, it turns out that the general H ∞ -problem can be reduced to a model-matching problem. 2.2. The frequency-domain stabilization and H ∞ problem The following result on characterization of stabilizing controllers is well known (see, e.g., [64] or [136, 137] for a more general setting). 11 G12 Theorem 2.1. Suppose that we are given a rational matrix function G = G G21 G22 of size (nZ + nY ) × (nW + nU ) with entries in C(z) as above. Assume that G is stabilizable, i.e., there exists a rational matrix function K of size nU × nY so that the nine transfer functions in (2.2) are all stable. Then a given rational matrix function K stabilizes G if and only if K stabilizes G22 , i.e., Θ(G, K) in (2.2) is

Control and Interpolation stable if and only if

21

I + K(I − G22 K)−1 G22 K(I − G22 K)−1 Θ(G22 , K) : = (I − G22 K)−1 G22 (I − G22 K)−1 (I − KG22 )−1 (I − KG22 )−1 K = G22 (I − KG22 )−1 I + G22 (I − KG22 )−1 K

is stable. Moreover, if we are given a double coprime factorization for G22 , i.e., N , X and Y so that the determinants of stable transfer matrices D, N , X, Y , D, ∞ D, D, X and X are all nonzero (in RH ) and D −N X N 0 InY −1 −1 (2.3) G22 = D N = N D , = 0 InU −Y X Y D (such double coprime factorizations always exists since RH ∞ is a Principal Ideal Domain), then the set of all stabilizing controllers K is given by either of the formulas + ΛN )−1 (Y + ΛD), Λ)−1 = (X K = (Y + DΛ)(X +N Λ) = 0 or where Λ is a free stable parameter from RH ∞ such that det(X + N + ΛN ) = 0. equivalently det(X

L(U ,Y)

Through the characterization of the stabilizing controllers, those controllers that, in addition, achieve performance can be obtained from the solutions of a Model-Matching/Sarason interpolation problem. Theorem 2.2. Assume that G ∈ C(z)(nZ +nY )×(nW +nU ) is stabilizable and that G22 admits a double coprime factorization (2.3). Let K ∈ C(z)nU ×nY . Then K is a solution to the standard H ∞ problem for G if and only if Λ)−1 = (X + ΛN )−1 (Y + ΛD), K = (Y + DΛ)(X +N = 0, or equivalently det(X + ΛN ) = 0, where Λ ∈ RH ∞ so that det(X + NΛ) L(U ,Y)

12 11 , G is any solution to the Model-Matching/Sarason interpolation problem for G 21 deﬁned by and G 12 := G12 D, 11 := G11 + G12 Y DG21 , G 21 := DG21 , G G i.e., so that

12 ΛG 21 ∞ ≤ 1. 11 + G G

21 is surjective on the unit circle, 12 is injective and G We note that in case G by absorbing outer factors into the free parameter Λ we may assume without loss 12 is inner (i.e., G 12 (z) is isometric for z on unit circle) and of generality that G G21 is co-inner (i.e., G21 (z) is coisometric for z on the unit circle). Let Γ : L2W

∗ H 2⊥ → L2 G 12 H 2 be the compression of multiplication by G 11 to the spaces G 21 U Z U ∗21 H 2⊥ and L2 G 12 H 2 , i.e., Γ = P 2 L2W G 2 G11 |L2 G ∗ H 2⊥ . Then, as a U Z U LZ G12 HU 21 Y W consequence of the Commutant Lifting theorem (see [63, Corollary 10.2 pages 40– 41]), one can see that the strict Model-Matching/Sarason interpolation problem

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J.A. Ball and S. ter Horst

posed in Theorem 2.2 has a solution if and only if Γop < 1. Alternatively, in 12 and G 21 are square and invertible on the unit circle, one can convert case G this Model-Matching/Commutant-Lifting problem to a bitangential NevanlinnaPick interpolation problem (see [26, Theorem 16.9.3]), a direct generalization of the connection between a model-matching/Sarason interpolation problem with Nevanlinna-Pick interpolation as given in [125, 65] for the scalar case, but we will not go into the details of this here. 2.3. The state-space approach We now restrict the classes of admissible plants and controllers to the transfer matrices whose entries are in C(z)0 , the space of rational functions without a pole at 0 (i.e., analytic in a neighborhood of 0). In that case, a transfer matrix F : U → Y with entries in C(z)0 admits a state-space realization: There exists a quadruple {A, B, C, D} consisting of matrices whose sizes are given by X X A B , (2.4) → : Y U C D where the state-space X is ﬁnite-dimensional, so that F (z) = D + zC(I − zA)−1 B for z in a neighborhood of 0. Sometimes we consider quadruples {A, B, C, D} of operators, of compatible size as above, without any explicit connection to a transfer matrix, in which case we just speak of a realization. Associated with the realization {A, B, C, D} is the linear discrete-time system of equations x(n + 1) = Ax(n) + Bu(n), Σ := (n ∈ Z+ ) y(n) = Cx(n) + Du(n). The system Σ and function F are related through the fact that F is the transferfunction of Σ. The two-by-two matrix (2.4) is called the system matrix of the system Σ. For the rest of this section we shall say that an operator A on a ﬁnitedimensional state space X is stable if all its eigenvalues are in the open unit disk, or, equivalently, An x → 0 as n → ∞ for each x ∈ X . The following result deals with two key notions for the stabilizability problem on the state-space level. Theorem 2.3. (I) Suppose that {A, B} is an input pair, i.e., A, B are operators with A : X → X and B : U → X for a ﬁnite-dimensional state space X and a ﬁnite-dimensional input space U. Then the following are equivalent: 1. {A, B} is operator-stabilizable, i.e., there exists a state-feedback operator F : X → U so that the operator A + BF is stable. 2. {A, B} is Hautus-stabilizable, i.e., the matrix pencil I − zA B is surjective for each z in the closed unit disk D.

Control and Interpolation

23

3. The Stein inequality AXA∗ − X − BB ∗ < 0 has a positive-deﬁnite solution X. Here Γ < 0 for a square matrix Γ means that −Γ is positive deﬁnite. (II) Dually, if {C, A} is an output pair, i.e., C, A are operators with A : X → X and C : X → Y for a ﬁnite-dimensional state space X and a ﬁnite-dimensional output space Y, then the following are equivalent: 1. {C, A} is operator-detectable, i.e., there exists an output-injection operator L : Y → X so that A + LC is stable. 2. {C, A} is Hautus-detectable, i.e., the matrix pencil I−zA is injective for C all z in the closed disk D. 3. The Stein inequality A∗ Y A − Y − C ∗ C < 0 has a positive deﬁnite solution Y . When the input pair {A, B} satisﬁes any one (and hence all) of the three equivalent conditions in part (I) of Theorem 2.3, we shall say simply that {A, B} is stabilizable. Similarly, if (C, A) satisﬁes any one of the three equivalent conditions in part (II), we shall say simply that {C, A} is detectable. Given a realization {A, B, C, D}, we shall say that {A, B, C, D} is stabilizable and detectable if {A, B} is stabilizable and {C, A} is detectable. In the state-space formulation of the internal stabilization/H ∞ -control problem, one assumes to be given a state-space realization for the plant G: D11 D12 C G(z) = + z 1 (I − zA)−1 B1 B2 (2.5) D21 D22 C2 where the system matrix has ⎡ A ⎣C1 C2

the form B1 D11 D21

⎡ ⎤ ⎤⎡ ⎤ B2 X X D12 ⎦ ⎣W ⎦ → ⎣ Z ⎦ . Y U D22

(2.6)

One then seeks a controller K which is also given in terms of a state-space realization K(z) = DK + zCK (I − zAK )−1 BK which provides internal stability (in the state-space sense to de deﬁned below) and/or H ∞ -performance for the closed-loop system. Well-posedness of the closedloop system is equivalent to invertibility of I − D22 DK . To keep various formulas aﬃne in the design parameters AK , BK , CK , DK , it is natural to assume that D22 = 0; this is considered not unduly restrictive since under the assumption of well-posedness this can always be arranged via a change of variables

24

J.A. Ball and S. ter Horst

(see [78]). Then the closed loop system Θ(G, K) admits a state space realization {Acl , Bcl , Ccl , Dcl } given by its system matrix ⎡ ⎤ A + B2 DK C2 B2 CK B1 + B2 DK D21 Acl Bcl ⎦ BK C2 AK BK D21 =⎣ (2.7) Ccl Dcl C1 + D12 DK C2 D12 CK D11 + D12 DK D21 internal stability (in the state-space sense) is taken to mean that Acl = and A+B2 DK C2 B2 CK should be stable, i.e., all eigenvalues are in the open unit disk. BK C2 AK The following result characterizes when a given G is internally stabilizable in the state-space sense. Theorem 2.4. (See Proposition 5.2 in [57].) Suppose that we are given a system matrix as in (2.6) with D22 = 0 with associated transfer matrix G as in (2.5). Then there exists a K(z) = DK + zCK (I − zAK )−1 BK which internally stabilizes G (in the state-spaces sense) if and only if {A, B2 } is stabilizable and {C2 , A} is detectable. In this case one such controller is given by the realization {AK , BK , CK , DK } with system matrix AK BK A + B2 F + LC2 −L = F 0 CK DK where F and L are state-feedback and output-injection operators chosen so that A + B2 F and A + LC2 are stable. In addition to the state-space version of the stabilizability problem we also consider a (strict) state-space H ∞ problem, namely to ﬁnd a controller K given by a state-space realization {AK , BK , CK , DK } of compatible size so that the transfer-function Tzw of the closed loop system, given by the system matrix (2.7), is stable (in the state-space sense) and has a supremum norm Tzw ∞ of at most 1 (less than 1). The deﬁnitive solution of the H ∞ -control problem in state-space coordinates for a time was the coupled-Riccati-equation solution due to Doyle-GloverKhargonekar-Francis [54]. This solution has now been superseded by the LMI solution of Gahinet-Apkarian [66] which can be stated as follows. Note that the problem can be solved directly without ﬁrst processing the data to the ModelMatching form. 1 D11 D12 Theorem 2.5. Let {A, B, C, D} = A, [ B1 B2 ] , C be a given realC2 , D21 0 ization. Then there exists a solution for the strict state-space H ∞ -control problem associated with {A, B, C, D} if and only if there exist positive-deﬁnite matrices X, Y satisfying the LMIs ⎤ ⎡ ∗ AY A∗ − Y AY C1∗ B1 Nc 0 ⎣ N 0 C1 Y A∗ C1 Y C1∗ − I D11 ⎦ c < 0, Y > 0, (2.8) 0 I 0 I ∗ B1∗ D11 −I ⎤ ⎡ ∗ A∗ XA − X A∗ XB1 C1∗ No 0 ⎣ 0 ∗ ⎦ No B1∗ XA B1∗ XB1 − I D11 < 0, X > 0, (2.9) 0 I 0 I C1 D11 −I

Control and Interpolation and the coupling condition

X I ≥ 0. I Y Here Nc and No are matrices chosen so that

25

Nc is injective and Im Nc = Ker B2∗ No is injective and Im No = Ker C2

(2.10) ∗ D12 and D21 .

We shall discuss the proof of Theorem 2.5 in Section 4.2 below in the context of a more general multidimensional-system H ∞ -control problem. The next result is the key to transferring from the frequency-domain version of the internal-stabilization/H ∞-control problem to the state-space version. Theorem 2.6. (See Lemma 5.5 in [57].) Suppose that the realization {A, B2 , C2 , 0} for the plant G22 and the realization {AK , BK , CK , DK } for the controller K are both stabilizable and detectable. Then K internally stabilizes G22 in the state-space sense if and only ifK stabilizes G22 in the frequency-domain sense, i.e., the closed2 DK C2 B2 CK loop matrix Acl = A+B is stable if and only if the associated transfer BK C2 AK matrix I DK DK C2 CK −1 B2 B2 DK +z (I − zAcl ) Θ(G22 , K) = 0 I C2 0 0 BK has all matrix entries in RH ∞ . 2.4. Notes In the context of the discussion immediately after the statement of Theorem 2.2, 21 drop rank at points on the unit circle, the Model-Matching 12 and/or G in case G problem in Theorem 2.2 may convert to a boundary Nevanlinna-Pick interpolation problem for which there is an elaborate specialized theory (see, e.g., Chapter 21 of [26] and the more recent [43]). However, if one sticks with the strictly suboptimal version of the problem, one can solve the problem with the boundary interpolation conditions if and only if one can solve the problem without the boundary interpolation conditions, i.e., boundary interpolation conditions are irrelevant as far as existence criteria are concerned. This is the route taken in the LMI solution of the H ∞ -problem and provides one explanation for the disappearance of any rank conditions in the formulation of the solution of the problem. For a complete analysis of the relation between the coupled-Riccati-equation of [54] versus the LMI solution of [66], we refer to [127].

3. The fractional representation approach to stabilizability and performance In this section we work in the general framework of the fractional representation approach to stabilization of linear systems as introduced originally by Desoer, Vidyasagar and coauthors [50, 137] in the 1980s and reﬁned only recently in the

26

J.A. Ball and S. ter Horst

work of Quadrat [118, 121, 122]. For an overview of the more recent developments we recommend the survey article [117] and for a completely elementary account of the generalized Youla-Kuˇcera parametrization with all the algebro-geometric interpretations stripped out we recommend [120]. The set of stable single-input single-output (SISO) transfer functions is assumed to be given by a general ring A in place of the ring RH ∞ used for the classical case as discussed in Section 2; the only assumption which we shall impose on A is that it be a commutative integral domain. It therefore has a quotient ﬁeld K := Q(A) = {n/d : d, n ∈ A, d = 0} which shall be considered as the set of all possible SISO transfer functions (or plants). Examples of A which come up include the ring Rs (z) of real rational functions of the complex variable z with no poles in the closed right half-plane, the Banach algebra RH ∞ (C+ ) of all bounded analytic functions on the right half-plane C+ which are real on the positive real axis, and their discrete-time analogues: (1) real rational functions with no poles in the closed unit disk (or closed exterior of the unit disk depending on how one sets conventions), and (2) the Banach algebra RH ∞ (D) of all bounded holomorphic functions on the unit disk D with real values on the real interval (−1, 1). There are also Banach subalgebras of RH ∞ (C+ ) or RH ∞ (D) (e.g., the Wiener algebra and its relatives such as the Callier-Desoer class – see [48]) which are of interest. In addition to these examples there are multivariable analogues, some of which we shall discuss in the next section. We now introduce some notation. We assume that the control-signal space U, the disturbance-signal space W, the error-signal space Z and the measurement signal space Y consist of column vectors of given sizes nU , nW , nZ and nY , respectively, with entries from the quotient ﬁeld K of A: U = KnU ,

W = KnW , Z = KnZ , Y = KnY . G11 G12 We are given a plant G = G21 G22 : W ⊕ U → Z ⊕ Y and seek to design a controller K : Y → U that stabilizes the system Σ(G, K) of Figure 1 as given in Section 2. The various matrix entries Gij of G are now matrices with entries from K (rather than RH ∞ as in the classical case) of compatible sizes (e.g., G11 has size nW × nU ) and K is a matrix over K of size nU × nY . Again v1 and v2 are tap signals used to detect stability properties of the internal signals u and y. Just as was explained in Section 2 for the classical w case, z the system Σ(G, K) is well posed if there is a well-deﬁned map from vv1 to uy and this happens 2

now is an element of A); exactly when det(I − G22 K) = 0 (where w the determinant z when this is the case, the map from vv1 to uy is given by 2 ⎡ ⎤ ⎡ ⎤ w z ⎣u⎦ = Θ(G, K) ⎣v1 ⎦ v2 y

where Θ(G, K) is given by (2.2). We say that the system Σ(G, K) is internally stable if Σ(G, K) is well posed and, in addition, if the map Θ(G, K) maps AnW ⊕

Control and Interpolation

27

AnU ⊕ AnY into AnZ ⊕ AnU ⊕ AnY , i.e., stable inputs w, v1 , v2 are mapped to stable outputs z, u, y. Note that this is the same as the entries of Σ(G, K) being in A. To formulate the standard problem of H ∞ -control, we assume that A is equipped with a positive-deﬁnite inner product making A at least a pre-Hilbert space with norm · A ; in the classical case, one takes this norm to be the L2 norm over the unit circle. Then we say that the system Σ(G, K) has performance if Σ(G, K) is internally stable and in addition the transfer function Tzw from w to z has induced operator norm bounded by some tolerance which we normalize to be equal to 1: Tzw op := sup{zAnZ : wAnW ≤ 1, v1 = 0, v2 = 0} ≤ 1. We say that the system Σ(G, K) has strict performance if in fact Tzw op < 1. The stabilization problem then is to describe all (if any exist) internally stabilizing controllers K for the given plant G, i.e., all K ∈ KnU ×nY so that the associated closed-loop system Σ(G, K) is internally stable. The standard H ∞ -problem is to ﬁnd all internally stabilizing controllers which in addition achieve performance Tzw op ≤ 1. The strictly suboptimal H ∞ -problem is to describe all internally stabilizing controllers which achieve strict performance Tzw op < 1. The H ∞ -control problem for the special case where G22 = 0 is the ModelMatching problem for this setup. With the same arguments as in Subsection 2.1 it follows that stabilizability forces G11 , G12 and G21 all to be stable (i.e., to have all matrix entries in A) and then K stabilizes exactly when also K is stable. 3.1. Parametrization of stabilizing controllers in terms of a given stabilizing controller 11 G12 We return to the general case, i.e., G = G G21 G22 : W ⊕ U → Z ⊕ Y. Now suppose we have a stabilizing controller K ∈ KnU ×nY . Set U = (I − G22 K)−1

and V = K(I − G22 K)−1 .

(3.1)

Then U ∈ AnY ×nY , V ∈ AnU ×nY , det U = 0 ∈ A, K = V U −1 and U − G22 V = I. Furthermore, Θ(G, K) can then be written as ⎡ ⎤ G11 + G12 V G21 G12 + G12 V G22 G12 V ⎦ . (3.2) V G21 I + V G22 V Θ(G, K) = Θ(G; U, V ) := ⎣ U G21 U G22 U It is not hard to see that if U ∈ AnY ×nY and V ∈ AnU ×nY are such that det U = 0, U − G22 V = I and (3.2) is stable, i.e., in A(nZ +nU +nY )×(nW +nU +nY ) , then K = V U −1 is a stabilizing controller. A dual result holds if we set = (I − KG22 )−1 U

and V = (I − KG22 )−1 K.

(3.3)

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= 0 ∈ A, K = U −1 V , U − V G22 = I ∈ AnU ×nU , V ∈ AnU ×nY , det U In that case U and we can write Θ(G, K) as ⎡ ⎤ G11 + G12 V G21 G12 U G12 V ⎥ , V ) = ⎢ Θ(G, K) = Θ(G; U (3.4) U V V G21 ⎣ ⎦, I + G22 V (I + G22 V )G21 G22 U = 0 and ∈ AnU ×nU and V ∈ AnU ×nY with det U while conversely, for any U −1 V is a U − V G22 = I and such that (3.4) is stable, we have that K = U stabilizing controller. This leads to the following ﬁrst-step more linear reformulation of the deﬁnition of internal stabilization. Theorem 3.1. A plant G deﬁned by a transfer matrix G ∈ K(nZ +nY )×(nW +nU ) is internally stabilizable if and only if one of the following equivalent assertions holds: 1. There exists L = [ VU ] ∈ A(nU +nY )+nY with det U = 0 such that: (a) The block matrix (3.2) is stable (i.e., has all matrix entries in A), and (b) −G22 I L = I. Then the controller K = V U −1 internally stabilizes the plant G and we have: U = (I − G22 K)−1 ,

V = K(I − G22 K)−1 .

= [ U −V ] ∈ AnU ×(nU +nY ) with det U = 0 such that: 2. There exists L (a) The block matrix (3.4) is stable (i.e., has all matrix entries in A), and I := [ U −V ] I = I. (b) L G22 G22 −1 V internally stabilizes the If this is the case, then the controller K = U plant G and we have: = (I − KG22 )−1 , U

V = (I − KG22 )−1 K.

With this result in hand, we are able to get a parametrization for the set of all stabilizing controllers in terms of an assumed particular stabilizing controller. Theorem 3.2. 1. Let K∗ ∈ KnU ×nY be a stabilizing controller for G ∈ K(nZ +nY )×(nW +nU ) . Deﬁne U∗ = (I − G22 K∗ )−1 and V∗ = K(I − G22 K∗ )−1 . Then the set of all stabilizing controllers is given by K = (V∗ + Q)(U∗ + G22 Q)−1 ,

(3.5)

where Q ∈ KnU ×nY is an element of the set ⎧ ⎫ ⎡ ⎤ G12 ⎨ ⎬ Ω := Q ∈ KnU ×nY : ⎣ I ⎦ Q G21 G22 I ∈ A(nZ +nU +nY )×(nW +nU +nY ) ⎩ ⎭ G22 (3.6) such that in addition det(U∗ + G22 Q) = 0.

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2. Let K∗ ∈ KnU ×nY be a stabilizing controller for G ∈ K(nZ +nY )×(nW +nU ) . ∗ = (I − K∗ G22 )−1 and V∗ = (I − K∗ G22 )−1 K∗ . Then the set of all Deﬁne U controllers is given by ∗ + QG22 )−1 (V∗ + Q), K = (U

(3.7)

where Q ∈ KnU ×nY is an element of the set Ω (3.6) such that in addition ∗ + QG22 ) = 0. det(U ∗ + QG22 ) = 0, Moreover, if Q ∈ Ω, that det(U∗ + G22 Q) = 0 if and only if det(U and the formulas (3.5) and (3.7) give rise to the same controller K. Proof. By Theorem 3.1, if K is a stabilizing controller for G, then K has the form K = V U −1 with L = [ VU ] as in part (1) of Theorem 3.1 and then Θ(G, K) is as in (3.2). Similarly Θ(G, K∗ ) is given as Θ(G; U∗ , V∗ ) in (3.2) with U∗ , V∗ in place of U, V . As by assumption Θ(G; U∗ , V∗ ) is stable, it follows that Θ(G; U, V ) is stable if and only if Θ(G; U, V )− Θ(G; U∗ , V∗ ) is stable. Let Q = V − V∗ ; as U = I + G22 V and U∗ = I + G22 V∗ , it follows that U − U∗ = G22 Q. From (3.2) we then see that the stable quantity Θ(G; U, V ) − Θ(G; U∗ , V∗ ) is given by ⎤ ⎡ G12 Θ(G; U, V ) − Θ(G; U∗ , V∗ ) = ⎣ I ⎦ Q G21 G22 I . G22 Thus K = V U −1 = (V∗ + (V − V∗ ))(U∗ + (U − U∗ ))−1 = (V∗ + Q)(U∗ + G22 Q)−1 , where Q is an element of Ω such that det(U∗ + G22 Q) = 0. Conversely, suppose K has the form K = (V∗ + Q)(U∗ + G22 Q)−1 where Q ∈ Ω and det(U∗ + G22 Q) = 0. Deﬁne V = V∗ + Q, U = U∗ + G22 Q. Then one easily checks that ⎡ ⎤ G12 Θ(G; U, V ) = Θ(G; U∗ , V∗ ) + ⎣ I ⎦ Q G21 G22 I G22 is stable and V V∗ Q −G22 I = −G22 I = I + 0 = I. + −G22 I U G22 Q U∗ So K = V U −1 stabilizes G by part (1) of Theorem 3.1. This completes the proof of the ﬁrst statement of the theorem. The second part follows in a similar way by using the second statement in Theorem 3.1 and Q = V − V∗ . Finally, since V = V and ∗ +QG22 ) = 0, V∗ = V∗ , we ﬁnd that indeed det(U∗ +G22 Q) = 0 if and only if det(U and the formulas (3.5) and (3.7) give rise to the same controller K. The drawback of the parametrization of the stabilizing controllers in Theorem 3.2 is that the set Ω is not really a free-parameter set. By deﬁnition, Q ∈ Ω if Q

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itself is stable (from the (1,3) entry in the deﬁning matrix for the Ω in (3.6)), but, in addition, the eight additional transfer matrices G12 QG21 , G12 QG22 , G12 Q, QG21 , QG22 , G22 QG21 , G22 QG22 , G22 Q should all be stable as well. The next lemma shows how the parameter set Ω can in turn be parametrized by a free stable parameter Λ of size (nU + nY )× (nU + nY ). Lemma 3.3. Assume that G is stabilizable and that K∗ is a particular stabilizing controller for G. Let Q ∈ KnU ×nY . Then the following are equivalent: (i) Q is anelement of the set Ω in (3.6), I (ii) Q G22 I is stable, G22 (iii) Q has the form Q = LΛL for a stable free-parameter Λ ∈ A(nU +nY )×(nU +nY ) , nU ×(nU +nY ) and L ∈ A(nU +nY )×nY are given by where L ∈ A −1 ∗) = (I − K∗ G22 )−1 −(I − K∗ G22 )−1 K∗ , L = −K∗ (I − G22 K−1 L . (I − G22 K∗ ) (3.8) I Proof. The implication (i) =⇒ (ii) is obvious. Suppose that Λ = G22 Q [ G22 I ] is stable. Note that I Q LΛL = (I − K∗ G22 )−1 −(I − K∗ G22 )−1 K∗ G22 −K∗ (I − G22 K∗ )−1 × G22 I (I − G22 K∗ )−1 = Q. Hence (ii) implies (iii). Finally assume Q = LΛL for a stable Λ. To show that Q ∈ Ω, as Λ is stable, it suﬃces to show that ⎤ ⎡ G12 is stable, and L2 := L G21 G22 I is stable. L1 := ⎣ I ⎦ L G22 from (3.8), gives Spelling out L1 , using the deﬁnition of L ⎡ ⎤ G12 L1 = ⎣ I ⎦ (I − K∗ G22 )−1 −(I − K∗ G22 )−1 K∗ . G22 We note that each of the six matrix entries of L1 are stable, since they all occur among the matrix entries of Θ(G, K∗ ) (see (2.2)) and K∗ stabilizes G by assumption. Similarly, each of the six matrix entries of L2 given by −K∗ (I − G22 K∗ )−1 G21 G22 I L2 = (I − G22 K∗ )−1 is stable since K∗ stabilizes G. It therefore follows that Q ∈ Ω as wanted.

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We say that K stabilizes G22 if the map [ vv12 ] → [ uy ] in Figure 1 is stable, i.e., the usual stability holds with w = 0 and z ignored. This amounts to the stability of the lower right 2 × 2 block in Θ(G, K): (I − KG22 )−1 (I − KG22 )−1 K . G22 (I − KG22 )−1 I + G22 (I − KG22 )−1 K The equivalence of (i) and (ii) in Lemma 3.3 implies the following result. Corollary 3.4. Assume that G is stabilizable. Then K stabilizes G if and only if K stabilizes G22 . Proof. Assume K∗ ∈ KnU ×nY stabilizes G. Then in particular the lower left 2 × 2 block in Θ(G, K∗ ) is stable. Thus K∗ stabilizes G22 . Moreover, K stabilizes G22 if and only if K stabilizes G when we impose G11 = 0, G12 = 0 and G21 = 0, that is, K is of the form (3.5) with U∗ and V∗ as in Theorem 3.2 and Q ∈ KnU ×nY is such that GI22 Q [ G22 I ] is stable. But then it follows from the implication (ii) =⇒ (i) in Lemma 3.3 that Q is in Ω, and thus, by Theorem 3.2, K stabilizes G (without G11 = 0, G12 = 0, G21 = 0). Combining Lemma 3.3 with Theorem 3.2 leads to the following generalization of Theorem 2.1 giving a parametrization of stabilizing controllers without the assumption of any coprime factorization. Theorem 3.5. Assume that G ∈ K(nZ +nY )×(nW +nU ) is stabilizable and that K∗ is one stabilizing controller for G. Deﬁne U∗ = (I − G22 K∗ )−1 , V∗ = K∗ (I − ∗ = (I − K∗ G22 )−1 and V∗ = (I − K∗ G22 )−1 K∗ . Then the set of all G22 K∗ )−1 , U stabilizing controllers for G are given by ∗ + QG22 )−1 (V∗ + Q), K = (V∗ + Q)(U∗ + G22 Q)−1 = (U and L are given by (3.8) and Λ is a free stable parameter where Q = LΛL where L ∗ + of size (nU + nY ) × (nU + nY ) so that det(U∗ + G22 Q) = 0 or equivalently det(U QG22 ) = 0. 3.2. The Youla-Kuˇcera parametrization There are two drawbacks to the parametrization of the stabilizing controllers obtained in Theorem 3.5, namely, to ﬁnd all stabilizing controllers one ﬁrst has to ﬁnd a particular stabilizing controller, and secondly, the map Λ → Q given in Part (iii) of Lemma 3.3 is in general not one-to-one. We now show that, under the additional hypothesis that G22 admits a double coprime factorization, both issues can be remedied, and we are thereby led to the well-known Youla-Kuˇcera parametrization for the stabilizing controllers. Recall that G22 has a double coprime factorization in case there exist stable N, X and Y so that the determinants of D, D, transfer matrices D, N , X, Y , D, X and X are all nonzero (in A) and D −N X N 0 InY −1 −1 . (3.9) G22 = D N = N D , = 0 InU −Y X Y D

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According to Corollary 3.4 it suﬃces to focus on describing the stabilizing controllers of G22 . Note that K stabilizes G22 means that (I − KG22 )−1 (I − KG22 )−1 K G22 (I − KG22 )−1 I + G22 (I − KG22 )−1 K nU ×nY is stable, or, by Theorem 3.2, that K is given by (3.5) or (3.7) for some Q ∈ K so that GI22 Q [ G22 I ] is stable. In case G22 has a double coprime factorization Quadrat shows in [120, Proposition 4] that the equivalence of (ii) and (iii) in Lemma 3.3 has the following reﬁnement. We provide a proof for completeness.

Lemma 3.6. Suppose that G22 has a double coprime factorization (3.9). Let Q ∈ for some Λ ∈ KnU ×nY . Then GI22 Q [ G22 I ] is stable if and only if Q = DΛD nU ×nY A . Proof. Let Q = DΛD for some Λ ∈ AnU ×nY . Then DΛN DΛD Q QG22 I = Q G22 I = ΛN N ΛD . G22 QG22 G22 Q G22 N Hence GI22 Q [ G22 I ] is stable. −1 QD−1 . Then Conversely, assume that GI22 Q [ G22 I ] is stable. Set Λ = D and Y the transfer matrices from the coprime factorization (3.9) we with X, Y , X have D −Y −Y N D Λ = Λ X X N DΛN DΛD −Y = X −Y ΛD X NΛN N QG22 Q −Y −Y = . X G22 QG22 G22 Q X Thus Λ is stable.

Lemma 3.7. Assume that G22 admits a double coprime factorization (3.9). Then K0 is a stabilizing controller for G22 if and only if there exist X0 ∈ AnY ×nY , 0 ∈ AnU ×nU and Y0 ∈ AnU ×nY with det(X0 ) = 0, det(X 0 ) = 0 so Y0 ∈ AnU ×nY , X −1 −1 that K0 = Y0 X0 = X0 Y0 and D −N 0 InY X0 N . 0 = 0 InU −Y0 X Y0 D −1 Y is a stabilizing controller for G22 , where In particular, K = Y X −1 = X Y come from the double coprime factorization (3.9) for G22 . X, Y, X,

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Proof. Note that if K is a stabilizing controller for G22 , then, in particular, −1 I −K (I − KG22 )−1 K(I − G22 K)−1 = (3.10) −G22 I (I − G22 K)−1 G22 (I − G22 K)−1 is stable. The above identity makes sense, irrespectively of K being a stabilizing and Y be the controller, as long as the left-hand side is invertible. Let X, Y , X −1 Y = Y X −1 . transfer matrices from the double coprime factorization. Set K = X Then we have −1 −1 0 −Y −Y −1 X 0 X X X = 0 D −1 0 D −N D −N D −1 −1Y I −K I −X = = . −G22 I I −D −1 N −1 Y D D and X −Y Since X, = X are stable, it follows that the right-hand side −N D N −1 Y = Y X −1 stabilizes G22 . of (3.10) is stable as well. We conclude that K = X Now let K0 be any stabilizing controller for G22 . It follows from the ﬁrst part −1 Y is stabilizing for G22 . Deﬁne V and U by of the proof that K = Y X −1 = X (3.1) and V and U by (3.3). Then, using Theorem 3.2 and Lemma 3.6, there exists a Λ ∈ AnU ×nY so that + QG22 )−1 (V + Q), K0 = (V + Q)(U + G22 Q)−1 = (U where Q = DΛD. We compute that (I − G22 K)−1 = (I − D−1 N Y X −1)−1 = X(DX − N Y )−1 D = XD

(3.11)

−1 Y N X D − Y N )−1 X D −1 )−1 = D( =D X. (I − KG22 )−1 = (I − X

(3.12)

and

Thus V = Y D,

U = XD,

Y , V = D

=D X. U

Therefore K0

ΛD)−1 (3.13) = (V + Q)(U + G22 Q)−1 = (Y D + DΛD)(XD +N −1 = (Y + DΛ)(X + NΛ)

K0

+ QG22 )−1 (V + Q) = (D X + DΛN Y + DΛD) = (U )−1 (D (3.14) −1 = (X + ΛN ) (Y + ΛD).

and

Set Y0 = (Y + DΛ),

Λ), X0 = (X + N

Y0 = (Y + ΛD),

0 = (X + ΛN ). X

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0 = 0, and we have Then certainly det X0 = 0 and det X Λ N D −N D −N X0 N X +N 0 + ΛN = −Y − ΛD X −Y0 X Y0 D Y + DΛ D D −N X N I 0 I 0 I 0 I 0 = = Λ I −Λ I Λ I −Λ I −Y X Y D I 0 . = 0 I Since any stabilizing controller for G is also a stabilizing controller for G22 , the following corollary is immediate. Corollary 3.8. Assume that G ∈ K(nZ +nY )×(nW +nU ) is a stabilizable and that G22 admits a double coprime factorization. Then any stabilizing controller K of G admits a double coprime factorization. Lemma 3.9. Assume that G is stabilizable and that G22 admits a double coprime factorization. Then there exists a double coprime factorization (3.9) for G22 so are stable. that DG21 and G12 D Proof. Let K be a stabilizing controller for G. Then K is also a stabilizing controller for G22 . Thus, according to Lemma 3.7, there exists a double coprime −1 Y −1 . Note that (3.9) implies =X factorization (3.9) for G22 so that K = Y X D −N Y = Y D and N X = XN . Moreover, = I. In particular, D that X N X −Y Y D

from the computations (3.11) and (3.12) we see that (I − G22 K)−1 = XD

X. and (I − KG22 )−1 = D

Inserting these identities into the formula for Θ(G, K), and using that K stabilizes G, we ﬁnd that ⎡ ⎤ X G12 D Y G11 + G12 Y DG21 G12 D ⎢ ⎥ X Y Θ(G, K) = ⎣ D D Y DG21 ⎦ is stable. XDG21 NX I + NY X G12 D Y is stable, and thus In particular G12 D N X G12 D Y G12 D = G12 D(X N − Y D) = G12 D −D Y DG21 is stable. Similarly, since XDG is stable, we ﬁnd that 21 Y DG21 −N D = (−N Y + DX)DG21 = DG21 XDG21 is stable.

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We now present an alternative proof of Corollary 3.4 for the case that G22 admits a double coprime factorization. Lemma 3.10. Assume that G is stabilizable and G22 admits a double coprime factorization. Then K stabilizes G if and only if K stabilizes G22 . Proof. It was already noted that in case K stabilizes G, then K also stabilizes G22 . Now assume that K stabilizes G22 . Let Q ∈ KnU ×nY so that K is given by (3.5). It suﬃces to show that Q ∈ Ω, with Ω deﬁned by (3.6). Since G is stabilizable, it follows from Lemma 3.9 that there exists a double coprime factorization (3.9) of are stable. According to Lemma 3.6, Q = DΛD G22 so that DG21 and G12 D for nU ×nY some Λ ∈ A . It follows that ⎡ ⎤ ⎤ ⎡ G D G12 12 ⎢ ⎣ I ⎦ Q G21 G22 I ⎥ = ⎣ D ⎦ Λ DG21 DG22 D G22 G22 D ⎡ ⎤ G12 D ⎢ ⎥ = ⎣ D ⎦ Λ DG21 N D N is stable. Hence Q ∈ Ω.

Combining the results from the Lemmas 3.6, 3.7 and 3.10 with Theorem 3.2 and the computations (3.13) and (3.14) from the proof of Lemma 3.7 we obtain the Youla-Kuˇcera parametrization of all stabilizing controllers. Theorem 3.11. Assume that G ∈ K(nZ +nY )×(nW +nU ) is stabilizable and that G22 admits a double coprime factorization (3.9). Then the set of all stabilizing controllers is given by + ΛN )−1 (Y + ΛD), Λ)−1 = (X K = (Y + DΛ)(X +N where Λ is a free stable parameter from AnU ×nY such that det(X + NΛ) = 0 or + ΛN ) = 0. equivalently det(X 3.3. The standard H ∞ -problem reduced to model matching. 11 G12 We now consider the H ∞ -problem for a plant G = G G21 G22 : W ⊕ U → Z ⊕ Y, i.e., we seek a controller K : Y → U so that not only Θ(G, K) in (2.2) is stable, but also G11 + G12 K(I − G22 K)−1 G21 op ≤ 1. Assume that the plant G is stabilizable, and that K∗ : Y → U stabilizes G. ∗ and V∗ as in Theorem 3.2. We then know that all stabilizing Deﬁne U∗ , V∗ , U controllers of G are given by ∗ + QG22 )−1 (V∗ + Q) K = (V∗ + Q)(U∗ + G22 Q)−1 = (U

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where Q ∈ KnU ×nY is any element of Ω in (3.6). We can then express the transfer matrices U and V in (3.1) in terms of Q as follows: U

= (I − G22 K)−1 = (I − G22 (V∗ − Q)(U∗ − G22 Q)−1 )−1 = (U∗ − G22 Q)(U∗ − G22 Q − G22 (V∗ − Q))−1 = (U∗ − G22 Q)(U∗ − G22 V∗ )−1 = (U∗ − G22 Q),

where we used that U∗ − G22 V∗ = I, and V = KU = V∗ − Q. Similar computations provide the formulas =U ∗ + QG22 and V = V∗ + Q U and V in (3.3). Now recall that Θ(G, K) can be exfor the transfer matrices U pressed in terms of U and V as in (3.2). It then follows that left upper block in Θ(G, K) is equal to G11 + G12 K(I − G22 K)−1 G21

= G11 + G12 V G21 (3.15) = G11 + G12 V∗ G21 − G12 QG21 . 11 := G11 + G12 V∗ G21 is stable, and The fact that K∗ stabilizes G implies that G thus G12 QG21 is stable as well. We are now close to a reformulation of the H ∞ problem as a model matching problem. However, to really formulate it as a model matching problem, we need to apply the change of design parameter Q → Λ deﬁned in Lemma 3.3, or Lemma 3.6 in case G22 admits a double coprime factorization. The next two results extend the idea of Theorem 2.2 to this more general setting. Theorem 3.12. Assume that G ∈ K(nZ +nY )×(nW +nU ) is stabilizable and let K ∈ KnU ×nY . Then K is a solution to the standard H ∞ problem for G if and only if ∗ + QG22 )−1 (V∗ + Q) K = (V∗ + Q)(U∗ + G22 Q)−1 = (U and L are deﬁned by (3.8), so that det(U∗ + G22 Q) = 0, with Q = LΛL, where L or equivalently det(U∗ + QG22 ) = 0, and Λ ∈ A(nU +nY )×(nU +nY ) is any solution 12 and G 21 deﬁned by 11 , G to the model matching problem for G 12 := G12 L, 21 := LG21 , 11 := G11 + G12 V∗ G21 , G G G i.e., so that

12 ΛG 21 op ≤ 1. 11 + G G

Proof. The statement essentially follows from Theorem 3.5 and the computation 11 , G 12 and G 21 satisfy (3.15) except that we need to verify that the functions G the conditions to be data for a model matching problem, that is, they should be 12 and G 21 are 11 is stable. The fact that G stable. It was already observed that G stable was shown in the proof of Lemma 3.3. We have a similar result in case G22 admits a double coprime factorization.

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Theorem 3.13. Assume that G ∈ K(nZ +nY )×(nW +nU ) is stabilizable and that G22 admits a double coprime factorization (3.9). Let K ∈ KnY ×nU . Then K is a solution to the standard H ∞ problem for G if and only if + ΛN )−1 (Y + ΛD), Λ)−1 = (X K = (Y + DΛ)(X +N Λ) = 0, or equivalently det(X + ΛN ) = 0, where Λ ∈ AnU ×nY so that det(X + N 12 and G 21 deﬁned by 11 , G is any solution to the model matching problem for G 11 := G11 + G12 Y DG21 , G

12 := G12 D, G

21 := DG21 , G

i.e., so that 12 ΛG 21 op ≤ 1. 11 + G G Proof. The same arguments apply as in the proof of Theorem 3.12, except that in 12 and G 21 are stable. this case Lemma 3.9 should be used to show that G 3.4. Notes The development in Section 3.1 on the parametrization of stabilizing controllers without recourse to a double coprime factorization of G22 is based on the exposition of Quadrat [120]. It was already observed by Zames-Francis [140] that Q = K(I − G22 K)−1 can be used as a free stable design parameter in case G22 is itself already stable; in case G22 is not stable, Q is subject to some additional interpolation conditions. The results of [120] is an adaptation of this observation to the general ring-theoretic setup. The more theoretical papers [118, 122] give module-theoretic interpretations for the structure associated with internal stabilizability. In particular, it comes out that every matrix transfer function G22 with entries in K has a double-coprime factorization if and only if A is a Bezout domain, i.e., every ﬁnitely generated ideal in A is principal; this recovers a result already appearing in the book of Vidyasagar [136]. A new result which came out of this module-theoretic interpretation was that internal stabilizability of a plant G22 is equivalent to the existence of a double-coprime factorization for G22 exactly when the ring A is projective-free, i.e., every submodule of a ﬁnitely generated free module over A must itself be free. This gives an explanation for the earlier result of Smith [130] that this phenomenon holds for the case where A is equal H ∞ over the unit disk or right half-plane. Earlier less complete results concerning parametrization of the set of stabilizing controllers without the assumption of a coprime factorization were obtained by Mori [102] and Sule [132]. Mori [103] also showed that the internal-stabilization problem can be reduced to model matching form for the general case where the 11 G12 plant has the full 2 × 2-block structure G = G G21 G22 . Lemma 3.10 for the classical case is Theorem 2 on page 35 in [64]. The proof there relies in a careful analysis of signal-ﬂow diagrams; we believe that our proof is more direct.

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4. Feedback control for linear time-invariant multidimensional systems 4.1. Multivariable frequency-domain formulation The most obvious multivariable analogue of the classical single-variable setting considered in the book of Francis [64] is as follows. We take the underlying ﬁeld to be the complex numbers C; in the engineering applications, one usually requires that the underlying ﬁeld be the reals R, but this can often be incorporated at the end by using the characterization of real rational functions as being those complex rational functions which are invariant under the conjugation operator s(z) → s(z). We let Dd = {z = (z1 , . . . , zd ) : |zk | < 1} be the unit polydisk in the d-dimensional complex space Cd and we take our ring A of stable plants to be the ring C(z)s of all rational functions s(z) = p(z) q(z) in d variables (thus, p and q are polynomials in the d variables z1 , . . . , zd where we set z = (z1 , . . . , zd )) such that s(z) is bounded on the polydisk Dd . The ring C[z] of polynomials in d variables is a unique factorization domain so we may assume that p and q have no common factor (i.e., that p and q are relatively coprime) in the fractional representation s = pq for any element of C(z1 , . . . , zd ). Unlike in the single-variable case, for the case d > 1 it can happen that p and q have common zeros in Cd even when they are coprime in C[z] (see [138] for an early analysis of the resulting distinct notions of coprimeness). It turns out that for d ≥ 3, the ring C(z)s is diﬃcult to work with since the denominator q for a stable ring element depends in a tricky way on the numerator p: if s ∈ C(z)s has coprime fractional representation s = pq , while it is the case that necessarily q has no zeros in the open polydisk Dd , it can happen that the zero variety of q touches the boundary ∂Dd as long as the zero variety of p also touches the same points on the boundary in such a way that the quotient s = pq remains bounded on Dd . Note that at such a boundary point ζ, the quotient s = p/q has no well-deﬁned value. In the engineering literature (see, e.g., [45, 131, 84]), such a point is known as a nonessential singularity of the second kind. To avoid this diﬃculty, Lin [92, 93] introduced the ring C(z)ss of structured stable rational functions, i.e., rational functions s ∈ C(z) so that the denominator q in any coprime fractional representation s = pq for s has no zeros in the closed d

noz [84], whenever polydisk D . According to the result of Kharitonov-Torres-Mu˜ s = pq ∈ C(z)s is stable in the ﬁrst (non-structured) sense, an arbitrarily small perturbation of the coeﬃcients of q may lead to the perturbed q having zeros in the open polydisk Dd resulting in the perturbed version s = pq of s being unstable; this phenomenon does does not occur for s ∈ C(z)ss , and thus structured stable can be viewed just as a robust version of stable (in the unstructured sense). Hence one can argue that structured stability is the more desirable property from an engineering perspective. In the application to delay systems using the systemsover-rings approach [46, 85, 83], on the other hand, it is the collection C(z)ss of structurally stable rational functions which comes up in the ﬁrst place.

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39

As the ring A = C(z)ss is a commutative integral domain, we can apply the results of Section 3 to this particular setting. It was proved in connection with work on systems-over-rings rather than multidimensional systems (see [46, 83]) that the ring C(z)ss is projective-free. As pointed out in the notes of Section 3 above, it follows that stabilizability of G22 is equivalent to the existence of a double coprime factorization for the plant G22 (see [119]), thereby settling a conjecture of Lin [92, 93, 94]. We summarize these results as follows. 11 G12 over the quoTheorem 4.1. Suppose that we are given a system G = G G21 G22 tient ﬁeld Q(C(z)ss ) of the ring C(z)ss of structurally stable rational functions in −1 Y which internally stad variables. If there exists a controller K = Y X −1 = X bilizes G, then G22 has a double coprime factorization and all internally stabilizing controllers K for G are given by the Youla-Kuˇcera parametrization. Following Subsection 3.3, the Youla-Kuˇcera parametrization can then be used to rewrite the H ∞ -problem in the form of a model-matching problem: Given T1 , T2 , T3 equal to matrices over C(z)ss of respective sizes nZ × nW , nW × nU and nY × nW , ﬁnd a matrix Λ over C(z)ss of size nU × nY so that the aﬃne expression S given by S = T1 + T2 ΛT3 (4.1) d

has supremum norm at most 1, i.e., S∞ = max{S(z) : z ∈ D } ≤ 1. For mathematical convenience we shall now widen the class of admissible solutions and allow Λ1 , . . . , ΛJ to be in the algebra H ∞ (Dd ) of bounded analytic functions on Dd . The unit ball of H ∞ (Dd ) is the set of all holomorphic functions S mapping the polydisk Dd into the closed single-variable unit disk D ⊂ C; we denote this space by Sd , the d-variable Schur class. While T1 , T2 and T3 are assumed to be in C(z)ss , we allow Λ in (4.1) to be in H ∞ (Dd ). Just as in the classical one-variable case, it is possible to give the modelmatching form (4.1) an interpolation interpretation, at least for special cases (see [73, 74, 32]). One such case is where nW = nZ = nY = 1 while nU = J. Then T1 and T3 are scalar while T2 = [ T2,1 ··· T2,J ] is a row. Assume in addition that T3 = 1. Then the model-matching form (4.1) collapses to S = T1 + T21 Λ1 + · · · + T2J ΛJ

(4.2)

where Λ1 , . . . ΛJ are J free stable scalar functions. Under the assumption that the d intersection of the zero varieties of T2,1 , . . . , T2,J within the closed polydisk D consists of ﬁnitely many (say N ) points z1 = (z1,1 , . . . , z1,d ), · · · , zN = (zN,1 , . . . , zN,d ) and if we let w1 , . . . , wN be the values of T1 at these points w1 = T1 (z1 ), . . . , wN = T1 (zN ), then it is not hard to see that a function S ∈ C(z)ss has the form (4.2) if and only if it satisﬁes the interpolation conditions S(zi ) = wi for i = 1, . . . , N.

(4.3)

40

J.A. Ball and S. ter Horst

In this case the model-matching problem thus becomes the following ﬁnite-point Nevanlinna-Pick interpolation problem over Dd : ﬁnd S ∈ C(z)ss subject to |S(z)| ≤ 1 for all z ∈ Dd which satisﬁes the interpolation conditions (4.3). Then the dvariable H ∞ -Model-Matching problem becomes: ﬁnd S ∈ Sd so that S(z1 ) = w1 for i = 1, . . . , N . A second case (see [32]) where the polydisk Model-Matching Problem can be reduced to an interpolation problem is the case where T2 and T3 are square (so nZ = nU and nY = nW ) with invertible values on the distinguished boundary of the polydisk; under these assumptions it is shown in [32] (see Theorem 3.5 there) how the model-matching problem is equivalent to a bitangential Nevanlinna-Pick interpolation problem along a subvariety, i.e., bitangential interpolation conditions are speciﬁed along all points of a codimension-1 subvariety of Dd (namely, the union of the zero sets of det T2 and det T3 intersected with Dd ). For d = 1, codimension-1 subvarieties are isolated points in the unit disk; thus the codimension-1 interpolation problem is a direct generalization of the bitangential Nevanlinna-Pick interpolation problem studied in [26, 58, 62]. However for the case where the number of variables d is at least 3, there is no theory with results parallel to those of the classical case. Nevertheless, if we change the problem somewhat there is a theory parallel to the classical case. To formulate this adjustment, we deﬁne the d-variable SchurAgler class SAd to consist of those functions S analytic on the polydisk for which the operator S(X1 , . . . , Xd ) has norm at most 1 for any collection X1 , . . . , Xd of d commuting strict contraction operators on a separable Hilbert space K; here S(X1 , . . . , Xd ) can be deﬁned via the formal power series for S: S(X1 , . . . , Xd ) = sn X n , if S(z) = sn z n n∈Zd +

n∈Zd +

where we use the standard multivariable notation n = (n1 , . . . , nd ) ∈ Zd+ ,

X n = X1n1 · · · Xdnd and z n = z1n1 · · · zdnd .

For the cases d = 1, 2, it turns out, as a consequence of the von Neumann inequality or the Sz.-Nagy dilation theorem for d = 1 and of the Andˆ o dilation theorem [17] for d = 2 (see [109, 34] for a full discussion), that the Schur-Agler class SAd and the Schur class Sd coincide, while, due to an explicit example of Varopoulos, the inclusion SAd ⊂ Sd is strict for d ≥ 3. There is a result due originally to Agler [1] and developed and reﬁned in a number of directions since (see [3, 35] and [4] for an overview) which parallels the one-variable case; for the case of a simple set of interpolation conditions (4.3) the result is as follows: there exists a function S in the Schur-Agler class SAd which satisﬁes the set of interpolation conditions S(zi ) = wi for i = 1, . . . , N if and only if there exist d positive semideﬁnite matrices P(1) , . . . , P(d) of size N × N so that 1 − wi wj =

d k=1

(k)

(1 − zi,k zj,k )Pi,j .

Control and Interpolation

41

1−w w N For the case d = 1, the Pick matrix P = 1−zii zjj i,j=1 is the unique solution of this equation, and we recover the classical criterion P ≥ 0 for the existence of solutions to the Nevanlinna-Pick problem. There is a later realization result of Agler [2] (see also [3, 35]): a given holomorphic function S is in the Schur-Agler class SAd (L(U, Y)) if and only if S has a contractive Givone-Roesser realization: d A B ] : (⊕d S(z) = D + C(I − Z(z)A)−1 Z(z)B where [ C k=1 Xk ⊕ U) → (⊕k=1 Xk ⊕ Y) z1 IX D 1

is contractive with Z(z) =

..

.

. zd IXd

Direct application of the Agler result to the bitangential Nevanlinna-Pick interpolation problem along a subvariety, however, gives a solution criterion involving an inﬁnite Linear Matrix Inequality (where the unknown matrices have inﬁnitely many rows and columns indexed by the points of the interpolation-node subvariety) – see [32, Theorem 4.1]. Alternatively, one can use the polydisk Commutant Lifting Theorem from [31] to get a solution criterion involving a Linear Operator Inequality [32, Theorem 5.2]. Without further massaging, either approach is computationally unattractive; this is in contrast with the state-space approach discussed below. In that setting there exists computable suﬃcient conditions, in terms of a pair of LMIs and a coupling condition, that in general are only suﬃcient, unless one works with a more conservative notion of stability and performance. 4.2. Multidimensional state-space formulation The starting point in this subsection is a quadruple {A,XB, C, D} consisting of X A B] : operators A, B, C and D so that [ C → W⊕U Z⊕Y and a partitioning D X = X1 ⊕ · · · ⊕ Xd of the space X . Associate with such a quadruple {A, B, C, D} is a linear state-space system Σ of Givone-Roesser type (see [67]) that evolves over Zd+ and is given by the system of equations ⎧ x (n+e ) x1 (n) 1 1 ⎪ ⎨ .. .. =A + Bu(n) . . (n ∈ Zd+ ), (4.4) Σ := xd (n+ed ) xd (n) ⎪ ⎩ y(n) = Cx(n) + Du(n) " with initial conditions a speciﬁcation of the state values xk ( j=k tj ej ) for t = (t1 , . . . , td ) ∈ Zd+ subject to tk = 0 where k = 1, . . . , d. Here ek stands for the kth x1 (n) .. unit vector in Cd and x(n) = . We call X the state-space and A the state . xd (n)

A B ] is referred to as the system operator. Moreover, the block operator matrix [ C D matrix. Following [81], the Givone-Roesser system (4.4) is said to be asymptotically stable in case, for zero input u(n) = 0 for n ∈ Zd+ and initial conditions with the property d # $ %# # # sup #xk tj ej # < ∞ for k = 1, . . . , d, t∈Zd + : tk =0

j=1

42

J.A. Ball and S. ter Horst

the state sequence x satisﬁes sup x(n) < ∞ n∈Zd +

and

lim x(n) = 0,

n→∞

where n → ∞ is to be interpreted as min{n1 ,...,nd } → ∞ when n = (n1 ,...,nd ) ∈ Zd+ . With the Givone-Roesser system (4.4) we associate the transfer function G(z) given by G(z) = D + C(I − Z(z)A)−1 Z(z)B, deﬁned at least for z ∈ Cd with z suﬃciently small, where ⎤ ⎡ z1 IX1 ⎥ ⎢ d .. Z(z) = ⎣ ⎦ (z ∈ C ). . zd IXd

(4.5)

(4.6)

We then say that {A, B, C, D} is a (state-space) realization for the function G, or if G is not speciﬁed, just refer to {A, B, C, D} as a realization. The realization {A, B, C, D}, or just the operator A, is said to be Hautus-stable in case the pencil d I − Z(z)A is invertible on the closed polydisk D . Here we only consider the case that X is ﬁnite-dimensional; then the entries of the transfer function G are in the quotient ﬁeld Q(C(z)ss ) of C(z)ss and are analytic at 0, and it is straightforward to see that G is structurally stable in case G admits a Hautus-stable realization. For the case d = 2 it has been asserted in the literature [81, Theorem 4.8] that asymptotic stability and Hautus stability are equivalent; presumably this assertion continues to hold for general d ≥ 1 but we do not go into details here. Given a realization {A, B, C, D} where the decomposition X = X1 ⊕ · · · ⊕ Xd is understood, our main interest will be in Hautus-stability; hence we shall say simply that A is stable rather than Hautus-stable. As before we consider controllers K in Q(C(z)ss ) of size nY × nU that we also assume to be given by a state-space realization K(z) = DK + CK (I − ZK (z)AK )−1 ZK (z)BK (4.7) K BK XK → XUK , a decomposition of the state-space with system matrix A CK DK : Y XK = X1,K ⊕ · · · ⊕ Xd,K and ZK (z) deﬁned analogous to Z(z) but with respect to the decomposition of XK . We now further specify the matrices B, C and D from the realization {A, B, C, D} as D11 D12 C1 B B2 , C = B= , D= (4.8) 1 C2 D21 D22 compatible with the decompositions Z ⊕ Y and W ⊕ U. We can then form the closed loop system Gcl = Σ(G, K) of the two transfer functions. The closed loop

Control and Interpolation

43

system Gcl = Σ(G, K) corresponds to the feedback connection ⎡ ⎤ ⎤ ⎡ ⎤⎡ A B1 B2 x x AK BK xK x K ⎣ C1 D11 D12 ⎦ ⎣ w ⎦ → ⎣ z ⎦ , : → CK DK uK yK C2 D21 D22 y u subject to x = Z(z) x,

xK = ZK (z) xK ,

uK = y

and yK = u.

This feedback loop is well posed exactly when I − D22 DK is invertible. Since, under the assumption of well-posedness, one can always arrange via a change of variables that D22 = 0 (cf., [78]), we shall assume that D22 = 0 for the remainder of the paper. In that case well-posedness is automatic and the closed loop system Gcl admits a state-space realization Gcl (z) = Dcl + Ccl (I − Zcl (z)Acl )−1 Zcl (z)Bcl with system matrix

Acl Ccl

⎡ A + B2 DK C2 Bcl BK C2 =⎣ Dcl C1 + D12 DK C2

and

Zcl (z) =

B2 CK AK D12 CK

Z(z) 0 0 ZK (z)

⎤ B1 + B2 DK D21 ⎦ BK D21 D11 + D12 DK D21

(4.9)

(4.10)

(z ∈ Cd ).

The state-space (internal) stabilizability problem then is: Given the realization {A, B, C, D} ﬁnd a compatible controller K with realization {AK , BK , CK , DK } so that the closed-loop realization {Acl , Bcl , Ccl , Dcl } is stable, i.e., so that I − d Zcl (z)Acl is invertible on the closed polydisk D . We also consider the strict state∞ space H -problem: Given the realization {A, B, C, D}, ﬁnd a compatible controller K with realization {AK , BK , CK , DK } so that the closed loop realization {Acl , Bcl , Ccl , Dcl } is stable and the closed-loop system Gcl satisﬁes Gcl (z) < 1 for all z ∈ Dd . State-space stabilizability. In the fractional representation setting of Section 3 it took quite some eﬀort to derive the result: “If G is stabilizable, then K stabilizes G if and only if K stabilizes G22 ” (see Corollary 3.4 and Lemma 3.10). For the state-space stabilizability problem this result is obvious, and what is more, one can drop the assumption that G needs to be stabilizable. Indeed, G22 admits the realization {A, B2 , C2 , 0} (assuming D22 = 0), so that the closed-loop realization for Σ(G22 , K) is equal to {Acl , 0, 0, 0}. In particular, both closed-loop realizations have the same state operator Acl , and thus K with realization {AK , BK , CK , DK } stabilizes G22 if and only if K stabilizes G, without any assumption on the stabilizability of G. The state-space stabilizability problem does not have a clean solution; To discuss the partial results which exist, we ﬁrst introduce some terminology.

44

J.A. Ball and S. ter Horst

Let {A, B, C, D} be a given realization as above with decomposition of B, C and D as in (4.8). The Givone-Roesser output pair {C2 , A} is said to be Hautusdetectable if the block-column matrix

I−Z(z)A C2 d

is of maximal rank nX (i.e., is

left invertible) for all z in the closed polydisk D . We say that {C2 , A} is operatordetectable in case there exists an output-injection operator L : Y → X so that A + LC2 is stable. Dually, the Givone-Roesser input pair {A, B2 } is called Hautusstabilizable if it is the case that the block-row matrix I − AZ(z) B2 has maxd

imal rank nX (i.e., is right invertible) for all z ∈ D , and operator-stabilizable if there is a state-feedback operator F : X → U so that A+B2 F is stable. Notice that both Hautus-detectability and operator-detectability for the pair (C, A) reduce to stability of A in case C = 0. A similar remark applies to stabilizability for an input pair (A, B). We will introduce yet another notion of detectability and stabilizability shortly, but in order to do this we need a stronger notion of stability. We ﬁrst deﬁne D to be the set

X & 1 .. : Xi : Xi → Xi , i = 1, , . . . , d , D= (4.11) . Xd

which is also equal to the commutant of {Z(z) : z ∈ Zd } in the C ∗ -algebra of bounded operators on X . We then say that the realization {A, B, C, D}, or just A, is scaled stable in case there exists an invertible operator Q ∈ D so that Q−1 AQ < 1, or, equivalently, if there exists a positive deﬁnite operator X (notation X > 0) in D so that AXA∗ − X < 0. To see that the two deﬁnitions coincide, take either X = QQ∗ ∈ D, or, when starting with X > 0, factor X as X = QQ∗ for some Q ∈ D. It is not hard to see that scaled stability implies stability. Indeed, assume there exists an invertible Q ∈ D so that Q−1AQ < 1. d Then Z(z)Q−1 AQ = Q−1 Z(z)AQ is a strict contraction for each z ∈ D , and thus d Q−1 (I − Z(z)A)Q = I − Z(z)Q−1 AQ is invertible on D . But then I − Z(z)A d is invertible on D as well, and A is stable. The converse direction, even though asserted in [111, 95], turns out not to be true in general, as shown in [16] via a concrete example. The output pair {C2 , A} is then said to be scaled-detectable if there exists an output-injection operator L : Y → X so that A+LC2 is scaled stable, and the input pair {A, B2 } is called scaled-stabilizable if there exists a state-feedback operator F : X → U so that A + B2 F is scaled stable. While a classical result for the 1-D case states that operator, Hautus and scaled detectability, as well as operator, Hautus and scaled stabilizability, are equivalent, in the multidimensional setting considered here only one direction is clear. Proposition 4.2. Let {A, B, C, D} be a given realization as above with decomposition of B, C and D as in (4.8).

Control and Interpolation

45

1. If the output pair {C2 , A} is scaled-detectable, then {C2 , A} is also operatordetectable. If the output pair {C2 , A} is operator-detectable, then {C2 , A} is also Hautus-detectable. 2. If the input pair {A, B2 } is scaled-stabilizable, then {A, B2 } is also operatorstabilizable. If the input pair {A, B2 } is operator-stabilizable, then {A, B2 } is also Hautus-stabilizable. Proof. Since scaled stability is a stronger notion than stability, the ﬁrst implications of both (1) and (2) are obvious. Suppose that L : Y → X is such that A+LC2 is stable. Then I − Z(z)A I −Z(z)L = I − Z(z)(A + LC2 ) C2 d

is invertible for all z ∈ D from which it follows that {C2 , A} is Hautus-detectable. The last assertion concerning stabilizability follows in a similar way by making use of the identity I I − AZ(z) B2 = I − (A + B2 F )Z(z). −F Z(z) The combination of operator-detectability together with operator-stabilizability is strong enough for stabilizability of the realization {A, B, C, D} and we have the following weak analogue of Theorem 2.4. Theorem 4.3. Let {A, B, C, D} be a given realization as above with decomposition of B, C and D as in (4.8) (with D22 = 0). Assume that {C2 , A} is operatordetectable and {A, B2 } is operator-stabilizable. Then {A, B, C, D} is stabilizable. Moreover, in this case one stabilizing controller is K ∼ {AK , BK , CK , DK } where AK BK A + B2 F + LC2 −L (4.12) = F 0 CK DK where L : Y → X and F : X → U are any operators chosen such that A + LC2 and A + B2 F are stable. Proof. It is possible to motivate these formulas with some observability theory (see [57]) but, once one has the formulas, it is a simple direct check that Acl Bcl A + B2 DK C2 B2 CK = Ccl Dcl BK C2 AK A B2 F . = −LC2 A + B2 F + LC2 It is now a straightforward exercise to check that this last matrix can be put 0 2 in the triangular form A+LC −LC2 A+B2 F via a sequence of block-row/block-column similarity transformations, from which we conclude that Acl is stable as required.

46

J.A. Ball and S. ter Horst

Remark 4.4. A result for the systems-over-rings setting that is analogous to that of Theorem 4.3 is given in [85]. There the result is given in terms of a Hautustype stabilizable/detectable condition; in the systems-over-rings setting, Hautusdetectability/stabilizability is equivalent to operator-detectability/stabilizability (see Theorem 3.2 in [83]) rather than merely suﬃcient as in the present setting (see Proposition 4.2 above). The Hautus-type notions of detectability and stabilizability in principle are checkable using methods from [80]: see the discussion in [83, p. 161]. The weakness of Theorem 4.3 for our multidimensional setting is that there are no checkable criteria for when {C2 , A} and {A, B2 } are operator-detectable and operator-stabilizable since the Hautus test is in general only necessary. An additional weakness of Theorem 4.3 is that it goes in only one direction: we do not assert that operator-detectability of {C2 , A} and operator-stabilizability for {A, B2 } is necessary for stabilizability of {A, B, C, D}. These weaknesses probably explain why apparently this result does not appear explicitly in the control literature. While there are no tractable necessary and suﬃcient conditions for solving the state-space stabilizability problem available, the situation turns out quite differently when working with the more conservative notion of scaled stability. The following is a more complete analogue of Theorem 2.4 combined with Theorem 2.3. Theorem 4.5. Let {A, B, C, D} be a given realization. Then {A, B, C, D} is scaledstabilizable, i.e., there exists a controller K with realization {AK , BK , CK , DK } so that the closed loop state operator Acl is scaled stable, if and only if the input pair {A, B2 } is scaled operator-stabilizable and the output pair {C2 , A} is scaled operator-detectable, i.e., there exist matrices F and L so that A+B2 F and A+LC2 are scaled stable. In this case the controller K given by (4.12) solves the scaledstabilization problem for {A, B, C, D}. Moreover: 1. The following conditions concerning the input pair are equivalent: (a) {A, B2 } is scaled operator-stabilizable. (b) There exists Y ∈ D satisfying the LMIs: ∗ (AY A∗ − Y )B2,⊥ < 0, B2,⊥

Y >0

(4.13)

where B2,⊥ any injective operator with range equal to Ker B2 . (c) There exists Y ∈ D satisfying the LMIs AY A∗ − Y − B2 B2∗ < 0,

Y > 0.

(4.14)

2. The following conditions concerning the output pair are equivalent: (a) {C2 A, } is scaled operator-detectable. (b) There exists X ∈ D satisfying the LMIs: ∗ (A∗ XA − X)C2,⊥ < 0, C2,⊥

X > 0.

(4.15)

where C2,⊥ any injective operator with range equal to Ker C2 . (c) There exists X ∈ D satisfying the LMIs A∗ XA − X − C2∗ C2 < 0,

X > 0.

(4.16)

Control and Interpolation

47

One of the results we shall use in the proof of Theorem 4.5 is known as Finsler’s lemma [61], which also plays a key role in [98, 78]. This result can be interpreted as a reﬁnement of the Douglas lemma [51] which is well known in the operator theory community. Lemma 4.6 (Finsler’s lemma). Assume R and H are given matrices of appropriate size with H = H ∗ . Then there exists a μ > 0 so that μR∗ R > H if and only if ∗ R⊥ HR⊥ < 0 where R⊥ is any injective operator with range equal to ker R. Finsler’s lemma can be seen as a special case of another important result, which we shall refer to as Finsler’s lemma II. This is one of the main underlying tools in the proof of the solution to the H ∞ -problem obtained in [66, 18]. Lemma 4.7 (Finsler’s lemma II). Given matrices R, S and H of appropriate sizes with H = H ∗ , the following are equivalent: 0 J∗ R < 0, (i) There exists a matrix J so that H + R∗ S ∗ S J 0 ∗ ∗ (ii) R⊥ HR⊥ < 0 and S⊥ HS⊥ < 0, where R⊥ and S⊥ are injective operators with ranges equal to ker R and ker S, respectively. The proof of Finsler’s Lemma II given in [66] uses only basic linear algebra and is based on a careful administration of the kernels and ranges from the various matrices. In particular, the matrices J in statement (i) can actually be constructed from the data. We show here how Finsler’s lemma follows from the extended version. Proof of Lemma 4.6 using Lemma 4.7. Apply Lemma 4.7 with R = S. Then (ii) ∗ reduces to R⊥ HR⊥ < 0, which is equivalent to the existence of a matrix J so that ∗ K = −(J + J) satisﬁes R∗ KR > H. Since for such a matrix K we have K ∗ = K, > H holds for K = μI as long as μI > K. it follows that R∗ KR With these results in hand we can prove Theorem 4.5. Proof of Theorem 4.5. We shall ﬁrst prove that scaled stabilizability of {A, B, C, D} is equivalent to the existence of solutions X and Y in D for the LMIs (4.15) and (4.13). Note that Acl can be written in the following aﬃne way: 0 B2 A 0 AK BK 0 I + . (4.17) Acl = 0 0 I 0 CK DK C2 0 Now let Xcl ∈ L(X ⊕ XK ) be an invertible matrix in Dcl , where Dcl stands for the commutant of {Zcl (z) : z ∈ Zd }. Let X be the compression of Xcl to X and −1 to X . Then X, Y ∈ D. Assume that Xcl > 0. Thus, in Y the compression of Xcl particular, X > 0 and Y > 0. Then A∗cl Xcl Acl − Xcl < 0 if and only if −1 Acl −Xcl < 0. (4.18) A∗cl −Xcl

48

J.A. Ball and S. ter Horst

Now deﬁne ⎡

−1 −Xcl

⎢ ∗ H=⎢ ⎣ A 0 and

0 0

A 0 0 0 −Xcl

⎤ ⎥ ⎥, ⎦

⎡

0 ⎢ 0 R∗ = ⎢ ⎣ 0 I

⎤ 0 0 ⎥ ⎥, C2∗ ⎦ 0

⎡

0 ⎢ B 2 S∗ = ⎢ ⎣ 0 0

⎤ I 0 ⎥ ⎥ 0 ⎦ 0

AK BK . J= CK DK Note that H, R and S are determined by the problem data, while J amounts to the system matrix of the controller to be designed. Then −1 ∗ 0 J∗ R Acl −Xcl ∗ S R . (4.19) = H + S J 0 A∗cl −Xcl K BK Thus, by Finsler’s lemma II, the inequality (4.18) holds for some J = A CK DK ∗ ∗ HR⊥ < 0 and S⊥ HS⊥ < 0, where without loss of generality we if and only if R⊥ can take ⎤ ⎤ ⎡ ⎡ I 0 0 0 0 0 ⎢ 0 I ⎢ B2,⊥ 0 0 ⎥ 0 ⎥ ⎥ ⎥ ⎢ R⊥ = ⎢ ⎣ 0 0 C2,⊥ ⎦ and S⊥ = ⎣ 0 I 0 ⎦ 0 0 I 0 0 0

with C2,⊥ and B2,⊥ as described in part (b) of statements 1 and 2. Writing out ∗ ∗ R⊥ HR⊥ we ﬁnd that R⊥ HR⊥ < 0 if and only if ⎤ ⎡ AC2,⊥ −1 −Xcl ⎣ ⎦<0 0 ∗ ∗ C2,⊥ A∗ 0 −C2,⊥ XC2,⊥ which, after taking a Schur complement, turns out to be equivalent to ∗ AC2,⊥ ∗ ∗ ∗ ∗ − C2,⊥ XC2,⊥ < 0. C2,⊥ (A XA − X)C2,⊥ = C2,⊥ A 0 Xcl 0 ∗ HS⊥ < 0 is equivalent to B2,⊥ (AY A∗ − A similar computation shows that S⊥ ∗ < 0. This proves the ﬁrst part of our claim. Y )B2,⊥ For the converse direction assume we have X and Y in D satisfying (4.15)– (4.13). Most of the implications in the above argumentation go both ways, and it suﬃces to prove that there exists an operator Xcl on X ⊕ XK in Dcl , with XK an arbitrary ﬁnite-dimensional Hilbert space with some partitioning XK = XK,1 ⊕ · · · ⊕ XK,d , so that Xcl > 0 and X and Y are the compressions to X of −1 , respectively. Since (4.15)–(4.13) hold with X and Y replaced by Xcl and Xcl ρX and ρY for any positive number ρ, we may assume without loss of generality I that [ X I Y ] > 0. The existence of the required matrix Xcl can then be derived from Lemma 7.9 in [57] (with nK = n). To enforce the fact that Xcl be in Dcl we decompose X = diag(X1 , . . . , Xd ) and Y = diag(Y1 , . . . , Yd ) as in (4.11) and −1 complete Xi and Yi to positive deﬁnite matrices so that [ X∗i ∗∗ ] = [ Y∗i ∗∗ ].

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To complete the proof it remains to show the equivalences of parts (a), (b) and (c) in both statements 1 and 2. The equivalences of the parts (b) and (c) follows immediately from Finsler’s lemma with R = B2 (respectively, R = C2∗ ) and H = AY A∗ − Y (respectively, H = A∗ XA − X), again using that X in (4.15) can be replaced with μX (respectively, Y in (4.13) can be replaced with μY ) for any positive number μ. We next show that (a) is equivalent to (b) for statement 1; for statement 2 the result follows with similar arguments. Let F : X → U, and let X ∈ D be positive deﬁnite. Taking a Schur complement it follows that (A∗ + F ∗ B2∗ )X(A + B2 F ) − X < 0 if and only if

−X −1 ∗ A + F ∗ B2∗

A + B2 F −X

(4.20)

< 0.

Now write

A + B2 F −X −1 A∗ + F ∗ B2∗ −X −1 −X B2 A = + A∗ 0 −X

Thus, applying Finsler’s lemma II with A −X −1 , R = B2∗ H= ∗ A −X

0

0 I

,

0 F∗

S=

F 0

0 I

B2∗ 0

0 I

.

and J = F, (4.21)

we ﬁnd that there exists an F so that (4.20) holds if and only if ∗ ∗ R⊥ HR⊥ < 0 and S⊥ HS⊥ < 0 B2,⊥ 0 I with now R⊥ = and S⊥ = [ 0 ]. The latter inequality is the same as 0 I ∗ −X −1 < 0 and thus vacuous. The ﬁrst inequality, after writing out R⊥ HR⊥ , turns out to be ∗ ∗ X −1 B2,⊥ B2,⊥ A −B2,⊥ < 0, −X A∗ B2,⊥ which, after another Schur complement, is equivalent to ∗ B2,⊥ (AX −1 A∗ − X −1 )B2,⊥ < 0.

Since scaled stability implies stability, it is clear that ﬁnding operators F and L with A + B2 F and A + LC2 scaled-stable implies that A + B2 F and A + LC2 are also stable. In particular, having such operators F and L we ﬁnd the coprime factorization of G22 via the functions in Theorem 4.3. While there are no known tractable necessary and suﬃcient conditions for operator-detectability/stabilizability, the LMI criteria in parts (iii) and (iv) of Theorem 4.5 for the scaled versions are considered computationally tractable. Moreover, an inspection of the last part of the proof shows how operators F and L so that A + B2 F and A + LC2 are scaled stable can be constructed from the solutions X and Y from the LMIs in

50

J.A. Ball and S. ter Horst

(4.13)–(4.16): Assume we have X, Y ∈ D satisfying (4.13)–(4.16). ∗ Deﬁne H, R and S as in (4.21), and determine a J so that H + [ R∗ S ∗ ] J0 J0 [ R S ] < 0; this is possible as the proof of Finsler’s lemma II is essentially constructive. Then take F = J. In a similar way one can construct L using the LMI solution Y . Stability versus scaled stability, μ versus μ . We observed above that the notion of scaled stability is stronger, and more conservative than the more intuitive notions of stability in the Hautus or asymptotic sense. This remains true in a more general setting that has proved useful in the study of robust control [98, 57, 107] and that we will encounter later in the paper. Let A be a bounded linear operator on a Hilbert space X . Assume that in addition we are given a unital C ∗ -algebra Δ which is realized concretely as a subalgebra of L(X ), the space of bounded linear operators on X . The complex structured singular value μΔ (A) of A (with respect to the structure Δ) is deﬁned as μΔ (A) =

1 . inf{σ(Δ) : Δ ∈ Δ, I − ΔA is not invertible}

(4.22)

Here σ(M ) stands for the largest singular value of the operator M . Note that this contains two standard measures for A: the operator norm A if we take Δ = L(X ), and ρ(A), the spectral radius of A, if we take Δ = {λIX : λ ∈ C}; it is not hard to see that for any unital C ∗ -algebra Δ we have ρ(A) ≤ μΔ (A) ≤ A. See [107] for a tutorial introduction on the complex structured singular value and [60] for the generalization to algebras of operators on inﬁnite-dimensional spaces. The C ∗ -algebra that comes up in the context of stability for the N -D systems studied in this section is Δ = {Z(z) : z ∈ Cd }. Indeed, note that for this choice of Δ we have that A is stable if and only if μΔ (A) < 1. In order to introduce the more conservative measure for A in this context, we write DΔ for the commutant of the C ∗ -algebra Δ in L(X ). We then deﬁne μ Δ (A) = inf{γ : Q−1AQ < γ for some invertible Q ∈ DΔ } = inf{γ : AXA∗ − γ 2 X < 0 for some X ∈ DΔ , X > 0}.

(4.23)

The equivalence of the two deﬁnitions again goes through the relation between X and Q via X = Q∗ Q. It is immediate that with Δ = {Z(z) : z ∈ Cd } we ﬁnd DΔ = D as in (4.11), and that A is scaled stable if and only if μ Δ (A) < 1. The state-space H ∞ -problem. The problems of ﬁnding tractable necessary and suﬃcient conditions for the strict state-space H ∞ -problem are similar to that for the state-space stabilizability problem. Here one also typically resorts to a more conservative ‘scaled’ version of the problem. We say that the realization {A, B, C, D} with decomposition (4.8) has scaled performance whenever there exists an invertible Q ∈ D so that # −1 # # Q # Q 0 0 A B # # < 1, (4.24) # 0 IW⊕U # 0 IZ⊕Y C D

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or, equivalently, if there exists an X > 0 in D so that ∗ A B X 0 X 0 A B < 0. − C D 0 IW⊕U 0 IW⊕U C D

(4.25)

The equivalence of the two deﬁnitions goes as for the scaled stability case through the relation X = QQ∗ . Looking at the left upper entry in (4.25) it follows that scaled performance of {A, B, C, D} implies scaled stability. Moreover, if (4.24) holds for Q ∈ D, then it is not hard to see that the transfer function G(z) in (4.5) is also given by G(z) = D + C (I − Z(z)A )−1 Z(z)B

where the system matrix −1

Q 0 A B Q A B

0 = 0 IW⊕U C D 0 IZ⊕Y C D is equal to a strict contraction. It then follows from a standard fact on feedback connections (see, e.g., Corollary 1.3 page 434 of [62] for a very general formulation) d that G(z) < 1 for z ∈ D , i.e., G has strict performance. The scaled H ∞ -problem is then to ﬁnd a controller K with realization {AK , BK , CK , DK } so that the closed loop system {Acl , Bcl , Ccl , Dcl } has scaled performance. The above analysis shows that solving the scaled H ∞ -problem implies solving that state-space H ∞ problem. The converse is again not true in general. Further elaboration of the same techniques as used in the proof of Theorem 4.5 yields the following result for the scaled H ∞ -problem; see [18, 66]. For the connections between the Theorems 4.8 and 4.5, in the more general setting of LFT models with structured uncertainty, we refer to [25]. Note that the result collapses to Theorem 2.5 given in the Introduction when we specialize to the single-variable case d = 1. Theorem 4.8. Let {A, B, C, D} be a given realization. Then there exists a solution for the scaled H ∞ -problem associated with {A, B, C, D} if and only if there exist X, Y ∈ D satisfying LMIs: ⎤ ⎡ ∗ AY A∗ − Y AY C1∗ B1 Nc 0 Nc 0 ⎣ < 0, Y > 0, (4.26) C1 Y A∗ C1 Y C1∗ − I D11 ⎦ 0 I 0 I ∗ B1∗ D11 −I ⎤ ⎡ ∗ A∗ XA − X A∗ XB1 C1∗ 0 No 0 ⎣ ∗ ⎦ No < 0, X > 0, (4.27) B1∗ XA B1∗ XB1 − I D11 0 I 0 I C1 D11 −I and the coupling condition

X I ≥ 0. I Y Here Nc and No are matrices chosen so that

Nc is injective and Im Nc = Ker B2∗ No is injective and Im No = Ker C2

(4.28) ∗ D12 and D21 .

52

J.A. Ball and S. ter Horst

Note that Theorem 4.8 does not require that the problem be ﬁrst brought into model-matching form; thus this solution bypasses the Nevanlinna-Pick-interpolation interpretation of the H ∞ -problem. 4.3. Equivalence of frequency-domain and state-space formulations In this subsection we suppose that we are given a transfer matrix G of size (nZ + nY ) × (nW + nU ) with coeﬃcients in Q(C(z)ss ) as in Section 4.1 with a given state-space realization as in Subsection 4.2: D11 D12 C G11 G12 = + 1 (I − Z(z)A)−1 Z(z) B1 B2 (4.29) G(z) = G21 G22 D21 D22 C2 where Z(z) is as in (4.6). We again consider the problem of ﬁnding stabilizing controllers K, also equipped with a state-space realization K(z) = DK + CK (I − ZK (z)AK )−1 ZK (z)BK ,

(4.30)

in either the state-space stability or in the frequency-domain stability sense. A natural question is whether the frequency-domain H ∞ -problem with formulation in state-space coordinates is the same as the state-space H ∞ -problem formulated in Section 4.2. For simplicity in the computations to follow, we shall always assume that the plant G has been normalized so that D22 = 0. In one direction the result is clear. Suppose that K(z) = DK + CK (I − Z(z)AK )−1 Z(z)BK is a stabilizing controller for G(z) in the state-space sense. It follows that the closed-loop state matrix A + B2 DK C2 B2 CK (4.31) Acl = BK C2 AK d

is stable, i.e., I − Zcl (z)Acl is invertible for all z in the closed polydisk D , with Zcl (z) as deﬁned in Subsection 4.2. On the other hand one can compute that the −1 I −K(z) transfer matrix Θ(G22 , K) := −G22 has realization (z) I ' (z) = I DK + DK C2 CK (I − Zcl (z)Acl )−1 Zcl (z) B2 B2 DK . W 0 I C2 0 0 BK (4.32) −1 has no singularities in the closed As the resolvent expression (I − Zcl (z)Acl ) d ' (z) has matrix entries in C(z)ss , and it follows that polydisk D , it is clear that W K stabilizes G22 in the frequency-domain sense. Under the assumption that G is internally stabilizable (frequency-domain sense), it follows from Corollary 3.4 that K also stabilizes G (frequency-domain sense). We show that the converse direction holds under an additional assumption. The early paper [88] of Kung-L´evy-Morf-Kailath introduced the notion of modal controllability and modal observability for 2-D systems. We extend these notions to N -D systems as follows. Given a Givone-Roesser output pair {C, A}, we say that {C, A} is modally observable if the block-column matrix I−Z(z)A has maximal C rank nX for a generic point z on each irreducible component of the variety det(I −

Control and Interpolation

53

Z(z)A) = 0. Similarly we say that the Givone-Roesser input pair {A, B} is modally controllable if the block-row matrix [ I−AZ(z) B ] has maximal rank nX for a generic point on each irreducible component of the variety det(I − AZ(z)) = det(I − Z(z)A) = 0. Then the authors of [88] deﬁne the realization {A, B, C, D} to be minimal if both {C, A} is modally observable and {A, B} is modally controllable. While this is a natural notion of minimality, unfortunately it is not clear that an arbitrary realization {A, B, C, D} of a given transfer function S(z) = D + C(I − Z(z)A)−1 Z(z)B can be reduced to a minimal realization {A0 , B0 , C0 , D0 } of the same transfer function S(z) = D0 + C0 (I − Z(z)A0 )−1 Z(z)B0 . As a natural modiﬁcation of the notions of modally observable and modally controllable, we now introduce the notions of modally detectable and modally stabilizable as follows. For {C, A} a Givone-Roesser pair, we say that {C, A} I−Z(z)Aoutput is modally detectable if the column matrix has maximal rank nX for a C generic point z on each irreducible component of the variety det(I − Z(z)A) = 0 d which enters into the polydisk D . Similarly, we say that the Givone-Roesser input pair {A, B} is modally stabilizable if the row matrix [ I−AZ(z) B ] has maximal rank nX for a generic point z on each irreducible component of the variety det(I − d Z(z)A) = 0 which has nonzero intersection with the closed polydisk D . We then have the following partial converse of the observation made above that state-space internal stabilization implies frequency-domain internal stabilization; this is an N -D version of Theorem 2.6 in the Introduction. Z Theorem 4.9. Let (4.29) and (4.30) be given realizations for G : [ W Y U ] → and K : Y → U. Assume that {C2 , A} and {CK , AK } are modally detectable and {A, B2 } and {AK , BK } are modally stabilizable. Then K internally stabilizes G22 in the state-space sense (and thus state-space stabilizes G) if and only if K stabilizes G22 in the frequency-domain sense (and G if G is stabilizable in the frequencydomain sense). Remark 4.10. As it is not clear that a given realization can be reduced to a modally observable and modally controllable realization for a given transfer function, it is equally not clear whether a given transfer function has a modally detectable and modally stabilizable realization. However, in the case that d = 1, such realizations always exists and Theorem 4.9 recovers the standard 1-D result (Theorem 2.6 in the Introduction). The proof of Theorem 4.9 will make frequent use of the following basic result from the theory of holomorphic functions in several complex variables. For the proof we refer to [128, Theorem 4 p. 176]; note that if the number of variables d is 1, then the only analytic set of codimension at least 2 is the empty set and the theorem is vacuous; the theorem has content only when the number of variables is at least 2. Theorem 4.11. Principle of Removal of Singularities. Suppose that the complexvalued function ϕ is holomorphic on a set S contained in Cd of the form S = D −E

54

J.A. Ball and S. ter Horst

where D is an open set in Cd and E is the intersection with D of an analytic set of codimension at least 2. Then ϕ has analytic continuation to a function holomorphic on all of D. We shall also need some preliminary lemmas. Lemma 4.12. 1. Modal detectability is invariant under output injection, i.e., given a GivoneRoesser output pair {C, A} (where A : X → X and C : X → Y) together with an output injection operator L : Y → X , then the pair {C, A} is modally detectable if and only if the pair {C, A + LC} is modally detectable. 2. Modal stabilizability is invariant under state feedback, i.e., given a GivoneRoesser input pair {A, B} (where A : X → X and B : U → X ) together with a state-feedback operator F : X → U, then the pair {A, B} is modally stabilizable if and only if the pair {A + BF, B} is modally stabilizable. Proof. To prove the ﬁrst statement, note the identity I − Z(z)(A + LC) I −Z(z)L I − Z(z)A . = C C 0 I Since the factor 0I −Z(z)L is invertible for all z, we conclude that, for each z ∈ Cd , I I−Z(z)A has maximal rank exactly when I−Z(z)(A+LC) has maximal rank, and C C hence, in particular, the modal detectability for {C, A} holds exactly when modal detectability for {C, A + LC} holds. The second statement follows in a similar way from the identity I 0 I − AZ(z) B = I − (A + BF )Z(z) B . −F Z(z) I Lemma 4.13. Suppose that the function W (z) is stable (i.e., all matrix entries of W are in C(z)ss ) and suppose that W (z) = D + C(I − Z(z)A)−1 Z(z)B

(4.33)

is a realization for W which is both modally detectable and modally stabilizable. Then the matrix A is stable, i.e., (I − Z(z)A)−1 exists for all z in the closed d polydisk D . Proof. As W is stable and Z(z)B is trivially stable, then certainly Z(z)B I − Z(z)A −1 (4.34) (I − Z(z)A) Z(z)B = W (z) − D C d is stable (i.e., holomorphic on D ). Trivially I−Z(z)A has maximal rank nX for C I−Z(z)A d all z ∈ D where det(I − Z(z)A) = 0. By assumption, has maximal rank C generically on each irreducible component of the zero variety of det(I − Z(z)A) d has maximal rank nX at all points which intersects D . We conclude that I−Z(z)A C

Control and Interpolation

55

d

of D except those in an exceptional set E which is contained in a subvariety, each irreducible component of which has codimension at least 2. In a neighborhood of d each such point z ∈ D − E, I−Z(z)A has a holomorphic left inverse; combining C this fact with the identity (4.34), we see that (I − Z(z)A)−1 Z(z)B is holomorphic d on D − E. By Theorem 4.11, it follows that (I − Z(z)A)−1 Z(z)B has analytic d continuation to all of D . We next note the identity Z(z) (I − Z(z)A)−1 Z(z)B = Z(z)(I − AZ(z))−1 I − AZ(z) B (4.35) d

where the quantity on the left-hand side is holomorphic on D by the result established above. By assumption {A, B} is modally stabilizable; by an argument analogous to that used above for the modally detectable pair {C, A}, we see that the pencil I − AZ(z) B has a holomorphic right inverse in the neighborhood d

of each point z in D − E where the exception set E is contained in a subvariety each irreducible component of which has codimension at least 2. Multiplication of the identity (4.35) on the right by this right inverse then tells us that d Z(z)(I −AZ(z))−1 is holomorphic on D −E . Again by Theorem 4.11, we conclude d that in fact Z(z)(I − AZ(z))−1 is holomorphic on all of D . d

We show that (I−Z(z)A)−1 is holomorphic on D as follows. Let Ej : X → Xj be the projection on the jth component of X = X1 ⊕ · · · ⊕ Xd . Note that the ﬁrst block row of (I − Z(z)A)−1 is equal to z1 E1 (I − Z(z)A)−1 . This is holomorphic d on the closed polydisk D . For z in a suﬃciently small polydisk |zi | < ρ for i = 1, . . . , d, (I − Z(z)A)−1 is analytic and hence z1 E1 (I − Z(z)A)−1 |z1 =0 = 0. By analytic continuation, it then must hold that z1 (E1 (I − Z(z)A)−1 = 0 for all z = (0, z2 , . . . , zd ) with |zi | ≤ 1 for i = 2, . . . , d. For each ﬁxed (z2 , . . . , zd ), we may use the single-variable result that one can divide out zeros to conclude that E1 (I − Z(z)A)−1 is holomorphic in z1 at z1 = 0. As the result is obvious for z1 = 0, we conclude that E1 (I − Z(z)A)−1 is holomorphic on the whole closed d polydisk D . In a similar way working with the variable zi , one can show that Ei (I − Z(z)A)−1 is holomorphic on the whole closed polydisk, and it follows that (I − Z(z)A)−1 = as wanted.

E1

.. .

(I − Z(z)A)−1 is holomorphic on the whole closed polydisk

Ed

We are now ready for the proof of Theorem 4.9. Proof of Theorem 4.9. Suppose that K stabilizes G22 in the frequency-domain ' given by (4.32) is holomorsense. This simply means that the transfer function W d To show that Acl is stable, by Lemma phic on the closed polydisk B2itBsuﬃces D . 4.13 C2 CK 2 DK to show that DK , , A is modally detectable and that A cl cl C2 0 0 BK is modally stabilizable.

56

J.A. Ball and S. ter Horst

To prove that (4.17) we note that

DK C2 C2

CK 0

, Acl

is modally detectable, from the deﬁnition

DK C2 CK . C2 0 C2 CK , Acl is equivalent By Lemma 4.12 we see that modal detectability of DK C 0 2 C2 CK A 0 to modal detectability of DK , . D IC 2 C20 0 0 AK D I DK C2 CK K K = with invertible, it is easily seen As C2 0 0 CK I 0 DK CI2 C0 K A 0 that modal detectability of the input pair , 0 AK is equivalent to C2 0 modal detectability of C02 C0K , A0 A0K . But the modal detectability of this last pair in turn follows from its diagonal form and the assumed modal detectability of {C2 , A} and {CK , AK }. 2 DK The modal stabilizability of Acl , B02 BB follows in a similar way by K making use of the identities B B D B2 D K B 2 DK I B2 C2 0 0 A 0 = 2 Acl = + 2 K , BK 0 BK 0 0 CK 0 BK I 0 0 AK D I K and noting that I 0 is invertible. Acl =

A 0 B2 + 0 AK 0

0 BK

In both the frequency-domain setting of Section 4.1 and the state-space setting of Section 4.2, the true H ∞ -problem is intractable and we resorted to some compromise: the Schur-Agler-class reformulation in Section 4.1 and the scaledH ∞ -problem reformulation in Section 4.2. We would now like to compare these compromises for the setting where they both apply, namely, where we are given both the transfer function G and the state-space representation {A, B, C, D} for the plant. 11 G12 is in model-matching form with Theorem 4.14. Suppose that G(z) = G G21 0 state-space realization G(z) = D + C(I − Z(z)A)−1 Z(z)B as in (4.29). Suppose that the controller K(z) = DK + CK (I − ZK (z)AK )−1 ZK (z)BK solves the scaled ' (z) as in (4.32) is a Schur-Agler-class H ∞ -problem. Then the transfer function W solution of the Model-Matching problem. ' (z) has a realProof. Simply note that, under the assumptions of the theorem, W −1 ' = Dcl + Ccl (I − Zcl (z)Acl ) Zcl (z)Bcl for which there is a state-space ization W change of coordinates Q ∈ D transforming the realization to a contraction: #

#

−1

# A B # 0 #

# < 1 where A B = Q 0 Acl Bcl Q . # C D # 0 I 0 I Ccl Dcl C D ' (z) = D + C (I − Zcl (z)A )−1 Zcl (z)B from which it follows Thus we also have W ' is in the strict Schur-Agler class, i.e., W ' (X) < 1 for any d-tuple X = that W (X1 , . . . , Xd ) of contraction operators Xj on a separable Hilbert space X . By ' necessarily has the model matching form W ' = G11 + G12 ΛG21 construction W with Λ stable.

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57

Remark 4.15. In general a Schur-Agler function S(z) can be realized with a colliA B ] which is not of the form gation matrix [ C D −1 Q A B 0 A B Q 0 = (4.36) C D 0 I 0 I C D A B equal to a strict contraction and Q ∈ D invertible. As an example, let with C D A be the block 2 × 2 matrix given by Anderson-et-al in [16]. This matrix has the 2 property that I − Z(z)A is invertible for all z ∈ D , but there is no Q ∈ D so that Q−1 AQ < 1. Here Z(z) and D are compatible with the block decomposition of A. Then for γ > 0 suﬃciently small the function S(z) = γ(I − Z(z)A)−1 has 2 S(z) ≤ ρ < 1 for some 0 < ρ < 1 and all z ∈ D . Hence S is a strict Schur-class function. As mentioned in Section 4.1, a consequence of the Andˆ o dilation theorem [17] is that the Schur class and the Schur-Agler class coincide for d = 2; it is not hard to see that this equality carries over to the strict versions and hence S is in the strict Schur-Agler class. As a consequence of the strict Bounded-Real-Lemma A B . However, in [29], S admits a strictly contractive state-space realization C D A B]= A A the realization [ C γI γI of S, obtained from the fact that D S(z) = γ(I − Z(z)A)−1 = γI + γ(I − Z(z)A)−1 Z(z)A, A B cannot relate to C as in (4.36) since that would imply the existence of an D invertible Q ∈ D so that Q−1 AQ = A is a strict contraction. Remark 4.16. Let us assume that the G(z) in Theorem 4.14 is such that G12 and G21 are square and invertible on the distinguished boundary Td of the polydisk Dd so that the Model-Matching problem can be converted to a polydisk bitangential Nevanlinna-Pick interpolation problem along a subvariety as in [32]. As we have seen, the solution criterion using the Agler interpolation theorem of [1, 35] then involves an LOI (Linear Operator Inequality or inﬁnite LMI). On the other hand, if we assume that weare given a stable state-space realization {A, B, C, D} for G11 (z) G12 (z) , we may instead solve the associated scaled H ∞ -problem G(z) = G21 (z) 0 associated with this realization data-set. The associated solution criterion in Theorem 4.8 remarkably involves only ﬁnite LMIs. A disadvantage of this state-space approach, however, is that in principle one would have to sweep all possible (similarity equivalence classes of) realizations of G(z); while each non-equivalent realization gives a distinct H ∞ -problem, the associated frequency-domain ModelMatching/bitangential variety-interpolation problem remains the same. 4.4. Notes In [92] Lin conjectured the result stated in Theorem 4.1 that G22 -stabilizability is equivalent to the existence of a stable coprime factorization for G22 . This conjecture was settled by Quadrat (see [122, 117, 120]) who obtained the equivalence of this property with projective-freeness of the underlying ring and noticed the applicability of the results from [46, 83] concerning the projective-freeness of C(z)ss .

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For the general theory of the N -D systems, in particular for N =2, considered in Subsection 4.2 we refer to [81, 55]. The suﬃciency of scaled stability for asymptotic/Hautus-stability goes back to [59]. Theorem 4.5 was proved in [98] for the more general LFT models in the context of robust control with structured uncertainty. The proof given here is based on the extended Finsler’s lemma (Lemma 4.7), and basically follows the proof from [66] for the solution to the scaled H ∞ -problem (Theorem 4.8). As pointed out in [66], one of the advantages of the LMI-approach to the state-space H ∞ problem, even in the classical setting, is that it allows one to seek controllers that solve the scaled H ∞ -problem with a given maximal order. Indeed, it is shown in [66, 18] (see also [57]) that certain additional rank constraints on the solutions X and Y of the LMIs (4.26) and (4.27) enforce the existence of a solution with a prescribed maximal order. However, these additional constraints destroy the convexity of the solution criteria, and are therefore usually not considered as a desirable addition. An important point in the application of Finsler’s lemma in the derivation of the LMI solution criteria in Theorems 4.5 and 4.8 is that the closed-loop system matrix Acl in (4.31) has an aﬃne expression in terms of the unknown design parameters {AK , BK , CK , DK }. This is the key point where the assumption D22 = 0 is used. A parallel simpliﬁcation occurs in the frequency-domain setting where the assumption G22 = 0 leads to the Model-Matching form. The distinction however is that the assumption G22 = 0 is considered unattractive from a physical point of view while the parallel state-space assumption D22 := G22 (0) = 0 is considered innocuous. There is a whole array of lemmas of Finsler type; we have only mentioned the form most suitable for our application. It turns out that these various Finsler lemmas are closely connected with the theory of plus operators and Pesonen operators on an indeﬁnite inner product space (see [44]). An engaging historical survey on all the Finsler’s lemmas is the paper of Uhlig [135]. The notions of modally detectable and modally stabilizable introduced in Subsection 4.3 along with Theorem 4.9 seem new, though of somewhat limited use because it is not known if every realization can be reduced to a modally detectable and modally stabilizable realization. We included the result as an illustration of the diﬃculties with realization theory for N -D transfer functions. We note that the usual proof of Lemma 4.13 for the classical 1-D case uses the pole-shifting characterization of stabilizability/detectability (see [57, Exercise 2.19]). The proof here using the Hautus characterization of stabilizability/detectability provides a diﬀerent proof for the 1-D case.

5. Robust control with structured uncertainty: the commutative case In the analysis of 1-D control systems, an issue is the uncertainty in the plant parameters. As a control goal, one wants the control to achieve internal stability

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(and perhaps also performance) not only for the nominal plant G but also for a whole prescribed family of plants containing the nominal plant G. A question then is whether the controller can or cannot have (online) access to the uncertainty parameters. In a state-space context it is possible to ﬁnd sufﬁcient conditions for the case that the controller cannot access the uncertainty parameters, with criteria that are similar to those found in Theorems 4.5 and 4.8 but additional rank constraints need to be imposed as well, which destroys the convex character of the solution criterion. The case where the controller can have access to the uncertainty parameters is usually given the interpretation of gainscheduling, and ﬁts better with the multidimensional system problems discussed in Section 4. In this section we discuss three formulations of 1-D control systems with uncertainty in the plant parameters, two of which can be given gain-scheduling interpretation, i.e., the controller has access to the uncertainty parameters, and one where the controller is not allowed to use the uncertainty parameters. 5.1. Gain-scheduling in state-space coordinates Following [106], we suppose that we are given a standard linear time-invariant input/state/output system ⎧ ⎪ ⎨ x(t + 1) = AM (δU )x(t) + BM1 (δU )w(t) + BM2 (δU )u(t) z(t) = CM1 (δU )x(t) + DM11 (δU )w(t) + DM12 (δU )u(t) (t ∈ Z+ ) (5.1) Σ: ⎪ ⎩ y(t) = CM2 (δU )x(t) + DM21 (δU )w(t) + DM22 (δU )u(t) but where the system matrix ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ X X AM (δU ) BM1 (δU ) BM2 (δU ) ⎣CM1 (δU ) DM11 (δU ) DM12 (δU )⎦ : ⎣ W ⎦ → ⎣ Z ⎦ Y U CM2 (δU ) DM21 (δU ) DM22 (δU ) is not known exactly but depends on some uncertainty parameters δU = (δ1 , . . . , δd ) in Cd . Here the quantities δi are viewed as uncertain parameters which the controller can measure and use in real time. The goal is to design a controller ΣK (independent of δU ) oﬀ-line so that the closed-loop system (with the controller accessing the current values of the varying parameters δ1 , . . . , δd as well as the value of the measurement signal y from the plant) has desirable properties for all admissible values of δU , usually normalized to be |δk | ≤ 1 for k = 1, . . . , d. The transfer function for the uncertainty parameter δU can be expressed as DM11 (δU ) DM12 (δU ) G(δ) = DM21 (δU ) DM22 (δU ) C (δ ) (5.2) + λ M1 U (IX − λAM (δU ))−1 BM1 (δU ) BM2 (δU ) CM2 (δU ) where we have introduced the aggregate variable δ = (δU , λ) = (δ1 , . . . , δd , λ).

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It is not too much of a restriction to assume in addition that the functional dependence on δU is given by a linear fractional map (where the subscript U suggests uncertainty and the subscript S suggests shift) ⎡ ⎤ ⎡ ⎤ AM (δU ) BM1 (δU ) BM2 (δU ) ASS BS1 BS2 ⎣ CM1 (δU ) DM11 (δU ) DM12 (δU ) ⎦ = ⎣ C1S D11 D12 ⎦ CM2 (δU ) DM21 (δU ) DM22 (δU ) C2S D21 D22 ⎤ ⎡ ASU + ⎣ C1U ⎦ (I − Z(δU )AUU )−1 Z(δU ) AUS BU 1 BU 2 , C2U where Z(δU ) is deﬁned analogously to Z(z) in (4.6) relative to a given decomposition of the “uncertainty” state-space XU = XU,1 ⊕ · · · ⊕ XU,d on which that state operator AUU acts. In that case the transfer function G(δ) admits a state-space realization D11 D12 C G11 G12 = + 1 (I − Z(δ)A)−1 Z(δ) B1 B2 (5.3) G(δ) = G21 G22 D21 D22 C2 with system matrix given by ⎡ ⎡ ⎤ AUU B2 A B1 ⎢ ASU ⎣C1 D11 D12 ⎦ = ⎢ ⎣ C1U C2 D21 D22 C2U

AUS ASS C1S C2S

BU 1 BS1 D11 D21

⎤ BU 2 BS2 ⎥ ⎥. D12 ⎦ D22

(5.4)

Here Z(δ) is again deﬁned analogously to (4.6) but now on the extended statespace Xext = XU ⊕ X . We can then consider this gain-scheduling problem as a problem of the constructed N -D system (with N = d + 1), and seek for a controller K with a statespace realization K(δ) = DK + CK (I − ZK (δ)AK )ZK (δ)BK

(5.5)

so that the closed loop system has desirable properties from a gain-scheduling perspective. Making a similar decomposition of the system matrix for the controller K as in (5.4), we note that K(δ) can also be written as K(δ) = DM,K (δU ) + λCM,K (δU )(I − λAM,K (δU ))−1 BM,K (δU ), where AM,K (δU ), BM,K (δU ), CM,K (δU ) and DM,K (δU ) appear as the transfer functions of N -D systems (with N = d), that is, K(δ) can be seen as the transfer function of a linear time-invariant input/state/output system xK (t + 1) = AM,K (δU )xK (t) + BM,K (δU )u(t) (n ∈ Z+ ) ΣK : u(t) = CM,K (δU )xK (t) + DM,K (δU )y(t) depending on the same uncertainty parameters δU = (δ1 , . . . , δd ) as the system Σ. Similarly, transfer function Gcl (δ) of the closed-loop system with system Acl Bcl the matrix Ccl Dcl as deﬁned in (4.10) also can be written as a transfer matrix Gcl (δ) = DM,cl (δU ) + λCM,cl (δU )(I − λAM,cl (δU ))−1 BM,cl (δU )

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with AM,cl (δU ), BM,cl (δU ), CM,cl (δU ) and DM,cl (δU ) transfer functions of N -D systems (with N = d), and the corresponding linear time-invariant input/state/output system x(t + 1) = AM,cl (δU )x(t) + BM,cl (δU )w(t) Σcl : (n ∈ Z+ ) z(t) = CM,cl (δU )x(t) + DM,cl (δU )w(t) also appears as the closed-loop system of Σ and ΣK . It then turns out that stability of Acl , that is, I − Zcl (δ)Acl invertible for d+1 (with Zcl as deﬁned in Subsection 4.2) corresponds precisely to all δ in D robust stability of Σcl , i.e., the spectral radius of AM,cl (δU ) is less than 1 for all δU = (δ1 , . . . , δd ) so that |δk | ≤ 1 for k = 1, . . . , d, and K with realization (5.5) solves the state-space H ∞ -problem for G with realization (5.3) means that the closed loop system Σcl has robust performance, i.e., Σcl is robustly stable and the transfer function Gcl satisﬁes Gcl (δ) ≤ 1 for all δ = (δ1 , . . . , δd , λ) ∈ D

d+1

.

We may thus see the state-space formulation of the gain-scheduling problems considered in this subsection as a special case of the N -D system stabilization and H ∞ -problems of Subsection 4.2. In particular, the suﬃciency analysis given there, and the results of Theorem 4.5 and 4.8, provide practical methods for obtaining solutions. As the conditions are only suﬃcient, solutions obtained in principle may be conservative. 5.2. Gain-scheduling: a pure frequency-domain formulation In the approach of Helton (see [73, 74]), one eschews transfer functions and statespace coordinates completely and supposes that one is given a plant G whose frequency response depends on a load with frequency function δ(z) at the discretion of the user; when the load δ is loaded onto G, the resulting frequency-response function has the form G(z, δ(z)) where G = G(·, ·) is a function of two variables. The control problem (for the company selling this device G to a user) is to design the controller K = K(·, ·) so that K(·, δ(·)) solves the H ∞ -problem for the plant G(·, δ(·)). The idea here is that once the user loads δ onto G with known frequency-response function, he is also to load δ onto the controller K (designed oﬀ-line); in this way the same controller works for many customers using many diﬀerent δ’s. When the dust settles, this problem reduces to the frequency-domain problem posed in Section 4.1 with d = 2; an application of the Youla-Kuˇcera parametrization (or simply using the function Q(z) = K(z)(I − G22 (z)K(z))−1 if the plant G itself is stable) reduces the problem of designing the control K to a Nevanlinna-Pick-type interpolation problem on the bidisk. 5.3. Robust control with a hybrid frequency-domain/state-space formulation We now consider a hybrid frequency-domain/state-space formulation of the problem considered in Subsection 5.1; the main diﬀerence is that in this case the controller is not granted access to the uncertainty parameters.

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Assume we are given a 1-D-plant G(λ) that depends on uncertainty parameters δU = (δ1 , . . . , δd ) via the linear fractional representation G11 (λ) G12 (λ) G(δU , λ) = G21 (λ) G22 (λ) G1U (λ) (I − Z(δU )GUU (λ))−1 Z(δU ) GU 1 (λ) GU 2 (λ) (5.6) + G2U (λ) with Z(δU ) as deﬁned in Subsection 5.1, and where the coeﬃcients are 1-D-plants independent of δU : ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ GUU (λ) GU 1 (λ) GU 2 (λ) XU XU Gaug (λ) = ⎣ G1U (λ) G11 (λ) G12 (λ) ⎦ : ⎣ W ⎦ → ⎣ Z ⎦ . G2U (λ) G21 (λ) G22 (λ) U Y In case Gaug (λ) is also given by a state-space realization, we can write G(δU , λ) as in (5.3) with δ = (δU , λ) and Z(δ) acting on the extended state-space Xext = XU ⊕ X . For this variation of the gain-scheduling problem we seek to design a controller K(λ) with matrix values representing operators from Y to U so that K solves the H ∞ -problem for G(δU , λ) for every δU with Z(δU ) ≤ 1, i.e., |δj | ≤ 1 for j = 1, . . . , d. For the sequel it is convenient to assume that Z = W. In that case, using the Main Loop Theorem [141, Theorem 11.7 p. 284], it is easy to see that this problem can be reformulated as: Find a single-variable transfer matrix K(·) K) given by (2.2), with G = G 11 G 12 in (2.2) taken to be so that Θ(G, 21 G 22 G ⎡ G (λ) G (λ) G (λ) ⎤ UU U1 U2 12 (λ) 11 (λ) G G ⎣ G1U (λ) G11 (λ) G12 (λ) ⎦ , = 22 (λ) 21 (λ) G G G2U (λ) G21 (λ) G22 (λ) is stable and such that % $ 12 (λ)(I − K(λ)G 22 (λ))−1 K(λ)G 21 (λ) < 1. 11 (λ) + G μΔ G Here μΔ is as deﬁned in (4.22) with Δ the C ∗ -algebra ( Z(δU ) 0 Δ= : δU ∈ Cd , T ∈ L(Z) ⊂ L(XU ⊕ Z). 0 T Application of the Youla-Kuˇcera parametrization of the controllers K that K) as in Subsection 3.3 converts the problem to the following: Given stabilize Θ(G, stable 1-variable transfer functions T1 (λ), T2 (λ), and T3 (λ) with matrix values representing operators in the respective spaces L(XU ⊕ W, XU ⊕ Z),

L(XU ⊕ U, XU ⊕ Z),

L(XU ⊕ W, XU ⊕ Y),

ﬁnd a stable 1-variable transfer function Λ(λ) with matrix values representing operators in L(XU ⊕ Y, XU ⊕ U) so that the transfer function S(λ) given by S(λ) = T1 (λ) + T2 (λ)Λ(λ)T3 (λ)

(5.7)

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has μΔ (S(λ)) < 1 for all λ ∈ D. If T2 (ζ) and T3 (ζ) are square and invertible for ζ on the boundary T of the unit disk D, the model-matching form (5.7) can be converted to bitangential interpolation conditions (see, e.g., [26]); for simplicity, say that these interpolation conditions have the form xi S(λi ) = yi ,

S(λ j )uj = vj for i = 1, . . . , k,

j = 1, . . . , k

(5.8)

for given distinct points λi , λ j in D, row vectors xi , yi and column vectors uj , vj . Then the robust H ∞ -problem (H ∞ rather than rational version) can be converted to the μ-Nevanlinna-Pick problem: ﬁnd holomorphic function S on the unit disk with matrix values representing operators in L(XU ⊕ W, XU ⊕ Z) satisfying the interpolation conditions (5.8) such that also μΔ (S(λ)) < 1 for all λ ∈ D. It is this μ-version of the Nevanlinna-Pick interpolation problem which has been studied from various points of view (including novel variants of the Commutant Lifting Theorem) by Bercovici-Foias-Tannenbaum (see [38, 39, 40, 41]) and Agler-Young (see [5, 7, 9, 11] and Huang-Marcantognini-Young [77]). These authors actually study only very special cases of the general control problem as formulated here; hence the results at this stage are not particularly practical for actual control applications. However this work has led to interesting new mathematics in a number of directions: we mention in particular the work of Agler-Young on new types of dilation theory and operator-model theory (see [6, 9]), new kinds of realization theorems [10], the complex geometry of new kinds of domains in Cd (see [8, 12, 13]), and a multivariable extension of the Bercovici-Foias-Tannenbaum spectral commutant lifting theorem due to Popescu [114]. 5.4. Notes In the usual formulation of μ (see [107, 141]), in addition to the scalar blocks δi Ini in Z(δ), it is standard to also allow some of the blocks to be full blocks of the form ⎡ (i) ⎤ (i) δ11 ··· δ1n

⎢ Δi = ⎣ .. .

(i) i1

δn

i

.. ⎥ . . ⎦

(i) ··· δn n

i i

The resulting transfer functions then have domains equal to (reducible) Cartan domains which are more general than the unit polydisk. The theory of the SchurAgler class has been extended to this setting in [15, 20]. More generally, it is natural also to allow non-square blocks. A formalism for handling this is given in [29]; for this setting one must work with the intertwining space of Δ rather than the commutant of Δ in the deﬁnition of μ in (4.23). With a formalism for such a non-square uncertainty structure available, one can avoid the awkward assumption in Subsection 5.3 and elsewhere that W = Z.

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6. Robust control with dynamic time-varying structured uncertainty 6.1. The state-space LFT-model formulation Following [97, 98, 96, 108], we now introduce a variation on the gain-scheduling problem discussed in Section 5.1 where the uncertainty parameters δU = (δ1 , . . . , δd ) become operators on 2 , the space of square-summable sequences of complex numbers indexed by the integers Z, and are to be interpreted as dynamic, timevarying uncertainties. To make the ideas precise, we suppose that we are given a system matrix as in (5.4). We then tensor all operators with the identity operator I2 on 2 to obtain an enlarged system matrix ⎡ ⎤ ⎤ ⎡ AUU AUS BU 1 BU 2 B2 A B1 ⎢ ASU ASS BS1 BS2 ⎥ ⎥ (6.1) M = ⎣C1 D11 D12 ⎦ ⊗ I2 = ⎢ ⎣ C1U C1S D11 D12 ⎦ ⊗ I2 , C2 D21 D22 C2U C2S D21 D22 which we also write as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A B1 (XU ⊕ XS ) ⊗ 2 (XU ⊕ XS ) ⊗ 2 B2 ⎦→⎣ ⎦. W ⊗ 2 Z ⊗ 2 M = ⎣C1 D11 D12 ⎦ : ⎣ 2 2 C2 D21 D22 U ⊗ Y ⊗

(6.2)

Given a decomposition XU = XU 1 ⊕ · · · ⊕ XUd of the uncertainty state space XU , we deﬁne the matrix pencil ZU (δ U ) with argument equal to a d-tuple δ U = (δ 1 , . . . , δ d ) of (not necessarily commuting) operators on 2 by ⎤ ⎡ IXU 1 ⊗ δ 1 ⎥ ⎢ .. ZU (δ U ) = ⎣ ⎦. . IXU d ⊗ δ d In addition we let S denote the bilateral shift operator on 2 ; we sometimes will also view S as an operator on the space of all sequences of complex numbers or on the subspace 2ﬁn of 2 that consists of all sequences in 2 with ﬁnite support. We obtain an uncertain linear system of the form ⎧ ∗ AM (δ U )x + BM1 (δ U )w + BM2 (δ U )u ⎨ S x = z = CM1 (δ U )x + DM11 (δ U )w + DM12 (δ U )u Σ: (6.3) ⎩ y = CM2 (δ U )x + DM21 (δ U )w + DM22 (δ U )u where the system ⎡ AM (δ U ) ⎣CM1 (δ U ) CM2 (δ U )

matrix ⎤ ⎤ ⎡ ⎤ ⎡ XS ⊗ 2ﬁn BM1 (δ U ) BM2 (δU ) XS ⊗ DM11 (δU ) DM12 (δ U )⎦ : ⎣ W ⊗ 2ﬁn ⎦ → ⎣ Z ⊗ ⎦ Y ⊗ DM21 (δU ) DM22 (δ U ) U ⊗ 2ﬁn

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is obtained from the feedback connection ⎤ ⎤ ⎡ ⎡ x xU U ⎥ ⎥ ⎢ ⎢ x ⎢ S ⎥ = M ⎢ xS ⎥ , subject to xU = ZU (δ U )x U , ⎣ w ⎣ z ⎦ ⎦ u y that is, ⎡ ⎤ ⎡ AM (δ U ) BM1 (δU ) BM2 (δ U ) ASS ⎣ CM1 (δ U ) DM11 (δ U ) DM12 (δ U ) ⎦ = ⎣ C1S CM2 (δ U ) DM21 (δ U ) DM22 (δ U ) C2S ⎤ ⎡ ASU + ⎣ C1U ⎦ (I − ZU (δ U )AUU )−1 ZU (δ U ) C2U

BS1 D11 D21

⎤ BS2 D12 ⎦ D22

AUS

BU 1

BU 2

.

(6.4) As this system is time-varying, due to the presence of the time-varying uncertainty parameters δU , it is not convenient to work with a transfer-function acting on the frequency-domain; instead we stay in the time-domain and work with the input-output operator which has the form DM11 (δ U ) DM12 (δ U ) CM1 (δ U ) G(δ) = + (6.5) DM21 (δ U ) DM22 (δ U ) CM2 (δ U ) × (IXS ⊗2 − (IXS ⊗ S)AM (δ U ))−1 (IXS ⊗ S) BM1 (δ U ) BM2 (δ U ) . Now write δ for the collection (δ U , S) of d+1 operators on 2 . Then the inputoutput operator G(δ) given by (6.5) has the noncommutative transfer-function realization G11 (δ) G12 (δ) D11 D12 C1 G(δ) = = + (I − Z(δ)A)−1 Z(δ) B1 B2 D21 D22 C2 G21 (δ) G22 (δ) (6.6) ZU (δ U ) 0 with system matrix as in (6.1) and Z(δ) = 0 IX ⊗S . In the formulas (6.4)– S

(6.6) the inverses may have to be interpreted as the algebraic inverses of the corresponding inﬁnite block matrices; in that way, the formulas make sense at least for the nominal plant, i.e., with δU = (0, . . . , 0). More generally, the transfer-function G can be extended to a function of d+1 variables in L(2 ) by replacing S with another variable δd+1 ∈ L(2 ). In that case, the transfer-function can be viewed as an LFT-model with structured uncertainty, as studied in [98, 57]. However, as a consequence of the Sz.-Nagy dilation theory, without loss of generality it is possible in this setting of LFT-models to ﬁx one of the variables to be the shift operator S; in this way the LFT-model results developed for d + 1 free variable contractions apply equally well to the case of interest where one of the variables is ﬁxed to be the shift operator.

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Such an input/state/output system Σ with structured dynamic time-varying uncertainty δ U is said to be robustly stable (with respect to the dynamic timevarying uncertainty structure ZU (δ U )) if the state-matrix AM (δ U ) is stable for all choices of δ U subject to ZU (δ U ) ≤ 1, that is, if IXS ⊗2 − (IXS ⊗ S)AM (δ U ) is invertible as an operator on XS ⊗ 2 for all δ U with ZU (δ U ) ≤ 1. Since AM (δ U ) = ASS + ASU (I − ZU (δU )AUU )−1 ZU (δ U )AUS , it follows from the Main Loop Theorem [141, Theorem 11.7 p. 284], that this condition in turn reduces to: IX − Z(δ)A is invertible for all δ = (δ U , S) with Z(δ) ≤ 1.

(6.7)

Note that this condition amounts to a noncommutative version of the Hautusstability criterion for the matrix A (where A = A ⊗ I2 ). We shall therefore call the state matrix A nc-Hautus-stable if (6.7) is satisﬁed (with nc indicating that we are in the noncommutative setting). The input/state/output system Σ is said to have nc-performance (with respect to the dynamic time-varying uncertainty structure ZU (δ U )) if it is robustly stable (with respect to this dynamic timevarying uncertainty structure) and in addition the input-output operator G(δ) has norm strictly less than 1 for all choices of δ = (δ U , S) with Z(δ) ≤ 1. One of the key results from the thesis of Paganini [108] which makes the noncommutative setting of this section more in line with the 1-D case is that, contrary to what is the case in Subsection 4.2, for operators A = A ⊗ I2 on X ⊗ 2 we do have μΔ (A) = μ Δ (A) when we take Δ to be the C ∗ -algebra ( ZU (δ U ) 0 2 Δ= : δ U = (δ 1 , . . . , δ d ), δ j ∈ L( ), j = 1, . . . , d + 1 . 0 IXS ⊗ δ d+1 (6.8) Write D for the commutant of Δ in L((XU ⊕ XS ) ⊗ 2 ). Then the main implication of the fact that μΔ (A) = μ Δ (A) is that nc-Hautus-stability of A is now the same as the existence of an invertible operator Q ∈ D so that Q−1 AQ < 1 or, equivalently, the existence of a solution X ∈ D to the LMIs A∗ XA − A < 0 and X > 0. However, it is not hard to see that X is an element of D if and only if X = X ⊗ I2 with X being an element of the C ∗ -algebra D in (4.11). Thus, in fact, we ﬁnd that A = A ⊗ I2 is nc-Hautus-stable precisely when A is scaled stable, i.e., when there exists a solution X ∈ D to the LMIs A∗ XA − A < 0 and X > 0. These observations can also be seen as a special case (when C2 = 0 and B2 = 0) of the following complete analogue of Theorem 2.3 for this noncommutative setting due to Paganini [108]. Proposition 6.1. Given a system matrix as in (6.1)–(6.2), then: (i) The output pair {C2 , A} is nc-Hautus-detectable, that is, for every δ = (δ 1 , . . . , δ d+1 ), with δ j ∈ L(2 ) for j = 1, . . . , d + 1, so that Z(δ) ≤ 1 the operator I − Z(δ)A (XU ⊕ XS ) ⊗ 2 2 : (XU ⊕ XS ) ⊗ → C2 Y ⊗ 2

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has a left inverse, if and only if {C2 , A} is nc-operator-detectable, i.e., there exists an operator L = L ⊗ I2 , with L : Y → X , so that A + LC2 is ncHautus-stable, if and only if there exists a solution X ∈ D to the LMIs A∗ XA − X − C2∗ C2 < 0,

X > 0.

(6.9)

(ii) The input pair {A, B2 } is nc-Hautus-stabilizable, that is, for every δ = (δ 1 , . . . , δ d+1 ), with δ j ∈ L(2 ) for j = 1, . . . , d + 1, so that Z(δ) ≤ 1 the operator (XU ⊕ XS ) ⊗ 2 I − AZ(δ) B2 : → (XU ⊕ XS ) ⊗ 2 U ⊗ 2 has a right inverse, if and only if {A, B2 } is nc-operator-stabilizable, i.e., there exists an operator F = F ⊗ I2 , with F : X → U, so that A + B2 F is nc-Hautus-stable, which happens if and only if there exists a solution Y ∈ D to the LMIs AY A∗ − Y − B2 B2∗ < 0, Y > 0. (6.10) In case the input/state/output system Σ is not stable and/or does not have performance, we want to remedy this by means of a feedback with a controller K, which we assume has on-line access to the structured dynamic time-varying uncertainty operators δU in addition to being dynamic, i.e., K = K(δ) = K(δ U , S). More speciﬁcally, we shall restrict to controllers of the form K(δ) = DK + CK (I − ZK (δ)AK )−1 ZK (δ)BK where ZK (δ) =

0 ZKU (δ U ) 0 IXKS ⊗ S

⎡ ⎢ , ZKU (δ U ) = ⎣

(6.11) ⎤

IXK1 ⊗ δ 1 ..

⎥ ⎦,

. IXKd ⊗ δ d

with system matrix MK of the form AK B K (XKU ⊕ XKS ) ⊗ 2 (XKU ⊕ XKS ) ⊗ 2 : → MK = C K DK Y ⊗ 2 U ⊗ 2

(6.12)

where XKU = XKU1 ⊕ · · · ⊕ XKUd , and where the matrix entries in turn have a tensor-factorization AK BK AK ⊗ I2 BK ⊗ I2 = . (6.13) CK DK CK ⊗ I2 DK ⊗ I2 If such a controller K(δ) is put in feedback connection with G(δ), where we impose the usual assumption D22 = 0 to guarantee well-posedness, the resulting closed-loop system input-output operator Gcl (δ), as a function of the operator uncertainty parameters δ U = (δ 1 , . . . , δ d ) and the shift S, has a realization which is formally exactly as in (4.9), that is Gcl (δ) = Dcl + Ccl (I − Zcl (δ)Acl )−1 Zcl (δ)Bcl

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with system matrix ⎡ A + B 2 DK C 2 Acl Bcl BK C2 =⎣ Ccl Dcl C1 + D12 DK C2

B 2 CK AK D12 CK

⎤ B1 + B2 DK D21 ⎦, BK D21 D11 + D12 DK D21

which is the same as the system matrix (4.10) tensored with I2 , and Z(δ) 0 where δ = (δ U , S). Zcl (δ) = 0 ZK (δ)

(6.14)

(6.15)

The state-space nc-stabilization problem (with respect to the given dynamic time-varying uncertainty structure δ U ) then is to design a controller K with statespace realization {AK , BK , CK , DK } as above so that the closed-loop system Σcl deﬁned by the system matrix (6.14) is robustly stable. The state-space nc-H ∞ problem is to design a controller K with state-space realization {AK , BK , CK , DK } as above so that the closed-loop system Σcl also has robust performance. Since the closed-loop state-operator Acl is equal to Acl ⊗ I2 with Acl deﬁned by (4.10), it follows as another implication of the fact that μΔ is equal to μ Δ for operators that are tensored with I2 (with respect to the appropriate C ∗ -algebra Δ) that Acl is nc-Hautus-stable precisely when Acl is scaled stable, i.e., we have the following result. Proposition 6.2. Let Σ and Σ be the systems given by (6.3) and (5.1), respectively, corresponding to a given system matrix (5.4). Then Σ is nc-Hautus-stabilizable if and only if Σ is scaled-stabilizable. Thus, remarkably, the solution criterion given in Section 4.2 for the scaled state-space stabilization problem turns out to be necessary and suﬃcient for the solution of the dynamic time-varying structured-uncertainty version of the problem. Theorem 6.3. Let Σ be the system given by (6.3) corresponding to a given system matrix (6.1). Then Σ is state-space nc-stabilizable if and only if the output pair {C2 , A} is nc-Hautus-detectable and the input pair {A, B2 } is nc-Hautus-stabilizable, i.e., if there exist solutions X, Y ∈ D, with D C ∗ -algebra given (4.11), Athe in K BK K BK to the LMIs (6.9) and (6.10). In this case K ∼ CK DK ⊗ I2 with A CK DK as in (4.12) is a controller solving the nc-Hautus stabilization problem for Σ. In a similar way, the state-space nc-H ∞ -problem corresponds to the scaled H -problem of Subsection (4.2). ∞

Theorem 6.4. Let Σ be the system given by (6.3) for a given system matrix (6.1). Then there exists a solution K, with realization (6.11), to the state-space nc-H ∞ problem for the non-commutative system Σ if and only if there exist X, Y ∈ D that satisfy the LMIs (4.27) and (4.26) and the coupling condition (4.28). Proof. Let Σ and Σ be the systems given by (6.3) and (5.1), respectively, corresponding to a given system matrix (5.4). Using the strict bounded real lemma from

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[29] in combination with similar arguments as used above for the nc-stabilizability problem, it follows that a transfer-function K with realization (6.11)–(6.13) is a solution to the state-space nc-H ∞ -problem for Σ if and only if the transfer function K with realization (4.7) is a solution to the scaled H ∞ -problem for the system Σ. The statement then follows from Theorem 4.8. 6.2. A noncommutative frequency-domain formulation In this subsection we present a frequency-domain version of the noncommutative state-space setup of the previous subsection used to model linear input/state/output systems with LFT-model for dynamic time-varying structured uncertainty. The frequency-domain setup here is analogous to that of Section 4.1 but the unit d d polydisk D is replaced by the noncommutative polydisk Dnc consisting of all d-tuples δ = (δ1 , . . . , δ d ) of contraction operators on a ﬁxed separable inﬁnitedimensional Hilbert space K. We need a few preliminary deﬁnitions. We deﬁne Fd to be the free semigroup consisting of all words α = iN · · · i1 in the letters {1, . . . , d}. When α = iN · · · i1 we write N = |α| for the number of letters in the word α. The multiplication of two words is given by concatenation: α · β = iN · · · i1 jM · · · j1 if α = iN · · · i1 and β = jM · · · j1 . The unit element of Fd is the empty word denoted by ∅ with |∅| = 0. In addition, we let z = (z1 , . . . , zd ) stand for a d-tuple of noncommuting indeterminates, and for any α = iN · · · i1 ∈ Fd − {∅}, we let z α denote the noncommutative monomial z α = ziN · · · zi1 , while z ∅ = 1. If α and β are two words in Fd , we multiply the associated monomials z α and z β in the natural way: z α · z β = z α·β . Given two Hilbert spaces U and Y, we let L(U, Y)z denote the " collection of α all noncommutative formal power series S(z) of the form S(z) = α∈Fd Sα z where the coeﬃcients Sα " are operators in L(U, Y) for each α ∈ Fd . Given a formal power series S(z) = α∈Fd Sα z α together with a d-tuple of linear operators δ = (δ 1 , . . . , δd ) acting on 2 , we deﬁne S(δ) by S(δ) = lim Sα ⊗ δ α ∈ L(U ⊗ K, Y ⊗ K) N →∞

α∈Fd : |α|=N

whenever the limit exists in the operator-norm topology; here we use the notation δ α for the operator δ α = δ iN · · · δ i1 if α = iN · · · i1 ∈ Fd − {∅} and δ ∅ = IK . We deﬁne the noncommutative Schur-Agler class SAnc,d (U, Y) (strict noncommutative Schur-Agler class SAonc,d (U, Y)) to consist of all formal power series in L(U, Y)z such that S(δ)) ≤ 1 (S(δ) < 1) whenever δ = (δ 1 , . . . , δ d ) is a

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d-tuple of operators on K with δ j < 1 (δj ≤ 1) for j = 1, . . . , d. Let Dnc,d := {δ = (δ 1 , . . . , δ d ) : δj ∈ L(K), δj < 1, j = 1, . . . , d}, Dnc,d := {δ = (δ 1 , . . . , δ d ) : δj ∈ L(K), δj ≤ 1, j = 1, . . . , d}. ∞,o (L(U, Y)) to consist of We then deﬁne the strict noncommutative H ∞ -space Hnc,d all functions F from Dnc,d to L(U ⊗ K, Y ⊗ K) which can be expressed in the form

F (δ) = S(δ) for all δ ∈ Dnc,d where ρ−1 S is in the strict noncommutative Schur-Agler class ∞ SAonc,d (U, Y) for some real number ρ > 0. We write Hnc,d (L(U, Y)) for the set of functions G from Dnc,d to L(U ⊗ K, Y ⊗ K) that are also of the form G(δ) = S(δ), but now for δ ∈ Dnc,d and ρ−1 S in SAnc,d (U, Y) for some ρ > 0. Note that SAnc,d (U, Y) amounts to SAnc,d (C, C) ⊗ L(U, Y). In the sequel we abbreviate the notation SAnc,d (C, C) for the scalar Schur-Agler class to simply SAnc,d . Similarly, ∞,o ∞,o ∞ we simply write SAonc,d , Hnc,d and Hnc,d instead of SAonc,d (C, C), Hnc,d (C, C) and ∞,o ∞,o ∞ Hnc,d (C, C), respectively. Thus we also have Hnc,d (L(U, Y)) = Hnc,d ⊗ L(U, Y), ∞,o of the etc. We shall be primarily interested in the strict versions SAonc,d and Hnc,d ∞ noncommutative Schur-Agler class and H -space. ∞,o We have the following characterization of the space Hnc,d (L(U, Y)). For the deﬁnition of completely positive kernel and more complete details, we refer to [30]. The formulation given here does not have the same form as in Theorem 3.6(2) of [30], but one can use the techniques given there to convert to the form given in the following theorem. Theorem 6.5. The function F : Dnc,d → L(U ⊗ K, Y ⊗ K) is in the strict noncom∞,o mutative H ∞ -space Hnc,d (L(U, Y)) if and only if there are d strictly completely positive kernels Kk : (Dnc,d × Dnc,d ) × L(K) → L(Y ⊗ K) for k = 1, . . . , d and a positive real number ρ so that the following Agler decomposition holds: ρ2 · (I ⊗ B) − S(δ) (I ⊗ B) S(τ )∗ =

d

Kk (δ, τ )[B − δ k Bτ ∗k ]

k=1

for all B ∈ L(K) and δ = (δ 1 , . . . , δ d ), τ = (τ 1 , . . . , τ d ) in Dnc,d . One of the main results of [28] is that the noncommutative Schur-Agler class has a contractive Givone-Roesser realization. Theorem 6.6. (See [28, 29].) A given function F : Dnc,d → L(U ⊗ K, Y ⊗ K) is in the strict noncommutative Schur-Agler class SAonc,d (U, Y) if and only if there exists a strictly contractive colligation matrix d d ⊕j=1 Xj ⊕j=1 Xj A B : M= → C D U Y

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for some Hilbert state space X = X1 ⊕ · · · ⊕ Xd so that the evaluation of F at δ = (δ 1 , . . . , δd ) ∈ Dnc,d is given by F (δ) = D ⊗ IK + (C ⊗ IK ((I − Z(δ)(A ⊗ IK ))−1 Z(δ)(B ⊗ IK ) where

⎡ IX1 ⊗ δ 1 ⎢ Z(δ) = ⎣

(6.16)

⎤ ..

⎥ ⎦.

. IXd ⊗ δ d

Hence a function F : Dnc,d → L(U ⊗ K, Y ⊗ K) is in the strict noncommutative ∞,o H ∞ -space Hnc,d (L(U, Y)) if and only if there is a bounded linear operator d d ⊕k=1 Xk ⊕k=1 Xk A B : → C D U Y such that

# # A # −1 # ρ C

# # # < 1 for some ρ > 0 ρ−1 D # B

so that F is given as in (6.16). If U and Y are ﬁnite-dimensional Hilbert spaces, we may view SAonc,d (U, Y) ∞,o (L(U, Y)) as matrices over the respective scalar-valued classes SAonc,d and Hnc,d ∞,o and Hnc,d . When this is the case, it is natural to deﬁne rational versions of SAonc,d ∞,o ∞,o and Hnc,d to consist of those functions in SAonc,d (respectively, Hnc,d ) for which the realization (6.16) can be taken with the state spaces X1 , . . . , Xd also ﬁnite∞,o dimensional; we denote the rational versions of SAonc,d and Hnc,d by RSAonc,d and ∞,o , respectively. We remark that as a consequence of Theorem 11.1 in [27], this RHnc,d ∞,o can be expressed intrinsically rationality assumption on a given function F in Hnc,d in terms of the ﬁniteness of rank for a ﬁnite collection of Hankel matrices formed from the power-series coeﬃcients Fα of F , i.e., the operators Fα ∈ L(U, Y) such that F (δ) = Fα ⊗ δ α . α∈Fd

In general, the embedding of a noncommutative integral domain into a skew∞,o , the embedding issue ﬁeld is diﬃcult (see, e.g., [75, 82]). For the case of RHnc,d ∞,o becomes tractable if we restrict to denominator functions D(δ) ∈ Hnc,d ∈ L(U) for which D(0) is invertible. If D is given in terms of a strictly contractive realization D(δ) = D + C(I − Z(δ)A)−1 Z(δ)B (where A = A ⊗ IK and similarly for B, C and D), then D(δ)−1 can be calculated, at least for Z(δ) small enough, via the familiar cross-realization formula for the inverse: D(δ)−1 = D−1 − D−1 C(I − Z(δ)A× )−1 Z(δ)BD−1 ∞,o where A× = A× ⊗ IK with A× = A − BD−1 C. We deﬁne Q(RHnc,d )(L(U, Y))0 to be the smallest linear space of functions from some neighborhood of 0 in Dnc,d (with respect to the Cartesian product operator-norm topology on Dnc,d ⊂ L(K)d )

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to L(U, Y) which is invariant under multiplication on the left by elements of ∞,o ∞,o (L(Y)) and by inverses of elements of RHnc,d (L(Y)) having invertible value RHnc,d at 0, and invariant under multiplication on the right by the corresponding set of functions with U in place of Y. Note that the ﬁnal subscript 0 in the notation ∞,o Q(RHnc,d )(L(U, Y))0 is suggestive of the requirement that functions of this class are required to be analytic in a neighborhood of the origin 0 ∈ Dnc,d . 0 (L(U, Y)) the space of functions deﬁned as follows: Let us denote by ROnc,d we say that the function G deﬁned on a neighborhood of the origin in Dnc,d with 0 values in L(U, Y) is in the space ROnc,d (L(U, Y)) if G has a realization of the form G(δ) = D + C(I − Z(δ)A)−1 Z(δ)B B for a colligation matrix M := [ A C D ] of the form M = M ⊗ IK where d d ⊕k=1 Xk ⊕k=1 Xk A B → : M= U Y C D for some ﬁnite-dimensional state-spaces X1 , . . . , Xd . Unlike the assumptions in the case of a realization for a Schur-Agler-class function in Theorem 6.6, there is no assumption that M be contractive or that A be stable. It is easily seen that ∞,o 0 (L(U, Y)))0 is a subset of ROnc,d (L(U, Y)); whether these two spaces Q(RHnc,d are the same or not we leave as an open question. We also note that the class 0 0 ROnc,d (L(U, Y)) has an intrinsic characterization: F is in ROnc,d (L(U, Y)) if and only if some rescaled version F (δ) = F (rδ) (where rδ = (rδ 1 , . . . , rδ d ) if δ = ∞,o (L(U, Y)) for (δ 1 , . . . , δ d )) is in the rational noncommutative H ∞ -class RHnc,d some r > 0 and hence has the intrinsic characterization in terms of a completely positive Agler decomposition and ﬁnite-rankness of a ﬁnite collection of Hankel ∞,o matrices as described above for the class RHnc,d (L(U, Y)). We may then pose the following control problems: Noncommutative polydisk internal-stabilization/H ∞ -control problem: We suppose that are given0 ﬁnite-dimensional spaces W, U, Z, Y and a block-matrix 11 we G12 G= G G21 G22 in ROnc,d (L(W ⊕ U, Z ⊕ Y)). We seek to ﬁnd a controller K in 0 (L(Y, U)) which solves the (1) internal stabilization problem, i.e. so that ROnc,d the closed-loop system is internally stable in the sense that all matrix entries of ∞,o , and which possibly also the block matrix Θ(G, K) given by (2.2) are in RHnc,d ∞ solves the (2) H -problem, i.e., in addition to internal stability, the closed-loop system has performance in the sense that Tzw = G11 + G12 K(I − G22 K)−1 G21 is in the rational strict noncommutative Schur-Agler class RSAonc,d (W, Z). 0 Even though our algebra of scalar plants ROnc,d is noncommutative, the parameterization result Theorem 3.5 still goes through in the following form; we leave it to the reader to check that the same algebra as used for the commutative case leads to the following noncommutative analogue. 0 (L(W ⊕ U, Z ⊕ Y)) is given and that G Theorem 6.7. Assume that G ∈ ROnc,d has at least one stabilizing controller K∗ . Deﬁne U∗ = (I − G22 K∗ )−1 , V∗ =

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∗ = (I − K∗ G22 )−1 and V∗ = (I − K∗ G22 )−1 K∗ . Then the set K∗ (I − G22 K∗ )−1 , U of all stabilizing controllers K for G is given by either of the two formulas K = (V∗ + Q)(U∗ + G22 Q)−1 subject to (U∗ + G22 Q)(0) is invertible, ∗ + QG22 )(0) is invertible, ∗ + QG22 )−1 (V∗ + Q) subject to (U K = (U and L are given by (3.8) and where in addition Q has the form Q = LΛL where L ∞,o Λ is a free stable parameter in Hnc,d (L(Y ⊕ U, U ⊕ Y)). Moreover, if Q = LΛL ∗ + QG22 )(0) is with Λ stable, then (U∗ + G22 Q)(0) is invertible if and only if (U invertible, and both formulas give rise to the same controller K. 0 (L(U, Y)), we say that G22 has a stable Given a transfer matrix G22 ∈ ROnc,d double coprime factorization if there exist transfer matrices D(δ), N (δ), X(δ), (δ), X(δ), Y (δ), D(δ), N and Y (δ) of compatible sizes with stable matrix entries ∞,o (i.e., with matrix entries in RHnc,d ) subject also to

D(0), D(0), X(0), X(0) all invertible so that the noncommutative version of condition (3.9) holds: (δ)D −1 (δ), G22 (δ) = D(δ)−1 N (δ) = N (δ) D(δ) −N (δ) X(δ) N 0 InY . = 0 InU −Y (δ) X(δ) Y (δ) D(δ)

(6.17)

Then we leave it to the reader to check that the same algebra as used for the commutative case leads to the following noncommutative version of Theorem 3.11. 0 Theorem 6.8. Assume that G ∈ ROnc,d is stabilizable and that G22 admits a double coprime factorization (6.17). Then the set of all stabilizing controllers is given by

(δ)Λ(δ))−1 (Y (δ) + D(δ)Λ(δ))(X(δ) +N = (X(δ) + Λ(δ)N (δ))−1 (Y (δ) + Λ(δ)D(δ)),

K(δ) =

∞,0 (0)Λ(0) where Λ is a free stable parameter from Hnc,d (L(U, Y) such that X(0) − N is invertible and X(0) + Λ(0)N (0) is invertible.

Just as in the commutative case, consideration of the H ∞ -control problem 0 for a given transfer matrix G ∈ ROnc,d (L(W ⊕ U, Z ⊕ Y)) after the change of the design parameter from the controller K to the free-stable parameter Λ in either of the two parameterizations of Theorems 6.7 and 6.8 leads to the following noncommutative version of the Model-Matching problem; we view this problem as a noncommutative version of a Sarason interpolation problem. Noncommutative-polydisk Sarason interpolation problem: Given matrices T1 , T2 , ∞,o , ﬁnd a matrix Λ (of appropriate size) over T3 of compatible sizes over RHnc,d ∞,o so that the matrix S = T1 +T2 ΛT3 is in the strict rational noncommutative RHnc,d Schur-Agler class RSAonc,d (W, Z).

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While there has been some work on left-tangential Nevanlinna-Pick-type interpolation for the noncommutative Schur-Agler class (see [22]), there does not seem to have been any work on a Commutant Lifting theorem for this setup or on how to convert a Sarason problem as above to an interpolation problem as formulated in [22]. We leave this area to future work. 6.3. Equivalence of state-space noncommutative LFT-model and noncommutative frequency-domain formulation In order to make the connections between the results in the previous two subsections, we consider functions as in Subsection 6.2, but we normalize the inﬁnitedimensional Hilbert space K to be 2 and work with d + 1 variables δ = (δ1 , . . ., δ d+1 ) in L(2 ) instead of d. As pointed out in Subsection 6.1, we may without loss of generality assume that the last variable δ d+1 is ﬁxed to be the shift operator S on 2 . The following is an improved analogue of Lemma 4.13 for the noncommutative setting. 0 (L(U, Y)) has a Theorem 6.9. Suppose that the matrix function W ∈ ROnc,d+1 ﬁnite-dimensional realization

W (δ) = D + C(I − Z(δ)A)−1 Z(δ)B, where A = A ⊗ I2 ,

B = B ⊗ I2 ,

C = C ⊗ I2 ,

D = D ⊗ I2 ,

which is both nc-Hautus-detectable and nc-Hautus-stabilizable. Then W is stable in the noncommutative frequency-domain sense (i.e., all matrix entries of W are ∞,o ) if and only if W is stable in the state-space sense, i.e., the matrix A in Hnc,d+1 is nc-Hautus-stable. Proof. If the matrix A is nc-Hautus-stable, it is trivial that then all matrix entries ∞,o of W are in Hnc,d+1 . We therefore assume that all matrix entries of W are in ∞,o . It remains to show that, under the assumption that {C, A} is nc-Hautus Hnc,d+1 detectable and that {A, B} is nc-Hautus stabilizable, it follows that A is nc-Hautus stable. The ﬁrst step is to observe the identity I − Z(δ)A Z(δ)B −1 S1 (δ) := (I − Z(δ)A) Z(δ)B = . (6.18) C W (δ) − D ∞,o (L(U, Y)) by assumption and trivially Z(δ)B is in Since W (δ) − D is in Hnc,d+1 ∞,o ∞,o (L(U, X ⊕ Y)). By the deHnc,d+1 (L(U, X )), it follows that S1 (δ) is in Hnc,d+1 tectability assumption and Proposition 6.1 it follows that there exists an operator L = L ⊗ I2 with L : Y → X so that A + LC is nc-Hautus-stable. Thus F1 (δ) = (I − Z(δ)(A + LC))−1 I −Z(δ)L

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∞,o (L(X ⊕ Y, X )). Note that F1 (δ)S1 (δ) = (I − Z(δ)A)−1 Z(δ)B. The is in Hnc,d+1 ∞,o fact that both F1 and S1 are transfer-functions over Hnc,d+1 implies that S2 (δ) = ∞,o −1 (I − Z(δ)A) Z(δ)B is in Hnc,d+1 (L(U, X )). We next use the identity Z(δ) S2 (δ) : = Z(δ) (I − AZ(δ))−1 Z(δ)B (6.19) = Z(δ)(I − AZ(δ))−1 I − AZ(δ) B .

Now the nc-Hautus-stabilizability assumption and the second part of Proposition ∞,o 6.1 imply in a similar way that S3 (δ) = Z(δ)(I −AZ(δ))−1 is in Hnc,d+1 (L(X , X )). Note that S3 in turn has the trivial realization S3 (δ) = D + C (I − Z(δ)A )−1 Z(δ)B

A B A B

I where C D = C D ⊗ I2 and C = [A I 0 ]. Thus (A , B , C , D ) = D (A, I, I, 0) is trivially GR-controllable and GR-observable in the sense of [27]. On A B the other hand, by Theorem 6.6 there exists a strictly contractive matrix C 0 so that S3 (δ) = r

C

(I − Z(δ)A

)−1 Z(δ)B

A

B

for some r

< ∞. Moreover, by the Kalman decomposition for noncommutative GR-systems given in [27], we may assume without loss of generality that (A

, B

, C

, 0) is GR-controllable and GR-observable. Then, by the main result of in [14], it is known that the function S(δ) = " Alpay-Kaliuzhnyi-Verbovetskyi " α α α∈Fd Sα ⊗ δ uniquely determines the formal power series S(z) = α∈Fd Sα z . It now follows from the State-Space Similarity Theorem for noncommutative GRsystems in [27] that there is an invertible block diagonal similarity transform Q ∈ L(X , X

) so that −1

Q A B

A I A 0 B

Q 0 . = := 0 I C 0 I 0 0 I r

C

0 In particular, A = Q−1 A

Q where A

is a strict contraction and Q is a structured similarity from which it follows that A is also nc-Hautus-stable as wanted. We can now obtain the equivalence of the frequency-domain and state-space formulations of the internal stabilization problems for the case where the statespace internal stabilization problem is solvable. Theorem 6.10. Suppose that we are given a realization G11 (δ) G12 (δ) D11 D12 C1 G(δ) = = + (I − Z(δ)A)−1 Z(δ) B1 G21 (δ) G22 (δ) D21 0 C2

B2

0 for an element G ∈ ROnc,d+1 (L(W ⊕ U, Z ⊕ Y)) such that the state-space internal stabilization problem has a solution. Suppose also that we are given a controller 0 K ∈ ROnc,d+1 (L(Y, U)) with state-space realization

K(δ) = DK + CK (I − ZK (δ)AK )−1 ZK (δ)BK .

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which is both nc-Hautus-stabilizable and nc-Hautus-detectable. Then the controller state-space K ∼ {AK , BK , CK , DK } solves C1 the D internal stabilization problem as11 D12 sociated with {A, B1 B2 , C2 , D21 0 } if and only if K(δ) solves the noncommutative frequency-domain internal stabilization problem associated with G (δ) G12 (δ) . G(δ) = G11 (δ) G (δ) 21 22 Proof. By Theorem 6.3, the assumption that the state-space internal stabilization problem is solvable means that {C2 , A} is nc-Hautus-detectable and {A, B2 } is ncHautus-stabilizable. We shall use this form of the standing assumption. Moreover, in this case, a given controller K ∼ {AK , BK , CK , DK } solves the state-space internal stabilization problem if and only if K stabilizes G22 . Suppose now that K ∼ {AK , BK , CK , DK } solves the state-space internal stabilization problem, i.e., the state operator Acl in (6.14) is nc-Hautus-stable. Note that the 3 × 3 noncommutative transfer matrix Θ(G, K) has realization Θ(G, K) = DΘ + CΘ (I − ZΘ (δ)AΘ )−1 ZΘ (δ)BΘ with ZΘ (δ) = Zcl (δ) as in (6.15) where AΘ BΘ AΘ BΘ = ⊗ I2 CΘ DΘ CΘ DΘ with A + B2 DK C2 B2 CK B1 + B2 DK D21 B2 B2 DK AΘ = , BΘ = , BK C2 AK BK D21 0 BK ⎡ ⎡ ⎤ ⎤ C1 + D12 DK C2 D12 CK D11 + D12 DK D21 D12 D12 DK DK C2 CK ⎦ , DΘ = ⎣ DK D21 I DK ⎦. CΘ = ⎣ C2 0 D21 0 I (6.20) Now observe that AΘ is equal to Acl , so that all nine transfer matrices in Θ(G, K) have a realization with state operator AΘ = Acl nc-Hautus-stable. Hence all ∞,o matrix entries of Θ(G, K) are in Hnc,d+1 . Suppose that K(δ) with realization K ∼ {AK , BK , CK , DK } internally stabilizes G in the frequency-domain sense. This means that all nine transfer ' := matrices in Θ(G, K) are stable. In particular, the 2 × 2 transfer matrix W ' has realizaΘ(G22 , K) − Θ(G22 , K)(0) is stable. From (6.20) we read oﬀ that W tion ' (δ) = DK C2 CK (I − ZΘ (δ)AΘ )−1 B2 B2 DK . W C2 0 0 BK to show By Theorem to show that Acl = AΘ is nc-Hautus-stable, it suﬃces DK C2 6.9, CK 2 DK , Acl is nc-Hautus-detectable and that Acl , B02 BB is that C2 0 K nc-Hautus-stabilizable. By using our assumption that {AK , BK , CK , DK } is both nc-Hautus-detectable and nc-Hautus-stabilizable, can nowfollow the argument K Cone 2 CK , Acl is noncommutative in the proof of Theorem 4.9 to deduce that DC 0 2 2 DK detectable and that Acl , B02 BB is noncommutative Hautus-stabilizable as K needed.

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We do not know as of this writing whether any given controller K in the 0 (L(Y, U)) has a nc-Hautus-detectable/stabilizable realization (see space ROnc,d+1 the discussion in the Notes below). However, for the Model-Matching problem, internal stabilizability in the frequency-domain sense means that all transfer ma∞,o trices T1 , T2 , T3 are stable (i.e., have all matrix entries in Hnc,d+1 ) and hence the T1 T2 standard plant matrix G = T3 0 has a stable realization. A given controller K solves the internal stabilization problem exactly when it is stable; thus we may work with realizations K ∼ {AK , BK , CK , DK } with AK nc-Hautus-stable, and hence a fortiori with both {CK , AK } nc-Hautus-detectable and {AK , BK } ncHautus-stabilizable. In this scenario Theorem 6.10 tells us that a controller K(δ) solves the frequency-domain internal stabilization problem exactly when any stable realization K ∼ {AK , BK , CK , DK } solves the state-space internal stabilization problem. Moreover, the frequency-domain performance measure matches with the state-space performance measure, namely: that the closed-loop transfer matrix Tzw = G11 + G12 (I − KG22 )−1 KG21 be in the strict noncommutative Schur-Agler class SAonc,d+1 (W, Z). In this way we arrive at a solution of the noncommutative Sarason interpolation problem posed in Section 6.2. Theorem Suppose that we are given a transfer matrix of the form G = T1 T2 6.11. ∞,o ∈ Hnc,d+1 (L(W ⊕ U, Z ⊕ Y)) with a realization T3 0 D11 T1 (δ) T2 (δ) = T3 (δ) 0 D21

D12 C1 (I − Z(δ)A)−1 Z(δ) B1 + 0 C2

(so C2 (I − Z(δ)A)−1 Z(δ)B = 0 for all δ) where ⎡ ⎤ ⎡ A B 1 B2 A B1 ⎣C1 D11 D12 ⎦ = ⎣C1 D11 C2 D21 0 C2 D21

B2

⎤ B2 D12 ⎦ ⊗ I2 0

∞,o as usual. Then there exists a K ∈ Hnc,d+1 so that T1 + T2 KT3 is in the strict noncommutative Schur-Agler class SAonc,d+1 if and only if there exist X, Y ∈ D, with D as in (4.11), satisfying LMIs: ⎤ ⎡ ∗ AY A∗ − Y AY C1∗ B1 Nc 0 Nc 0 ⎣ ∗ ∗ ⎦ C1 Y A C1 Y C1 − I D11 < 0, Y > 0, 0 I 0 I ∗ B1∗ D11 −I ⎤ ⎡ ∗ A∗ XA − X A∗ XB1 C1∗ 0 No 0 ⎣ ∗ ∗ ∗ ⎦ No B1 XA B1 XB1 − I D11 < 0, X > 0, 0 I 0 I C1 D11 −I

and the coupling condition

X I

I Y

≥ 0.

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Here Nc and No are matrices chosen so that

Nc is injective and Im Nc = Ker B2∗ No is injective and Im No = Ker C2

∗ D12 and D21 .

6.4. Notes Δ (A) where Δ is as in (6.8) appears in Paganini’s 1. The equality of μΔ (A) with μ thesis [108]; as mentioned in the Introduction, results of the same ﬂavor have been given in [37, 42, 60, 99, 129]. Ball-Groenewald-Malakorn [29] show how this result is closely related to the realization theory for the noncommutative Schur-Agler class obtained in [28]. There it is shown that μΔ (A) ≤ μΔ (A) = μ Δ (A), where μΔ (A) is a uniform version of μΔ (A). The fact that μΔ (A) = μΔ (A) is the content of Theorem B.3 in [108]. Paganini’s analysis is carried out in the more general form required to obtain the result of Proposition 6.1. The thesis of Paganini also includes some alternate versions of Proposition 6.1. Speciﬁcally, rather than letting each δ j be an arbitrary operator on 2 , one may restrict to such operators which are causal (i.e., lower-triangular) and/or slowly time-varying in a precise quantitative sense. With any combination of these reﬁned uncertainty structures in force, all the results developed in Section 6 continue to hold. With one or more of these modiﬁcations in force, it is more plausible to argue that the assumption made in Section 6.1 that the controller K has on-line access to the uncertainties δ i is physically realistic. Δ (A) < 1 can be conThe replacement of the condition μΔ (A) < 1 by μ sidered as a relaxation of the problem: while one really wants μΔ (A) < 1, one is content to analyze μ Δ (A) < 1 since μ Δ (A) is easier to compute. Necessary and suﬃcient conditions for μ Δ (A) < 1 then provide suﬃcient conditions for μΔ (A) < 1 (due to the general inequality μΔ (A) ≤ μ Δ (A)). In the setting of the enhanced uncertainty structure discussed in this section, by the discussion immediately preceding Proposition 6.1 we see in this case that the relaxation is exact in the sense that μ Δ (A) < 1 is necessary as well as suﬃcient for μΔ (A) < 1. In Remark 1.2 of the paper of Megretsky-Treil [99], it is shown how the μ-singularvalue approach can be put in the following general framework involving quadratic constraints (called the S-procedure for obscure reasons). One is given quadratic functionals σ0 , σ1 , . . . , σ deﬁned on some set L and one wants to know when it is the case that σj (x) ≥ 0 for j = 1, . . . , =⇒ σ0 (x) ≤ 0 for x ∈ L.

(6.21)

A computable suﬃcient condition (the relaxation) is the existence of nonnegative real numbers τ1 , . . . , τ (τj ≥ 0 for j = 1, . . . , ) so that σ0 (x) +

τj σj (x) ≤ 0 for all x ∈ L.

(6.22)

j=1

The main result of [99] is that there is a particular case of this setting (where L is a linear shift-invariant subspace of vector-valued L2 (0, ∞) (or more generally

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L2loc (0, ∞)) and the quadratic constraints are shift-invariant) where the relaxation is again exact (i.e., where (6.21) and (6.22) are equivalent); this result is closely related to Proposition 6.1 and the work of [108]. A nice survey of the S-procedure and its applications to a variety of other problems is the paper of P´ olik-Terlaky [112]. 2. It is of interest to note that the type of noncommutative system theory developed in this section (in particular, nc-detectability/stabilizability and nccoprime representation as in (6.17)) has been used in the work of Beck [36] and Li-Paganini [89] in connection with model reduction for linear systems with LFTmodelled structured uncertainty. 3. We note that Theorem 6.8 gives a Youla-Kuˇcera-type parametrization for 0 the set of stabilizing controllers for a given plant G ∈ ROnc,d (L(W ⊕ U, Z ⊕ Y)) under the assumption that G22 has a double coprime factorization. In connection with this result, we formulate a noncommutative analogue of the conjecture of 0 Lin: If G ∈ ROnc,d (L(W ⊕ U, Z ⊕ Y)) is stabilizable, does it follow that G22 has a double-coprime factorization? If G22 has a realization G22 (δ) = C2 (I − Z(δ)A)−1 Z(δ)B2 B AB with [ A C 0 ] = [ C 0 ] ⊗ I2 nc-Hautus stabilizable and nc-Hautus detectable, then one can adapt the state-space formulas for the classical case (see [104, 85]) to arrive at state-space realization formulas for a double-coprime factorization of G22 . If it is the case that one can always ﬁnd a nc-Hautus stabilizable/detectable realization for G22 , it follows that G22 in fact always has a double-coprime factorization and hence the noncommutative Lin conjecture is answered in the aﬃrmative. However, we do not know at this time whether nc-Hautus stabilizable/detectable realizations 0 always exist for a given G22 ∈ ROnc,d (L(U, Y)). From the results of [27], it is known that minimal, i.e., controllable and observable realizations exist for a given G22 . However, here controllable is in the sense that a certain ﬁnite collection of control operators be surjective and observable is in the sense that a certain ﬁnite collection of observation operators be injective. It is not known if this type of controllability is equivalent to nc-Hautus controllability, i.e., to the operator pencil I − Z(δ)A B being surjective for all δ ∈ L(2 )d+1 (not just δ in the noncommutative polydisk Dnc,d ). Thus it is unknown if controllable implies ncHautus stabilizable in this context. Dually, we do not know if observable implies nc-Hautus detectable. 4. Theorem 6.9 can be viewed as saying that, under a stabilizability/detectability hypothesis, any stable singularity of the noncommutative function W must show up internally as a singularity in the resolvent (I − Z(δ)A)−1 of the state matrix A. A variant on this theme is the well-known fact for the classical case that, under a controllability/observability assumption, any singularity (stable or not) of the rational matrix function W (λ) = D + λC(I − λA)−1 B necessarily must show up internally as a singularity in the resolvent (I−λA)−1 of the state matrix A. A version of this result for the noncommutative case has now appeared in the paper of Kaliuzhnyi-Verbovetskyi-Vinnikov [82]; however the notion of controllable and

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observable there is not quite the same as the notion of controllable and observable for non-commutative Givone-Roesser " systemsn as given in [27]. 5. Given a function S(z) = n∈Zd Sn z (where z = (z1 , . . . , zd ) is the varid

+

able in the commutative polydisk D and we use the standard multivariable notation z n = z1n1 · · · zdnd if n = (n1 , . . . , nd ) ∈ Zd+ ), we know from the results of [2, 3, 35] that S has a contractive realization S(z) = D + C(I − Z(z)A)Z(z)B. In light of the work of [28], we see that any such contractive system matrix d A B ] : (⊕d [C k=1 Xk ⊕ U) → (⊕k=1 Xk ⊕ Y) can also be used to deﬁne an element D S of the noncommutative Schur-Agler class SAnc,d (U, Y): S(δ) = D + C(I − Z(δ)A)−1 Z(δ)B B A B where [ A C D ] = [ C D ] ⊗ I2 . Thus a choice of contractive realization {A, B, C, D} for the commutative Schur-Agler-class function S can be viewed as a choice of noncommutative lifting to a noncommutative Schur-Agler-class function S(δ); the lifting property is that d

S(zI) = S(z) ⊗ I2 where zI = (z1 I2 , . . . , zd I2 ) ∈ Dnc,d if z = (z1 , . . . , zd ) ∈ D . While the realization for the commutative function is highly non-unique, the realization for the noncommutative function is unique up to state-space similarity if arranged to be minimal (i.e., controllable and observable as in [27]). Philosophically one can say that evaluation of the function on the commutative polydisk Dd does not give enough frequencies to detect the realization; enlarging the frequency domain (or points of evaluation) to the noncommutative polydisk Ddnc,d does give enough frequencies to detect the realization in an essentially unique way. Acknowledgement The authors thank Quanlei Fang and Gilbert Groenewald for the useful discussions in an early stage of preparation of the present paper. We also thank the two anonymous reviewers for their thorough readings of the ﬁrst version and constructive suggestions for the preparation of the ﬁnal version of this paper.

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[133] S. Treil, The gap between the complex structures singular value μ and its upper bound is inﬁnite, preprint. [134] H.L. Trentelman and J.C. Willems, H∞ control in a behavioral context: the full information case, IEEE Trans. Automat. Control 44 (1999) No. 3, 521–536. [135] F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions: a survey, Linear Algebra and its Applications 25 (1979), 219–237. [136] M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, 1985. [137] M. Vidyasagar, H. Schneider and B.A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Automat. Control 27 (1982) No. 4, 880–894. [138] D.C. Youla and G. Gnavi, Notes on n-dimensional system theory, IEEE Trans. Circuits and Systems 26 (1979) No. 2, 105–111. [139] G. Zames, Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Trans. Automat. Control 26 (1981) No. 2, 301–320. [140] G. Zames and B.A. Francis, Feedback, minimax sensitivity, and optimal robustness, IEEE Trans. Automat. Control 28 (1983) No. 5, 585–601. [141] K. Zhou, J.C. Doyle and K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1996. Joseph A. Ball and Sanne ter Horst Department of Mathematics Virginia Tech Blacksburg VA 24061, USA e-mail: [email protected] [email protected] Received: April 24, 2009 Accepted: June 17, 2009

Operator Theory: Advances and Applications, Vol. 203, 89–98 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Absence of Existence and Uniqueness for Forward-backward Parabolic Equations on a Half-line P. Binding and I.M. Karabash Abstract. We consider the “forward-backward” parabolic equation dψ/dx = −JLψ, 0 < x < ∞, where L is a self-adjoint Sturm-Liouville operator in the |r|-weighted space L2|r| , and J is the operator of multiplication by sgn r in the same space. It is assumed that the weight function r changes sign. The equation is equipped with a half-range boundary condition at x = 0 and a growth/decay condition as x → +∞. In situations where the operator L has some negative spectrum, we show that a general existence and uniqueness theorem cannot be obtained just by appropriate choice of the above growth/decay condition. Mathematics Subject Classiﬁcation (2000). Primary 35K70, 47B50; Secondary 35M10, 35K90. Keywords. Forward-backward parabolic equations, existence and uniqueness theorem, Riesz basis property, two parameter eigencurves.

1. Introduction Consider the equation dψ = −JLψ(x) (0 < x < X ≤ ∞), (1.1) dx where L and J are operators in an abstract Hilbert space H such that L is a self-adjoint operator and J is a signature operator in H. The latter means that J = J ∗ = J −1 , and therefore J = P+ ⊕ P− , where P± are orthogonal projections on the mutually complementary subspaces H± := ker(J ∓ I), H = H+ ⊕ H− . We The work of PB was partly supported by NSERC of Canada. The work of IK was partly supported by a PIMS Postdoctoral Fellowship at the University of Calgary. Communicated by L. Rodman.

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study strong solutions of (1.1), i.e., functions ψ : [0, X) → H that are continuous on [0, X), strongly diﬀerentiable on (0, X), and satisfy (1.1). We are mainly interested in the case when H is the |r|-weighted space L2|r| (a, b), −∞ ≤ a 0, and J is the operator of multiplication by sgn r. Thus the operator A := JL is a J-self-adjoint Sturm-Liouville operator associated with the diﬀerential expression ay := 1r (−y

+ qy), J is a signature operator and a fundamental symmetry in the Krein space L2r (a, b). One type of boundary condition that has both physical sense (see, e.g., [11, 21, 2, 10, 20] and the references therein) and tractability from the mathematical point of view is the class of “half-range” conditions, which for X = +∞ take the form P+ ψ(0) = ϕ+ ∈ H+

(1.3)

with a certain growth (or decay) condition on ψ(x) as x → +∞. With such choice of L and J, (1.1) is a so-called “forward-backward” parabolic equation. Also, it belongs to the wider class of second order equations with nonnegative characteristic form (see [16]). “Half-range” boundary problems for such equations have been studied under some positivity assumptions on the coeﬃcients (see [17, 2, 18, 21], [10, Chapter 10], [20]) and for (1.1) they usually involve L ≥ 0. It is considered known by experts that if the negative part of the spectrum of L is nonempty, then a “good” existence and uniqueness theorem does not exist. In this note we aim to give conditions attaching a precise meaning to this statement and to discuss situations illustrating these conditions. The case σ(L) ∩ (−∞, 0) = ∅ for abstract equations close in form to (1.1) was considered in [11, 9] and [10, Chapter 4] in connection with transport problems in a multiplying medium. In these papers, the decay condition at x = +∞ was chosen as ψ(x) = o(1) (or ψ(x) = O(1) ) as x → ∞, since these conditions are reasonable for some applied problems. The question of non-uniqueness or non-existence of solutions was studied via non-uniqueness and non-completeness indices, which could be made zero by additional orthogonality conditions imposed on the boundary value ϕ+ . In other circumstances, orthogonality conditions appeared in [18] (see also the references therein). In what follows, we consider more general growth/decay conditions of the type ψ(x) = o(γ(x))

(x → +∞).

(1.4)

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91

for appropriate functions γ. From the parabolic equations point of view, the case ψ(x) = o(xβ e−αx ) as x → +∞, with α, β ∈ R, is reasonable, and simple conditions of this type were mentioned in [8, 12]. Clearly, a change in the constants α, β provides a change in the non-uniqueness and non-completeness indices. For example, in the case (a, b) = R, r = sgn, A = JL = (sgn)(−d2 /dx2 + k sgn), where k ∈ R is a constant, the equation (1.1) equipped with “boundary” conditions (1.3) and ψ(x) = o(e−αx ) has a unique solution for any ϕ+ ∈ L2 (0, +∞) if and only if α = k (this follows easily from the arguments of [15, 13]). Under certain conditions on A (set up as (C1)–(C4) in Section 2), we show in Theorem 3.1 that a general existence and uniqueness theorem cannot be obtained just by appropriate choice of a growth/decay condition at +∞. Indeed we show that there is no scalar function γ : R+ → R+ such that problem (1.1), (1.3), (1.4) has a unique solution for arbitrary ϕ+ ∈ H+ (the same statement is valid with O(γ) instead of o(γ)). We also show in Section 4 that the conditions (C1)–(C3) mentioned above are not exceptional, but hold for a large class of operators of the form J(L + M ) where L arises from (1.2) and M corresponds to multiplication by a constant from a suitable range. (C4) involves extra conditions on the function r, and several authors have studied this question under the heading of “half-”, “partial-” or “full”range completeness – see Section 4 for an explicit condition, and [4] for a recent comparison of several conditions in the literature, some of which are shown there to be equivalent. A simple example satisfying (C1)–(C4), and detailed in Section 4, is given by (Jf )(x) = (sgn x)f (x), L = −d2 /dx2 with Dirichlet boundary conditions y(±1) = 0, and M f = μf with μ in the range (−π 2 , −π 2 /4).

2. Preliminaries Consider a complex Hilbert space H with a scalar product (·, ·) and norm h = ) (h, h). Suppose that H = H+ ⊕ H− , where H+ and H− are subspaces of H (all subspaces will be closed). Denote by P± the orthogonal projections from H onto H± . Let J = P+ − P− and [·, ·] := (J·, ·). Then the pair K = (H, [·, ·]) is called a Krein space (see also [1, 14]). A subspace H1 ⊂ H is called non-negative (non-positive) if [h, h] ≥ 0 (≤ 0, resp.) for all h ∈ H1 . A subspace H0 ⊂ H will be called indeﬁnite if there exist h± ∈ H0 such that ±[h± , h± ] > 0. A non-negative (non-positive) subspace H1 is called maximal non-negative (non-positive) if for any non-negative (non-positive) ˙ 3 if H1 admits a subspace H ⊃ H1 we have H = H1 . We write H1 = H2 [+]H decomposition into the direct sum of two J-orthogonal subspaces H2 and H3 . Suppose that two subspaces H± possess the properties (i) H+ is non-negative, H− is non-positive, (ii) (H+ , [·, ·]) and (H− , −[·, ·]) are Hilbert spaces;

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˙ − ; then this decomposition is called a canonical and suppose that H = H+ [+]H decomposition of the Krein space K. Evidently, H = H+ ⊕ H− is a canonical decomposition, although it is not the only one (see [14]). Proposition 2.1 (e.g., Theorems I.4.1 and I.4.5 in [1]). Let H = H+ ⊕ H− be a canonical decomposition and let P+ and P− be corresponding mutually complementary projections on H+ and H− , respectively. If H1 is a maximal non-negative subspace in H, then the restriction P+ H1 : H1 → H+ is a homeomorphism, that is, it is bijective, continuous, and the inverse mapping (P+ H1 )−1 : H+ → H1 is also continuous. The conditions that we shall impose on the operator A will be as follows: (C1) A is a J-self-adjoint operator in H. (C2) The spectrum σ(A) of A is discrete. +∞ (C3) The set of eigenvalues (counted by multiplicity) of A takes the form {λ+ n }n=0 ∪ − +∞ {λn }n=0 . The eigenvalues and eigenvectors satisfy the following conditions Re λ± 0 = η, λ± n

∈ R,

n ∈ N,

and

, yλ+ ] > 0, [yλ+ n n

Im λ± 0 = ±ζ, ··· <

λ− 2

<

λ− 1

ζ > 0,

<η<

[yλ− , yλ− ] < 0, n n

λ+ 1

(2.1) <

λ+ 2

n ∈ N,

< ···,

(2.2) (2.3)

where, for each eigenvalue λ, yλ denotes a ﬁxed eigenvector with yλ = 1. (n ≥ 0) form a Riesz basis of H. (C4) The eigenvectors yλ± n In [5], [6] and [7], one can ﬁnd Sturm-Liouville operators where most of these conditions are demonstrated: a fairly general situation satisfying them all will be studied in Section 4. (C1)–(C3) imply that A is deﬁnitisable (see [1, 14]). In fact, − [p(A)f, f ] ≥ 0 for all f ∈ dom(A3 ) if p(z) = (z − λ+ 0 )(z − λ0 )(z − η).

3. Absence of existence and uniqueness We write f (x) = O(g(x)) as x → x0 if f /g is bounded in a neighbourhood of x0 , f (x) g(x) if both f /g and g/f are bounded, and f (x) = o(g(x))) (as x → x0 ) if limx→x0 f (x)/g(x) = 0. Our abstract non-existence and -uniqueness result is as follows. Theorem 3.1. Assume that the operator A satisﬁes conditions (C1)–(C4). Then there is no scalar function γ : R+ → R+ such that the “half-range” boundary problem (1.3), (1.4) for the equation dψ = −Aψ(x) (0 < x < ∞), (3.1) dx has a unique solution for arbitrary ϕ+ ∈ H+ . The same statement holds true with condition ψ(x) = O(γ(x)) instead of (1.4).

(x → +∞)

(3.2)

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93

The proof is given below, but we start with two preliminary lemmas, assuming (C1)–(C4) throughout. : n ≥ 0} is maximal non-negative. Lemma 3.2. The subspace H1 := span{yλ+ n Proof. [14, Formula (II.2.10)], [14, Proposition I.3.2], and [14, Theorem II.3.1] imply the fact that eigenvectors yλ+ , yλ+ , yλ+ , . . . , are mutually J-orthogonal. 0 1 2 [14, Proposition I.3.2] and the remark after this proposition implies that [yλ+ , yλ+ ] = 0, 0

0

[yλ− , yλ− ] = 0 . 0

(3.3)

0

Taking (2.3) into account, we see that H1 is a non-negative subspace. Assume that a subspace H is such that H1 H . Then there exists h ∈

H \ H1 and h = cλ− yλ− + h1 , h1 ∈ span{yλ− , yλ− , . . . }. If h1 = 0, then, as before, 0 0 1 2 the second equality in (3.3) and (C3) imply [h, h] = [h1 , h1 ] < 0. If instead h1 = 0, then H0 := span{yλ− , yλ+ } ⊂ H . Since the direct sum in [14, formula (II.2.10)] 0 0 is J-orthogonal, H0 has zero isotropic part. Hence {H0 , [·, ·]} is a Krein space (see [14, Section 1.2 (c)]). (3.3) shows that {H0 , [·, ·]} is indeﬁnite, that is there exist h+ , h− ∈ H0 such that ±[h± , h± ] > 0. Since h− ∈ H , we again see that H is indeﬁnite. 1 := span{y − , y + , y + , . . . } is also a maximal nonRemark 3.3. It is clear that H λ0 λ1 λ2 1 are invariant subspaces of A. Note negative subspace, and that both H1 and H that we also established indeﬁniteness of H2 := span{yλ+ , yλ− , yλ+ , yλ+ , . . . }. 0

0

1

2

Consider equation (3.1) together with the initial condition ψ(0) = ϕ,

ϕ ∈ H.

(3.4)

Property (C4) is equivalent to the fact that an arbitrary h ∈ H admits the spectral decomposition cλ (h) yλ , where cλ (h) ∈ C are certain constants, h= λ∈σ(A)

and K1

|cλ (h)|2 ≤ h2 ≤ K2

λ∈σ(A)

|cλ (h)|2 ,

(3.5)

λ∈σ(A)

where K1 and K2 are positive constants. Lemma 3.4. Assume that ψ(x), x ≥ 0, is a solution of the initial value problem (3.1), (3.4). Then: (i) ψ(x) = o(e−αx ), x → ∞, α ∈ R, if and only if ϕ= cλ (ϕ) yλ . λ∈σ(A) Re λ>α

(3.6)

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P. Binding and I.M. Karabash

(ii) ψ(x) = O(e−αx ), x → ∞, α ∈ R, if and only if ϕ= cλ (ϕ) yλ .

(3.7)

λ∈σ(A) Re λ≥α

(iii) If ϕ = 0 and α(ϕ) := inf {Re λ : cλ (ϕ) = 0} > −∞,

(3.8)

ψ(x) e−xα(ϕ) .

(3.9)

then

Proof. It is clear that ψ(x) =

cλ (ϕ) e−λx yλ .

λ∈σ(A)

Combining this equality with (3.5), one easily obtains (i), (ii), and (iii).

Proof of Theorem 3.1. We start with problem (3.1), (1.3), (1.4) and as before we deﬁne ϕ = ψ(0). Let αγ := inf{α ∈ R : e−αx = o(γ(x)), x → +∞} (if {α ∈ R : e−αx = o(γ(x))} = ∅, we put αγ = +∞). Consider the case when αγ > −∞. Then Lemma 3.4 (i), (iii) shows that Hγ := span{yλ : Re λ ≥ αγ } is the set of ϕ such that problem (3.1), (3.4), (1.4) has a solution ψ. If αγ > η of (2.1), then Lemma 3.4 (i) yields Hγ H1 of Lemma 3.2. Therefore Proposition 2.1 implies that P+ Hγ H+ , so there exists ϕ+ ∈ H+ \ P+ Hγ such that problem (3.1), (1.3), (1.4) has no solutions. If αγ ≤ η, then Hγ ⊃ H2 H1 (the subspace H2 was deﬁned in Remark 3.3). Thus Proposition 2.1 implies that there exists ϕ ∈ Hγ \ {0} such that P+ ϕ = 0. Clearly, in this case, problem (3.1), (1.3), (1.4) has inﬁnitely many solutions for any ϕ+ ∈ H+ . γ of ϕ such that Finally, consider the case when αγ = −∞. Then the set H problem (3.1), (3.4), (1.4) has a solution is a linear manifold and includes H2 (as well as any subspace of the type span{yλ : Re λ ≥ α}, α ∈ R). The arguments for the case αγ ≤ η now show that problem (3.1), (1.3), (1.4) has inﬁnitely many solutions for any φ+ ∈ H+ . One can modify this proof for condition (3.2) using Lemma 3.4 (ii).

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4. Satisfaction of (C1)–(C4) We assume −∞ < a 0 a.e. and r is indeﬁnite, i.e., takes positive and negative values on sets of *b positive measure. We write H = L2|r| with norm given by y2 = a |ry 2 |, and Jy = (sgn r)y. For μ ∈ R, we let L(μ) be the (for simplicity, Dirichlet) operator in H satisfying 1 L(μ)y = (−y

+ (q + μ)y) |r| on the domain D(L(μ)) = {y ∈ H : y, y ∈ AC, L(μ)y ∈ H, y(a) = y(b) = 0} and we deﬁne A(μ) := JL(μ). Our next result shows that (C1) and (C2) always hold for such A(μ), and that (C3) holds for a suitable μ interval. We remark that the choice of (self-adjoint) boundary conditions does not aﬀect this result, although the proof is simpler in the separated case. Theorem 4.1. The operator A(μ) satisﬁes (C1) and (C2) for all real μ, and (C3) for a nonempty real μ interval. Proof. Since q + μ, |r| ∈ L1 (a, b), it follows from, e.g., [22] that L(μ) (and also L − λJ for any real λ) (a) are self-adjoint in H (b) are bounded below and (c) have compact resolvents. Then (C1) for A(μ) follows from (a). From (b), there exists μ+ such that L+ := L(μ+ ) > 0. Thus (c) shows that A+ := JL+ has a compact inverse L−1 + J, and hence A(μ) = A+ + (μ − μ+ )J has a compact resolvent. This proves (C2) for A(μ). To establish (C3) we shall use two parameter spectral theory, noting that the pencil L(μ) − λJ = L + μI − λJ satisﬁes the conditions of [3], and we brieﬂy summarise some properties that we shall need from that reference. From (C2), λ ∈ σ(A(μ)) if and only if μ is the nth eigenvalue μn (λ) (indexed in decreasing order with n ≥ 0) of λJ − L for some n. The graph of μn is called the nth eigencurve, and its slope satisﬁes μ n (λ) = (y, Jy) = [y, y]

(4.1)

where y = yλ,μ belongs to the null space of L + μI − λJ and is of unit norm in H. Moreover, μn (λ) is analytic in λ in a neighbourhood of any real λ and μn (λ) → +∞ as |λ| → +∞

(4.2)

– in fact μn (λ)/λ → ±1 as λ → ±∞. Let μ∗ be the minimum value of μ0 (λ), achieved at λ = λ∗ , say. From [3, Theorem 2.5] (translated to (λ∗ , μ∗ ) as origin), the eigenvalues of A(μ) are real

(4.3)

(λ − λ∗ )μ n (λ) > 0

(4.4)

and

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whenever μ > μ∗ (e.g., when μ = μ+ ). For such μ, μn (λ∗ ) < 0 so (4.2) and (4.4) ± show that the nth eigencurve has ordinate μ at two points λ± n = λn (μ), where

± ±(λ± − λ ) > 0 and ±μ (λ ) > 0. Using this and (4.1), we see that A(μ) satisﬁes ∗ n n n (C3) with η = λ∗ , except that (2.1) has been replaced by satisfaction of (2.2) for all n ≥ 0. We now vary μ near μ∗ , noting that the eigenvalues (real and nonreal, counted by multiplicity) of A(μ) are continuous in μ (see [3, Section 3] for details). In particular, if μn has a k-tuple zero at λ± n (μ∗ ), then there are k branches λ(μ) ∈ C of simple eigenvalues of A(μ), continuous in μ near μ∗ , with fractional power expansions in μ − μ∗ . Starting with n = 0, we note the estimate

1/2 λ± + o(|μ − μ∗ |)1/2 0 (μ) = λ∗ ± (2(μ − μ∗ )/μ0 (λ∗ ))

where μ

0 (λ∗ ) > 0, in [3, Corollary 6.2]. Thus the two eigenvalues λ± 0 (μ) are real for μ > μ∗ (as above), and become nonreal for μ < μ∗ (but near μ∗ ). Then (2.1), with μ-dependent (η, ζ) near (λ∗ , 0), follows from continuity in μ and the fact that such eigenvalues occur in conjugate pairs (see, e.g., [3, Corollary 3.4]). For n > 0, we claim that

± (λ± n (μ∗ ) − λ∗ )μn (λn (μ∗ )) > 0

(4.5)

λ± n (μ)

so by the implicit function theorem, the eigenvalues remain real and satisfy (C3) for an interval In of the form (μ∗ − δn , μ∗ ), say, with δn > 0. Indeed, > in (4.5) cannot be < without violating (4.4) so it remains to consider the case when ∗

∗ μn has a zero of order k > 1 at one of λ± n (μ∗ ), say λ . If k = 2 and μn (λ ) > 0 then again we violate (4.4), while in all other cases there must be nonreal eigenvalues for μ > μ∗ , contradicting (4.3) and establishing our claim. Finally, we need to show that the above intervals In contain a nonempty interval independent of n. If this fails then there is an unbounded sequence λn (positive or negative) where μ n (λn ) = 0 and μn (λn ) → 0 as n → ∞. On the other hand, [3, Theorem 2.7] (with a translation of origin) shows that there can be only ﬁnitely many such λn , and this contradiction completes the proof. Turning to (C4), we assume the following suﬃcient condition (see [6, Deﬁnition 3.1, Theorem 3.6 and Proposition 4.1]): r has ﬁnitely many turning points xj in neighbourhoods of which one-sided estimates of the form r(x) = |x − xj |p ρ(x)

(4.6)

hold for (j-dependent) p > −1 and ρ ∈ C 1 . Conditions which are more general but less simple to check can be found in, e.g., [19]. Then, for such r, Theorem 3.1 holds for A = A(μ) in a suitable interval of μ values given by Theorem 4.1. Example 4.2. We consider the family of eigenvalue problems (sgn x)(−y

(x) + μy(x)) = λy(x),

x ∈ (−1, 1);

y(±1) = 0

parametrized by μ ∈ R. Here L(μ) is the operator in L (−1, 1) corresponding to the diﬀerential expression (−d2 /dx2 + μ) and Dirichlet boundary conditions 2

Forward-backward Parabolic Equations

97

at x = ±1, and A(μ) corresponds to (sgn x)(−d2 /dx2 + μ) in the Krein space L2r (−1, 1) with r(x) = sgn x. The conditions of (4.6) are met at the single turning point x = 0 with p = 0 and ρ(x) = sgn x on each side. It is clear that the conditions of Theorem 4.1 at the start of this section are satisﬁed, and indeed, [5, Section 3] (where μ is replaced by −μ) gives an explicit parametrization of a pure imaginary pair λ± 0 (μ) in the interval μ ∈ (−π 2 , −π2 /4), as well as an illustration of the (λ, μ) eigencurves, which can be used to generate λ± n (μ) for n ≥ 1.

References [1] T.Ya. Azizov, I.S. Iokhvidov, Linear operators in spaces with an indeﬁnite metric. John Wiley and Sons, 1989. [2] R. Beals, Indeﬁnite Sturm-Liouville problems and half-range completeness. J. Diﬀerential Equations 56 (1985), 391–407. [3] P. Binding, P.J. Browne, Applications of two parameter spectral theory to symmetric generalised eigenvalue problems. Applic. Anal. 29 (1988), 107–142. [4] P. Binding, A. Fleige, Conditions for an indeﬁnite Sturm-Liouville Riesz basis property. Oper. Theory Adv. Appl., to appear. [5] P. Binding, H. Volkmer, Eigencurves for two-parameter Sturm-Liouville equations. SIAM Review 38 (1996), 27–48. ´ [6] B. Curgus, H. Langer, A Krein space approach to symmetric ordinary diﬀerential operators with an indeﬁnite weight function. J. Diﬀerential Equations 79 (1989), 31–61. [7] K. Daho, H. Langer, Sturm-Liouville operators with an indeﬁnite weight function. Proc. Royal Soc. Edinburgh Sect. A 78 (1977) 161–191. [8] A. Ganchev, W. Greenberg, C.V.M. van der Mee, A class of linear kinetic equations in a Krein space setting. Integral Equations Operator Theory 11 (1988), no.4, 518– 535. [9] W. Greenberg, C.V.M. van der Mee, Abstract kinetic equations relevant to supercritical media. J. Funct. Anal. 57 (1984), 111–142. [10] W. Greenberg, C.V.M. van der Mee, V. Protopopescu, Boundary value problems in abstract kinetic theory. Oper. Theory Adv. Appl. Vol. 23, Birkh¨ auser, 1987. [11] H.G. Kaper, C.G. Lekkerkerker, J. Hejtmanek, Spectral methods in linear transport theory. Oper. Theory Adv. Appl. Vol. 5, Birkh¨ auser, 1982. [12] I.M. Karabash, Stationary transport equations; the case when the spectrum of collision operators has a negative part. Proc. of the XVI Crimean Autumn Math. School– Symposium, Simferopol, Spectral and evolution problems 16 (2006), 149–153. [13] I.M. Karabash, Abstract kinetic equations with positive collision operators. Oper. Theory Adv. Appl. Vol. 188, Birkh¨ auser, Basel, 2008, 175–195. [14] H. Langer, Spectral functions of deﬁnitizable operators in Krein space. Lecture Notes in Mathematics 948, Springer (1982), 1–46. [15] C.V.M. van der Mee, Exponentially dichotomous operators and applications. Oper. Theory Adv. Appl. Vol. 182, Birkh¨ auser, 2008.

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[16] O.A. Ole˘ınik, E.V. Radkeviˇc, Second order equations with nonnegative characteristic form. Plenum Press, 1973. [17] C.D. Pagani, On forward-backward parabolic equations in bounded domains. Bollettino U.M.I. (5) 13-B (1976), 336–354. [18] S.G. Pyatkov, On the solvability of a boundary value problem for a parabolic equation with changing time direction. Dokl. Akad. Nauk SSSR 285 (1985), 1327–1329 (Russian); translation in Sov. Math. Dokl. 32 (1985), 895–897. [19] S.G. Pyatkov, Interpolation of some function spaces and indeﬁnite Sturm-Liouville problems. Oper. Theory Adv. Appl. Vol. 102, Birkh¨ auser, 1998. [20] S.G. Pyatkov, Operator Theory. Nonclassical Problems. Utrecht, VSP 2002. [21] S.A. Tersenov, Parabolic equations with changing time direction. Novosibirsk, Nauka 1985 (Russian). [22] J. Weidmann, Spectral theory of ordinary diﬀerential operators. Lecture Notes in Mathematics 1258, Springer (1987). P. Binding Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary T2N 1N4 Alberta, Canada e-mail: [email protected] I.M. Karabash Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary T2N 1N4 Alberta, Canada and Department of PDE Institute of Applied Mathematics and Mechanics R. Luxemburg str. 74 Donetsk 83114, Ukraine e-mail: [email protected] [email protected] Received: February 28, 2009 Accepted: March 26, 2009

Operator Theory: Advances and Applications, Vol. 203, 99–113 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Bounds for Eigenvalues of the p-Laplacian with Weight Function of Bounded Variation P.A. Binding and H. Volkmer Abstract. Pr¨ ufer angle methods are used to establish bounds for eigenvalues of the equation −(|y |p−2 y ) = λ(p − 1)r(x)|y|p−2y involving the p-Laplacian. The bounds are expressed in terms of a generalized total variation of the coeﬃcient r. An application of Kronecker’s theorem shows that the bounds are optimal in generic cases. Mathematics Subject Classiﬁcation (2000). 34B24, 26A45. Keywords. p-Laplacian, eigenvalue bounds, Pr¨ ufer angle, total variation, Kronecker’s theorem.

1. Introduction We study the diﬀerential equation −(|y |p−2 y ) = λ(p − 1)r(x)|y|p−2 y,

a≤x≤b

(1.1)

involving the p-Laplacian, which has attracted much attention in recent years. Boundary conditions of either Dirichlet or periodic/antiperiodic type will be imposed. We assume throughout that p > 1 and that r is a positive and integrable function on [a, b]. Initially we assume Dirichlet conditions y(a) = y(b) = 0.

(1.2)

For continuous r, Elbert [8] showed that the eigenvalues of (1.1), (1.2) form an increasing sequence 0 < λ1 < λ2 < λ3 < · · · , and that an eigenfunction y corresponding to λn has exactly n − 1 zeros in (a, b). Moreover, Elbert proved that the eigenvalues obey the asymptotic formula b 1/p πp π 2π λn = csc . (1.3) , cp := r1/p , πp := lim n→∞ n cp p p a Communicated by L. Rodman.

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Since then several authors have discussed eigenvalue asymptotics for the p-Laplacian, in one and several variables – see, for example, [1, 3, 7, 9]. We note that (1.3) can be expressed in the form Rp,n = o(n) where Rp,n := cp λ1/p n − nπp .

(1.4)

Binding and Dr´ abek [1] generalized this to integrable r and also showed that Rp,n = o(1) if r is absolutely continuous. Roughly, we aim for situations between these two, with r (in general) discontinuous but of bounded variation, related to estimates of the form Rp,n = O(1). (1.5) Indeed we give conditions allowing an explicit bound for the O(1) term, and examples showing that if our conditions fail then so may (1.5). In [7], Eberhard and Elbert gave results related to ours. While they aimed at indeﬁnite r (with a ﬁnite number of turning points) the deﬁnite version of [7, Theorem 2.4] would give (1.5) under conditions like ours but with extra diﬀerentiability of r. In [3], Bonder and Pinasco discussed the counting function N (λ) (i.e., the number of eigenvalues ≤ λ) for a collection of intervals, but for comparison we specialise to one interval. Then their Theorem 1.2 gives N (λ) = λ1/p (πp−1 cp + o(1)) as λ → ∞ for continuous r, corresponding to Elbert’s estimate (1.3). Their Theorem 1.6 requires a mean oscillation type assumption on r but their conclusion corresponds to a version of (1.5) with O(1) replaced by O(nα ) for some α > 0. In the case when p = 2 and ln r is of bounded variation, Hald and McLaughlin [10] have given the explicit bound |R2,n | ≤ 14 Tab (ln r),

(1.6)

Tab (f )

where denotes the total variation of a function f on [a, b]. They also showed that this inequality is “best possible” if r is continuous, but if r is allowed to have jumps then an estimate sharper than (1.6) is available. In Sections 2 and 3, we generalize (1.6) and its modiﬁcations to the case p = 2. For general p > 1, we show that (1.6) becomes |Rp,n | ≤ p−1 p−1/p q −1/q Tab (ln r), 1 p

1 q

(1.7)

where q is the conjugate of p (i.e., + = 1.) We also obtain improved estimates involving a generalized total variation (introduced in Section 2) which may also be used in some situations when the usual deﬁnition does not apply, as we shall see. In Section 4 we use a theorem of Kronecker in number theory to show that our estimates are optimal if p is a generic step function. Section 5 provides examples showing (even when p = 2) that (1.5) may fail when our assumptions are not satisﬁed, for example when r is not of bounded variation (but r and 1/r are bounded) or when r is of bounded variation but 1/r is unbounded. In Section 6 we consider periodic/antiperiodic boundary conditions for (1.1). In this case the spectrum can be considerably more complicated than under Dirichlet (or even general separated) conditions, but it is known [2, Theorems 3.7, 4.1]

p-Laplacian

101

that the eigenvalues with oscillation count n form a nonempty compact set Λn , say. (As in the case p = 2, oscillation is counted relative to the half open interval [a, b).) Our results bracket the Λn explicitly, and when p = 2, they provide estimates for “instability interval” lengths and criteria for boundedness of periodic and antiperiodic solutions extended over the real line. Many authors have considered such questions, and we shall compare our results with those of [6, 11] and [13] below. For general p, Brown and Eastham [4] have recently given asymptotics for the “rotational” eigenvalues introduced by Zhang [15]. We discuss further relations between these works and ours in Section 6.

2. Generalized total variation Let f : [a, b] → X be a function with values in a metric space (X, d). For a partition a = x0 < x1 < x2 < · · · < xm = b the variation of f is deﬁned as m

(2.1)

d(f (xj ), f (xj−1 )).

j=1

The total variation is Tab (f, d) := sup

m

d(f (xj ), f (xj−1 )),

j=1

where the supremum is taken over all possible partitions of [a, b]. The function f is said to have bounded variation if Tab (f, d) is ﬁnite. These notions have appeared frequently in the literature, for example, see Chistyakov [5]. We consider the special case X = R with metric d(u, v) := G(|u − v|),

(2.2)

where G : [0, ∞) → [0, ∞) is twice continuously diﬀerentiable and has the properties: G(0) = 0, G(x) > 0 for x > 0, and G (x) ≥ 0, G

(x) ≤ 0 for x ≥ 0 (so G (0) > 0.) From G(s) ≤ G (0)s for all s ≥ 0 it follows that Tab (f, d) ≤ G (0)Tab (f ),

(2.3)

Tab (f )

where denotes the total variation of f with respect to the usual (Euclidean) metric in R. It is easy to see that Tab (f, d) is ﬁnite if and only if Tab (f ) is ﬁnite. Moreover, Tab (f, d) = G (0)Tab (f ) < ∞ holds if and only if f is a continuous function of bounded variation. If f is of bounded variation but has jumps of large height then Tab (f, d) may be much smaller than G (0)Tab (f ). The following lemma will be useful in the next section. Lemma 2.1. Let f : [a, b] → R be a function of bounded variation. There is a sequence of step functions fk : [a, b] → R such that fk → f uniformly on [a, b] and Tab (fk , d) → Tab (f, d) with d as in (2.2).

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Proof. Let k ∈ N. Writing f as a diﬀerence of nondecreasing functions we ﬁnd a partition (2.1) such that |f (t) − f (s)| <

1 k

for s, t ∈ (xj−1 , xj ), j = 1, 2, . . . , m.

We deﬁne fk (xj ) := f (xj ), j = 0, 1, 2 . . . , m, and fk (x) := f ( 12 (xj−1 + xj ))

for x ∈ (xj−1 , xj ).

Then fk is a step function and |fk (x) − f (x)| < deﬁnition of fk ,

1 k

for all x ∈ [a, b]. Moreover, by

Tab (fk , d) ≤ Tab (f, d). Since fk (x) → f (x) for every x ∈ [a, b] it follows easily that Tab (fk , d) → Tab (f, d). In connection with equation (1.1) we will work with the metric dp (u, v) := Gp (|u − v|), where Gp is the function deﬁned by $ %1/p $ %1/q σ σ sinh 2q 1 s sinh 2p dσ, Gp (s) := p 0 sinh σ2

(2.4)

(2.5)

and q is the conjugate of p > 1. For example, if p = 2 then s G2 (s) = arctan sinh . 4 Then Gp has all the properties we required of G. This is easy to see except for the following fact. Lemma 2.2. We have G

p (s) < 0 for s > 0. Proof. Since G p (s) > 0 for s ≥ 0 it will be suﬃcient to show that the logarithmic derivative of G p (s) is negative for s > 0. We calculate s2

s2 s2 s s2 s s d ln G p (s) = 2 coth + 2 coth − coth . ds 2p 2p 2q 2q 2 2

This expression is negative for s > 0 provided that h(t) := t2 coth t has the subadditivity property h(t1 + t2 ) < h(t1 ) + h(t2 ) for t1 , t2 > 0. Subadditivity follows from the fact that h

(t) > 0 for t > 0 which in turn follows from 1 d2 sinh3 t 2 (t2 coth t) = sinh t(cosh t sinh t − t) + t(t cosh t − sinh t) 2 dt > 0 + 0 = 0.

p-Laplacian

103

Lemma 2.2 yields Gp (s) ≤ G p (0)s = p−1 p−1/p q −1/q s

for s ≥ 0.

We note that this inequality also follows from H¨older’s inequality applied to (2.5). Therefore, we have (2.6) Tab (f, dp ) ≤ p−1 p−1/p q −1/q Tab (f ).

3. Estimates of eigenvalues Consider ﬁrst equation (1.1) with λr = 1. It has the solution y = Sp (x) introduced by Elbert [8]. This function is odd and has period 2πp . Moreover, Sp is continuously diﬀerentiable, Sp (0) = 0, Sp (0) = 1 and |Sp (x)|p + |Sp (x)|p = 1

for x ∈ R.

(3.1)

It follows that the eigenvalues of (1.1), (1.2) with r = 1 are given by πp =n λ1/p , n b−a i.e., Rp,n = 0 in the notation of (1.4). Next, consider equation (1.1) with a step function r : [a, b] → (0, ∞). We suppose that we are given a partition (2.1) such that rj := r(x) > 0 is constant on (xj−1 , xj ). For λ > 0 and j = 1, 2, . . . , m, we consider the Pr¨ ufer-type transformation (with μ = λ1/p > 0 for notational simplicity) y = ρj Sp (θj ),

y = μrj ρj Sp (θj ); 1/p

see [1, §2]. For θj we obtain the ﬁrst-order diﬀerential equation $ % −1+1/p 1/p θj = μ rj r(x)|Sp (θj )|p + rj |Sp (θj )|p

(3.2)

(3.3)

of Carath´eodory type for θj (x, μ) on [a, b], where we impose the initial condition θj (a, μ) = 0. It is known that the eigenvalue λn of (1.1), (1.2) is the unique solution λ = μp of (3.4) θm (b, μ) = nπp . In fact m may be replaced by any j between 1 and m in (3.4) but we will use the equation as stated. Lemma 3.1. Let r be a positive step function. Then |Rp,n | ≤ Tab (ln r, dp )

(3.5)

for all n, where Rp,n is deﬁned in (1.4) and dp in (2.4). Proof. In this proof μ > 0 is ﬁxed and so the dependence of θj on μ is suppressed. If x ∈ (xj−1 , xj ) then equation (3.3) simpliﬁes to θj = μrj

1/p

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by virtue of (3.1), so 1/p

θj (xj ) = θj (xj−1 ) + μrj (xj − xj−1 ) for j = 1, 2, . . . , m.

(3.6)

Summing these equations we obtain m

θj (xj ) − θj (xj−1 ) = cp μ.

(3.7)

j=1

Familiar reasoning shows that, for every x ∈ [a, b], there is an integer k (deπ π pending on x but independent of j) such that θj (x) ∈ [k 2p , (k + 1) 2p ). Therefore, πp π the angles ψ = θj+1 (xj ) and φ = θj (xj ) both lie in [k 2 , (k + 1) 2p ) for some integer k and are connected by Tp (ψ) = uTp (φ), where

u := uj :=

rj+1 rj

1/p ,

and Tp (θ) :=

Sp (θ) . Sp (θ)

We consider the maximum of |ψ − φ| as a function of φ, and in order to do this, π it will be suﬃcient to restrict φ to the interval [0, 2p ]. Consider the function ψ − φ = f (φ) = Ap (uTp(φ)) − φ,

φ ∈ [0,

πp 2 ).

Here

πp πp , ) is the inverse function of Tp 2 2 π (see [8]) and we extend f (φ) to f ( 2p ) = 0 by continuity. Using Ap : R → (−

Tp (φ) = 1 + |Tp (φ)|p ,

A p (t) =

(3.8)

1 , 1 + |t|p

we calculate f (φ) =

u − 1 + (u − up )|Tp (φ)|p . 1 + |uTp (φ)|p π

Therefore, the function f (φ) vanishes at the end points φ = 0 and φ = 2p , and has only one critical point φ∗ determined by (Tp (φ∗ ))p = uu−1 p −u . This gives |θj+1 (xj ) − θj (xj )| ≤ |Fp (uj )|, where

+ + 1/p , 1/p , u−1 u−1 − Ap . Fp (u) := Ap u up − u up − u

(3.9)

p-Laplacian

105

Using (3.7), (3.9) we obtain m m−1 θj (xj ) − θj (xj−1 ) + θj+1 (xj ) − θj (xj ) − cp μ , |cp μ − θm (b)| = j=1 j=1 ≤

m−1

|Fp (uj )|.

j=1

A somewhat lengthy calculation yields |Fp (es/p )| = Gp (|s|) so |cp μ − θm (b)| ≤ Tab (ln r, dp ).

Together with (3.4), this yields (3.5). Lemma 2.1 allows us to extend estimate (3.5) to more general r.

Theorem 3.2. Suppose that r : [a, b] → (0, ∞) and ln r is of bounded variation. Then the nth eigenvalue λn of (1.1), (1.2) satisﬁes |Rp,n | ≤ Tab (ln r, dp ),

(3.10)

where Rp,n is deﬁned in (1.4) and dp in (2.4). Proof. By Lemma 2.1, there is a sequence of step functions fk such that fk → ln r uniformly on [a, b], and Tab (fk , dp ) → Tab (ln r, dp ). Set rk (x) := exp(fk (x)). Let λn,k denote the nth eigenvalue of (1.1) with r = rk and boundary conditions (1.2). It is easy to show that λn,k tends to λn as k → ∞. Let

b

cp,k := a

1/p

rk .

As k → ∞, cp,k converges to cp , and, by choice of rk , Tab (ln rk , dp ) converges to Tab (ln r, dp ). Therefore, by applying Lemma 3.1 to the eigenvalue λn,k and letting k → ∞ we obtain the desired inequality (3.10). In view of (2.6) we obtain the following corollary. Corollary 3.3. Under the assumptions of Theorem 3.2 we also have |Rp,n | ≤ p−1 p−1/p q −1/q Tab (ln r), where q is the conjugate of p.

(3.11)

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4. Optimality of bounds I In this section we consider (1.1), (1.2), again setting λ = μp . We assume that r is a positive step function: r(x) = rj > 0 for xj−1 ≤ x < xj , r(b) = rm , where a = x0 < x1 < · · · < xm = b is a partition of [a, b]. Lemma 3.1 yields the estimate −M ≤ Rp,n ≤ M

for n = 1, 2, . . . ,

(4.1)

where M:=

m−1

Mj ,

(4.2)

j=1

Mj := |Ap (uj vj ) − Ap (vj )| ,

(4.3)

in the notation of (3.8), and uj :=

rj+1 rj

+

1/p ,

vj :=

uj − 1 upj − uj

,1/p .

(4.4)

The following theorem shows that the bounds −M and M in (4.1) cannot replaced by tighter ones (independent of n) for “generic” step functions r. Theorem 4.1. For any sequence rj , such that the system xj 1/p r1/p = (xj − xj−1 )rj ej := xj−1

is linearly independent over the ﬁeld of rational numbers Q, we have lim sup Rp,n = M

(4.5)

n→∞

and lim inf Rp,n = −M.

(4.6)

n→∞

Proof. We have cp = e1 + · · · + em and we set τn := c−1 p (nπp + M ). " j By assumption, the system c−1 p i=1 ei , j = 1, 2, . . . , m, is linearly independent over Q. Therefore, by Kronecker’s theorem [12, Theorem 442], there is a sequence n1 < n2 < . . . of positive integers such that, for all j = 1, 2, . . . , m − 1, as k → ∞, +

, j j−1 Ap (vj ) if uj ≤ 1 τnk ei − Mi mod πp → (4.7) A p (uj vj ) if uj > 1 i=1 i=1 Therefore,

+ Tp

τnk

j i=1

ei −

j−1 i=1

, Mi

→

vj uj vj

if uj ≤ 1, if uj > 1.

(4.8)

p-Laplacian

107

We again use the modiﬁed Pr¨ ufer angles θj (x) := θj (x, μ), j = 1, 2 . . . , m, deﬁned 1/p by (3.2) and with constant derivative θj = μrj on [xj−1 , xj ]. Moreover, θ1 (a) = 0 and (4.9) Tp (θj+1 (xj )) = uj Tp (θj (xj )) πp πp with θj (xj ), θj+1 (xj ) ∈ [ij 2 , (ij + 1) 2 ) for some integers ij . It follows that θ1 (x1 , τnk ) = e1 τnk , and, by (4.8) with j = 1,

vj if uj ≤ 1 Tp (θ1 (x1 , τnk )) → uj vj if uj < 1. Therefore, by (4.9) with j = 1, θ2 (x1 , τnk ) − e1 τnk → −M1 and so θ2 (x2 , τnk ) − (e1 + e2 )τnk → −M1 . By (4.8) and (4.9) with j = 2 we get θ3 (x2 , τnk ) − (e1 + e2 )τnk → −M1 − M2 . Continuing in this way we obtain θm (b, τnk ) − τnk

m

ej → −

j=1

m−1

Mj = −M,

j=1

or, equivalently, (4.10) θm (b, τnk ) − nk πp → 0 as k → ∞. We know that θm (xm−1 , μ) is an increasing and diﬀerentiable function of μ. Since θm (b, μ) = θm (xm−1 , μ) + em μ we obtain ∂θm (b, μ) ≥ em . ∂μ 1/p

Therefore, (4.10) implies that τnk − λnk → 0, i.e., Rp,nk → M which, with (4.1), proves (4.5). The proof of (4.6) is similar. We just interchange the cases uj ≤ 1 and uj > 1 in (4.7).

5. Optimality of bounds II We continue with the problem (1.1), (1.2). In this section we show, even when p = 2, that our assumption that ln r is of bounded variation cannot be relaxed very far. We take r as an integrable “inﬁnite step function” r, that is, r(x) = rj > 0 for xj−1 ≤ x < xj for each j = 1, 2, . . . , where a = x0 < x1 < x2 < · · ·

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is an increasing sequence converging to b. We write μ = λ2 , and deﬁne Mj as in the previous section (with p = 2), but now M :=

∞

Mj

(5.1)

j=1

may be inﬁnite. First we prepare with the following Lemma 5.1. Let R : [a, b] → (0, 1] be a measurable function. For a < d ≤ b let μn (d) be the nth positive eigenvalue of −y

= μ2 R(x)y, y(a) = y(d) = 0. Then μn (b) ≤ μn (d) ≤ μn (b) + (b − d)δ −1 L−1 e(b−a)μn (d) μn (d), whenever R(x) ≥ δ > 0 on an interval of length L. Proof. Let θ(x, μ) be the (modiﬁed) Pr¨ ufer angle satisfying θ = μ(cos2 θ + R sin2 θ),

θ(a, μ) = 0.

(5.2)

Then θ(d, μn (d)) = nπ. It follows from (5.2) that θ(b, μn (d)) ≤ nπ + (b − d)μn (d).

(5.3)

The derivative of θ(b, μ) with respect to μ is given by b ∂θ (b, μ) = eL(t,μ) h(t, μ) dt, ∂μ a where h(x, μ) = cos2 θ(x, μ) + R(x) sin2 θ(x, μ), (x, μ) = μ(R(x) − 1) sin(2θ(x, μ)), b (t, μ) dt. L(x, μ) = x

It follows that ∂θ (b, μ) ≥ δLe−(b−a)μ . ∂μ Combining this estimate with (5.3), we obtain δLe−(b−a)μn (d) (μn (d) − μn (b)) ≤ θ(b, μn (d)) − θ(b, μn (b)) ≤ (b − d)μn (d). This proves the lemma.

The following theorem shows that if M = ∞ then the sequence R2,n can become unbounded with n.

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Theorem 5.2. Let {rj }∞ j=1 be a positive and bounded sequence such that M = ∞, where M is deﬁned by (5.1), (4.3), (4.4). Then there is an inﬁnite step function r attaining values rj such that the corresponding eigenvalues λn of (1.1), (1.2) satisfy lim inf R2,n = −∞

(5.4)

lim sup R2,n = ∞.

(5.5)

n→∞

and n→∞

Proof. We will assume without loss of generality that 0 < rj ≤ 1 for all j. We set ∞ ∞ x0 = 0, 0 = 2 and recursively construct sequences {xm }∞ m=1 , {nm }m=1 , {km }m=1 , ∞ {m }m=1 enjoying the following properties for each m = 1, 2, . . . : 1. 0 < xm − xm−1 ≤ 12 m−1 . 1/2 2. The system ej := (xj − xj−1 )rj , j = 1, 2, . . . , m, is linearly independent over Q. 3. Set r(x) = rj for xj−1 ≤ x < xj , j = 1, 2, . . . , m. The nm th eigenvalue μ ˜nm > 0 of −y

= μ2 r(x)y, y(0) = y(xm ) = 0, satisﬁes xm m−1 μ ˜nm r1/2 − nm π > Mj − 1, 0

while

μ ˜km

xm

j=1

r1/2 − km π < −

0

m−1

Mj + 1.

j=1

4. 0 < m ≤ 12 m−1 , e3˜μnm m ≤ r1 where μ ˜km m ≤ 1. We begin the deﬁnition by setting x1 = 1, n1 = k1 = 0 and choosing 1 > 0 so small that (4) holds. Suppose that x1 , . . . , xm−1 , n1 , . . . , nm−1 , k1 , . . . , km−1 and 1 , . . . , m−1 with properties (1),(2),(3),(4) (with m replaced by m − 1) are already constructed for some m ≥ 2. We choose xm such that (1) and (2) are satisﬁed. By Theorem 4.1, we ﬁnd positive integers nm and km such that (3) is true. Finally, we choose m > 0 so small that (4) holds. This completes our recursive deﬁnition. We now set a = 0, b = limm→∞ xm ≤ 2, and r(x) = rj on [xj−1 , xj ) for 1/2 j = 1, 2, . . . Employing the notation (0 <)μn = λn , we see that μkm ≤ μ ˜km . Therefore, referring to properties (1), (3), (4), we obtain b b 1/2 r ≤μ ˜ km r1/2 μkm a a xm ≤μ ˜ km r1/2 + μ ˜km m 0

≤ km π −

m−1 j=1

As m → ∞, (5.4) follows.

Mj + 2.

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Let R : [a, b] → (0, 1] be deﬁned by R(x) = r(x) for a ≤ x ≤ xm and R(x) = 1 otherwise. For a < d ≤ b let μn (d) be the nth positive eigenvalue of −y

= μ2 R(x)y, y(a) = y(d) = 0. Applying Lemma 5.1 with δ = r1 , L = 1 and noting properties (1), (4), we obtain 0≤μ ˜nm − μnm ≤ μnm (xm ) − μnm (b) ≤ r1−1 (b − xm )e3˜μnm ≤ r1−1 m e3˜μnm ≤ 1. Consequently, in the notation of (4.3) and Theorem 4.1, b b 1/2 r ≥ (˜ μnm − 1) r1/2 μnm a

a

≥μ ˜ nm

m

ej − 2

j=1

≥ nm π +

m−1

Mj − 3

j=1

by (3). As m → ∞, (5.5) follows.

Remark. We can apply Theorem 5.2 to any sequence {rj } of positive numbers converging to zero. In particular, we can ﬁnd a function r which is positive and non-increasing (and therefore of bounded variation) such that the corresponding eigenvalues satisfy (5.4), (5.5). We may also apply the theorem to oscillating sequences like rj = 2 + (−1)j , for which r and 1/r both remain bounded.

6. The periodic p-Laplacian We consider the equation −(|y |p−2 y ) = λ(p − 1)r(x)|y|p−2 y,

−∞ < x < ∞

(6.1)

and assume that r : R → R is a locally integrable, positive function with period ω > 0. We are interested in those real values of λ for which (6.1) admits a nontrivial periodic or antiperiodic solution y, i.e., for which y(x + ω) = y(x) or y(x + ω) = −y(x), respectively. For n ∈ N0 , we denote by Λn the set of all real values of λ for which a nontrivial periodic or antiperiodic solution exists having exactly n zeros in [0, ω). This number n is even for periodic solutions and odd for antiperiodic solutions. For general p the precise structure of Λn is not fully understood, but it follows from [2, Theorem 4.1] that Λ0 is a singleton and since 0 ∈ Λ0 is evident we have Λ0 = {0}. Using [2, Theorem 3.7] as well, we see that the Λn form nonempty disjoint compact sets ordered by n, and, in particular, Λn ⊂ (0, ∞) for n > 0. − + We write λ− n = min Λn (λ0 := −∞), λn = max Λn .

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Theorem 6.1. Suppose that ln r is of bounded variation on [0, ω]. Set ω r1/p , T := T0ω ( 14 ln r, dp ). c :=

(6.2)

0

Then Λn has the following lower and upper bounds: −p λ− (nπp − T )p provided nπp ≥ T , and n ≥ c −p (nπp + T )p for all n. λ+ n ≤c

Proof. Let λ ∈ Λn and let y be a corresponding eigenfunction. Since the case n = 0 is without interest, we may assume that y(a) = 0 for some a ∈ R. By periodicity/antiperiodicity of y, y(a + ω) = 0 and y has exactly n − 1 zeros in (a, a+ω). Therefore, λ is the nth eigenvalue for equation (6.1) subject to boundary conditions y(a) = y(a + ω) = 0. The results now follow from Theorem 3.2. As mentioned in Section 1, Brown and Eastham [4] gave asymptotics for socalled “rotational” eigenvalues introduced by Zhang [15]. From [2, Theorem 4.4] these eigenvalues are precisely the λ± n , so [4] implicitly brackets the Λn as well. On the other hand, [4] considers (6.1) with r = 1 and a “potential” denoted there by q. Since, however, we know of no analogue of Liouville’s transformation for the p-Laplacian, these formulations are not directly comparable. + + − Let us write In := (λ− n , λn ) and n := λn − λn . From Theorem 6.1 we obtain Corollary 6.2. If nπp ≥ T then n ≤ 2p T c−p (nπp )p−1 when p ≥ 2 n ≤ 2pT c−p(nπp )p−1 when p ≤ 2. nπ

T Proof. From Theorem 6.1, n ≤ ( c p )p f ( nπ ), where f (t) = (1 + t)p − (1 − t)p . p Since f is convex (resp. concave) on [0, 1] for p ≥ 2 (resp. p ≤ 2), the result follows from chordal and tangent approximations to f on [0, 1].

In the case p = 2 (when both estimates in Corollary 6.2 coincide) [14, Theorem 13.10] shows that Λn consists of one or two elements, and the In are usually called instability intervals. Their lengths have been studied by many authors, and for example Eastham [6] and Ntinos [13] have given results similar to ours for a class of r which is piecewise continuous but diﬀerentiable between the jumps. They also raised the issue of optimality of the jump term, and (using Liouville’s transformation) they established it if r has one jump, but is otherwise piecewise C 2 . In Section 4, for general p, we established optimality for generic step functions with any number of jumps. − Now let Sn denote the interval (λ+ n , λn+1 ). Evidently such intervals contain no periodic/antiperiodic eigenvalues. Theorem 6.1 has the immediate Corollary 6.3. Suppose that ln r is of bounded variation on [0, ω]. If T < −p

(c

p

(nπp + T ) , c

−p

((n + 1)πp − T ) ) ⊂ Sn . p

πp 2

then

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In particular, any λ satisfying nπ < λ1/p c − T ≤ λ1/p c + T < (n + 1)π

(6.3)

must belong to Sn . When p = 2, Sn is called the nth stability interval since all solutions of the periodic equation (6.1) are known to be bounded for λ ∈ Sn . Thus a suﬃcient condition for stability of (6.1) (when p = 2) is that (6.3) be satisﬁed for some n. Hochstadt [11] gave a similar result for a parameterless equation (in which we put λ = 1) with a weight function r which is even and diﬀerentiable. To conclude, we show by a simple example how the metric d2 can be used even when the Euclidean one does not apply in Corollary 6.3. Example 6.4 We take r(x) = u > 0 for 0 ≤ x < 12 and r(x) = v > u for 12 ≤ x < 1 and extend r to a function of period 1. Then v T01 ( 14 ln r) = 14 ln u while v T := T01 ( 14 ln r, d2 ) = arctan sinh 14 ln . u If we choose u = e−4 , v = e4 then T01 ( 14 ln r) = 2 > π2 and Corollary 6.3 does not apply if T is calculated via the Euclidean metric instead of d2 . On the other hand, (6.2) gives T = arctan sinh 2 = 1.30 · · · < π2 so Corollary 6.3 shows that for c := u1/2 + v 1/2 , each interval (c−2 (nπ + T )2 , c−2 ((n + 1)π − T )2 ), of positive length is contained in some stability interval of −y

= λry.

References [1] P. Binding and P. Dr´ abek, Sturm-Liouville theory for the p-Laplacian, Studia Sci. Math. Hungar. 40 (2003), 375–396. [2] P. Binding and B.P. Rynne, Oscillation and interlacing for various spectra of the p-Laplacian, Nonlin. Anal., 71 (2009), 2780–2791. [3] J.F. Bonder and J.P. Pinasco, Asymptotic behaviour of the eigenvalues of the onedimensional weighted p-Laplace operator, Ark. Mat. 41 (2003), 267–280. [4] B.M. Brown and M.S.P Eastham, Titchmarsh’s asymptotic formula for periodic eigenvalues and an extension to the p-Laplacian, J. Math. Anal. Appl. 338 (2008), 1255–1266. [5] V.V. Chistyakov, On mappings of bounded variation with values in a metric space. (Russian) Uspekhi Mat. Nauk 54 (1999), no. 3(327), 189–190; translation in Russian Math. Surveys 54 (1999), 630–631 [6] M.S.P Eastham, Results and problems in the spectral theory of periodic diﬀerential equations, Spectral theory and diﬀerential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), 126–135. Lecture Notes in Math. 448, Springer, Berlin, 1975.

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113

´ Elbert, On the eigenvalues of half-linear boundary value prob[7] W. Eberhard and A. lems, Math. Nachr. 213 (2000), 57–76. ´ Elbert, A half-linear second order diﬀerential equation, Coll. Math. Soc. J. Bolyai [8] A. 30, In: Qualitative theory of diﬀerential equations (Szeged, 1979), 153–179. [9] L. Friedlander, Asymptotic behavior of the eigenvalues of the p-Laplacian, Comm. Partial Diﬀ. Equ. 14 (1989), 1059–1069. [10] O. Hald and J. McLaughlin, Inverse problems: recovery of BV coeﬃcients from nodes, Inverse Problems 14 (1998), 245–273. [11] H. Hochstadt, A class of stability criteria for Hill’s equation. Quart. Appl. Math. 20 (1962/1963) 92–93. [12] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, Fifth Edition, Clarendon Press, Oxford 1979. [13] A.A. Ntinos, Lengths of instability intervals of second order periodic diﬀerential equations. Quart. J. Math. Oxford (2) 27, (1976), 387–394. [14] J. Weidmann, Spectral Theory of Ordinary Diﬀerential Operators, Lecture Notes in Math 1258, Springer-Verlag, 1987. [15] M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc. 64 (2001), 125–143. P.A. Binding Department of Mathematics and Statistics University of Calgary University Drive NW Calgary, T2N 1N4 Alberta, Canada e-mail: [email protected] H. Volkmer Department of Mathematical Sciences University of Wisconsin – Milwaukee P.O. Box 413 Milwaukee, WI 53201, USA e-mail: [email protected] Received: March 8, 2009 Accepted: June 19, 2009

Operator Theory: Advances and Applications, Vol. 203, 115–136 c 2010 Birkh¨ auser Verlag Basel/Switzerland

The Gelfand-Levitan Theory for Strings Amin Boumenir Dedicated to Professor I. Gohberg

Abstract. In this note we extend the classical Gelfand-Levitan theory to allow spectral functions with power growth ρ(λ) ≈ λκ where κ ∈ (0, 1) ∪ (1, 2). The classical theory √ deals √with Sturm-Liouville operators whose spectral functions behave like λ or λ λ only. The main tool here is the Gohberg-Krein special factorization of operators close to the identity, which gives a better insight on the existence of transformation operators of Volterra type. Mathematics Subject Classiﬁcation (2000). 34A55, 34L05. Keywords. Inverse spectral theory, Gelfand and Levitan theory.

1. Introduction In this work, we are interested in extending the Gelfand-Levitan theory, G-L for short, to deal with spectral functions with power growth λκ , where κ ∈ (0, 1) ∪ (1, 2). Recall that G-L reconstructs a real locally integrable potential q and a real constant h associated with the self-adjoint diﬀerential operator H deﬁned by x≥0 H(y) := −y

(x, λ) + q(x)y(x, λ) = λy(x, λ) (1.1) y (0, λ) − hy(0, λ) = 0 from its given spectral function ρ, [16]. The method is based on integral equations. For the existence of a potential q that has m-locally integrable derivatives, ρ is required to satisfy two conditions: ﬁrst we need 2 (1.2) |Fcos (f ) (λ)| dρ (λ) = 0 ⇒ f (x) = 0 in L2 (0, ∞)

Communicated by J.A. Ball.

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A. Boumenir

2 to hold for any $ √f %∈ L (0, ∞) with compact support and where Fcos (f ) (λ) = *∞ f (x) cos x λ dx. Next if λ+ = max (λ, 0) then we need the functions 0

$ √ % 2) cos x λ d ρ(λ) − λ+ ΦN (x) = π −∞

N

(1.3)

to converge boundedly, as N → ∞, to a function that has m + 1 locally integrable derivatives, [26, Theorem 1.5.1, p. 22]. The results in [26] improved the original G-L in [16] as they closed a gap between the necessary and suﬃcient conditions. In [4], the authors revisited G-L, [26], and showed that only the second condition is needed, since it automatically implies the ﬁrst. The key to G-L is the use of transformation operators of Volterra type, that map eigensolutions of two similar operators $ √ % x $√ % y(x, λ) = cos x λ + K(x, t) cos t λ dt. (1.4) 0

The secret why we need to have Volterra type operators is unveiled by the GohbergKrein factorization theorem. This is crucial to the inverse spectral theory of differential operators. Also one needs to observe that these transformation operators V = 1 + K, as deﬁned by (1.4), act in rigged Hilbert spaces,$ or Gelfand triplets √ % [17, Vol. 4, Section 4.1, p. 103], since the eigenfunctional cos x λ ∈ / L2 (0, ∞) . It is shown that when they exist, they satisfy a factorization theorem [2], which is at the heart of the G-L theory, and also known as the nonlinear integral equation. Marchenko has shown that the spectral function ρ must satisfy the growth condition, [27] $√ % 2√ λ+o λ as λ → ∞, (1.5) ρ(λ) = π which implies that G-L applies to one particular class of Sturm-Liouville problems. In this note, instead of (1.5), we want to allow spectral functions with power growth at inﬁnity ρ (λ) ≈ cλk where k ∈ (0, 1) ∪ (1, 2)

(1.6)

by making use of weighted Sturm-Liouville operators −1 d2 f (x) + q(x)f (x). xα dx2

(1.7)

An alternative, and certainly more direct way, would be to use M.G. Krein inverse spectral theory for the string, [24, 14], to recover the mass of the string M (x) associated with the string operator −d d+ dM dx+

for x ∈ [0, l),

(1.8)

which is symmetric in L2dM . Recall that all is needed from a spectral function in

order to recover (1.8) under the boundary condition y− (0, λ) = 0, is to satisfy,

The Gelfand-Levitan Theory for Strings

117

[23, Theorem 11.1, p. 75] or [14, Section 5.8, p. 194] ∞ 1 dΓ (λ) < ∞ which allows Γ (λ) ≈ cλk with k ∈ (0, 1). 1 + λ −0 The purpose of this note is to show that we can easily extend G-L to cover (1.6), by using the Gohberg-Krein theory on the factorization of operators close to unity [19, Theorem 2.1, Chapter IV, p. 160] and also how to use G-L to recover a string. Observe that the classical G-L theory compares two close operators, with identical principal part, i.e., −D2 and −D2 + q(x), which explains the restriction on the growth of the spectral function as λ → ∞. On the other hand, a key idea in the spectral theory of the string, is the behavior of a spectral function ρ(λ) as λ → ∞ depends mainly of the behavior of the mass M (x) as x → 0, [22]. Statement of the problem: Given a nondecreasing, right continuous function, ρ(λ) subject to ρ(λ) ≈ cκ λκ as λ → ∞, and κ ∈ (0, 1) ∪ (1, 2)

(1.9)

ﬁnd a function w(x) ≥ 0 such that ρ(λ) is the spectral function associated with a selfadjoint extension of an operator generated by L(f )(x) :=

−1 d2 f (x) w(x) dx2

for x ≥ 0.

(1.10)

The ﬁrst step is to ﬁnd an operator whose spectral function is close to the given power in (1.9). To this end, by making use of Bessel functions, it is shown that d2 2 the spectral function of − x1α dx 2 acting in Lxα dx (0, ∞) is precisely 1 1± α+2

ρ(λ) = cα λ+

where the ± accounts for either the Dirichlet or Neumann boundary condition at x = 0. Thus, in the spirit of the G-L theory to match the principal part we must have: 1 where α > −1. κ=1± α+2 We now outline the procedure. Given ρ ∼ cλκ+ , where κ ∈ (0, 1) ∪ (1, 2), we 1 for α > −1 and the sign ± would then indicate the nature of solve κ = 1 ± α+2 the boundary condition to impose at x = 0, say Dirichlet or Neumann. Thus we d2 ﬁrst start by building the principal part −1 xα dx2 and the boundary condition. Next the G-L theory would recover a potential, q from the given spectral function ρ to complete the operator −1 d2 −1 d2 →→ G-L →→ + q(x). xα dx2 xα dx2 In other words xα takes care of the behavior of ρ as λ → ∞, whereas q for ﬁnite λ. For the ﬁnal step to obtain a string, we use a special transformation operator,

118

A. Boumenir

which by shifting the spectral function ρ(λ) into ρ(λ−γ), see [23, p. 91], transforms the operator into a string with mass density w, i.e. −1 d2 −1 d2 + q(x) →→ . α 2 x dx w(x) dx2

*x The last operator represents the sought string whose mass M (x) = 0 w(η)dη and whose spectral function is the given ρ. The veriﬁcation that the newly reconstructed operator is in the limit-point case at inﬁnity is also easier to show. Early works that used G-L for a string are by Dym and Kravitsky [12, 13]. They rewrote the string as an integral operator and then split the measure m(x) = m1 (x) + m2 (x), which led to a perturbation argument and a linear integral equation of the G-L type. This was the ﬁrst time, where the factorization principle of Gohberg and Krein was used as a basic tool in an inverse problem and this opened the door to more general inverse problems and applications since there was no restriction on the growth of spectral functions. Here we give another application, that helps extend and bridge G-L and M.G. Krein inverse spectral theory for the string, [14]. It is powerful enough to avoid the use of DeBranges spaces that are essential in the recovery of the string, [14, Chapter 6]. Another advantage of G-L is the use of simple integral equations which gives a handle on the smoothness of the newly reconstructed function in terms of the closeness of the spectral functions. The Gohberg-Krein theory of factorization of operators close to unity explains beautifully and gives a deeper insight on why only transformation operators of Volterra type can link eigenfunctions such as in (1.4).

2. Notation In all that follows we assume that α > −1 and deﬁne ν = spaces, Lpdμ =

f measurable :

1 α+2 .

Denote the weighted

( p |f (x)| dμ(x) < ∞

where dμ is a Lebesgue-Stieltjes measure. If q is a real-valued function such that q ∈ L1,loc xα dx [0, ∞) then we can deﬁne the diﬀerential expressions m0 and m1 for x≥0 −1 d2 −1 d2 and m1 [f ](x) = α 2 f (x) + q(x)f (x). (2.1) m0 [f ](x) = α 2 f (x) x dx x dx *∞ / L2xα dx , and so the minimal Since 1 x2+α dx = ∞ we conclude the solution x ∈ + operator generated by m0 is in the limit-point case at x = ∞, [23, p. 70]. At the same time, it is regular at x = 0 and therefore only one boundary condition there sin(βπ/2)f (0) − cos (βπ/2) f (0) = 0

(2.2)

where β ∈ (−1, 1], is suﬃcient to deﬁne a self-adjoint extension of the minimal operators, say M0 [30, Section 17, p. 58]. Similarly for the minimal operator generated by m1 we need to assume that the function q is chosen such that m1 is also

The Gelfand-Levitan Theory for Strings

119

in the limit-point case at x = ∞. In this case (2.2) is also enough to generate a self-adjoint extension M1 . Observe that β = 0, 1 corresponds respectively to the Neumann and Dirichlet case. We shall not consider the case α < −1, as it leads to a singular operator at x = 0, and so falls outside the standard G-L. Once the selfadjoint extensions are obtained, let us deﬁne their normalized eigenfunctions by ⎧ ⎧ ⎨ M1 [ϕβ (x, λ)] = λϕβ (x, λ) ⎨ M0 [yβ (x, λ)] = λyβ (x, λ) yβ (0, λ) = cos (βπ/2) ϕβ (0, λ) = cos (βπ/2) ⎩

⎩

yβ (0, λ) = sin(βπ/2) ϕβ (0, λ) = sin(βπ/2). (2.3) The solutions yβ (·, λ) and ϕβ (·, λ), are well deﬁned, and represent respectively eigenfunctionals, of the self-adjoint extensions of M0 and M1 under condition (2.2). Denote the y-transform of M0 , by ∞ f (x)yβ (x, λ)xα dx in L2dρ0 (2.4) F0,β (f )(λ) = 0

and its inverse transform is then given by ∞ f (x) = F0,β (f )(λ)yβ (x, λ)dρ0,β (λ), 0

in L2xα+ dx ,

where ρ0,β denotes the spectral functions associated with M0 and (2.2). For other properties of transforms, see [14, Section 5.7, p. 185] or [23, Section 3.2, p. 37]. Similarly we can deﬁne ∞ F1,β (f )(λ) = f (x)ϕβ (x, λ)xα dx in L2dρ1 , (2.5) 0

and its inverse transform is then given by ∞ F1,β (f )(λ)ϕβ (x, λ)dρ1,β (λ), f (x) = 0

in L2xα+ dx .

3. The spectral function ρ0,β T

Let T be the isometry L2 (0, ∞) → L2xα dx (0, ∞) , deﬁned by $ % −α α+2 T [f ](x) = x 4 f 2νx 2 . Using Bessel functions, one can express the various solutions of y0

(x, λ) + λxα y0 (x, λ) = 0,

(3.1)

as follows [25, formula 5.4.12 p. 106, ]: [N]: Neumann boundary condition (β = 0). Thus the conditions are y0 (0, λ) = 1 and y0 (0, λ) = 0, and the solution is $ % α+2 √ ν √ where Bν = ν ν Γ (1 − ν) , y0 (x, λ) = Bν λ 2 xJ−ν 2νx 2 λ

120 in other words

A. Boumenir

ν/2 √ √ λ λ T xJ (x λ) (x). y0 x, 2 = Bν −ν 4ν 4ν 2

(3.2)

From the Hankel inversion formula, see [25, 5.14.11, p. 130] ∞ ∞ √ √ √ √ √ √ f (x) = yJ−ν (y λ) f (y)dy x J−ν (x λ) λd λ 0

0

we deduce % the spectral function, dρ(λ) associated with the functional $ √that √ xJ−ν x λ is √ ) dρ(λ) = λd λ+ . By comparing the measures, see [2], we deduce that ρ0,0 (λ) =

2ν 2 λ1−ν + Bν2 1 − ν

where 1 − ν ∈ (0, 1).

(3.3)

[D]: Dirichlet boundary conditions (β = 1). In this case we have y1 (0, λ) = 0, y1 (0, λ) = 1, and $ % −ν √ α+2 √ (3.4) y1 (x, λ) = Bν λ 2 xJν 2νx 2 λ . Similarly

−ν/2 $ √ % √ λ λ T xJ x λ (x) y1 x, 2 = Bν ν 4ν 4ν 2

(3.5)

and hence

2ν 2 λ1+ν + where 1 + ν ∈ (1, 2). Bν2 1 + ν In both cases, we have no negative spectrum, i.e. ρ0,1 (λ) =

(3.6)

ρ0,0 (λ) = ρ0,1 (λ) = 0 if λ < 0. We recall that a spectral function, ρ say, is said to be asymptotic to a power, ρ(λ) ∼ λκ as λ → ∞ if ∞ ∞ κ f (λ)dρ(λ) = f (λ)dλκ +o (1) as ξ → ∞. for any f ∈ Ldλ (1, ∞) then ξ

ξ

In all that follows we assume that the given ρ(λ) is asymptotic to a power, ρ(λ) ∼ λκ as λ → ∞. In order to match the given spectral function with the correct operator and its boundary condition, we need to distinguish two cases, namely either 1 1 κ=1− or κ = 1 + holds for α > −1. α+2 α+2 In other words 1 • if κ ∈ (0, 1) then κ = 1 − α+2 and we must look for M1 , with β = 0, i.e., Neumann BC. 1 • if κ ∈ (1, 2) then κ = 1 + α+2 and we must look for M1 , with β = 1, i.e., Dirichlet BC.

The Gelfand-Levitan Theory for Strings

121

Recall some useful asymptotics associated with Bessel functions, as x → ∞, [25, 5.11.6 and 5.11.8 p. 123] ⎧ % $ α+2 √ ν 1 1 α 2 π π ⎨ 2 − 4 x 2 − 4 cos 2 λ≥0 B λ 2νx λ + ν − ν π 2 4 y0 (x, λ) ≈ (3.7) α+2 √ ν 1 −α − ⎩ √Bν x 4 (−λ) 2 4 e2νx 2 −λ λ < 0. 8ν In the next section, the following proposition plays a crucial role. Proposition 3.1. Assume the following identity holds for x > 0 and for some α > −1 x A(x) + B(x)y0 (x, λ) + C(x, t)y0 (t, λ)tα dt = 0 for all λ ∈ R 0

where C(x, .) ∈

L2tα dt (0, x),

then

C(x, .) = 0 in L2tα dt (0, x) and A(x) = B(x) = 0. (3.8) *x α 2 Proof. For any ﬁxed x > 0, we have 0 C(x, t)y0 (t, .)t dt ∈ Ldρ0,0 which means ∞ 2 (A(x) + B(x)y0 (x, λ)) dρ0,0 (λ) < ∞, N

i.e., for large N, we have ∞ 1 2 (A(x) + B(x)y0 (x, λ)) dλ1− α+2 < ∞.

(3.9)

N

There are three cases. If ν < 12 then from (3.7) we have y0 (x, λ) → 0 as λ → ∞ and for large N, (3.9) yields ∞ 1 A(x)2 dλ1− α+2 < ∞ ⇒ A(x) = 0. (3.10) N

Now (3.9) reduces to ∞ $ α+2 √ π α π% √ cos2 2νx 2 λ + ν − B(x)2 x1− 2 d λ < ∞ ⇒ B(x) = 0. 2 4 N $ % α+2 √ ν 1 In case ν > 12 then A(x) + B(x)y0 (x, λ) ∼ λ 2 − 4 B(x) cos 2νx 2 λ + ν π2 − π4 leads to ∞ $ α+2 √ π π% √ 2 B (x) d λ<∞ cos2 2νx 2 λ + ν − 2 4 N which implies that again B(x) = 0. Going back, we have again the case (3.10) and so A(x) = 0. The$ remaining case is ν =% 12 , which leads to A(x) + B(x)y0 (x, λ) ∼ √ α+2 √ A(x) + B(x) cos 2νx 2 λ + ν π2 − π4 , i.e., periodic in the variable λ and so we have A(x) = B(x) = 0. Since we * xalways have A(x) = B(x) = 0, we use the inverse transform, to conclude that 0 C(x, t)y0 (t, λ)tα dt = 0 for all λ ≥ 0, which yields C(x, .) = 0 in L2tα dt (0, ∞).

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A. Boumenir For the Dirichlet case, we have a similar result.

Proposition 3.2. Assume that the following identity holds x C(x, t)y1 (t, λ)tα dt + B(x)y1 (x, λ) + A(x) = 0 for all λ ∈ R 0

where C(x, .) ∈ L2tα dt (0, x) then C(x, .) = 0 in L2tα dt(0, x) and B(x) = A(x) = 0. We end this section with two facts. Observe that if f ∈ C[0, ∞) then for −1 < α we have t−α f, f ∈ L2,loc (3.11) tα dt [0, ∞). Also if g is bounded as x → 0 then for −1 < α we also have x α α x2 g(t)t 2 dt → 0 as x → 0. 0

4. The transformation operator For our purpose we only need to express the eigenfunctionals ϕβ (x, λ) in terms of the eigenfunctionals yβ (x, λ) and this is achieved with the help of a transformation operator, [7, 8, 28]. As in [16], and also as dictated by the factorization of operators close to unity, we need to use a Volterra type operator to connect these solutions and we ﬁrst start with the Neumann case. A] Neumann case, (β = 0), we have for x > 0 x K(x, t)y0 (t, λ)tα dt ϕ(x, λ) = y0 (x, λ) + 0 x H(x, t)ϕ0 (t, λ)tα dt. y(x, λ) = ϕ0 (x, λ) +

(4.1)

0

The kernels of the above transformation operators are deﬁned in the following sector Ω := (x, t) ∈ R2 : 0 ≤ t ≤ x, 0 < x < ∞ . Let us try to ﬁnd some necessary conditions on K in the Neumann case, so that ϕ(x, λ) = ϕ0 (x, λ). Proposition 4.1. Assume that K ∈ C 2 (Ω), q ∈ C[0, ∞), α = 0, α > −1 then x ϕ(x, λ) := y0 (x, λ) + K(x, t)y0 (t, λ)tα dt (4.2) 0

is solution of (2.3) with β = 0, if and only if ⎧ 1 K (x, t) − t1α Ktt (x, t) = q(x)K(x, t) ⎪ ⎨ xα xx α d α q(x) = 2x− 2 dx (x 2 K(x, x)) ⎪ ⎩ Kt (x, 0) = 0.

0 < t < x, (4.3)

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123

Proof. Since K(x, t) is smooth, diﬀerentiating the representation (4.2) yields: x

α Kx (x, t)y0 (t, λ)tα dt ϕ (x, λ) = y0 (x, λ) + K(x, x)y0 (x, λ)x + 0

ϕ (x, λ) =

y0

(x, λ)

d (K(x, x)y0 (x, λ)xα ) + Kx (x, x)y0 (x, λ)x + dx x Kxx (x, t)y0 (t, λ)tα dt. + α

0

Integration by parts together with the conditions y0 (0, λ) = 1 and y0 (0, λ) = 0 lead to x K(x, t)y0 (t, λ)tα dt = −K(x, t)y0 (t, λ) λ 0 x t=x Ktt (x, t)y0 (t, λ)dt + Kt (x, t)y0 (t, λ)|t=0 − 0

= −K(x, x)y0 (x, λ) + Kt(x, x)y0 (x, λ) − Kt (x, 0) x Ktt (x, t)y0 (t, λ)dt. − 0

Expressing now ϕ(x, λ) in terms of y0 (x, λ), as in (4.2), means that ϕ(x, λ) is a solution of (2.3) if and only if x xα α Kxx (x, t) − q(x)x K(x, t) − α Ktt (x, t) y0 (t, λ)tα dt − xα Kt (x, 0) t 0 ( d dK(x, x) α α α + (x K(x, x)) − x q(x) = 0 +y0 (x, λ) x dx dx where we have used dK(x, x) = Kt (x, x) + Kx (x, x). dx α By (3.11) since xα Ktt (x, t) is continuous and xtα Ktt (x, .), q(x)K(x, .), Kxx (x, .) ∈ L2tα dt (0, x), Proposition 3.1 implies ⎧ 1 1 ⎪ ⎨ xα Kxx (x, t) − tα Ktt(x, t) = q(x)K(x, t) 0 < t < x, d d K(x, x) + x−α dx (xα K(x, x)) q(x) = dx ⎪ ⎩ Kt (x, 0) = 0. Hence ϕ(x, λ) deﬁned by (4.2) satisﬁes the diﬀerential equation in (2.3) if and only if K(x, t) satisﬁes (4.3). For the boundary conditions observe that when α > −1 then q ∈ C 0 [0, ∞) =⇒ xα q(x) ∈ L1,loc[0, ∞). The initial conditions for ϕ follow from (4.2) and the boundedness of K in Ω ϕ(0, λ) = 1.

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+ αx−1 K(x, x) we have Also from q(x) = 2 dK(x,x) dx αxα K(x, x) = xα+1 q(x) − 2xα+1 and if α = 0, then K(x, x)xα → 0 as x → 0 and

ϕ (x, λ) =

y0 (x, λ)

dK(x, x) dx

(4.4)

x

α

Kx (x, t))y0 (t, λ)tα dt,

+ K(x, x)x y0 (x, λ) + 0

yields

ϕ (0, λ) = 0 and so ϕ(x, λ) = ϕβ (x, λ).

Remark 4.2. Observe that when α = 0, the transformation operator maps only Dirichlet into Dirichlet and Neumann into Neumann problems. The only case when diﬀerent boundary conditions are allowed is when α = 0. Indeed then K(0, 0) = 0 and ϕ (0, λ) = y0 (0, λ)+K(0, 0)y0(0, λ) which obviously is the classical G-L theory. d2 The α-parameter family of operators m0 = −1 , and their eigensolutions provide xα dx2 −d2 a natural extension of the classical operator dx2 . Even the formula for q in (4.3) depends analytically on α, and yields the G-L case when α = 0 as a particular case. Similarly for the second equation in (4.1) we have: Proposition 4.3. Assume that H ∈ C 2 (Ω), q ∈ C[0, ∞), α = 0, α > −1 then x y(x, λ) = ϕ0 (x, λ) + H(x, t)ϕ0 (t, λ)tα dt (4.5) 0

is solution of (2.3) with β = 0, if and only if ⎧ 1 t) ⎨ xα Hxx (x, t) − t1α H tt (x, t) = −q(t)H(x, α d α x 2 H(x, x) q(x) = −2x− 2 dx ⎩ Ht (x, 0) = 0.

0
The proof is similar to the one of Proposition 4.1 and so is omitted. Next we brieﬂy state the Dirichlet case. B] The Dirichlet case (β = 1). There are very few changes to add to the Neumann case to obtain the Dirichlet case, i.e., β = 1, in (2.2). For example the analog of Proposition 4.1 for the Dirichlet case reads as: Proposition 4.4. Assume that K ∈ C 2 (Ω) , q ∈ C[0, ∞) then x K(x, t)y1 (t, λ)tα dt ψ(x, λ) := y1 (x, λ) +

(4.7)

0

is solution of (2.3) with β = 1 if and only if ⎧ 1 ⎨ xα Kxx (x, t) − t1α Ktt (x, t) = q(x)K(x, t) d d K(x, x) + x−α dx (xα K(x, x)) q(x) = dx ⎩ K(x, 0) = 0.

0 < t < x, (4.8)

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125

Proof. The proof is essentially similar to Proposition 4.1, except for the boundary conditions. The boundedness of K implies that ψ(0, λ) = 0. Also since K(x, x)xα/2 is bounded and y1 (x, λ) = O (x) we deduce from x

α ψ (x, λ) = y1 (x, λ) + K(x, x)x y1 (x, λ) + Kx (x, t))tα y1 (t, λ)dt 0

that ψ (0, λ) = y1 (0, λ) = 1.

Remark 4.5. The existence of K(x, t) and H(x, t) as solutions of a hyperbolic system (4.6) is diﬃcult to prove, see for example [9] and a new proof using operator theory can be found in [5]. An alternative proof here is to see that they factor a certain operator close to unity, which is treated in the next section.

5. Existence of the transformation operator The next step in the G-L procedure would be to solve the linear integral equation x F (x, t) + K(x, t) + K(x, s)F (s, t)sα ds = H(t, x) (5.1) 0

for K, where

∞

F (x, t) = −∞

yβ (x, λ)yβ (t, λ)d [ρ1,β − ρ0,β ] (λ)

(5.2)

for β = 0, 1. An equivalent, and more direct way is to use the Gohberg–Krein factorization of operators close to unity, [19]. Our strategy is, to show that if V is any operator mapping eigenfunctionals, such that yβ (x, λ) = V ϕβ (x, λ),

(5.3)

holds for all λ ∈ R , then V satisﬁes a factorization identity. Assume the existence of an operator V, not necessarily of Volterra type, such that (5.3) holds. For the setting of this operator in Gelfand’s triplets, see [2, 18, 17]. For the sake of simplicity let f ∈ C00 [0, ∞), i.e., a continuous function with compact support, then ∞ ∞ f (x)yβ (x, λ)xα dx = f (x)V ϕβ (x, λ)xα dx (5.4) F0,β (f )(λ) = 0 0 ∞ = V f (x) ϕβ (x, λ)xα dx 0

= F1,β (V f )(λ). In (5.4), the integrals should be understood as duality brackets in rigged spaces and V is the adjoint, [2]. Now use the following identities, where f, ψ ∈ C00 [0, ∞) and

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A. Boumenir

we dropped β since it appears in all transforms F0 (f )(λ)F0 (ψ)(λ)dρ1 (λ) = F1 (V f )(λ)F1 (V ψ)(λ)dρ1 (λ) ∞ = V f (x)V ψ(x)xα dx 0 ∞ V

V f (x)ψ(x)xα dx. =

(5.5)

0

To proceed further assume that ρ1 is absolutely continuous with respect to dρ0 , that is (5.6) dρ1 (λ) = τ (λ)dρ0 (λ) where τ ∈ L1,loc dρ0 . Since τ is ﬁnite almost everywhere with respect to dρ0 , the multiplication by τ operator is densely deﬁned in L2dρ0 and one can deﬁne a function of the self-adjoint operator M0 , say τ (M0 ) through the transforms F0 (τ (M0 ) f ) = τ (λ)F0 (f ). The domain of τ (M0 ) is the set in L2xα dx (0, ∞) such that τ (λ)F0 (f ) ∈ L2dρ0 . Thus we can write F0 (f )(λ)F0 (ψ)(λ)dρ1 (λ) = τ (λ)F0 (f )(λ)F0 (ψ)(λ)dρ0 (λ) (5.7) = F0 (τ (M0 ) f )(λ)F0 (ψ)(λ)dρ0 (λ) ∞ = τ (M0 ) f (x)ψ(x)xα dx. 0

Combining (5.6), (5.5) and (5.7) yields τ (M0 ) = V

V

C00 [0, ∞)

(5.8)

is dense in ∞) . When the symbol τ is close to unity, i.e., since ρ1,β and ρ0,β are close enough at inﬁnity, then the solution of (5.8) is precisely V = 1 + H where H is a Volterra type operator in (4.1). Then from (4.1), once we have the kernel H(x, t) and α, Proposition 4.3 yields L2xα dx (0,

α d (H(x, x)x 2 ). dx In case we do not assume (5.6), then for any ψ ∈ C00 [0, ∞), we have F0 (f )(λ)F0 (ψ)(λ)dρ1 (λ) = F0 (f )(λ) y(x, λ) ψ(x)xα dx dρ1 (λ) = F0 (f )(λ)y(x, λ) dρ1 (λ) ψ(x)xα dx.

q(x) = −2x− 2

α

Combining (5.5) and (5.9) yields F0 (f )(λ)y(x, λ) dρ1 (λ) = V

V f (x).

(5.9)

The Gelfand-Levitan Theory for Strings If we denote by

127

T(f )(x) =

F0 (f )(λ)y(x, λ) d (ρ1 − ρ0 ) (λ),

(5.10)

then instead of (5.8) we have 1 + T = V

V .

(5.11)

Remark 5.1. In case it is known that V, as an operator acting in the space L2xα dx (0, ∞), is densely deﬁned and closed, then we have V

= V, see [11, Corollary 1.8, p. 305], and so (5.11) would reduce to 1 + T = V V . We can summarize the above identities in the following proposition. Proposition 5.2. Assume that V is a transformation operator between the two self1,loc 1 adjoint operators M0 and M1 , then (5.11) holds. If moreover dρ dρ0 ∈ Ldρ0 then (5.8) holds. In order to use the Gohberg-Krein factorization theorem it is necessary that T is a Hilbert-Schmidt compact operator in L2xα dx (0, ∞). To this end we obtain the kernel of the integral operator T, T(f )(x) = F0 (f )(λ)y(x, λ) d (ρ1 − ρ0 ) (λ) (5.12) ∞ = f (t)y(t, λ)ta dt y(x, λ) d (ρ1 − ρ0 ) (λ) 0 ∞ ∞ a y(t, λ) y(x, λ) d (ρ1 − ρ0 ) (λ) f (t) t dt = F (x, t)f (t) ta dt = 0 0 * where F (x, t) = y(x, λ)y(t, λ) d (ρ1 − ρ0 ) (λ) and f ∈ C00 (0, ∞). Thus for the convergence of the integral, we need to choose ρ1β close enough to ρ0β as λ → ∞ ρ10 (λ) ≈ ρ00 (λ) = such that

2ν 2 λ1−ν + Bν2 1 − ν

∞ 0

∞

or ρ11 (λ) ≈ ρ01 (λ) = 2

|F (x, t)| ta dt xa dx < ∞.

2ν 2 λ1+ν + Bν2 1 + ν

(5.13)

(5.14)

0

Proposition 5.3 (Gohberg-Krein). Assume that T, in (5.10) is a Hilbert-Schmidt compact operator acting in L2xα dx (0, ∞), i.e., (5.14) holds and that equation F0 (f )(λ)y(x, λ) dρ1 (λ) = g(x) has a unique solution in L2xα dx (0, ξ) for any ξ > 0 then equation (5.11) has a unique solution operator V such that ∞ H(x, t)f (t)tα dt (5.15) V (f )(x) = f (x) + x

and where the kernel H is Hilbert-Schmidt in L2xα dx (0, ∞).

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A. Boumenir

Proof. The original result [19, p. 184], [20, Theorem 5.1, Chapter XXII, p. 531] is about the factorization of operators acting in L2 (a, b), i.e., α = 0 which is a particular case of α > −1. Recall that if A is Hilbert-Schmidt in L2 (a, b) then I + A = (I + Y− ) (I + Y+ ) where Y+ and Y− are respectively upper and lower Volterra operators if and only if f (x) + Af (x) = 0 has only the trivial solution in L2 (0, ξ) for any ξ > 0. Furthermore in case I + A is strictly positive then Y− = Y+ , i.e., . / I + A = (I + Y− ) I + Y− . In order to use the above result we ﬁrst need to recast the equation F0 (f )(λ)y(x, λ) dρ1 (λ) = g(x) as

ξ

F (x, t) f (t) ta dt = g(x) in L2xα dx (0, ξ)

f (x) +

(5.16)

0 M

and then set (5.16) in L2 (0, ∞) . To this end denote the isometry: L2xα dx (0, ∞) → L2 (0, ∞) deﬁned by the multiplication operator M (f )(x) = xα/2 f (x)

(5.17)

2

and so we can recast (5.16) in L (0, ξ) as ξ xα/2 F (x, t)tα/2 tα/2 f (t)dt = xα/2 g(x) xα/2 f (x) + 0

or

ξ

M (f )(x) +

F(x, t)M (f )(t)dt = M (g)(x)

0

2 * ∞ * ∞ where F(x, t) = xα/2 F (x, t)tα/2 satisﬁes 0 0 F(x, t) dxdt < ∞, by (5.14). be the Hilbert-Schmidt operator in L2 (0, ∞) whose kernel is F . Since Now let T is also self-adjoint, then the Gohberg-Krein factorization theorem [19, Theorem T ' deﬁned by 5.2, p. 175], implies the existence of a triangular operator W ∞ x)f (t)dt, ' f (x) = f (x) + H(t, W x

acting in L2 (0, ∞), with

0

and such that

∞

∞

x

2 H(t, x) dtdx < ∞

=W ' W '. 1+T

(5.18)

(5.19)

The Gelfand-Levitan Theory for Strings

129

To conclude observe that = M TM −1 , T where T is deﬁned by (5.10), yields ' W '. 1 + M TM −1 = W ' W 'M 1 + T = M −1 W ' M M −1 W ' M. 1 + T = M −1 W

(5.20)

Finally deﬁne 'M V = M −1 W which means that for f ∈

(5.21)

L2xα dx (0, ∞) ,

' M f (x) M V f (x) = W

∞

x)M f (t)dt H(t, ∞

−α/2 x)tα/2 f (t)dt H(t, V f (x) = f (x) + x x ∞ x)t−α/2 f (t)tα dt = f (x) + x−α/2 H(t, x ∞ H(t, x)f (t)tα dt, = f (x) + = M f (x) +

x

x

where x)t−α/2 H(t, x) = x−α/2 H(t, satisﬁes

∞ ∞

0

2

|H(t, x)| xα tα dxdt < ∞

t

by (5.18). Use (5.20) and (5.21) to deduce the factorization 1 + T = V

V

since M = M −1 yields $ %

. / 'M = M W ' M −1 = M −1 W ' M V

= M −1 W

(5.22)

The Gohberg-Krein factorization theorem tells us that in order to solve the inverse spectral problem for M1 and so have Volterra type transformation operators such as equation (4.1), we need the operator T in (5.10) to be a Hilbert Schmidt operator acting in L2xα dx (0, ∞). In other words, the triangular form of V , which is essential for the construction of q in (4.6), is a direct consequence of the GohbergKrein theorem.

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A. Boumenir

6. Smoothness If we multiply (5.22) by V

−1 , then we obtain the linear equation (5.1). We now can use (5.1) to get the smoothness of q. Recall that for 0 < t < x we must have H(t, x) = 0, i.e., x K(x, s)F (s, t)sα ds = 0 (6.1) F (x, t) + K(x, t) + 0 α/4

which when multiplied by (xt)

yields x α xα/4 F (x, t)tα/4 + xα/4 K(x, t)tα/4 + xα/4 K(x, s)sα/4 sα/4 F (s, t)tα/4 s 2 d s = 0. 0

(6.2)

Using the change of variables X = 2νx

α+2 2

,

T = 2νt

K(X, T ) = xα/4 K(x, t)tα/4 and

α+2 2

,

(6.3)

F(X, T ) = xα/4 F (x, t)tα/4

equation (6.2) reduces to a simple form with no weight X F(X, T ) + K(X, T ) + K(X, S)F(S, T )dS = 0

0 < T < X.

(6.4)

0

We have then Proposition 6.1. Assume that ∀X > 0, the homogeneous equation associated with (6.4) has the trivial solution only and F(X, T ) ∈ C (m+1) ([0, ∞) × [0, ∞)) then K(X, T ) ∈ C (m+1) (Ω). Proof. The equation in (6.4) has no weight and so the result follows from the well-known Lemma (1.2.2) in [26]. Remark 6.2. The variable X in (6.3) has already been used in the solution (3.2). It basically transforms (3.1) into a Bessel type operator which is singular at x = 0 which falls outside the spirit of the G-L theory.

7. The inverse problem We now need to verify that the operator V obtained (5.15) can be used to deﬁne the sought potential q, whose smoothness is measured through a new function Q, α+2 deﬁned by Q (X) = q(x) where X = 2νx 2 , and similarly the smoothness of F (x, t) is through F (X, T ) = F (x, t), as in (6.3). We now prove the following theorem. Theorem 7.1. Assume for either β = 0 or β = 1, α > −1, and ρ1,β is a given nondecreasing, right continuous function with power growth at inﬁnity ρ1,β (λ) ≈ 1 c(α)λ1± α+2 , and α > −1,

The Gelfand-Levitan Theory for Strings

131

[i] For any f ∈ L2tα dt (0, ∞), with compact support, we have ∞ | F0,β (f )(λ) |2 dρ1,β (λ) = 0 ⇒ | f (t) |2 tα dt = 0 0

[ii] F(X, T ) ∈ C

(m+1)

([0, ∞) × [0, ∞)),

then there exists a function q(x) such that Q ∈ C (m) [0, ∞) and ρ1,β is a spectral function associated with the self-adjoint problem, ⎧ x>0 ⎨ −ϕ

(x, λ) + xα q(x)ϕ(x, λ) = λxα ϕ(x, λ) ϕ (0, λ) = 0 (ϕ(0, λ) = 1) if β = 1 (7.1) ⎩ ϕ(0, λ) = 0 (ϕ (0, λ) = 1) if β = 0. Proof. By condition [i] the Fredholm alternative holds for (6.1) and we have the uniqueness of a solution K(x, .) in L2tα dt (0, x). Let us agree to denote by Ξx = −1 ∂ 2 xα ∂x2 and deﬁne / −α d . α q(x) = 2x 2 x 2 K(x, x) . dx In case m ≥ 1, then simple diﬀerentiation of the linear equation (6.1) yields x (Ξx − Ξt − q(x))K + (Ξx − Ξt − q(x))K(x, s)F (s, t)sα ds = 0. 0

Since by [i], the above integral equation has only the trivial solution, and we get ⎧ 1 K (x, t) − t1α Ktt (x, t) = q(x)K(x, t) ⎪ ⎨ xα xx −α α d (7.2) q(x) = 2x 2 dx [x 2 K(x, x)] ⎪ ⎩ Kt (x, 0) = 0 (β = 0) or K(x, 0) = 0 (β = 1). For the last condition, if β = 1 we have the Dirichlet case, and so y(0, λ) = 0. This implies that F (s, 0) = 0 and t → 0 in (6.1) leads to K(x, 0) = 0. Similarly if β = 0, i.e., Neumann case, then Ft (s, 0) = 0, and diﬀerentiating (6.1) with respect to the variable t yields Kt (x, 0) = 0. Finally the case F(X, T ) ∈ C 1 ([0, ∞)×[0, ∞)) follows by approximating F by a sequence Fn ∈ C 2 and then taking the limit. Thus by Propositions 4.1 and 4.4, we can construct q and so construct the operator M1 . It remains to check that ρ1,β is indeed the spectral function corresponding to newly reconstructed operator. To this end we need to check that the Parseval relation hold, i.e., if g ∈ L2xα dx (0, ∞) then ∞ 2 2 |g(x)| xα dx. (7.3) |F1,β (g)(λ)| dρ1β (λ) = 0

Given g ∈

L2xα dx (0, ∞)

there exists f ∈

L2xα dx

(0, ∞) such that

V f = g. This follows from (5.4) and ρ1,β (λ) ≈ ρ0,β (λ) as λ → ∞ F0,β (f )(λ) = F1,β (V f )(λ) = F1,β (g)(λ).

(7.4)

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A. Boumenir

Thus 2 2 2 |F1,β (g)(λ)| dρ1β (λ) = |F1,β (V f )(λ)| dρ1β (λ) = |F0,β (f )(λ)| dρ1β (λ). For the sake of simplicity, let us assume that ρ1β is absolutely continuous with respect to dρ0β , i.e., dρ1β (λ) = τ (λ)dρ0β (λ), then from (5.8) and (7.4), we deduce 2 2 |F1,β (g)(λ)| dρ1β (λ) = |F0,β (f )(λ)| τ (λ)dρ0β (λ) = τ (λ)F0,β (f )(λ)F0,β (f )(λ)dρ0β (λ) = F0,β (τ (M0 ) f ) (λ)F0,β (f )(λ)dρ0β (λ) ∞ τ (M0 ) f (x) f (x) xα dx = 0 ∞ V

V f (x) f (x)xα dx = 0 ∞ = V f (x) V f (x)xα dx 0 ∞ 2 |g(x)| xα dx. = 0

This concludes the proof that ρ1 is the spectral function of the newly reconstructed operator M1 .

8. The limit-point case At the end of an inverse problem, one also needs to check that the recovered operator is in the limit-point at x = ∞. Otherwise, we need to ﬁnd a new boundary condition there for the self-adjoint extension to take place. Consider the operator T, deﬁned by (5.10) and acting in L2tα dt (0, ∞), and assume that there exists a transformation operator V, not necessarily of Volterra type. In both cases, Neumann or Dirichlet, i.e., β = 0, 1, the factorization holds, see (5.11) 1 + T = V

V

Using the polar decomposition of 1 + T we have: Proposition 8.1. The operator T is bounded in L2tα dt (0, ∞) if and only if the transformation operator V is bounded in L2tα dt (0, ∞) . Proof. It √is readily seen that ||V f ||2 = ([1 + T] f, f ) from which follows that ||V || = || 1 + T||. Proposition 8.2. If the operator T is bounded in L2xα dx (0, ∞) , then the operator M1 is in the limit point case at x = ∞.

The Gelfand-Levitan Theory for Strings

133

Proof. Assume that M1 is in the limit circle, then all solutions ϕ(x, λ) ∈ L2xα dx (0, ∞) . Recall that the relation x y(x, λ) = ϕ(x, λ) + H(x, t)ϕ(t, λ)tα dt = Vϕ(x, λ) 0

holds for all complex λ. Since 1 + T is also bounded, then we would have y(·, λ) ∈ L2xα dx (0, ∞) , which is not possible since y(x, i) ∈ / L2xα dx (0, ∞) by (3.7), as we already know that M0 is in the limit point case. We now look for a suﬃcient condition for T to be a bounded operator acting in L2xα dx (0, ∞). Proposition 8.3. Assume that ρ1 is absolutely continuous with respect to ρ0 , then dρ1 (λ). ||1 + T|| = sup λ∈σ0 dρ0 Proof. Let f ∈ L2xα dx (0, ∞) and with compact support then, with the understanding that we could be are either in the Neumann or Dirichlet case, dρ1 [1 + T] f (x) = F0 (f )(λ)y(x, λ)dρ1 (λ) = (λ)F0 (f )(λ)y(x, λ)dρ0 (λ). dρ0 Hence by Parseval equality # # # dρ1 dρ1 # dρ1 # # (λ)F0 (f )(λ)# ≤ sup (λ) F0 (f )(λ) = sup (λ) f [1 + T] f = # dρ0 λ≥0 dρ0 λ≥0 dρ0 We now recall that ρ0β is given explicitly by (3.3) and (3.6), and so for λ > 0 we have 2ν 2 2ν 2 ρ 00 (λ) = 2 λ−ν and ρ 01 (λ) = 2 λν Bν Bν and therefore we end up with a suﬃcient explicit condition for T to be a bounded operator acting in L2tα dt (0, ∞) sup λ≥0

dρ1,0 (λ) = c sup λν ρ 1β (λ) < ∞ dρ0,0 λ≥0

or sup λ≥0

dρ1,1 (λ) = c sup λ−ν ρ 1β (λ) < ∞. dρ0,1 λ≥0

9. The string We now show how to construct a string from a given spectral function, ρ1 (λ) ≈ λκ as λ → ∞ where κ ∈ (0, 1) ∪ (1, 2) by the G-L method. Since a string has a positive spectrum, and for the sake of simplicity, let us assume that suppdρ1 ⊂ [δ, ∞) where δ > 0. Let us begin with the Neumann case ﬁrst, i.e., β = 0 and so κ ∈ (0, 1). By Theorem 7.1 we can recover an operator −1 G-L ρ1 (λ) ⇒⇒ M1 [ϕ] := α ϕ

(x, λ) + q(x)ϕ(x, λ) = λϕ(x, λ) x

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where ϕ (0, λ) = 0 and (ϕ(0, λ) = 1) . Since the spectrum is strictly positive, the solution ϕ(x, 0) has no zeros. We now follow the procedure outlined in [23, Section 14, p. 91], which starts with the following change of variables, ς(x) = ϕ(x, 0) > 0 so that ϕ(·, λ) satisﬁes a new equation without q, i.e., d d ϕ(x, λ) ϕ(x, λ) 2 2 ς (x) ς (x) + λxα ς 4 (x) = 0. dx dx ς(x) ς(x) Thus if we recast (9.1) with a new variable x 1 dt so we have ξ(x) = 2 (t) ς 0

(9.1)

d d = ς 2 (x) dξ dx

and a new function χ χ(ξ(x), λ) =

ϕ(x, λ) ς(x)

(9.2)

then χ(ξ, λ) satisﬁes d2 χ(ξ, λ) + λw(ξ)χ(ξ, λ) = 0 dξ 2 where the density of the string is w(ξ(x)) = xα ς 4 (x).

(9.3)

(9.4)

It remains to check the new boundary condition and how the spectrum was modiﬁed. From (9.2), setting x = 0 yields χ(0, λ) =

ϕ(0, λ) ϕ(0, λ) = = 1. ς(0) ϕ(0, 0)

Diﬀerentiating χ (ξ(x), λ)

1 ς 2 (x)

=

ϕ (x, λ)ς(x) − ϕ(x, λ)ς (x) ς 2 (x)

yields χ (0, λ) = ϕ (0, λ)ϕ(0, 0) − ϕ(0, λ)ϕ (0, 0) = 0. Thus the string we have constructed is also in the Neumann case ⎧ d d ⎨ dM(ξ) dξ χ(ξ, λ) + λχ(ξ, λ) = 0 χ(0, λ) = 1 ⎩

χ (0, λ) = 0 and its mass is

t

M (t) =

t∗

w(ξ)dξ =

0

0

= 0

w(ξ(x))ξ (x)dx

t∗

1 dx = xα ς 4 (x) 2 ς (x)

t∗

xα ς 2 (x)dx 0

(9.5)

The Gelfand-Levitan Theory for Strings

135

where w is given by (9.4) and ξ(t∗ ) = t. Next we examine the spectral function of the newly reconstructed string (9.5), which we denote by Γ. From [23, equation 14.14, p. 92], we have ρ1 (λ) = Γ (λ) − Γ(0) and so dρ1 (λ) = dΓ (λ), i.e., the same measure. For the Dirichlet case, we simply need to use the same function ς(x) since it is positive and satisﬁes the equation with the same q. Acknowledgment The author thanks Professor Norrie Everitt for the many interesting discussions on spectral theory and support while he visited him in Birmingham, UK. The author also sincerely thanks the referee for his valuable comments.

References [1] Akhiezer, N.I. and Glazman I.M., Theory of Linear Operators in Hilbert Space, Dover, 1993. [2] Boumenir, A., Comparison Theorem for Self-adjoint Operators, Proc.Amer.Math. Soc., Vol. 111, Number 1, (1991), 161–175. [3] Boumenir, A. and Nashed, M. Z., Paley-Wiener type theorems by transmutations. J. Fourier Anal. Appl. 7 (2001), 395–417. [4] Boumenir, A. and Tuan, V., The Gelfand-Levitan theory revisited. J. Fourier Anal. Appl. 12 (2006), 257–267. [5] Boumenir, A. and Tuan, V., Existence and construction of the transmutation operator. J. Math. Phys. 45 (2004), 2833–2843. [6] Boumenir, A and Zayed, A., Sampling with a string. J. Fourier Anal. Appl. 8 (2002), 211–231. [7] Carroll, R.W., Transmutation theory and applications, Mathematics Studies, Vol. 117, North-Holland, 1985. [8] Carroll, R.W., Transmutation and Operator Diﬀerential Equations, Mathematics Studies, Vol. 37, North-Holland, 1979. [9] Carroll, R.W. and Showalter, R.E., Singular and Degenerate Cauchy Problem, Mathematics. In Science and Engineering, Vol. 127, Academic Press, 1976. [10] Chadan, K. and Sabatier, P.C., Inverse Problems in Quantum Scattering Theory, Springer-Verlag, 1989. [11] Conway, J.B., A Course in Functional Analysis, Graduate Texts in Mathematics, second edition, Springer-Verlag, 1990. [12] Dym, H. and Kravitsky, N., On recovering the mass distribution of a string from its spectral function, Topics in Functional Analysis (Essays dedicated to M.G. Krein on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., 3, Academic Press, New York-London, 45–90, 1978. [13] Dym, H. and Kravitsky, N., On the inverse spectral problem for the string equation, Integral Equations Operator Theory 1, 2, (1978), 270–277. [14] Dym, H. and McKean, H., Gaussian processes, Function theory and inverse spectral problem, Dover, 2008.

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[15] Everitt,W.N. and Halvorsen, S.G., On the Asymptotic form of the Titchmarsh-Weyl Coeﬃcient, Applicable Analysis; Vol. 8, (1978), 153–169. [16] Gelfand, I.M. and Levitan, B.M., On the determination of a diﬀerential equation from its spectral function, Amer. Math. Transl. (2) Vol. 1, (1951), 239–253. [17] Gelfand, I.M. and Shilov, G.E., Generalized functions. Vol. 3, 4. Theory of diﬀerential equations, Academic Press New York-London, 1967. [18] Gelfand, I.M. and Kostyuchenko, A.G., Eigenfunction expansions of diﬀerential and other operators, Dokl. Akad. Nauk SSSR 103 (1955), 349–352. [19] Gohberg, I. and Krein, M.G., Theory and Applications of Volterra Operators in Hilbert Spaces, Amer. Math. Transl. Mono. Vol. 24, 1970. [20] Gohberg, I. Goldberg, S. and Kaashoek, M., Classes of Linear Operators, Vol. II. Operator Theory: Advances and Applications, 63 Birkh¨auser Verlag, Basel, 1993. [21] Kac, I.S., The Existence of Spectral Functions of Generalized second-order diﬀerential systems, Amer. Math. Soc. Transl.(2) Vol. 62, (1966), 204–262. [22] Kac, I.S., Power asymptotics estimates for spectral functions of generalized boundary value problem of the second order Sov. Math. Dokl. Vol. 13, 2, (1972), 453–457. [23] Kac, I.S. and Krein, M.G., Spectral function of the string, Amer. Math. Soc. Transl. (2). Vol. 103, (1970), 19–103. [24] Krein, M.G., Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum, Dokl. Akad. Nauk. SSSR 76, (1951), 345–348. [25] Lebedev, N.N., Special functions and their applications, Dover 1972. [26] Levitan, B.M. and Gasymov, M.G., Determination of a diﬀerential equation by two of its spectra, Russ. Math. Surveys. Vol. 2, (1964), 2–62. [27] Levitan, B.M. Remark on a theorem of V.A. Marchenko, Amer. Math. Soc. Transl (2) 101, (1973), 105–106 [28] Marchenko, V.A., Sturm-Liouville operators and applications, OT22, Birkh¨auser, 1986. [29] McLaughlin, J.R., Analytical methods for recovering coeﬃcients in diﬀerential equations from spectral data. SIAM Rev. 28 no. 1, (1986), 53–72. [30] Naimark, M.A., Linear diﬀerential operators in Hilbert spaces, Eng. trans., Part 2, Ungar, New York, 1968. [31] Titchmarsh, E.C., Eigenfunction expansions associated with second-order diﬀerential equations. Part I. Second Edition Clarendon Press, Oxford, 1962. Amin Boumenir Department of mathematics, University of West Georgia, 1601 Maple street, Carrollton, GA 30118, USA e-mail: [email protected] Received: December 12, 2008 Accepted: April 14, 2009

Operator Theory: Advances and Applications, Vol. 203, 137–164 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On the Uniqueness of a Solution to Anisotropic Maxwell’s Equations T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili Dedicated to Israel Gohberg, the outstanding teacher and scientist, on his 80th birthday anniversary

Abstract. In the present paper we consider Maxwell’s equations in an anisotropic media, when the dielectric permittivity ε and the magnetic permeability μ are 3 × 3 matrices. We formulate relevant boundary value problems, investigate a fundamental solution and ﬁnd a Silver-M¨ uller type radiation condition at inﬁnity which ensures the uniqueness of solutions when permittivity and permeability matrices are real-valued, symmetric, positive deﬁnite and proportional ε = κμ, κ > 0. Mathematics Subject Classiﬁcation (2000). Primary 78A40; Secondary 35C15, 35E05, 35Q60. Keywords. Maxwell’s equations, Anysotropic media, Radiation condition, Uniqueness, Green’s formula, Integral representation, Fundamental solution.

Introduction In the paper we analyse the uniqueness of solutions to the time harmonic exterior three-dimensional boundary value problems (BVPs) for anisotropic Maxwell’s equations. It is well known that in the electro-magnetic wave scattering theory the most important question is the formulation of appropriate radiation conditions at inﬁnity, which are crucial in the study of uniqueness questions. In the case of isotropic Maxwell’s equations such conditions are the Silver-M¨ uller radiation conditions which are counterparts of the Sommerfeld radiation conditions for the Helmholtz equation. In view of the celebrated Rellich-Vekua lemma it follows The investigation was supported by the grant of the Georgian National Science Foundation GNSF/ST07/3-175. Communicated by J.A. Ball.

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T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili

that the Helmholtz equation and isotropic Maxwell’s equations do not admit nontrivial solutions decaying at inﬁnity as O(|x|−1−δ ) with δ > 0. This property plays an essential role in the study of direct and inverse acoustic and electro-magnetic wave scattering (see, e.g., [CK1, Eo1, HW1, Jo1, Le1, Ne1, Ve1] and the references therein). Investigation of the same type problems for the general anisotropic case proved to be much more diﬃcult and only few results are worked out so far. The main problem here consists in ﬁnding the appropriate radiation conditions at inﬁnity, which, in turn, is closely related to the asymptotic properties of the corresponding fundamental solutions (see, e.g., [Va1, Wi1, Na1, Ag1] for special classes of strongly elliptic partial diﬀerential equations). As we will see below anisotropic Maxwell’s equations, as well as the isotropic one, has not a strongly elliptic symbol and its characteristic surface represents a self-intersecting two-dimensional manifold, in general. In the present paper, we consider a special case of anisotropy when the electric permittivity ε = [εkj ]3×3 and the magnetic permeability μ = [μkj ]3×3 are realvalued, symmetric, positive deﬁnite and proportional matrices ε = κμ, κ > 0. For this particular case we explicitly construct fundamental matrices, formulate the corresponding Silver-M¨ uller type radiation conditions and prove the uniqueness theorems for the exterior BVPs.

1. Basic boundary value problems for Maxwell’s equations Throughout the paper we denote by Ω a domain, which can be bounded or unbounded, while the notation Ω+ stands for a bounded domain and Ω− := R3 \ Ω+ . Maxwell’s equations

curl H + iωεE = 0 , in Ω ⊂ R3 (1) curl E − iωμH = 0 , for ω > 0 govern the scattering of time-harmonic electromagnetic waves with frequency ω in a domain Ω. E = (E1 , E2 , E3 ) and H = (H1 , H2 , H3 ) are 3 vector-functions, representing the scattered electric and magnetic waves respectively. Here and in what follows the symbol (·) denotes transposition and ⎤ ⎡ 0 −∂3 ∂2 ⎢ 0 −∂1 ⎥ curl := ⎣ ∂3 ⎦. −∂2 ∂1 0 System (1) can also be written in matrix form iωεI3 E = 0, M(D) := M(D) H curl D := −i(∂1 , ∂2 , ∂3 ) ,

∂j :=

∂ , ∂xj

curl −iωμI3

j = 1, 2, 3.

,

(2)

On the Uniqueness of a Solution

139

The scope of the present investigation is to consider an anisotropic case when relative dielectric permittivity ε = [εjk ]3×3 and relative magnetic permeability μ = [μjk ]3×3 in (1) are real-valued symmetric positive deﬁnite constant matrices, i.e., εξ, ξ ≥ c|ξ|2 ,

μξ, ξ ≥ d|ξ|2 ,

∀ξ ∈ C3

(3)

with some positive constants c > 0, d > 0 and where η, ξ :=

3

ηj ξ j ,

η, ξ ∈ C3 .

j=1

Consequently, these matrices admit the square roots ε1/2 , μ1/2 . In some models of anisotropic media the positive deﬁniteness (3) is a consequence of the energy conservation law (cf., e.g., [BDS1]). By solving E from the ﬁrst equation in (1) and introducing the result into the second one we obtain an equivalent system

curl ε−1 curl H − ω 2 μH = 0 , in Ω , (4) E = i(ωε)−1 curl H or, by ﬁrst solving H from the second equation and introducing the result into the ﬁrst one we obtain another equivalent system

curl μ−1 curl E − ω 2 εE = 0 , in Ω . (5) H = −i(ωμ)−1curl E Since div curl = 0, after applying the divergence operator div to the ﬁrst equations of the systems (4) and (5), we get div(μ H) = div(ε E) = 0 .

(6)

Here we will only investigate the system (5). Results for the system (4) can be worked out analogously. For a rigorous formulation of conditions providing the unique solvability of the formulated boundary value problems we use the Bessel potential Hrp (Ω), Hrp (S ), Hrp,loc (Ω), Hrp,com (Ω) and Besov Brp,q (Ω), Brp,p (S ) spaces, −∞ < r < ∞, 1 < p, q < ∞, when Ω ⊂ R3 is a domain and S is the suﬃciently smooth boundary surface of Ω. Note that, for an unbounded domain Ω, the space Hrp,loc (Ω) comprises all 3 distributions u for which ψ u ∈ Hrp (Ω) where ψ ∈ C∞ 0 (R ) is arbitrary. As usual, r r r r for the spaces H2 (Ω), H2 (S ), H2,loc (Ω), H2,com (Ω) we use the notation Hr (Ω), Hr (S ), Hrloc (Ω), Hrcom (Ω). r−1/p r−1/p It is well known that Wp (S ) = Bp,p (S ) (Sobolev-Slobodetski space) r is a trace space for Hp (Ω), provided r > 1/p. If C is an open smooth subsurface of r (C ). The space Hr (C ) a hypersurface S in R3 , we use the spaces Hrp (C ) and H p p comprises those functions ϕ which have extensions to functions φ ∈ Hrp (S ). The

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r (C ) comprises functions ϕ ∈ Hr (S ) which are supported in C (funcspace H p p tions with “vanishing traces on the boundary ∂C ”). For detailed deﬁnitions and properties of these spaces we refer to, e.g., [Hr1, HW1, Tr1]). Finally, as usual for the Maxwell’s equations, we need the following special space H(curl; Ω) := {U ∈ L2 (Ω) : curl U ∈ L2 (Ω)}. We also use the notation Hloc (curl; Ω), meaning the Fr´echet space of all locally integrable vector functions U and curl U instead of global integrability if the underlying domain Ω is unbounded, and the space H(curl; Ω) if Ω is bounded. Note that H1 (Ω) is a proper subspace of H(curl; Ω). Indeed, U + grad ψ ∈ H(curl; Ω) for a vector function U ∈ H (Ω) and a scalar function ψ ∈ H1 (Ω) but, in general, U + grad ψ ∈ H1 (Ω). Next we recall basic boundary value problems for Maxwell’s equations written for the electric ﬁeld: 1

I. The “magnetic” BVP

curl μ−1 curl E − ω 2 εE = 0 // . . γS ν × μ−1 curl E = e

in Ω ⊂ R3 , on S := ∂Ω ,

E ∈ Hloc (curl; Ω),

(7a)

e ∈ H−1/2 (S ),

where γS is the trace operator on the boundary and the symbol × denotes the vector product of vectors; II. The “electric” BVP

in Ω ⊂ R3 , curl μ−1 curl E − ω 2 εE = 0 (7b) γS (ν × E) = f on S , E ∈ Hloc (curl; Ω), III. The “mixed” BVP ⎧ −1 E − ω 2 εE = 0 ⎪ ⎨ curl .μ curl // . −1 γSN ν × μ curl E = eN ⎪ ⎩ γSD (ν × E) = fD E ∈ Hloc (curl; Ω),

eN ∈ H−1/2 (SN ),

f ∈ H1/2 (S ); in Ω ⊂ R3 , on SN , on SD ,

(7c)

fD ∈ H1/2 (SD ),

where SD and SN are disjoint parts of the boundary surface S := S N ∪S D . If S is an orientable, smooth, open surface in R3 with a boundary Γ := ∂S , it has two faces S − and S + , which diﬀer by the orientation of the normal vector ﬁeld ν(x), which points from S + to S − . The natural BVPs for scattering of electromagnetic ﬁeld by an open surface S in R3 \ S are the following:

On the Uniqueness of a Solution I. The crack type “magnetic-magnetic” BVP

curl μ−1 curl E − ω 2 εE = 0 // . . γS ± ν × μ−1 curl E = e± E ∈ Hloc (curl; R3 \ S ), II. The screen type “electric-electric” BVP

curl μ−1 curl E − ω 2 εE = 0 γS ± (ν × E) = f ± E ∈ Hloc (curl; R3 \ S ), III. The “magnetic-electric” BVP

curl μ−1 curl E − ω 2 εE = 0 // . . γS + ν × μ−1 curl E = e+ , E ∈ Hloc (curl; R3 \ S ),

in R3 \ S , on S ,

E ∈ Hloc (curl; R3 \ S ),

(8a)

e± ∈ H−1/2 (S ); in R3 \ S , on S ,

(8b)

f ± ∈ H1/2 (S );

γS − (ν × E) = f −

in R3 \ S , on S ,

e+ ∈ H−1/2 (S ),

f − ∈ H1/2 (S );

IV. The “mixed-mixed” type BVP ⎧ −1 E − ω 2 εE = 0 ⎪ ⎨ curl μ curl / . −1 γS ± ν × μ curl E = e± N, N ⎪ ⎩ γ ± [ν × E] = f ± SN

141

D

−1/2 e± (SN± ), N ∈ H

(8c)

in R3 \ S , on SN± , on

(8d)

SD± ,

± fD ∈ H1/2 (SD± ),

where SN± ∪ SD± = S and SN+ ∩ SD+ = ∅, SN− ∩ SD− = ∅. All BVPs (8a)–(8d) and BVPs (7a)–(7c) for an unbounded domain Ω should be endowed with a special condition at inﬁnity. If the medium is isotropic, i.e., the permeability and the permittivity coeﬃcients are scalar constants, the radiation conditions are well known (cf., e.g., [CK1, Eo1, Jo1, Ne1] etc.). For example, the classical radiation condition imposed on the electric ﬁeld reads . / ∂E(x) − iσkE = O R−2 for R = |x| → ∞, (9) ∂R √ where k = ω εμ and either σ = −1 for incoming waves or σ = +1 for outgoing waves. Similar condition can also be imposed on the magnetic ﬁeld H. The SilverM¨ uller radiation condition is imposed on both ﬁelds either √ . / √ εE(x) × x ˆ + μH(x) = O R−2 for R = |x| → ∞ (10) or

√ / . εE(x) − √μH(x) × x ˆ = O R−2

x where x ˆ := . |x|

for R = |x| → ∞,

(11)

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The basic boundary value problems for the magnetic ﬁeld H and the diﬀerential equation (4) are formulated similarly to (7a)–(7c) and (8a)–(8d). Remark 1.1. We can derive solutions to the screen type (the “electric”) BVP for electric E ﬁeld indirectly, provided we can solve the crack type (the “magnetic”) BVP for the magnetic ﬁeld H and vice versa. Indeed, let H be a solution to the “magnetic” boundary value problem with a boundary data h for the magnetic ﬁeld H. Due to the second equations in (4), we get . . // i i γS (ν × E) = γS ν × ε−1curl H = h. ω ω Therefore the vector ﬁeld E = i(ωε)−1 curl H is a solution to the “electric” BVP i (7b) with the boundary data f = h. ω The same is true, due to the second equations in (5) and (4), for the all three remaining BVPs for the magnetic H and the electric E vector ﬁelds. Radiation conditions for the matrix coeﬃcients ε and μ are unknown so far. In §5 a radiation condition for anisotropic Maxwell’s equations is derived when the permittivity and permeability matrices ε and μ are real-valued, positive deﬁnite, symmetric and proportional ε = κμ. The radiation conditions ensure the uniqueness of a solution. As a ﬁrst step to the investigation let us simplify the main object, namely, the system (1). Let ε1 , ε2 , ε3 , μ1 , μ2 , μ3 be the eigenvalues of the permittivity and permeability matrices. Due to (3) they are positive εj > 0, μj > 0, j = 1, 2, 3. Consider following Maxwell’s equations

curl H ∗ + iωε∗ E ∗ = 0 , in Ω∗ ⊂ R3 , (12) curl E ∗ − iωμ∗ H ∗ = 0 , with the diagonal permittivity ⎡ ε1 0 ε∗ = ⎣ 0 ε2 0 0

and permeability matrices ⎡ ⎤ μ1 0 0 0 ⎦, μ∗ = ⎣ 0 μ2 ε3 0 0

⎤ 0 0 ⎦. μ3

Lemma 1.2. Let the permittivity ε and the permeability μ be real-valued, positive deﬁnite and proportional matrices ε = κμ,

κ > 0.

(13)

Then there exists an orthogonal matrix R : R3 −→ R3 ,

|R x| = |x|,

R −1 = R ,

which establishes the following equivalence between Maxwell’s equations (1) and (12): Ω∗ := R Ω and E ∗ (x∗ ) := R E(R x∗ )

H ∗ (x∗ ) := R H(R x∗ ),

∀x∗ := Rx ∈ Ω∗ .

(14)

On the Uniqueness of a Solution

143

Proof. The proof is based on the following well-known result (see, e.g., [Me1, § 7.5] and [Ga1, § IX.10]): a matrix A ∈ Cn×n is unitarily similar to a diagonal matrix D, i.e., A = U DU with U U = I, if and only if the matrix A is normal, i.e., commutes with its adjoint A ∗ A = A A ∗ . Since the matrices ε and μ are real-valued, positive deﬁnite and proportional matrices there exists an orthogonal, i.e., real-valued and unitary, matrix R which reduces them to the diagonal (Jordan) form simultaneously ε = R ε∗ R,

μ = R μ∗ R.

(15)

By introducing the representations (15) into the system (1), applying the transformation R to both sides of equations and changing the variable to a new one x∗ = R x, we obtain the following:

curl∗ H ∗ (x∗ ) + iωε∗ E ∗ (x∗ ) = 0 , (16) x∗ ∈ Ω∗ , curl∗ E ∗ (x∗ ) − iωμ∗ H ∗ (x∗ ) = 0 , where curl∗ U (x∗ ) := R curl R U (x). Let R1 , R2 , R3 be the vector columns of the transposed matrix R . Then R = (R1 , R2 , R3 ), and we ﬁnd

Rj , Rk = δjk ,

(17)

⎞ R 1 ⎠ ∇x × (R1 , R2 , R3 )U curl∗ U = Rcurl R U = ⎝ R 2 R3 ⎛

= [Rj , ∇x × Rk ]3×3 U = − [Rj × Rk , ∇x ]3×3 U ⎤ ⎡ 0 −R3 , ∇x R2 , ∇x 0 −R1 , ∇x ⎦ U = ⎣ R3 , ∇x −R2 , ∇x R1 , ∇x 0 ⎤ ⎡ ∂x∗2 0 −∂x∗3 0 −∂x∗1 ⎦ U , = ⎣ ∂x∗3 −∂x∗2 ∂x∗1 0

(18)

since the variables after transformation are x∗j = Rj , x, j = 1, 2, 3. The last three equalities in (18) follow with the help of the formulae: Rj , ∇ × Rk = −Rj × Rk , ∇ = −εjkm Rm , ∇, R1 × R2 = R3 ,

R 2 × R3 = R1 ,

R3 × R 1 = R2 ,

where εjkm is the Levi-Civita symbol (the permutation sign), j, k, m = 1, 2, 3. The equality (18) accomplishes the proof.

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Remark 1.3. Hereafter, if not stated otherwise, we will assume that ε and μ are real-valued, positive deﬁnite, proportional (cf. (13)) and diagonal matrices ⎤ ⎡ ⎤ ⎡ μ1 0 ε1 0 0 0 μ = ⎣ 0 μ2 0 ⎦ . (19) ε = ⎣ 0 ε2 0 ⎦ , 0 0 ε3 0 0 μ3 Remark 1.4. Finally, let us note that for a complex-valued wave frequency Im ω = 0 and arbitrary real-valued, symmetric and positive deﬁnite matrices μ and ε, a fundamental solution to Maxwell’s operator exists and decays at inﬁnity exponentially. Moreover, each above-formulated basic BVPs for Maxwell’s equations has a unique solution in the class of polynomially bounded vector-functions, represented by layer potentials and actually these solutions decay exponentially at inﬁnity. For real-valued frequencies matters are diﬀerent and we consider the case in the next section.

2. A fundamental solution to Maxwell’s operator The equation M μ (D)F (x) = δ(x)I3 ,

M μ (D) := curl μ−1 curl , F = (F1 , F2 , F3 ) ,

(20)

x ∈ R3 ,

(cf. (5)), where I3 is the identity matrix, has no fundamental solution. In fact, the determinant of the symbol (the characteristic polynomial) of this operator vanishes identically, det M μ (ξ) = det σcurl (ξ) det μ−1 det σcurl (ξ) ≡ 0, where σcurl (ξ) is the symbol of the operator curl: ⎡ ⎤ 0 iξ3 −iξ2 0 iξ1 ⎦ . σcurl (ξ) := ⎣ −iξ3 iξ2 −iξ1 0

(21)

(22)

The absence of the fundamental solution is a consequence of the following theorem. " pα ∂ α with constant Theorem 2.1. A partial diﬀerential operator P(D) = |α|≤m

matrix coeﬃcients pα ∈ CN ×N has a fundamental solution FP ∈ S (Rn ) if and only if the determinant of the symbol P (ξ) = σP (ξ) := pα (−iξ)α , ξ ∈ Rn , |α|≤m

does not vanish identically.

On the Uniqueness of a Solution Proof. Let det P (ξ) ≡ 0 and consider the formal co-factor matrix of P(D) AP (D) := Ajk (D) N ×N , Ajk (D) = (−1)j+k Mkj (D) ,

145

(23)

where Mkj (D) are the (N − 1)-dimensional minors of P(D). Then AP (D)P(D) = P(D)AP (D) = diag{det P(D), . . . , det P(D)} . The distribution FP := AP (D)diag{Fdet P , . . . , Fdet P }, where Fdet P is the fundamental solution of the scalar equation det P(D)F (x) = δ(x) (cf. Malgrange-Ehrenpreis theorem; cf. [Hr1]) is the claimed fundamental solution of P(D). Next we assume that the determinant vanishes identically, i.e., det P (ξ) ≡ 0. Then det P(D) = 0 and the rows of the operator matrix are linearly dependent. There exists a non-singular permutation N × N matrix H with constant entries, such that the ﬁrst row of the matrix-operator P(D) = H P(D) is identically 0. If we assume that a fundamental solution exists, i.e., P(D)FP = δIN , we get the following equality . / . / (0, c2 , . . . , cN ) = H P(D) FP u = H P(D)FP u = H δu = H u(0) for all u ∈ S(Rn ). Since the test vector-function u is arbitrary and the matrix H is invertible, the latter equality is a contradiction. In contrast to equations (20) the corresponding spectral equation M e (D)Φe = δI3 ,

M e (D) := M μ (D) − ω 2 μI

(24)

has a fundamental solution. Theorem 2.2. The fundamental solution of the equation in (24) is given by Φe = M # e (D) Fdet M e I3

(25)

M# e (D)

where denotes the formal co-factor matrix operator of M e (D) and Fdet M e is a fundamental solution of the equation det M e (D) Fdet M e = δ . Proof. Due to Theorem 2.1 the fundamental solution Fdet M e exists and implies the existence of the fundamental solution Φe for M e (D): M e (D)Φe = M e (D)M # e (D) Fdet M e I3 = det M e (D) Fdet M e I3 = δI3 .

Remark 2.3. The symbol Me (ξ) of the operators M e (D) in (24) is not elliptic 1 n n and even not hypoelliptic. To be hypoelliptic (of the class HLm,m ρ,0 (R × R ) for m1 , m ∈ N0 , m1 ≤ m), the principal symbol σA (x, ξ) of a matrix diﬀerential (or a pseudodiﬀerential) operator A(x, D) needs, by deﬁnition, to meet the following two conditions [Hr1]:

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i. there exist positive constants C1 and C2 , such that the inequalities C1 |ξ|m1 ≤ | det σA (x, ξ)| ≤ C2 |ξ|m

∀ x, ξ ∈ Rn

(26)

hold; ii. for arbitrary α, β ∈ Rn , |α| + |β| = 0, there exist positive constants Cα,β and ρ > 0, such that (α) −1 (x, ξ)(σA )(β) (x, ξ)] ≤ Cα,β |ξ|−ρ|α| ∀ x, ξ ∈ Rn , (27) det [σA (α)

where (σA )(β) (x, ξ) := ∂xβ ∂ξα σA (x, ξ). If the indices coincide m1 = m, the symbol σA (x, D) is elliptic from the n n H¨ormander class HLm ρ,0 (R × R ). To show that the symbol Me (ξ) is not hypoelliptic we will check that the second condition (27) fails for it. In fact: det Me (ξ) = det σcurl (ξ)μ−1 σcurl (ξ) − ω 2 ε = ω 2 P4 (ξ) + ω 4 P2 (ξ) − ω 6 det ε .

(28)

Here Pk (ξ) is a homogeneous polynomial of order k = 2, 4. Then −1 = 0, ord Me (ξ) − ω 2 ε −1 ord Me (ξ) − ω 2 ε ∂j Me (ξ) − ω 2 ε = +1,

(29)

and the condition (27) fails. The next proposition is well known (cf. [Ne1], [CK1]). Proposition 2.4. Either of the following functions e±ik|x| 1 e±ik|x| I3 + . ∇∇ 2 4π|x| 4πk |x| is a fundamental solution of the equation Φ± M (x) =

(30)

MΦM := curl 2 ΦM − k 2 ΦM = δI3 .

(31)

Proof. The fundamental solution is equal to the inverse Fourier transform of the inverse symbol −1 −1 M (ξ) . ΦM (x) = Fξ→x (32) Since the symbol equals (cf. (22)) ⎤2 ⎡ 0 iξ3 −iξ2 0 iξ1 ⎦ − k 2 I3 = (|ξ|2 − k 2 )I3 − ξξ , M(ξ) = ⎣ −iξ3 iξ2 −iξ1 0 let us look for the inverse in the form 1 I3 − αξξ , |ξ|2 − k 2 where α is an unknown scalar function. M−1 (ξ) :=

ξ ∈ R3 ,

(33)

On the Uniqueness of a Solution

147

Since ξ ξ = |ξ|2 , the condition M−1 (ξ)M(ξ) ≡ I3 provides the equality α(ξ) =

1 k 2 (|ξ|2

− k2 )

,

which is well deﬁned outside the sphere |ξ|2 = k2 . Then, 1 1 −1 I3 − 2 ξξ , |ξ| = k. M (ξ) := 2 |ξ| − k 2 k

(34)

To regularize the singular integral, let us temporarily replace k by a complexvalued parameter k ± iθ, where θ > 0 is small. By inserting (34) (with k ± iθ) into (32) and by applying the identity −ix,ξ = −ξξ e−ix,ξ , ∇x ∇ x, ξ ∈ R3 , x e we proceed as follows: 1 e−ix,ξ dξ e−ix,ξ ξξ dξ ± (35) I3 − ΦM (x) = 3 lim 2 2 2 2π θ→0+ R3 |ξ|2 − (k ± iθ)2 R3 (k ± iθ) |ξ| − (k ± iθ) 1 = lim θ→0+ 2π 3

R3

1 e−ix,ξ dξ I3 + ∇∇ |ξ|2 − (k ± iθ)2 (k ± iθ)2

R3

e−ix,ξ dξ . |ξ|2 − (k ± iθ)2

To calculate the integral in (35) it is convenient to introduce the spherical coordinates ξ = ρη, η ∈ S2 ⊂ R3 and apply the residue theorem. After a standard manipulation we get the following: ∞ e−ix,ξ dξ ρ sin(ρ|x|) 1 1 lim G± (x) := lim = dρ (2π)3 θ→0+ R3 |ξ|2 − (k ± iθ)2 2π2 |x| θ→0+ 0 ρ2 − (k ± iθ)2 =

e±ik|x| , 4π|x|

x ∈ R3 .

(36)

By inserting the obtained integral in (35) we arrive at (30).

Theorem 2.5. Let coeﬃcients ε and μ be diagonal and proportional (see (13) and Remark 1.3). Then the fundamental solution Φe in (24) is written in explicit form −1 −1 (37) Φ± e (x) = Fξ→x Me (ξ, ω) ⎡ 2 ⎤ 2 −∂1 ∂2 −∂1 ∂3 ∂1 + ω κμ2 μ3 1 2 2 ⎣ ⎦ = −∂ ∂ ∂ + ω κμ μ −∂2 ∂3 1 2 1 3 2 4πω 2 κ(det μ)3/2 2 2 −∂ ∂ −∂ ∂ ∂ + ω κμ μ ×

√ ±iω κ det μ | x|

e

| x|

=

e

1 3 √ ±iω κ det μ| x|

4π| x|

2 3

1 2

3

. / Φe,∞ ( x) + O |x|−2

as

|x| → ∞,

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T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili

xj 2 , x 3 ) , x j := √ , j = 1, 2, 3, and the matrix where x := ( x1 , x μj ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ x) := ⎢ Φe,∞ ( ⎢ ⎢ ⎢ ⎣

| x|2 − x 21 √ μ1 | x|2 det μ √ μ3 x 1 x 2 2 | x| det μ √ μ2 x 1 x 3 2 | x| det μ

√ μ3 x 1 x 2 2 | x| det μ 22 | x|2 − x √ μ2 | x|2 det μ √ μ1 x 2 x 3 2 | x| det μ

√ μ2 x 1 x 3 2 | x| det μ √ μ1 x 2 x 3 2 | x| det μ 2 | x|2 − x √ 3 2 μ3 | x| det μ

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(38)

is known as the far ﬁeld pattern. Proof. If ε and μ are diagonal (cf. (19)), the operator M e (D) in (24) acquires the form: Me (ξ, ω) = σcurl (ξ) μ−1 σcurl (ξ) − ω 2 εI ⎤ ⎡ −1 ⎤⎡ ⎡ μ1 0 0 0 0 iξ3 −iξ2 ⎦ ⎣ 0 iξ1 ⎦ ⎣ 0 −iξ μ−1 = ⎣ −iξ3 0 3 2 iξ2 −iξ1 0 iξ2 0 0 μ−1 3 ⎡ −1 2 2 2 −μ−1 μ3 ξ2 + μ−1 2 ξ3 − ω ε1 3 ξ1 ξ2 ⎢ −1 −1 2 2 =⎢ −μ3 ξ1 ξ2 μ3 ξ12 + μ−1 1 ξ3 − ω ε2 ⎣ −μ−1 2 ξ1 ξ3

−μ−1 1 ξ2 ξ3

⎤ −iξ2 iξ1 ⎦ − ω 2 εI 0 ⎤ −μ−1 2 ξ1 ξ3 ⎥ ⎥. −μ−1 1 ξ2 ξ3 ⎦

iξ3 0 −iξ1

−1 2 2 2 μ−1 2 ξ1 + μ1 ξ2 − ω ε3

We have: det Me (ξ, ω) −1 2 −1 2 −1 2 −1 2 −1 2 2 2 2 2 = μ−1 3 ξ2 + μ2 ξ3 − ω ε1 μ3 ξ1 + μ1 ξ3 − ω ε2 μ2 ξ1 + μ1 ξ2 − ω ε3 −1 2 2 2 −1 −1 2 2 2 −2 2 2 μ3 ξ2 + μ−1 −2μ−1 1 μ2 μ3 ξ1 ξ2 ξ3 − μ1 2 ξ3 − ω ε1 ξ2 ξ3 −1 2 2 2 −1 2 2 2 −2 2 2 2 2 −μ−2 μ3 ξ1 + μ−1 μ2 ξ1 + μ−1 2 1 ξ3 − ω ε2 ξ1 ξ3 − μ3 1 ξ2 − ω ε3 ξ1 ξ2 = −ω 2 ε1 ξ12 + ε2 ξ22 + ε3 ξ32 − ω 2 ε1 ε3 μ2 −1 2 −1 −1 2 −1 −1 2 −1 2 × μ−1 2 μ3 ξ1 + μ1 μ3 ξ2 + μ1 μ2 ξ3 − ω ε2 μ2 −1 −1 −1 −1 2 +ω 4 [ε22 μ−1 ξ2 2 + ε1 ε3 μ1 μ2 μ3 − ε1 ε2 μ1 − ε2 ε3 μ3 ξ 2 ξ2 ξ22 ξ32 ξ22 ξ32 1 1 2 2 = −ω 2 det ε + + − ω + + − ω μ2 ε3 ε1 μ2 μ3 ε2 μ1 ε2 μ3 μ−1 μ1 ε2 ε1 μ2 ε3 ε−1 2 2 −1 −1 −1 −1 2 4 2 −1 +ω ε2 μ2 + ε1 ε3 μ1 μ2 μ3 − ε1 ε2 μ1 − ε2 ε3 μ3 ξ2 .

On the Uniqueness of a Solution

149

For diagonal and proportional matrices (see (13) and (19)), we get the following simpliﬁcation Me (ξ, ω) = σcurl (ξ) μ−1 σcurl (ξ) − ω 2 κμI ⎡ −1 2 2 2 μ3 ξ2 + μ−1 −μ−1 2 ξ3 − ω1 μ1 3 ξ1 ξ2 ⎢ −1 −1 2 2 2 =⎣ −μ3 ξ1 ξ2 μ3 ξ1 + μ−1 1 ξ3 − ω1 μ2 −μ−1 2 ξ1 ξ3

−μ−1 1 ξ2 ξ3

−μ−1 2 ξ1 ξ3 −μ−1 1 ξ2 ξ3 −1 2 2 μ2 ξ1 + μ−1 1 ξ2 −

2 2 ω 2 det ε ξ22 ξ32 ξ1 2 + + − κω det Me (ξ, ω) = − κ2 μ2 μ3 μ1 μ3 μ1 μ2 2 ω 2 det ε =− 2 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − κω 2 det μ κ det μ 2 = −ω12 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , ω12

2

(39) ⎤ ⎥ ⎦, ω12 μ3

(40) (41)

where := ω κ. It is easy to see that all minors of the matrix (39) have the factor μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ. Indeed, we have / . −1 2 −1 2 −1 2 −2 2 2 2 2 2 Me (ξ, ω) 11 = [μ−1 3 ξ1 + μ1 ξ3 − ω1 μ2 ][μ2 ξ1 + μ1 ξ2 − ω1 μ3 ] − μ1 ξ2 ξ3 2 ξ1 ξ2 ξ2 = (ξ12 − ω 2 κμ2 μ3 ) + 2 + 3 − ω2κ μ2 μ3 μ1 μ3 μ1 μ2 2 2 ξ − ω κμ2 μ3 = 1 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , det μ 2 / . ξ1 ξ2 ξ2 + 2 + 3 − ω2κ Me (ξ, ω) 22 = (ξ22 − ω 2 κμ1 μ3 ) μ2 μ3 μ1 μ3 μ1 μ2 2 2 ξ − ω κμ1 μ3 = 2 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , det μ 2 . / ξ1 ξ22 ξ32 2 2 2 Me (ξ, ω) 33 = (ξ3 − ω κμ1 μ2 ) + + −ω κ μ2 μ3 μ1 μ3 μ1 μ2 ξ 2 − ω 2 κμ1 μ2 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , = 3 det μ . / . / −1 2 −1 2 −1 −1 2 2 Me (ξ, ω) 12 = Me (ξ, ω) 21 = μ−1 3 ξ1 ξ2 [μ2 ξ1 + μ1 ξ2 − ω1 μ3 ] − μ1 μ2 ξ1 ξ2 ξ3 2 ξ1 ξ2 ξ2 = −ξ1 ξ2 + 2 + 3 − ω2κ μ2 μ3 μ1 μ3 μ1 μ2 ξ1 ξ2 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , =− det μ 2 . / . / ξ1 ξ2 ξ2 Me (ξ, ω) 13 = Me (ξ, ω) 31 = −ξ1 ξ3 + 2 + 3 − ω2κ μ2 μ3 μ1 μ3 μ1 μ2 ξ1 ξ3 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ , =− det μ

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/ . / Me (ξ, ω) 23 = Me (ξ, ω) 32 = −ξ2 ξ3

.

=−

ξ2 ξ3 det μ

ξ2 ξ2 ξ12 + 2 + 3 − ω2κ μ2 μ3 μ1 μ3 μ1 μ2 μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − ω12 det μ .

Applying the variable transformation mation formula (36), we easily obtain −1 Fξ→x

√ μj ξj = ηj and the Fourier transfor-

√ e±iω|x| κ det μ 1 √ = , μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − κω 2 det μ ± i0 4π| x| det μ

where x is deﬁned in (38). From the obtained expressions for the determinant, minors and the latter formula for Fourier transformation we easily derive formula (38) −1 −1 Φ± e (x) = Fξ→x Me (ξ, ω) 1 1 −1 =− 2 F κω det μ ξ→x μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − κω 2 det μ ± i0 ⎤⎤ ⎡ 2 −ξ1 ξ2 −ξ1 ξ3 ξ1 − ω 2 κμ2 μ3 ⎦⎦ −ξ1 ξ2 ξ22 − ω 2 κμ1 μ3 −ξ2 ξ3 ×⎣ 2 2 −ξ1 ξ3 −ξ2 ξ3 ξ3 − ω κμ1 μ2 ⎤ ⎡ 2 −∂1 ∂2 −∂1 ∂3 ∂1 + ω 2 κμ2 μ3 1 ⎦ ⎣ −∂1 ∂2 ∂22 + ω 2 κμ1 μ3 −∂2 ∂3 = κω 2 det μ 2 2 −∂1 ∂3 −∂2 ∂3 ∂3 + ω κμ1 μ2 1 −1 ×Fξ→x μ1 ξ12 + μ2 ξ22 + μ3 ξ32 − κω 2 det μ ± i0 ⎡ 2 ∂1 + ω 2 κμ2 μ3 1 ⎣ = −∂1 ∂2 κω 2 det μ −∂1 ∂3

−∂1 ∂2 ∂22 + ω 2 κμ1 μ3 −∂2 ∂3

where the variable x is deﬁned in (38).

⎤ √ −∂1 ∂3 e±iω|x| κ det μ ⎦ √ , −∂2 ∂3 4π| x| det μ ∂32 + ω 2 κμ1 μ2

Remark 2.6. It can be checked that the necessary and suﬃcient condition for the polynomial det Me (ξ, ω) to be factored into two second degree polynomials, det Me (ξ, ω) = P1 (ξ)P2 (ξ), is the condition that one of the following equalities hold: ε1 ε2 ε1 ε3 ε2 ε3 = , = , = μ1 μ2 μ1 μ3 μ2 μ3

On the Uniqueness of a Solution

Fig. 1: Outer characteristic ellipsoid

151

Fig. 2: Section of the characteristic surface

If (13) is not fulﬁlled, then the equations Pi (ξ) = 0, i = 1, 2, determine two diﬀerent ellipsoidal surfaces with two touching points at the endpoints of common axes (see Fig.1 and Fig.2). If conditions (13) and (19) hold, the ellipsoids coincide.

3. Green’s formulae Here we apply the results of [Du1] and derive Green’s formulae for Maxwell’s equations (5), needed for our analysis. For convenience we also use the notation ± U = U± γS Lemma 3.1. For a domain Ω+ ⊂ R3 with a smooth boundary S := ∂Ω+ the following Green’s formula holds (curl U , V )Ω+ − (U , curl V )Ω+ = (ν × U + , V + )S = −(U + , ν × V + )S ,

(42)

U , V ∈ H1 (Ω+ ) , where U + = (U1+ , U2+ , U3+ ) denotes the trace on the boundary S , (U , V )G := U (x), V (x) dx. G

In particular, (curl U , ∇ v)Ω+ = −(U + , MS v + )S ,

U ∈ H1 (Ω+ ),

v ∈ H1 (Ω+ ) ,

(43)

where the brackets (·, ·)S denotes the duality between adjoint spaces Hs (S ) and H−s (S ), MS := ν × ∇ = (M23 , M31 , M12 ) , (44) and Mjk = νj ∂k − νk ∂j are Stoke’s tangential diﬀerentiation operators on the boundary surface S .

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Proof. Formula (42) is a simple consequence of the Gauss integration by parts formula (∂j u, ψ)Ω+ = (νj u+ , ψ + )S − (u, ∂j ψ)Ω+ ,

u, ψ ∈ H1 (Ω+ )

(45)

In fact, (curl U , V )Ω+ (∂2 U3 − ∂3 U2 )V1 + (∂3 U1 − ∂1 U3 )V2 + (∂1 U2 − ∂2 U1 )V3 dx = + Ω = (ν2 U3+ − ν3 U2+ )V1+ + (ν3 U1+ − ν1 U3+ )V2+ + (ν1 U2+ − ν2 U1+ )V3+ dS S

+ Ω+

(∂2 V3 − ∂3 V2 )U1 + (∂3 V1 − ∂1 V3 )U2 + (∂1 V2 − ∂2 V1 )U3 dx

= (ν × U + , V + )S + (U , curlV )Ω+ . Since ν × U can be interpreted as the application of ⎡ 0 −ν3 ν ×U =N U, N := ⎣ ν3 0 −ν2 ν1

(46) the skew symmetric matrix ⎤ ν2 −ν1 ⎦ = −N , 0

we get (ν × U + , V + ) = (N U + , V + ) = −(U + , N V + ) = −(U + , ν × V + ) , and this accomplishes the proof of (42). To prove (43) ﬁrst note that ν × (∇ v)+ = (ν × ∇ v)+ = (MS v)+ = MS v+

∀ v ∈ H2 (Ω+ ),

(47)

because Mjk are tangential derivatives (cf. (44)) and therefore it commutes with the trace operator . /+ MS v = MS v + . (48) Moreover, due to equality (48) it is suﬃcient to suppose v ∈ H1 (Ω+ ) in (47): if v ∈ H1 (Ω+ ) then v + ∈ H1/2 (S ) by the classical trace theorem and therefore . /+ MS v := MS v+ ∈ H−1/2 (S ). Equation (43) is a consequence of (42). In fact, (curl U , ∇ v)Ω+ = −(U + , ν × (∇ v)+ )S + (U , curl∇ v)Ω+ . /+ = −(U + , MS v )S = −(U + , MS v + )S since curl∇ = 0.

On the Uniqueness of a Solution

153

For anisotropic Maxwell’s equations we have the following. Theorem 3.2. The operator M e = curl μ−1 curl − ω 2 εI (cf. (6)) is formally self adjoint hold

M ∗e

(49)

= M e and the following Green’s formulae

(M e U , V )Ω+ = (ν × (μ−1 curl U )+ , V + )S + (μ−1 curl U , curl V )Ω+ − ω 2 (ε U , V )Ω+

(50a)

= −((μ−1 curl U )+ , ν × V + )S + (μ−1 curl U , curl V )Ω+ − ω 2 (ε U , V )Ω+ , (50b) (M e U , V )Ω+ − (U , M e V )Ω+ = (ν × (μ−1 curl U )+ , V + )S − (U + , ν × (μ−1 curl V )+ )S

(50c)

= −((μ−1 curl U )+ , ν × V + )S + (ν × U + , (μ−1 curl V )+ )S

(50d)

−1 (Ω+ ) in (50a) and (50b), provided U , V ∈ H1 (Ω+ ), and additionally, M e U ∈ H −1 + while M e U , M e V ∈ H (Ω ) in (50c) and (50d).

Proof. The claimed formulae follow from Lemma 3.1.

4. Representation of solutions and layer potentials In the present section we continue to apply the results of [Du1] to Maxwell’s equations (also see [CK1]). For simplicity we suppose that the boundary S = ∂Ω is a C ∞ smooth surface. Let us consider the following operators, related to the Maxwell systems (4) and (5): Newton’s potential Φe (x − y)U (y) dy, x ∈ R3 , NeΩ U (x) := (51) Ω

the single layer potential Ve U (x) :=

4 S

Φe (x − τ )U (τ ) dS,

and the double layer potential 4 e [(γN Φe )(x − τ )] U (τ ) dS, We U (x) := S

x ∈ R3 \ S ,

x ∈ R3 \ S ,

(52)

(53)

+ where Φe denotes one of the fundamental solutions Φ− e or Φe and e V (τ ) := ν(τ ) × μ−1 curl V (τ ), γN

denotes the “magnetic” trace operator.

τ ∈ S,

(54)

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Theorem 4.1. Let Ω+ be a bounded domain with inﬁnitely smooth boundary S = ∂Ω+ and Ω− := R3 \ Ω+. The potential operators NeΩ+

:

Hsp (Ω+ ) → Hsp (Ω+ ),

Ve

:

p Hsp (S ) → Hp,loc

:

Hsp (S ) → Hp

:

p Hsp (S ) → Hp,loc

:

Hsp (S ) → Hp

γS V e

:

Hsp (S ) → Hs−1 p (S ),

γS We

:

Hsp (S ) → Hs−2 p (S ),

We

s+ 1 −1 1 s+ p −1

s+ 1 −2 1 s+ p −2

(Ω− ), (Ω+ ), (Ω− ),

(55)

(Ω+ ),

are continuous for all 1 < p < ∞, s ∈ R. Here (γS Ψ)(x) is the Dirichlet trace operator on the boundary S = ∂Ω+ . Proof. The operators NeΩ+ , γS Ve , and γS We are all pseudodiﬀerential (abbreviated as ΨDO; cf. [DNS1, DNS2]). The symbol NeΩ+ (ξ) = Fx→ξ [Φe (x)] of the pseudodiﬀerential operator NeΩ+ coincides with the inverse symbol Me−1 (ξ) of the initial operator M e , which is a rational function uniformly bounded at inﬁnity (cf. Theorem 2.2). Therefore the ΨDO NeΩ+ has order 0, has the transmission property (as a ΨDO with a rational symbol), which implies the mapping property (55) for NeΩ+ . For the potential operators Ve and We the proof is based on the aboveproved property of ΨDOs NeΩ+ and the trace theorem and follows the proof of [Du1, Theorem 3.2]. Let us consider the following surface δ-function 4 (56) (g ⊗ δS , v)R3 := g(τ )γS v(τ )dS, g ∈ C ∞ (S ), v ∈ C0∞ (R3 ). S

Obviously, supp(g ⊗ δS ) = supp g ⊂ S . The deﬁnition (56) is extendible to less regular functions. More precisely, the following holds: Let 1 < p < ∞, s < 0, g ∈ Wsp (S ). Then s− p1

g ⊗ δS ∈ Hp

s−

1

p (S ) ⊂ Hp,com (R3 ) ,

(57)

where p = p/(p − 1) (cf. [Du1, Lemma 4.9]). The layer potential V e can be written in the form 4 e Φe (x − τ )U (τ ) dS = Φe (x − y)(U ⊗ δS )(y) dy, V U (x) := S

=

NeΩ (U

Ω

⊗ δS )(x),

(58)

where Ω is compact and S ⊂ Ω, and can be interpreted as a pseudodiﬀerential operator. Assume, for simplicity, Ω is compact. From the inclusion (57) and the

On the Uniqueness of a Solution

155

mapping property of the pseudodiﬀerential operator NeΩ in (55) we derive the mapping property of V e in (55): # # # # # # # # # e s− p1 # s− p1 # # e s+ p1 −1 # # # # #V U Hp (Ω)# = #NΩ (U ⊗ δS ) Hp (Ω)# ≤ C1 #(U ⊗ δS ) Hp (Ω)# # # # # ≤ C2 #U Hsp(S )# , provided s < 0. The layer potential We is written in the form 4 [T (Dy , ν)Φe (x − τ )] U (τ ) dS W e U (x) =

S

[T (Dy , N (y))Φe (x − y)] (U ⊗ δS )(y) dy

= Ω

=

DeΩ (U ⊗ δS )(x),

S ⊂Ω

(59)

and the principal symbol of the ΨDO DeΩ is e DΩ (x, ξ) := N (x)μ−1 σcurl (ξ)NΩe (ξ) = N (x)μ−1 σcurl (ξ)Fx→ξ [Φe (x)] , (60) ⎤ ⎡ 0 −N3 (x) N2 (x) 0 −N1 (x) ⎦ , N (x) := ⎣ N3 (x) −N2 (x) N1 (x) 0

where (N1 (x), N2 (x), N3 (x)) is some smooth extension of the normal vector ﬁeld ν(x) from S onto the domain Ω. Therefore, ord DeΩ = +1 and this pseudodiﬀerential operator has the following mapping property DeΩ

s (Ω) → Hs−1 (R3 ). : H p p,loc

(61)

From the inclusion (57) and the mapping property (61) we derive, as above, the mapping property of We in (55) provided s < 0. For the case s ≥ 0 we quote a similar proof in [Du1, Theorem 3.2] and drop the details since it needs some auxiliary assertions, proved in [Du1]. The mapping properties of ΨDOs γS Ve and γS We , which are the traces of the potential operators Ve and We , follow immediately due to the generalized trace theorem (see, e.g., [Se1]). Theorem 4.2. Solutions of Maxwell’s equations (1) in a compact domain Ω+ with diagonal and proportional coeﬃcients ε and μ (see (13) and Remark 1.3) are represented as follows e E(x) = W e (γD E)(x) − V e (γN E)(x),

x ∈ Ω+ .

(62)

e Here γN E is the “magnetic” trace operators (cf. (54)) and (γD E)(x) := E + (x) is the “electric” trace operator on the boundary S = ∂Ω+ .

156

T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili

Proof. By introducing the substitution (M e U , V )Ω+ − (U , M e V )Ω+ = (ν × (μ−1 curl U )+ , V + )S − (U + , ν × (μ−1 curl V )+ )S in the Green formula, where U is the fundamental solution U = Φe and V is the electric ﬁeld V = E, we obtain the representation of the solution E of the system (5). If we take into account that the Newton potential eliminates since we deal with a homogeneous system (U , M e V )Ω+ = (Φe , M e E)Ω+ = 0. Remark 4.3. The case of an unbounded domain Ω− will be treated in Theorem 5.1 after we establish asymptotic properties of fundamental solutions. For non-homogeneous Maxwell’s equations

curl H + iωεE = f , curl E − iωμH = g

in Ω

the equivalent systems are

curl ε−1 curl H − ω 2 μH = ω −1 curl(ε−1 f ) + g , E = i(ωε)−1 curl H − i(ωε)−1 f and

curl μ−1 curl E − ω 2 εE = f − ω −1 curl(μ−1 g) , H = −i(ωμ)−1 curl E + i(ωμ)−1 g

(63)

in Ω

(64)

in Ω .

(65)

Theorem 4.4. Solutions of Maxwell’s equations (63) in a domain Ω+ with diagonal and proportional coeﬃcients ε and μ (see (13) and Remark 1.3) are represented as follows −1 m H(x) = Nm curl(ε−1 f ) + g (x) + Wm (γD H)(x) − Vm (γN H)(x), Ω+ ω e E(x) = NeΩ+ f − ω −1 curl(μ−1 g) (x) + W e (γD E)(x) − V e (γN E)(x), x ∈ Ω+ . Proof. The proof is analogous to the proof of the foregoing Theorem 4.2 with a single diﬀerence: Newton’s potential does not disappear (U , M e V ) Ω+ = (Φm , M e E) Ω+ = NeΩ+ f − ω −1 curl(μ−1 g) (cf. equation (65)).

5. The uniqueness of a solution A solution E of the system (1) is called radiating in an unbounded domain Ω− if the asymptotic condition xj E(x) = O(|x|−2 ) as |x| → ∞ , j = 1, 2, 3, (66) ∂j E(x) − i κe μj | x| ) x1 x2 x3 κe := ω κ det μ, x , (67) := √ , √ , √ μ1 μ2 μ3

On the Uniqueness of a Solution

157

holds uniformly in all directions x∗ /|x∗ |, where $x x x % 1 2 3 x∗ = (x∗1 , x∗2 , x∗3 ) := . , , μ1 μ2 μ3

(68)

A radiating solution H of the system (1) is deﬁned similarly. Without loss of generality we assume that the origin of the co-ordinate system belongs to the bounded domain Ω+ and R is a suﬃciently large positive number, such that the domain Ω+ lies inside the ellipsoid Ψ(x) :=

x21 x22 x23 | x|2 = + + = 1. R2 μ1 R2 μ2 R2 μ3 R2

(69)

− Further, let BR denote the interior of the ellipsoid and Ω− R := Ω ∩ BR . Note that the exterior unit normal vector to the ellipsoidal surface ΣR := ∂BR deﬁned by equation (69) at the point x ∈ ΣR reads as ∇Ψ(x) 1 $ x1 x2 x3 % , , ν(x) = (ν1 (x), ν2 (x), ν3 (x)) := , (70) = ∗ |∇Ψ(x)| |x | μ1 μ2 μ3

(cf. (68) for x∗ ), where, νj (x) = νj ( x) =

x∗j xj = , j = 1, 2, 3. μj |x∗ | |x∗ |

(71)

Theorem 5.1. Let E, H ∈ H1loc (Ω− ) be radiating solutions to Maxwell’s equations (1) with diagonal and proportional anisotropic coeﬃcients ε and μ (cf. (13) and (19)) in an exterior domain Ω− . Then m H)(x), H(x) = W m (γD H)(x) − V m (γN e E)(x), E(x) = W e (γD E)(x) − V e (γN

x ∈ Ω− .

(72)

Proof. We prove this proposition for the electric ﬁeld E and fundamental solution − Φ+ e ; the proof for other cases (for Φe , for the ﬁeld H and fundamental solutions Φ± ) are similar. m First note that the radiation condition (66) implies / . |x∗ | curl E − iκe as |x| → ∞, (73) [ν(ˆ x) × E] = O |x|−2 | x| and further

κ 2 |x∗ |2 μ−1 curl E, curl E + e 2 μ−1 (ν × E), ν × E | x| ∂BR ( κe |x∗ | −1 Imμ curlE, ν × E dS +2 | x|

2 1 1 |x∗ | [ν × E] = O(|x|−4 ) as = μ− 2 curl E − iμ− 2 κe | x| 1

where μ− 2 is a square root of μ−1 .

|x| → ∞,

158

T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili Using the fact that c1 ≤

|x∗ | ≤ c2 with some positive constants c1 and c2 for | x|

all x ∈ R3 \ {0}, we obtain κ 2 |x∗ | −1 | x| −1 μ curl E, curl E + e μ (ν × E), ν × E ∗ |x | | x| ∂BR +2κe Imμ−1 curlE, ν × E dS → 0 as R → ∞.

(74)

Green’s formula in the domain Ω− R gives us (μ−1 curl E, curl E)Ω− − ω 2 (εE, E)Ω− + (μ−1 curl E, ν × E)S R

R

= (μ−1 curl E, ν × E)∂BR . Now taking the imaginary part of the last equation and applying (74) we ﬁnd that ( κe2 |x∗ | −1 | x| −1 μ curl E, curl E + μ (ν × E), ν × E dS |x∗ | | x| ∂BR = −2κe Im μ−1 curlE, ν × EdS. (75) ∂S

Since both summands in the left-hand side of (75) are nonnegative, they are bounded at inﬁnity: |ν × E|2 ds = O(1) as R → ∞. (76) ∂BR

Write the representation formula (62) in the bounded domain Ω− R: 4 4 e + e E(x) = [(γN Φe )(x − τ )] (γD E)(τ ) dS − Φ+ e (x − τ )(γN E)(τ ) dS, ∂BR ∪S

e = W e (γD E)(x) − V e (γN E)(x) + IR ,

where

∂BR ∪S

(77)

4 4 e + + e (γN Φe )(x − τ ) (γD E)(τ ) dS IR = − Φe (x − τ )(γN E)(τ ) dS + ∂BR ∂BR 4 −1 + ν × Φe (x − τ ) (μ curl E)(τ ) dS = ∂BR 4 −1 (μ curl Φ+ − (ν × E)(τ ) dS e )(x − τ ) ∂BR 4 −1 |x∗ | −1 (x − τ ) curl E(τ ) − iκ (ν(τ ) × E(τ )) dS μ ν(τ ) × Φ+ μ = e e | x| ∂BR 4 −1 |x∗ | + + − ν(τ ) × Φe (x − τ ) curl Φe (x − τ ) − iκe μ (ν × E)(τ ) dS. | x| ∂BR

On the Uniqueness of a Solution

159

−1 ) at inﬁnity, due to (79), (73), (76) and Schwartz Since Φ+ e (x) = O(|x| inequality both integrals on the right-hand side vanish as R → ∞ and the claimed representation for E in (72) follows from (77).

Corollary 5.2. Radiating solutions to Maxwell’s equations (1) with anisotropic coeﬃcients ε and μ as in (13) and (19) in an exterior domain Ω− have the following asymptotic behaviour: % $ % $ as |x| → ∞, (78) H(x) = O |x|−1 , E(x) = O |x|−1 Proof. The proof follows immediately from the representation formulae (72) since the potential operators have the indicated asymptotic behaviour automatically. Clearly, each column of the fundamental matrix Φ+ e (x) is a radiating vector due to the asymptotic formulae (38). Moreover, we have the following asymptotic relations for suﬃciently large |x| Φ+ e (x) =

1 ei κe |x| Φe, ∞ ( x) + O(|x|−2 ), 4π | x|

∂j Φ+ e (x) =

1 i κe xj i κe |x| e Φe, ∞ ( x) + O(|x|−2 ), 4π | x| μj | x|

∂j Φ+ e (x) − i κe

xj Φ+ (x) = O(|x|−2 ), j = 1, 2, 3, μj | x| e

(79)

where κe and x are given by (67), x = x/|x| and Φe, ∞ ( x) is deﬁned by (38). Further, if y belongs to a compact set and |x| is suﬃciently large then we have | x − y| = | x| − | x|−1 x, y + O(|x|−1 ) , x|−1 + O(|x|−2 ) , | x − y|−1 = | −1

ei κe |x−y| = ei κe |x| e−i κe |x|

x, y

+ O(|x|−1 ) ,

whence it follows that Φ+ e (x − y) =

−1 1 ei κe |x| e−i κe |x| x,y Φe, ∞ ( x) + O(|x|−2 ), 4π | x|

∂ j Φ+ e (x − y) =

1 i κe xj i κe |x| −i κe |x|−1 x,y e e Φe, ∞ ( x) + O(|x|−2 ), 4π | x| μj | x|

∂ j Φ+ e (x − y) − i κe

xj Φ+ (x − y) = O(|x|−2 ), μj | x| e

j = 1, 2, 3.

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T. Buchukuri, R. Duduchava, D. Kapanadze and D. Natroshvili

These formulae can be diﬀerentiated arbitrarily many times |α+β| α+β x i κe α β + |β| Φ+ (x − y) = O(|x|−2 ) ∂x ∂y Φe (x − y) − (−1) | x| μ ˜α+β e ∀ α , β ∈ N30

as |x| → ∞,

(80)

|y| ≤ M < ∞,

˜ := (μ1 , μ2 , μ3 ), μ ˜α := where besides standard notation xα and ∂xα we use μ α1 α2 α3 μ1 μ2 μ3 . Applying the above asymptotic relations and taking into account that radiating solutions to the homogeneous equation Me (D)E(x) = 0 in the outer domain Ω− are representable by linear combination of the single and double layer potentials (see Theorem 5.1) we easily derive E(x) =

ei κe |x| x) + O(|x|−2 ), E ∞ ( | x|

∂j E(x) =

x =

x , |x|

(81)

i κe xj ei κe |x| E ∞ ( x) + O(|x|−2 ), j = 1, 2, 3, μj | x| | x|

(82)

x) = (E1, ∞ ( x), E2, ∞ ( x), E3, ∞ ( x)) is the far ﬁeld pattern of the radiwhere E ∞ ( ating vector E, cf. (38). Note that these asymptotic relations can be diﬀerentiated arbitrarily many times as well (cf. (80)): |α| α x i κe ∂ α E(x) − E ∞ ( x) = O(|x|−2 ) ∀ α ∈ N30 . (83) | x| μ ˜α Now we prove the uniqueness theorems for the above-formulated exterior boundary value problems. Theorem 5.3. Let E be a radiating solution to the homogeneous equation Me (D) E = curl μ−1 curl E − ω 2 ε E = 0

(84)

in Ω− satisfying the homogeneous boundary conditions for the “electric”, “magnetic” or “mixed” problems on ∂Ω− , cf. (8a)–(8d). Then E vanishes identically in Ω− . Proof. Let U be a solution of the homogenous exterior “electric”, “magnetic” or “mixed” problem. By Green’s formula (50b) for the domain Ω− R with vectors U = E and V = E, we obtain −1 −1 2 − μ curlE,[ν × E ]dΣR + μ curlE,curlEdx − ω εE,Edx = 0, ΣR

Ω− R

Ω− R

(85)

where ν is the exterior unit normal vector to ΣR . Note that the surface integral over S expires due to the homogenous boundary conditions. Since the matrices

On the Uniqueness of a Solution

161

μ and ε are positive deﬁnite the second and third summands in the left-hand side expression of (85) are real and we conclude (86) Im μ−1 curl E, [ ν × E ] dΣR = 0 . ΣR

In view of (68) the radiation condition (82) can be rewritten as ∂j E(x) =

i κe ei κe |x| ∗ x) + O(|x|−2 ), xj E ∞ ( | x| | x|

j = 1, 2, 3.

(87)

Therefore for suﬃciently large R and for x ∈ ΣR by (71) we have curl E(x) = ∇ × E(x) = =

i κe i κe |x| ∗ e [ x × E ∞ ( x) ] + O(|x|−2 ) | x|2

i κe |x∗ | i κe |x| e [ ν( x) × E ∞ ( x) ] + O(|x|−2 ), | x |2

j = 1, 2, 3. (88)

Take into account the asymptotic formulae (81) and (88) and transform equation (86) i κe |x∗ | −1 Im μ [ ν( x) × E ∞ ( x) ], [ ν( x) × E ∞ ( x) ] dΣR + O(R−1 ) = 0 . (89) | x|3 ΣR

It can be easily veriﬁed that the integrand in (89) does not depend on R. Furtherx| = R for x ∈ ΣR and dΣR = R2 dΣ1 , by more, since μ−1 is positive deﬁnite, | passing to the limit in (89) as R → ∞ we ﬁnally arrive at the relation |x∗ | μ−1 [ ν( x) × E ∞ ( x) ], [ ν( x) × E ∞ ( x) ] dΣ1 = 0 , (90) Σ1

where Σ1 = ∂B1 is the ellipsoidal surface deﬁned by (69) with R = 1 and the −1/2 −1/2 −1/2 integrand is non-negative. Note that |x∗ | ≥ min{μ1 , μ2 , μ3 } > 0 for x ∈ Σ1 in view of (68). Therefore from (90) it follows that x) × E ∞ ( x) ], [ ν( x) × E ∞ ( x) ] = 0 μ−1 [ ν( which implies x) = 0, ν( x) × E ∞ (

i.e.,

x∗ × E ∞ ( x) = 0,

where x∗ is given by (68). Now from (88) we get curl E(x) = O(|x|−2 ),

(91)

which leads to the asymptotic relation ∂ α E(x) = O(|x|−2 ) for arbitrary multi-index α = (α1 , α2 , α3 ),

(92)

due to equation (84) and since we can diﬀerentiate (91) any times with respect to the variables xj , j = 1, 2, 3.

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To show that E vanishes identically in Ω− we proceed as follows. From (41) and (84) it is clear that 2 det Me (D) := κ ω 2 μ1 ∂12 + μ2 ∂22 + μ3 ∂32 + κe2 and det Me (D)E(x) = 0

in Ω− .

Therefore Λ2 (D) E(x) = 0, Λ(D) := μ1 ∂12 + μ2 ∂22 + μ3 ∂32 + κe2 . Let us introduce new variables zk , √ xk = μk zk , and set

k = 1, 2, 3,

√ √ √ E(x) = E( μ1 z1 , μ2 z2 , μ3 z3 ) =: V (z).

It can be easily shown that the components of the vector function V solves the homogeneous equation [ Δ + κe2 ]2 V (z) = 0 for

|z| > R1 ,

where R1 is some positive number and Δ is the Laplace operator. Moreover, in view of (92) we have ∂ α V (z) = O(|z|−2 ) for arbitrary multi-index α. Thus, W (z) := [ Δ + equation and for suﬃciently large |z|

(93) κe2

] V (z) solves the Helmholtz

W (z) = [ Δ + κe2 ] V (z) = O(|z|−2 ), i.e., there holds the equality

lim

A→∞ |z|=A

|W (z)|2 dS = 0.

Therefore, due to the well-known Rellich-Vekua theorem W (z) vanishes identically for |z| > R1 , cf. [Ve1], [CK1], W (z) = [ Δ + κe2 ] V (z) = 0

for |z| > R1 .

Again with the help of the asymptotic behavior (93) and the Rellich-Vekua theorem we conclude that V (z) vanishes for |z| > R1 . In turn this yields that E(x) vanishes for |x| > R2 with some positive number R2 . Since E(x) is real analytic vector function with respect to the real variable x ∈ Ω− , we ﬁnally conclude that E = 0 in Ω− .

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163

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R. Kress, Scattering by obstacles. In: E.R. Pike, P.C. Sabatier (Eds.): Scattering. Scattering and inverse Scattering in Pure and Applied Science. Vol 1, Part 1. Scattering of waves by macroscopic targets, Academic Press, London, 2001, 52–73.

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R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner, Stuttgart, 1986.

[Me1]

C.D. Meyer, Matrix Analysis and Applied Linear Algebra. Book and Solutions Manual, Philadelphia, PA: SIAM, 2000.

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[Na1]

D. Natroshvili, Boundary integral equation method in the steady state oscillation problems for anisotropic bodies, Math. Methods in Applied Sciences 20, No. 2, 1997, 95–119. J.-C. Nedelec, Acoustic and Electromagnetic Equations. Applied mathematical Sciences 114, Springer Verlag, New York, Berlin, Heidelberg, 2001. R.T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88, No.4, 1966, 781–809. H. Triebel, Interpolation Theory, Function Spaces, Diﬀerential Operators, 2nd edition, Johann Ambrosius Barth Verlag, Heidelberg–Leipzig 1995. B.R. Vainberg, Principals of radiation, limiting absorption, and limiting amplitude in the general theory of partial diﬀerential equations, Uspekhi Mat. Nauk 21, No. 3, 1966, 115–194. I. Vekua, On metaharmonic functions, Proc. Tbilisi Mathem. Inst. of Acad. Sci. Georgian SSR 12, 1943, 105–174 (in Russian). C.H. Wilcox, Steady state propagation in homogeneous anisotropic media, Arch. Rat. Mech. Anal. 25, 3, 1967, 201–242.

[Ne1] [Se1] [Tr1] [Va1]

[Ve1] [Wi1]

T. Buchukuri, R. Duduchava and D. Kapanadze Andrea Razmadze Mathematical Institute 1, M. Alexidze str. Tbilisi 0193, Georgia e-mail: t [email protected] [email protected] [email protected] D. Natroshvili Department of Mathematics Georgian Technical University 77 M. Kostava st. Tbilisi 0175, Georgia e-mail: [email protected] Received: February 28, 2009 Accepted: August 24, 2009

Operator Theory: Advances and Applications, Vol. 203, 165–174 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Dichotomy and Boundedness of Solutions for Some Discrete Cauchy Problems Constantin Bu¸se and Akbar Zada In Honor of Israel Gohberg on the occasion of his 80th Birthday

Abstract. Let us denote by Z+ the set of all nonnegative integer numbers. We prove that a square size matrix A of order m having complex entries is dichotomic (i.e., its spectrum does not intersect the set {z ∈ C : |z| = 1}) if and only if there exists a projection P on Cm which commutes with A, and for each number μ ∈ R and each vector b ∈ Cm the solutions of the following two Cauchy problems are bounded:

xn+1 = Axn + eiμn P b, n ∈ Z+ x0 = 0 and

yn+1 = A−1 yn + eiμn (I − P )b,

n ∈ Z+

y0 = 0 . The result is also extended to bounded linear operators acting on arbitrary complex Banach spaces. Mathematics Subject Classiﬁcation (2000). Primary 35B35. Keywords. Stable and dichotomic matrices; discrete Cauchy problem.

1. Introduction It is clear that if a nonzero solution of the scalar diﬀerence equation xn+1 = axn , n ∈ Z+

(a)

is asymptotically stable, then each other solution has the same property, and this happens if and only if |a| < 1 or if and only if for each real number μ and each Communicated by L. Rodman.

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complex number b, the solution of the discrete Cauchy problem

zn+1 = azn + eiμn b, n ∈ Z+ z0 = 0

(a, μ, b, 0)0

is bounded. For a similar problem in the continuous case, see for instance [1], [3], [2], [8] and the references therein. In this note we prove that a m × m matrix A having complex entries is dichotomic, i.e., its spectrum does not intersect the unit circle if and only if for each real number μ and each b ∈ Cm the solutions of two Cauchy problems, like (A, μ, b, 0)0 , are bounded. This result is also extended to bounded linear operators acting on a complex Banach space X. The proofs in the ﬁnite-dimensional case are independent, elementary and use only linear algebra settings. For the general theory of dichotomy of inﬁnite-dimensional systems and its connection with evolution semigroups we refer the reader to the book [5] and the references therein.

2. Preliminary results The discrete Cauchy problem, associated with a square size matrix A of order m is z(n + 1) = Az(n), z(n) ∈ Cm , n ∈ Z+ (A, z0 )0 z(0) = z0 . Obviously the solution of (A, z0 )0 is given by zn = An z0 , where z(n) is denoted by zn . We could know much more about this solution if we have in our hands the eigenvalues of the matrix A, see Theorem 1 below. For now we state two elementary lemmas which are useful later. Lemma 1. The expression Ek (n) = 1k + 2k + · · · + nk , with k a given natural number is a polynomial in n of degree (k + 1). This lemma is well known. Its proof can easily be given by induction on k. We omit the details. In order to state the second lemma, let us denote zn+1 − zn by Δzn , Δ(Δzn ) by Δ2 zn , and so on. Lemma 2. Let N ≥ 1 be a natural number. If ΔN qn = 0 for all n = 0, 1, 2 . . . , then q is a Cm -valued polynomial of degree less than or equal to N − 1. Proof. We argue by induction on N. For N = 1, Δqn = 0 implies that qn+1 − qn = 0, for all n ∈ Z+ , and then qn is a constant polynomial. For N ≥ 2 let us suppose that if ΔN −1 qn = 0, then q is a polynomial of degree less than or equal to N − 2. We shall prove that the same fact is true for N . Indeed, if ΔN qn = 0, then ΔN −1 (Δqn ) = 0. Using

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the induction assumption we get that Δqn is a polynomial of degree less than or equal to N − 2, i.e., Δqn = qn − qn−1 = bN −2 nN −2 + bN −3 nN −3 + · · · + b1 n + b0 = PN −2 (n). Similarly, one has qn−1 − qn−2 = PN −2 (n − 1), qn−2 − qn−3 = PN −2 (n − 2), and ﬁnally, we get q2 − q1 = PN −2 (2). These equalities yield: qn = q1 + PN −2 (2) + PN −2 (3) + · · · + PN −2 (n). Now Lemma 1 implies that qn is a polynomial of degree N − 1 and ends the proof. Let pA be the characteristic polynomial associated with the matrix A and let σ(A) = {λ1 , λ2 , . . . , λk }, k ≤ m, be its spectrum. There exist integer numbers m1 , m2 , . . . , mk ≥ 1 such that pA (λ) = (λ − λ1 )m1 (λ − λ2 )m2 . . . (λ − λk )mk ,

m1 + m2 + · · · + mk = m.

Let j ∈ {1, 2, . . . , k} and Yj := ker(A − λj I) . The next theorem is well known. For its generalization to compact self-adjoint operators we refer the reader to the monograph [6], pages 105–124. mj

Theorem 1. Let A be an invertible m × m matrix. For each z ∈ Cm there exist yj ∈ Yj (j ∈ {1, 2, . . . , k}) such that An z = An y1 + An y2 + · · · + An yk . Moreover, An yj ∈ Yj for all n ∈ Z+ and there exist Cm -valued polynomials qj (n) with deg (qj ) ≤ mj − 1 such that An yj = λnj qj (n),

n ∈ Z+ , j ∈ {1, 2, . . . , k}.

Proof. Indeed, using the Hamilton-Cayley theorem and the well-known fact that ker[pq(A)] = ker[p(A)] ⊕ ker[q(A)], whenever the complex-valued polynomials p and q are relative prime, we obtain the decomposition (1) Cm = Y1 ⊕ Y2 ⊕ · · · ⊕ Yk . Let z ∈ Cm . For each j ∈ {1, 2, . . . , k} there exists a unique yj ∈ Yj such that z = y1 + y2 + · · · + yk and then An z = An y1 + An y2 + · · · + An yk ,

n ∈ Z+ .

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Let qj (n) = λ−n j yj (n). Successively, one has : Δqj (n)

= Δ(λ−n j yj (n)) n = Δ(λ−n j A yj ) −(n+1)

n An+1 yj − λ−n j A yj

−(n+1)

(A − λj I)An yj .

= λj = λj Taking again Δ, we obtain Δ2 qj (n) =

Δ[Δqj (n)] −(n+1)

=

Δ[λj

=

λj

=

λj

(A − λj I)An yj ]

−(n+2)

(A − λj I)A(n+1) yj − λj

−(n+1)

−(n+2)

(A − λj I)2 An yj .

(A − λj I)An yj

−(n+m )

J (A − λj I)mj An yj . But An yj Continuing up to mj , we get Δmj qj (n) = λj mj belongs to Yj for each n ∈ Z+ and thus Δ qj (n) = 0. Using Lemma 2 we can say that the degree of the polynomial qj (n) is less than or equal to mj − 1. The proof is complete.

3. Dichotomy and boundedness Let us denote Γ1 = {z ∈ C : |z| = 1}, Γi := {z ∈ C : | z| < 1}, Γe := {z ∈ C : | z| > 1}. Clearly C = Γ1 ∪ Γi ∪ Γe . A square matrix A of order m is called (i) stable if σ(A) is a subset of Γi or, equivalently, if there exist two positive constants N and ν such that An ≤ N e−νn for all n = 0, 1, 2 . . . , (ii) expansive if σ(A) is a subset of Γe and (iii) dichotomic if σ(A) does not intersect the set Γ1 . It is clear that any expansive matrix A whose spectrum consists of λ1 , λ2 , . . . , λk is an invertible one and its inverse is stable, because ( 1 1 1 −1 ⊂ Γi . , ,..., σ(A ) = λ1 λ2 λk Our ﬁrst result reads as follows. Theorem 2. The matrix A is stable if and only if for each μ ∈ R and each b ∈ Cm the solution of the discrete Cauchy problem

yn+1 = Ayn + eiμn b, n ∈ Z+ (A, μ, b, 0)0 y0 = 0, is bounded.

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Proof. Necessity: Let μ ∈ R and b ∈ Cm . The solution of (A, μ, b, 0)0 is given by yn = [eiμ(n−1) I + eiμ(n−2) A + eiμ(n−3) A2 + · · · + eiμ An−2 + An−1 ]b.

(2)

But e ∈ / σ(A) and thus (e I − A) is an invertible matrix. So equation (2) may be shortened to yn = (eiμ I − A)−1 [(eiμn b − An b)]. (3) Passing to the norm in (3), we get iμ

iμ

yn ≤ (eiμ I − A)−1 b + (eiμ I − A)−1 An b and be applying Theorem 1, we obtain An b = λn1 q1 (n) + λn2 q2 (n) + · · · + λnk qν (n), where q1 , q2 , . . . , qν are some Cm -valued polynomials. The previous representation of An b holds for all n ≥ m. Indeed, if λj = 0 then Anyj = 0 for all n ≥ m. Thus (yn ) is bounded. Suﬃciency: Suppose for the contrary that the matrix A is not stable, i.e., there exists ν ∈ {1, 2, . . . , k} such that |λν | ≥ 1. We are going to consider two cases. Case 1: σ(A)∩Γ1 = ∅. Let λj ∈ σ(A)∩Γ1 and choose μ ∈ R such that λj = eiμ . For each eigenvector b associated to λj , we have that An b = eiμn b. Thus the equation (2) yields: yn = [eiμ(n−1) + eiμ(n−1) + · · · + eiμ(n−1) ]b = neiμ(n−1) b. Therefore, (yn ) is an unbounded sequence and we arrive at a contradiction. Case 2: σ(A) does not intersect Γ1 but it intersects Γe . Let λj ∈ σ(A) ∩ Γe . Having in mind that dim(Yj ) ≥ 1, we may choose b = yj ∈ Yj \ {0}. By applying again Theorem 1, we obtain An b = λnj pj (n), n ∈ Z+ , m pj being a nonzero C -valued polynomial of degree less than or equal to mj − 1. Formula (3) still can be applied because eiμ ∈ / σ(A), and thus, the solution can be written as yn = (eiμ I − A)−1 eiμ b − (eiμ I − A)−1 λnj pj (n), n ∈ Z+ . This representation indicates that (yn ) is an unbounded sequence, being a sum of the bounded sequence given by zn = (eiμ I − A)−1 eiμn b and an unbounded one. Indeed, (eiμ I − A)−1 λnj pj (n) = |λnj |(eiμ I − A)−1 pj (n) → ∞ when n → ∞.

Corollary 1. A square size matrix A of order m is expansive if and only if it is invertible and for each μ ∈ R and each b ∈ Cm the solution of the discrete Cauchy problem 5 yn+1 = A−1 yn + eiμn b, n ∈ Z+ y0 = 0, is bounded.

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Proof. Apply Theorem 2 to the inverse of A.

We recall that a linear map P acting on Cm (or a square size matrix of order m) is called projection if P 2 = P . Theorem 3. The matrix A is dichotomic if and only if there exists a projection P having the property AP = P A such that for each μ ∈ R and each vector b ∈ Cm the solutions of the following two discrete Cauchy problems are bounded,

xn+1 = Axn + eiμn P b, n ∈ Z+ x0 = 0 and

yn+1 = A−1 yn + eiμn (I − P )b,

n ∈ Z+

y0 = 0 . Proof. Necessity: Working under the assumption that A is a dichotomic matrix we may suppose that there exists ν ∈ {1, 2, . . . , k} such that |λ1 | ≤ |λ2 | ≤ · · · ≤ |λν | < 1 < |λν+1 | ≤ · · · ≤ |λk |. Having in mind the decomposition of Cm given in (1), consider X1 = Y1 ⊕ Y2 ⊕ · · · ⊕ Yν ,

X2 = Yν+1 ⊕ Yν+2 ⊕ · · · ⊕ Yk .

Then C = X1 ⊕ X2 . Deﬁne P : C → Cm , by P x = x1 , where x = x1 + x2 , x1 ∈ X1 and x2 ∈ X2 . It is clear that P is a projection. Moreover for all x ∈ Cm and all n ∈ Z+ , we may write m

m

P An x = P (An (x1 + x2 )) = P (An (x1 ) + An (x2 )) = An (x1 ) = An P x, where the fact that X1 is an An -invariant subspace was used. Then P An = An P for all n ∈ Z+ . Now, we have xn = (eiμ I − A)−1 [(eiμn P b − P An b)]. Passing to the norm on both sides in the previous equality, we get xn ≤ (eiμ I − A)−1 P b + (eiμ I − A)−1 P Anb. Now from Theorem 1, it follows P An b = λn1 q1 (n) + λn2 q2 (n) + · · · + λnν qν (n), where q1 , q2 , · · · qν are polynomials. Then the sequence (xn ) is bounded. Our next goal is to prove that the solution of the second Cauchy problem is bounded. We have again yn = (eiμ I − A−1 )−1 [(eiμn (I − P )b − A−n (I − P )b)]. Passing to the norm on both sides of the previous equality, we get yn ≤ (eiμ I − A−1 )−1 (I − P )b + (eiμ I − A−1 )−1 A−n (I − P )b.

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First we prove that A−n v2 → 0 as n → ∞ for any v2 ∈ X2 . Since (I − P )b ∈ X2 the assertion would follow. On the other hand X2 = Yν+1 ⊕ Yν+2 ⊕ · · · ⊕ Yk , so each vector from X2 can be represented as a sum of k − ν vectors yν+1 , yν+2 , . . . , yk . It is suﬃcient to prove that A−n yj → 0, for any j ∈ {ν + 1, . . . , k}. Let Y ∈ {Yν+1 , Yν+2 , . . . , Yk }, say instantly that Y = ker(A − λI)ρ , where ρ ≥ 1 is an integer number and |λ| > 1. Consider w1 ∈ Y \ {0} such that (A − λI)w1 = 0 and let w2 , w3 , . . . , wρ be given by (A − λI)wj = wj−1 , j = 2, 3, . . . , ρ. Then B := {w1 , w2 , . . . , wρ } is a basis in Y. See, for instance, [7]. It is then suﬃcient to prove that A−n wj → 0 for any j = 1, 2, . . . , ρ. For j = 1 we have that A−n w1 = 1 w → 0. For j = 2, 3, . . . , ρ let us denote Xn = A−n wj . Then (A − λI)ρ Xn = 0, λn 1 i.e., (4) Xn − Cρ1 Xn−1 α + Cρ2 Xn−2 α2 + . . . Cρρ Xn−ρ αρ = 0 for all n ≥ ρ, where α = λ1 . Passing to the components in (4) it results that there exists a Cm -valued polynomial Pρ having degree at most ρ − 1 and such that Xn = αn Pρ (n). Thus Xn → 0, when n → ∞, i.e., A−n wj → 0 for any j ∈ {1, 2, . . . , ρ}. Suﬃciency: Suppose for a contradiction that the matrix A is not dichotomic. Then there exists j ∈ {1, 2, . . . , k} such that |λj | = 1. Let b ∈ Cm be a ﬁxed nonzero vector. We are going to analyze also two cases. Case 1: P b = 0. Choose μ ∈ R such that λj = eiμ . Then AP b = eiμ P b and An P b = eiμn P b, which yield xn = [eiμ(n−1) + eiμ(n−1) + · · · + eiμ(n−1) ]P b = neiμ(n−1) P b. Thus (xn ) is an unbounded sequence, in contradiction to the hypothesis. Case 2: P b = 0. In this case (I − P )b = 0. Let μ ∈ R such that λj = e−iμ . Then A−1 (I − A)P b = eiμ (I − A)P b and A−n (I − A)P b = eiμn (I − A)P b, hence yn = [eiμ(n−1) + eiμ(n−1) + · · · + eiμ(n−1) ](I − P )b = neiμ(n−1) (I − P )b. Thus (yn ) is an unbounded sequence. This completes the proof.

We remark that in the enunciation of Theorem 3 we did not impose the condition that the matrix A is invertible. In fact, if viewing A as a map acting on Cm , then A|X2 is an injective map and we may work with the inverse of this restriction instead of the global inverse of A.

4. The case of operators acting on Banach spaces Let X be a complex Banach space. By L(X) we denote the set of all bounded linear operators acting on X. Endowed with the operator norm, L(X) becomes a Banach algebra. Recall that the spectrum of a bounded linear operator A, denoted by σ(A), consists of all complex scalars λ for which λI − A is not an invertible operator.

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With our notation, the result contained in [[4], Theorem 1] may be reformulated as follows. Proposition 1. Let A in L(X). The following three statements concerning on the operator A are equivalent: (i) An → 0 in the norm of L(X). (ii) The spectral radius of A, i.e., 1

1

r(A) := lim ||An || n = inf n≥1 ||An || n = sup{|z| : z ∈ σ(A)}, n→∞

is less than 1. (iii) For each μ ∈ R and each b ∈ X, the solution of (A, μ, b, 0)0 , is bounded. The equivalence between (i) and (ii) is well known. Clearly, the second condition implies the third one. Let us now suppose that the statement (iii) is fulﬁlled. From (2), we have: n−1 (e−iμ A)k b. yn = eiμ(n−1) k=0

Now, the boundedness of the solution (yn ) and the principle of boundedness, yield: # # #n−1 # # −iμ k # (e A) # < ∞. sup # # n≥1 # k=0

The assertion in (ii) follows now directly from [[4], Lemma 1]. We are in the position to state the last result of this note. It reads as follows: Theorem 4. A bounded linear operator A acting on the complex Banach space X is dichotomic if and only if there exists a projection P on X that commutes with A and such that for each real number μ and each vector b ∈ X the solutions of the following two Cauchy problems are bounded,

xn+1 = Axn + eiμn P b, n ∈ Z+ (A, P b, x0 , 0)0 x0 = 0 and

yn+1 = A−1 yn + eiμn (I − P )b, y0 = 0 .

n ∈ Z+

(A−1 , (I − P )b, y0 , 0)0

Proof. Assume that A is dichotomic. Let K1 := {λ ∈ σ(A) : |λ| ≤ 1} and K2 := {λ ∈ σ(A) : |λ| ≥ 1}. Clearly K1 and K2 are compact and disjoint sets. It is well known that there exists a unique pair (X1 , X2 ) of closed subspaces of X having the properties: X = X1 ⊕ X2 ,

AX1 ⊂ X1 and AX2 ⊂ X2 .

Moreover, σ(A) = K1 ∪ K2 and if we denote A1 = A|X1 and A2 = A|X2 , then σ(A1 ) = K1 and σ(A2 ) = K2 . Let P1 be the spectral projection (Riesz projection) corresponding to K1 and P2 the Riesz projection corresponding to K2 . Then

Dichotomy and Boundedness

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P1 + P2 = I, P1 (X) = X1 , and P2 (X) = X2 . We may apply successively Proposition 1 to the pairs (A1 , P1 ) respectively (A−1 2 , P2 ) in order to prove that the Cauchy problems (A, P b, x0 , 0)0 respectively (A−1 , (I − P )b, y0 , 0)0 , with P =: P1 , have bounded solutions. Conversely, if both above Cauchy problems have bounded solutions, then Proposition 1 gives that the restriction of A to the range of P and the restriction of A−1 to the range of I − P have spectral radius less than 1. Hence A is dichotomic. Remark 1. Our result in ﬁnite dimensions, i.e., Theorem 3 above, is more informative then Theorem 4. Indeed, a careful inspection of the proof of Theorem 3 reveals that in the case when the matrix A is not dichotomic, for every projection P that commutes with A and for every nonzero vector b the solution of at least one of the Cauchy problems (A, μ, P b, 0)0 or (A−1 , μ, (I − P )b, 0)0 , with given real number μ, grows at the rate no slower than max{||P b||, ||(I − P )b||} × n, and moreover there exists a projection P commuting with A such that for each nonzero vector b, the solution of at least one of the above two discrete Cauchy problems grows at the rate exactly (constant) × n. Remark 2. Let X be a Banach space and A be a compact linear operator acting on X. Since each nonzero λ ∈ σ(A) is an isolated eigenvalue of ﬁnite multiplicity, the statements from the previous remark remain true in this more general framework. Acknowledgement The authors would like to thank the anonymous referees for their comments and suggestions on preliminary versions of this paper, which have led to a substantial improvement in its readability. In particular, we have completed the last section of this note at the suggestion of referees. The authors thank Professor Leiba Rodman for helpful and useful comments on the second version of this paper.

References [1] C. Bu¸se, D. Barbu, Some remarks about the Perron condition for strongly continuous semigroups, Analele Univ. Timisora, Vol. 35, fasc 1 (1997), 3–8. [2] C. Bu¸se, M. Reghi¸s, On the Perron-Bellman theorem for strongly continuous semigroups and periodic evolutionary processes in Banach spaces, Italian Journal of Pure and Applied Mathematics, No. 4 (1998), 155–166. [3] C. Bu¸se, M.S. Prajea, On Asymptotic behavior of discrete and continuous semigroups on Hilbert spaces, Bull. Math. Soc. Sci. Roum. Tome 51 (99), No. 2 (2008), 123–135. [4] C. Bu¸se, P. Cerone, S.S. Dragomir and A. Sofo, Uniform stability of periodic discrete system in Banach spaces, J. Diﬀerence Equ. Appl. 11, No. 12 (2005), 1081–1088. [5] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Diﬀerential Equations, Amer. Math. Soc., Math. Surv. and Monographs, No. 70(1999). [6] I. Gohberg, S. Goldberg, Basic Operator Theory, Birkh¨ auser, Boston-Basel, 1981.

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[7] P.D. Lax, Linear Algebra, Wiley-Interscience Publication, John Wiley and Sons Inc, (1996). [8] A. Zada, A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems, Electronic Journal of Diﬀerential Equations, No. 94 (2008), 1–5. Constantin Bu¸se West University of Timisoara Department of Mathematics Bd. V. Parvan No. 4 300223 Timisoara, Romania and Government College University Abdus Salam School of Mathematical Sciences (ASSMS) Lahore, Pakistan e-mail: [email protected] Akbar Zada Government College University Abdus Salam School of Mathematical Sciences (ASSMS) Lahore, Pakistan e-mail: [email protected] Received: February 16, 2009 Accepted: March 31, 2009

Operator Theory: Advances and Applications, Vol. 203, 175–193 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Control Laws for Discrete Linear Repetitive Processes with Smoothed Previous Pass Dynamics Bla˙zej Cichy, Krzysztof Galkowski and Eric Rogers Abstract. Repetitive processes are a distinct class of two-dimensional (2D) systems (i.e., information propagation in two independent directions occurs) of both systems theoretic and applications interest. In particular, a repetitive process makes a series of sweeps or passes through dynamics deﬁned on a ﬁnite duration. At the end of each pass, the process returns to the starting point and the next pass begins. The critical feature is that the output on the previous pass acts as a forcing function on, and hence contributes to, the current pass output. There has been a considerable volume of proﬁtable work on the development of a control theory for such processes but more recent applications areas require models with terms that cannot be controlled using existing results. This paper develops substantial new results on a model which contains some of these missing terms in the form of stability analysis and control law design algorithms. The starting point is an abstract model in a Banach space description where the pass-to-pass coupling is deﬁned by a bounded linear operator mapping this space into itself and the analysis is extended to obtain the ﬁrst results on robust control. Mathematics Subject Classiﬁcation (2000). Primary 99Z99; Secondary 00A00. Keywords. Stability analysis, control law design, robustness.

1. Introduction The unique characteristic of a repetitive, or multipass [12], process is a series of sweeps, termed passes, through a set of dynamics deﬁned over a ﬁxed ﬁnite duration known as the pass length. In particular, a pass is completed and then the process is reset before the start of the next one. On each pass, an output, This work has been partially supported by the Ministry of Science and Higher Education in Poland under the project N N514 293235. Communicated by J.A. Ball.

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termed the pass proﬁle, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass proﬁle. This, in turn, leads to the unique control problem where the output sequence of pass proﬁles generated can contain oscillations that increase in amplitude in the pass-to-pass direction. Physical examples of these processes include longwall coal cutting and metal rolling operations [12]. Also in recent years applications have arisen where adopting a repetitive process setting for analysis has distinct advantages over alternatives. Examples of these so-called algorithmic applications include classes of iterative learning control schemes [8] and iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle [9]. In this last case, for example, use of the repetitive process setting provides the basis for the development of highly reliable and eﬃcient solution algorithms and in the former it provides a stability theory which, unlike alternatives, provides information concerning an absolutely critical problem in this application area, i.e., the trade-oﬀ between convergence and the learnt dynamics. Recently iterative learning control algorithms designed in the repetitive process setting have been experimentally tested with results that clearly show how this trade-oﬀ can be treated in this setting [7]. Attempts to control these processes using standard (or 1D) systems theory/algorithms fail (except in a few very restrictive special cases) precisely because such an approach ignores their inherent 2D systems structure. In particular, information propagation occurs from pass-to-pass and along a given pass and also the initial conditions are reset before the start of each new pass. In this paper we study, motivated by physical examples, a model for discrete linear repetitive processes which capture features of the dynamics excluded from those previously studied. The new feature here is that the pass proﬁle at any point on the current pass depends on the complete previous pass proﬁle which has applications relevance since, for example, in longwall coal cutting the machine, weighting up to 5 tonnes, rests on the previous pass proﬁle during the production of the next one. Hence it clear that the complete previous pass proﬁle (weighted contributions from each point along this pass) substantially inﬂuences the pass proﬁle any point on the current pass [13]. Such behavior is sometimes termed inter-pass smoothing [13, 11]. For discrete linear repetitive processes without inter-pass smoothing, it is possible to check stability by using tests developed for 2D discrete linear systems described by Roesser/Fornasini Marchesini [10, 5] state-space models, such as in [4]. This is not possible in the presence of inter-pass smoothing. Preliminary work [6, 2] has also shown that existing algorithms for control law design can only be extended to the case when inter-pass smoothing is present for a weak form of stability that is unlikely to be adequate in many cases. The route is via an equivalent standard, also termed 1D, linear systems state-space model of the repetitive process dynamics but no extension to the case of uncertainty in the process model is possible In this paper we develop a new general approach to the stability analysis and control law design for discrete linear repetitive processes with inter-pass smoothing.

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Starting from the abstract model based stability theory in a Banach space setting we use suitably deﬁned Lyapunov functions to obtain stability conditions that can be computed using Linear Matrix Inequalities (LMIs). These results are then extended to allow the design of control laws, including the case when there is uncertainty associated with the process model. Throughout this paper, the null matrix and the identity 6matrix with the required dimensions are denoted by 0 and I, respectively. Also, (and ⊕) denotes direct sum of matrices and ⊗ denotes the Kronecker product of matrices, M > 0 (< 0) denotes a real symmetric positive (negative) deﬁnite matrix, X ≤ Y is used to represent the case when X − Y is a negative semi-deﬁnite matrix, and denotes a block entry in a symmetric matrix. The analysis in this paper will make extensive use of the well-known Schur’s complement formula for matrices and the following result. Lemma 1.1. [4] Given matrices X, Σ1 , Σ2 of compatible sizes, suppose that there is an > 0 so that X + −1 Σ1 ΣT1 + ΣT2 Σ2 < 0. Then there exist a matrix F with F T F ≤ I so that X + Σ1 F Σ2 + ΣT2 F T ΣT1 < 0.

(1.1)

2. Preliminaries and the new model Consider the case of discrete dynamics along the pass and let α < ∞ denote the pass length and k ≥ 0 the pass number or index. Then such processes evolve over the subset of the positive quadrant in the 2D plane deﬁned by {(p, k) : 0 ≤ p ≤ α − 1, k ≥ 0}, and most basic state-space model for their dynamics has [12] the following form xk+1 (p + 1) = Axk+1 (p) + Buk+1 (p) + B0 yk (p) yk+1 (p) = Cxk+1 (p) + Duk+1 (p) + D0 yk (p).

(2.1)

Here on pass k, xk (p) ∈ Rn is the state vector, yk (p) ∈ Rm is the pass proﬁle vector, and uk (p) ∈ Rr is the vector of control inputs. This state-space model has strong similarities with the well-known Givone-Roesser and Fornasini-Marchesini state-space models for 2D discrete linear systems. This means that some, but by no means all, systems theoretic questions for discrete linear repetitive processes described by this state-space model can be solved by exploiting these similarities. There are, however, important systems theoretic questions for these processes which cannot be answered in this way. For example, so-called pass proﬁle controllability requiring that processes described by (2.1) produce a pre-deﬁned pass proﬁle vector either on some pass or with the pass number also pre-deﬁned has no 2D Givone-Roesser or Fornasini-Marchesini state-space model interpretation. A comprehensive discussion of this general area can be found in [11] and the relevant cited references.

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In order to complete the process description it is necessary to specify the boundary conditions, that is, the pass state initial vector sequence and the initial pass proﬁle and the simplest form of these is xk+1 (0) = dk+1 , k ≥ 0 y0 (p) = f (p),

(2.2)

0 ≤ p ≤ α − 1,

where the n× 1 vector dk+1 has known constant entries and f (p) is an m× 1 vector whose entries are known functions of p. The stability theory [12, 11] for linear repetitive processes is based on an abstract model in a Banach space setting which includes a wide range of examples as special cases, including those described by (2.1) and (2.2). In terms of their dynamics it is the pass-to-pass coupling (noting again their unique feature) which is critical. This is of the form yk+1 = Lα yk , where yk ∈ Eα (Eα a Banach space with norm || · ||) and Lα is a bounded linear operator mapping Eα into itself. (In the case considered here Lα is a discrete convolution operator.) Stability is then deﬁned in bounded-input bounded-output terms and characterized in terms of properties of Lα . This has two forms termed asymptotic and along the pass respectively where the former demands this property over the ﬁnite and ﬁxed pass length α for a given example and the latter for all possible pass lengths. The structure of the boundary conditions and, in particular, the state initial vector sequence {xk+1 (0)}k≥0 is critical to the stability properties of the example considered since, unlike other classes of linear systems, these alone can cause instability. For example, if xk+1 (0) is a function of points along the previous pass, such as xk+1 (0) = dk+1 + K1 yk (α − 1) where K1 is an n × m matrix, then [11] an example which is stable with K1 = 0 could be unstable when K1 = 0. In applications there is therefore a critical need to adequately model this sequence. Inter-pass smoothing arises in the longwall coal cutting application since the cutting machine rests on the previous pass proﬁle as it cuts or machines the next pass proﬁle. On any pass the dynamics of the cutting machine in the along the pass direction can be approximated by a diﬀerence equation but, as the machines used in this application area can be up to 5 tonnes in weight, it is unrealistic to assume that at any point along the current pass the only previous pass proﬁle contribution is from a single point as in the model of (2.1). One alternative in this case is to use a model [2] of the following form over k ≥ 0 and 0 ≤ p ≤ α − 1 xk+1 (p + 1) = Axk+1 (p) + Buk+1 (p) +

α−1

Bl yk (l)

l=0

yk+1 (p) = Cxk+1 (p) + Duk+1 (p) +

α−1

(2.3) Dl yk (l)

l=0

with the same notation and boundary conditions as (2.1). In this last model, the inﬂuence of the previous pass proﬁle vector yk (l); l = 0, 1, . . . , α − 1 is the same at all points along the current pass and this again could

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be an inadequate, certainly in the longwall coal cutting example. An alternative model is considered here of the form α−1 Bi yk (i) + Eyk (p) xk+1 (p + 1) = Axk+1 (p) + Buk+1 (p) + i=0

yk+1 (p) = Cxk+1 (p) + Duk+1 (p) +

α−1

(2.4) Di yk (i) + F yk (p)

i=0

with again the same notation and boundary conditions as (2.1) and (2.2).

3. Stability analysis It is routine to show that the model of (2.4) can be written in the abstract model form. In particular, as noted in the previous section, the pass-to-pass coupling can be written in the form yk+1 = Lα yk , where yk ∈ Eα (Eα a Banach space with norm || · ||) and Lα is a bounded linear operator mapping Eα into itself. The method is a routine example of the construction in [11] for the case of processes described by (2.1) with Eα = m 2 [0, α], that is, the space of all real m × 1 vectors of length α, and hence the details are omitted here. As noted in the previous section, the stability theory consists of two forms where, with ||·|| denoting the induced operator norm, asymptotic stability demands the existence of ﬁnite real scalars Mα > 0 and λα ∈ (0, 1) such that ||Lkα || ≤ Mα λkα , k ≥ 0, which, in turn, is equivalent to r(Lα ) < 1 where r(·) denotes the spectral radius. Also if this property holds the strong limit y∞ := limk→∞ yk is termed the limit proﬁle and is the unique solution of the linear equation y∞ = Lα y∞ + b∞ . Moreover, asymptotic stability can be interpreted as a form of bounded-input bounded-output stability over the ﬁnite and constant pass length for the example considered. In the case of processes described by (2.1) and (2.2) it is known [12, 11] that asymptotic stability holds if, and only if, r(D0 ) < 1 and that the resulting limit proﬁle is described by a 1D discrete linear system state-space model with state matrix A + B0 (I − D0 )−1 C and hence can be unstable as the simple case when A = −0.5, B0 = 0.5 + β, C = 1, D = 0, and D0 = 0, where β is a real scalar with |β| ≥ 1, demonstrates. To prevent examples such as the one given above from arising, stability along the pass demands the bounded-input bounded-output property for all possible values of the pass length. This requires the existence of ﬁnite real scalars M∞ > 0 and λ∞ ∈ (0, 1), which are independent of α, such that ||Lkα || ≤ M∞ λk∞ , k ≥ 0. In the case of processes described by (2.1) and (2.2), the resulting abstract model based conditions can be reﬁned to ones that can be tested by direct application of 1D linear systems tests. Such tests, however, do not easily extend to control law design and one alternative is to use a Lyapunov function approach.

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In the case of processes described by (2.1) and (2.2), introduce the Lyapunov function as V = V1 (k, p) + V2 (k, p), (3.1) where V1 (k, p) =

xTk+1 (p)P1 xk+1 (p)

(3.2)

V2 (k, p) =

ykT (p)P2 yk (p)

(3.3)

with P1 > 0 and P2 > 0. Deﬁne also the associated increment as ΔV (k, p) = V1 (k, p + 1) − V1 (k, p) + V2 (p, k + 1) − V2 (p, k).

(3.4)

Then we have the following result [11]. Theorem 3.1. A discrete linear repetitive process described by (2.1) and (2.2) is stable along the pass if ΔV (k, p) < 0 (3.5) holds for all possible values of the pass length. Note that the structure of the Lyapunov function here is a measure of the process energy as the sum of quadratic terms in the current pass state and previous pass proﬁle vectors respectively. The result here states that stability along the pass requires that this energy decreases from pass-to-pass. It can also be interpreted as the repetitive process version of quadratic stability [1]. In the case of processes described by (2.4) and (2.2) consider matrices Qi > 0, i = 0, 1, . . . , α−1, and Pi > 0, i = 0, 1, . . . , α, and introduce the Lyapunov function V (k) = V1 (k) + V2 (k),

(3.6)

where V1 (k) =

α−1

ykT (i)Qi yk (i)

(3.7)

xTk (i)Pi xk (i),

(3.8)

i=0

and V2 (k) =

α−1 i=0

The term V1 (k) captures the pass-to-pass energy change and V2 (k) the change in energy along a pass. Also introduce V2 (k) =

α

xTk (i)Pi xk (i).

(3.9)

i=1

Then the associated increment for the Lyapunov function here is ΔV (k) = V1 (k + 1) − V1 (k) + V2 (k + 1) − V2 (k + 1)

(3.10)

and the proof of the result given next follows by routine extensions to that for Theorem 3.1 and hence the details are omitted.

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Theorem 3.2. A discrete linear repetitive process described by (2.4) and (2.2) is stable along the pass if ΔV (k) < 0 (3.11) for all possible values of the pass length. Note that if Pi = P,

i = 0, 1, . . . , α

(3.12)

then V2 (k+1)−V2 (k+1) = xTk (α)P xk (α)−xTk (0)P xk (0), that is, the diﬀerence between the current pass state energy at the start and end of the pass. A similar interpretation holds for the pass-to-pass energy change when Qi = Q, i = 0, 1, . . . , α−1. To develop a computationally feasible test for the condition of Theorem 3.2, introduce = I ⊗ A, C = I ⊗ C, E = I ⊗ E, F = I ⊗ F A = Q

α−1 6

Qi ,

P1 =

i=0

and also

⎡

B0 ⎢ .. B=⎣ . B0

··· .. . ···

α−1 6

Pi ,

P2 =

i=0

⎤ Bα−1 .. ⎥ , . ⎦ Bα−1

α 6

Pi

(3.13)

i=1

⎡ D0 ⎢ .. D=⎣ . D0

··· .. . ···

⎤ Dα−1 .. ⎥ . . ⎦

(3.14)

Dα−1

Then we have the following result. Theorem 3.3. A discrete linear repetitive process described by (2.4) and (2.2) is stable along the pass if ∃ matrices Qi > 0, i = 0, 1, . . . , α − 1, and Pi > 0, i = 0, 1, . . . , α, such that the following LMI holds +C T Q C − P1 T P2 A A T C (B + E) P2 A + (D + F )T Q (3.15) + E) +C T Q( D + F ) T P2 (B A + E) + (D + F)T Q( D + F) − Q < 0. + E) T P2 (B (B Proof. It is straightforward to show, using the process state-space model, that the condition of Theorem 3.2 is equivalent to the LMI of this theorem and hence the details are omitted. The previous result cannot be used in the case when there is uncertainty associated with the process state-space model since the resulting stability condition would not be in LMI form. The following result can, however, be used in such cases. Theorem 3.4. [3] A discrete linear repetitive process described by (2.4) and (2.2) ˜ i > 0, i = 0, 1, . . . , α − 1, and P˜i > 0, is stable along the pass if ∃ matrices Q

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i = 0, 1, . . . , α, such that the following LMI holds −P1 P1 AT < 0, AP1 −P2 where

(3.16)

¯ P1 = P¯1 ⊕ Q,

B +E A A= C D + F

¯ P2 = P¯2 ⊕ Q,

(3.17)

and ¯= Q

α−1 6

˜ i, Q

P¯1 =

i=0

α−1 6

P¯2 =

P˜i ,

i=0

α 6

P˜i .

(3.18)

i=1

Proof. Follows immediately on applying the Schur’s complement formula and appropriate congruence transforms and change of decision variables to the result of the previous theorem.

4. Stabilization The structure of repetitive processes means that control laws for them need, in almost all cases, to include contributions from both the current and previous pass data. For the processes with the inter-pass smoothing considered here one such law has the following form over k ≥ 0 and 0 ≤ p ≤ α − 1 uk+1 (p) = Kx (p)xk+1 (p) +

α−1 i=0

= Kx (p) K0 · · · Kα−1

Ki yk (i) + Ky (p)yk (p) ⎡

xk+1 (p) yk (0) .. .

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Ky (p) ⎢ ⎥, ⎥ ⎢ ⎣yk (α − 1)⎦ yk (p)

(4.1)

where the matrix functions Kx (p) and Ky (p) depend on the position p along the pass, 0 ≤ p ≤ α − 1. For processes described by (2.1), this control law simpliﬁes to uk+1 (p) = K1 xk+1 (p) + K2 yk (p). The controlled process state-space model after applying the control law (4.1) to (2.4) is α−1 . / . / . / Bi + BKi yk (i) + E + BKy (p) yk (p) xk+1 (p + 1) = A + BKx (p) xk+1 (p) + i=0

. / yk+1 (p) = C + DKx (p) xk+1 (p) +

α−1

.

/ . / Di + DKi yk (i) + F + DKy (p) yk (p).

i=0

(4.2)

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183

To apply the result of Theorem 3.4, introduce the following notation = I ⊕ B, B α−1 6

x = K

Kx (p),

y = K

p=0

x = N

α−1 6

=I ⊕D D α−1 6

Ky (p),

(4.3)

= K

p=0

Nx (p),

y = N

p=0

α−1 6

B

··· .. . ···

⎤ B .. ⎥ , .⎦ B

Ki

(4.4)

Ni

(4.5)

i=0

Ny (p),

= N

p=0

⎡ B ⎢ .. ¯ B=⎣.

α−1 6

α−1 6 i=0

⎡ D ⎢ .. ¯ D=⎣. D

··· .. . ···

⎤ D .. ⎥ .⎦

(4.6)

D

together with +B K x, AK = A

+B K y , EK = E

+B ¯K BK = B

+D K x, CK = C

K y , FK = F + D

+D ¯K DK = D

(4.7)

and +B ¯K +E +B K y X = BK + EK = B +D ¯K + F + D K y. Y = DK + F K = D The result of applying (3.16) to this case can be written as ⎡ ⎤ −P¯1 ¯ ⎢ 0 −Q ⎥ ⎢ ⎥<0 ¯ −P¯2 ⎣AK P¯1 X Q ⎦ ¯ ¯ CK P¯1 Y Q 0 −Q or

⎡

−P¯1 ⎢ 0 ⎢ K x P¯1 P¯1 + B ⎣A K x P¯1 P¯1 + D C

¯ −Q ¯ ¯ ¯ Q ¯+B K yQ ¯ BQ + BK Q + L Q ¯+D ¯K Q ¯ + F Q ¯+D K yQ ¯ D

−P¯2 0

⎤ ⎥ ⎥ < 0. ⎦ ¯ −Q

(4.8)

(4.9)

(4.10)

This condition is not in LMI form, and hence there are no eﬀective computational methods available to compute the control law matrices. The following result removes this diﬃculty. Theorem 4.1. Suppose that a control law of the form (4.1) is applied to a discrete linear repetitive process described by (2.4) and (2.2). Then the resulting controlled ˜ i > 0, i = 0, 1, . . . , α − 1, P˜i > 0, process is stable along the pass if ∃ matrices Q

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i = 0, 1, . . . , α, Nx (p), Ny (p), and Ni , p, i = 0, 1, . . . , α − 1, such that the following LMI holds ⎡

−P¯1 ⎢ 0 ⎢ N x P¯1 + B ⎣A P¯1 + D N x C

¯ −Q Q ¯+B ¯N +L Q ¯ +B N y B Q ¯+D ¯N + F Q ¯+D N y D

−P¯2 0

⎤ ⎥ ⎥ < 0. 0 ⎦ ¯ −Q

(4.11)

If this condition holds, stabilizing control law matrices are given by x P¯ −1 , x = N K 1

y = N y Q ¯ −1 , K

=N Q ¯ −1. K

(4.12)

Proof. Follows immediately on noting that substituting x P¯1 = N x , K

y Q ¯=N y , K

Q ¯=N K

(4.13)

into (4.10) yields (4.11).

5. Robustness In this section we consider the case when there is uncertainty associated with the process model. A natural place to begin work in this area is to impose an uncertainty structure on the matrices which deﬁne the state-space model of the process under consideration. One such case is when the uncertainty is modeled as additive perturbations to the nominal state-space model matrices. First, introduce the notation B ¯ B 0 B B = B 1 B2 , B 1 = , B 2 = (5.1) ¯ . 0 D D D Then it is assumed that the matrices A, given by (3.17), and B constructed from process state-space model matrices are perturbed by additive terms as follows Ap = A + ΔA, where

+ ΔA 7 A ΔA = 7 C + ΔC

Bp = B + ΔB,

+ ΔB 8 +E + ΔE 7 B 8 7 D + ΔD + F + ΔF

(5.2)

ΔB = ΔB1 ΔB2 + ΔB 7 B + ΔB 7 ¯ + ΔB B 0 B ΔB1 = , ΔB2 = ¯ + ΔD . + ΔD 7 D + ΔD 7 0 D D

(5.3) (5.4) (5.5)

with 7 = I ⊗ ΔA, ΔA

7 = I ⊗ ΔC, ΔC

7 = I ⊗ ΔE ΔE

7 = I ⊗ ΔF, ΔF

7 = I ⊗ ΔB, ΔB

7 = I ⊗ ΔD ΔD

(5.6)

Control Laws for Repetitive Processes and also

⎡ ΔB0 ⎢ .. 8 ΔB =⎣ .

··· .. . ···

ΔB0 ⎡ ΔB ⎢ ΔB = ⎣ ... ΔB

⎤ ΔBα−1 .. ⎥ , . ⎦

⎡ ΔD0 ⎢ .. 8 ΔD =⎣ .

ΔBα−1

ΔD0 ⎡ ΔD ⎢ ΔD = ⎣ ...

··· .. . ···

⎤

ΔB .. ⎥ , . ⎦ ΔB

ΔD

··· .. . ··· ··· .. . ···

185 ⎤ ΔDα−1 .. ⎥ . ⎦

(5.7)

ΔDα−1 ⎤ ΔD .. ⎥ . ⎦

(5.8)

ΔD

with ΔA = H1 Z1 VA ,

ΔC = H2 Z2 VC

ΔB = H1 Z1 VB , ΔE = H1 Z1 VE ,

ΔD = H2 Z2 VD ΔF = H2 Z2 VF

ΔBi = H1 Z1 VBi ,

ΔDi = H2 Z2 VDi ,

(5.9)

i = 0, . . . , α − 1.

Moreover, the unknown matrices Z1 and Z2 (with compatible dimensions) are required to satisfy Z1T Z1 ≤ I, Now write

V = V1

(5.10)

ΔB = HZV,

ΔA

where

Z2T Z2 ≤ I.

(5.11)

V A V B + VE V 2 , V1 = , V2 = V21 VC VD + VF VB VB 0 VB¯ , V22 = V21 = VD VD 0 VD¯

V22

H = H1 ⊕ H2 = (I ⊗ H1 ) ⊕ (I ⊗ H2 ) Z = Z1 ⊕ Z2 = (I ⊗ Z1 ) ⊕ (I ⊗ Z2 ) VA = I ⊗ VA , VC = I ⊗ VC , VE = I ⊗ VE , VF = I ⊗ VF VB = I ⊗ VB , and

⎡ VB0 ⎢ .. VB = ⎣ .

··· .. . ···

VB0 ⎡ VB ⎢ .. VB¯ = ⎣ . VB

⎤ VBα−1 .. ⎥ , . ⎦ VBα−1 ⎤ · · · VB .. ⎥ , .. . . ⎦ ···

VB

VD = I ⊗ VD ⎡

VD0 ⎢ .. VD = ⎣ .

··· .. . ···

VD0 ⎡ VD · · · ⎢ .. .. VD¯ = ⎣ . . VD · · ·

⎤ VDα−1 .. ⎥ . ⎦ VDα−1 ⎤

VD .. ⎥ . . ⎦ VD

Then it follows immediately that Z T Z ≤ I.

(5.12)

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Now we can apply Theorem 3.4 to conclude that stability along the pass holds in this case provided ∃ matrices P1 > 0 and P2 > 0 of the from given in (3.17) such that −P1 P1 (A + HZV1 )T < 0. (5.13) (A + HZV1 )P1 −P2 The diﬃculty now is that Z has unknown entries and hence it is not applicable as a computable stability test. To remove this diﬃculty, we have the following result as an obvious consequence of Lemma 1.1. Theorem 5.1. A discrete linear repetitive processes described by (2.4) and (2.2) with uncertainty of the form deﬁned by (5.2)–(5.10) is stable along the pass if ∃ matrices Q i > 0, i = 0, 1, . . . , α − 1, and Pi > 0, i = 0, 1, . . . , α, and a real scalar > 0 such that the following LMI holds ⎡ ⎤ 1 AT V1T −P 0 ⎢ A 2 −P 0 H ⎥ ⎢ ⎥ < 0, (5.14) ⎣ V1 0 −I 0 ⎦ 0 HT 0 −I where 1 = Pˆ1 ⊕ Q, ˆ P

2 = Pˆ2 ⊕ Q ˆ P

(5.15)

and ˆ= Q

α−1 6

Q i ,

Pˆ1 =

i=0

α−1 6

Pi ,

Pˆ2 =

i=0

α 6

Pi .

(5.16)

i=1

Proof. First pre- and post-multiply (5.13) by P1−1 ⊕ I to obtain −P1−1 (A + HZV1 )T < 0. A + HZV1 −P2 Use of Lemma 1.1 with the substitutions 0 F → Z, Σ1 → , Σ2 → V1 H1 yields that

−P1−1 A

T V1 V1 AT + 0 −P2

0

, X→

−P1−1 A

AT −P2

0 < 0 for some > 0 −1 HHT

(5.17)

is suﬃcient for the existence of a Z satisfying (5.12) and (5.13). Further, (5.17) can be written in the form T V1 0 0 V1 0 −1 I −P1−1 AT + < 0. (5.18) 0 HT 0 H 0 −1 I A −P2

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187

Applying the Schur’s complement formula to (5.18) now gives ⎡ ⎤ 0 −P1−1 AT V1T ⎢ A −P2 0 H ⎥ ⎢ ⎥ < 0. ⎣ V1 0 −I 0 ⎦ 0 HT 0 −I

(5.19)

Obvious substitutions now yield (5.14) and the proof is complete.

Suppose now that a control law of the form (4.1) is applied to this uncertain process model. Then routine manipulations show that the resulting controlled process state-space model is of the form to which Theorem 3.4 can be applied. Hence we have that this process is stable along the pass if there exists matrices P1 > 0 and P2 > 0 as deﬁned in (3.17) such that −P1 % $ < 0, (5.20) (A + HZV1 ) + (B + HZV2 )K P1 −P2 where K1 , K= K2

x K K1 = 0

0 y , K

K2 =

0 0 0 K

(5.21)

x, K y and K deﬁned in (4.4). with K The remaining diﬃculty with (5.20) is that the matrix Z has unknown entries and hence it is not in the form of a computable stability test. To remove this diﬃculty, we have the following result that is again an obvious consequence of Lemma 1.1. Theorem 5.2. Suppose that a control law of the form (4.1) is applied to a discrete linear repetitive processes described by (2.4) and (2.2) with uncertainty of the form deﬁned by (5.2)–(5.10). Then the resulting controlled process is stable along the ˜ i > 0, pass under the action of a control law of the form (4.1) if ∃ matrices Q i = 0, 1, . . . , α − 1, P˜i > 0, i = 0, 1, . . . , α, Nx (p), Ny (p), and Ni where p, i = 0, 1, . . . , α − 1, and a real scalar > 0, such that the following LMI holds ⎡ ⎤ −P1 ⎢ AP1 + BN −P2 ⎥ ⎢ ⎥ < 0, (5.22) ⎣V1 P1 + V2 N 0 −I ⎦ T 0 H 0 −I where the matrices P1 , P2 and A are deﬁned in (3.17), the matrix B is deﬁned in (5.1), and x 0 0 0 N1 N N = , N1 = (5.23) , y , N2 = 0 N N2 0 N

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y and N are deﬁned in (4.5). If (5.22) holds, stabilizing control law x , N where N matrices are given by K = N P1−1 ,

(5.24)

where K is deﬁned in (5.21). Proof. First rewrite (5.20) as −P1 % $ A + HZV1 + BK + HZV2 K P1

−P2

< 0.

Now pre- and post-multiply this last expression by P1−1 ⊕ I to obtain −P1−1 <0 A + HZV1 + BK + HZV2 K −P2 or 0 −P1−1 + <0 HZ(V1 + V2 K) 0 A + BK −P2 and hence 0 0 Z 0 V1 + V 2 K 0 −P1−1 + H 0 0 Z 0 0 A + BK −P2 T T T T 0 0 HT V + K V2 0 Z < 0. + 1 0 0 0 ZT 0 0 Lemma 1.1 can now be applied to obtain 0 0 0 HT −P1−1 + H 0 0 0 A + BK −P2 T V + KT V2T 0 V1 + V2 K + −1 1 0 0 0

0 <0 0

for some > 0. Consequently −1 0 (V1 + V2 K)T (V1 + V2 K) −P1−1 + <0 0 HHT A + BK −P2 and it is easy to show that (5.30) can be written as 0 (V1 + V2 K)T −P1−1 + 0 H A + BK −P2 −1 V1 + V 2 K I 0 0 −1 I 0

0 < 0. HT

Now we can apply the Schur’s complement formula to (5.31) and obtain ⎤ ⎡ −P1−1 ⎢ A + BK −P2 ⎥ ⎥ < 0. ⎢ ⎣V1 + V2 K 0 −I ⎦ 0 HT 0 −I

(5.25)

(5.26)

(5.27)

(5.28)

(5.29)

(5.30)

(5.31)

(5.32)

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189

Next, pre- and post-multiply (5.32) by P1 ⊕ I ⊕ I ⊕ I to obtain ⎡

−P1 ⎢ AP1 + BKP1 ⎢ ⎣V1 P1 + V2 KP1 0

−P2 0 HT

−I 0

⎤ ⎥ ⎥ < 0. ⎦ −I

(5.33)

Substituting N = KP1

(5.34)

now gives (5.22) and the proof is complete.

6. Numerical example Consider the case of (2.4) when 0.259 0.279 −0.526 0.716 0.571 , B0 = ,B= A= , B1 = −0.074 0.296 0.584 −0.659 −0.008 −0.398 −0.051 −0.616 −0.914 , B3 = , B4 = , B5 = B2 = 0.059 −0.974 0.205 −0.295 0.512 0.897 0.908 0.762 0.796 B6 = , B7 = , B8 = , B9 = ,E= 0.674 0.996 −0.309 0.007 0.931 C = −0.271 0.353 , D = 0.753, D0 = 0.143, D1 = 0.973, D2 = 0.11 D3 = −0.766, D4 = 0.696, D5 = 0.551, D6 = 0.188, D7 = −0.184 D8 = −0.065, D9 = −0.362, F = −0.324 with the following matrices deﬁning the uncertainty structure 0.064 0.174 0.092 0.054 0.171 , VB = , VB0 = , VB1 = 0.118 0.14 0.094 0.046 0.188 0.003 0.02 0.159 0.011 , VB3 = , VB4 = , VB5 = VB2 = 0.054 0.067 0.134 0.005 0.166 0.003 0.014 0.18 0.048 VB6 = , VB7 = , VB8 = , VB9 = , VE = 0.165 0.043 0.166 0.035 0.171 VC = 0.07 0.196 , VD = 0.153, VD0 = 0.138, VD1 = 0.197, VD2 = 0.09 VD3 = 0.188, VD4 = 0.108, VD5 = 0.046, VD6 = 0.072, VD7 = 0.033

VA =

VD8 = 0.173, VD9 = 0.199, VF = 0.101

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Figure 1. Pass proﬁles generated by the uncontrolled process with zero input on each pass.

and 0.08 0.104 , 0.084 0.07

H1 =

H2 = 0.035

with boundary conditions T xk+1 (0) = 1 1 , y0 (p) = 1,

k≥0 0 ≤ p ≤ α − 1.

(6.1)

The response of the uncontrolled process is shown in Figure 1 with zero control input. Clearly this example is not stable along the pass. Application of Theorem 5.2 does provide a stabilizing control law but this almost zeros the current pass state vector matrix in the controlled process with 0.999 0 , Z2 = 0.999. Z1 = 0 0.999 The result is dead-beat type control and if this is not acceptable an alternative is to extend a well known result for the 1D systems case [14] to the repetitive process setting as follows. Lemma 6.1. Suppose that a control law of the form (4.1) is applied to a discrete linear repetitive processes described by (2.4) and (2.2) with uncertainty of the form deﬁned by (5.2)–(5.10). Then the controlled process is stable along the pass if ∃ ˜ i > 0, i = 0, 1, . . . , α − 1, P˜i > 0, i = 0, 1, . . . , α, Nx (p), Ny (p), and Ni matrices Q where p, i = 0, 1, . . . , α − 1, and real scalars > 0, κ1 > 0 and κ2 > 0 such that

Control Laws for Repetitive Processes

191

the following generalized optimization problem has solutions minimize κ1 + κ2 − subject to ⎤ ⎡ −P1 ⎢ AP1 + BN −P2 ⎥ ⎥<0 ⎢ ⎣V1 P1 + V2 N 0 −I ⎦ 0 HT 0 −I T κ2 I I −κ1 I N < 0, ¯ > 0, I Q N −I

(6.2)

where the matrices P1 , P2 and A are deﬁned in (3.17), the matrix B in (5.1), and the matrix N in (5.23). If (6.2) holds, stabilizing matrices in the control law of (4.1) are given by (5.24). Applying this last result yields the following control κ2 − = −1.7199 × 105 Kx (0) = 0.359 −0.4696 , Kx (1) = 0.3599 Kx (2) = 0.3587 −0.4698 , Kx (3) = 0.3598 Kx (4) = 0.3598 −0.4688 , Kx (5) = 0.3597 Kx (6) = 0.3592 −0.4693 , Kx (7) = 0.3592 Kx (8) = 0.358 −0.4711 , Kx (9) = 0.3596

law matrices for κ1 + −0.4688 −0.4688 −0.4689 −0.4694 −0.469

and Ky (0) = −0.6025, Ky (1) = 0.0485, Ky (2) = −0.5008, Ky (3) = 0.9212 Ky (4) = −0.127, Ky (5) = −0.2465, Ky (6) = −0.5865, Ky (7) = 1.2558 Ky (8) = 0.9956, Ky (9) = 1.2069 and also K0 = −0.1717, K1 = −0.9185, K2 = −0.1339, K3 = 0.8108, K4 = −0.7588 K5 = −0.6286, K6 = −0.225, K7 = 0.224, K8 = 0.0819, K9 = 0.4294. Figure 2 shows the controlled process response and Figure 3 the corresponding pass control inputs.

7. Conclusions and further work Control law design algorithms have been developed for discrete linear repetitive processes with inter-pass smoothing eﬀects. The resulting algorithms can be computed using LMIs and an illustrative example has been given. Ongoing work includes replacing the current pass state vector component in the control law by a current pass proﬁle term since the law used in this paper would require an observer unless all current pass state vector terms are directly measurable.

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Figure 2. Pass proﬁles generated by the controlled process.

Figure 3. Control inputs used to generate the pass proﬁles of Figure 2.

References [1] S. Boyd, L.E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, volume 15 of SIAM Studies in Applied Mathematics. SIAM, Philadelphia, 1994. [2] B. Cichy, K. Galkowski, E. Rogers, and A. Kummert. Discrete linear repetitive process with smoothing. In The Fifth International Workshop on Multidimensional Systems (NDS 07), Aveiro, Portugal, 2007. [3] B. Cichy, K. Galkowski, E. Rogers, and A. Kummert. Stability of a class of 2D linear systems with smoothing. In Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications, pages 47–52, Xi’an, China, 25–27 May, 2009. [4] C. Du and L. Xie. Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46:1371–1374, 1999.

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[5] E. Fornasini and G. Marchesini. Doubly indexed dynamical systems: state-space models and structural properties. Mathematical System Theory, 12:59–72, 1978. [6] K. Galkowski, E. Rogers, S. Xu, J. Lam, and D.H. Owens. LMIs – a fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(6):768–778, 2002. [7] L . Hladowski, Z. Cai, K. Galkowski, E. Rogers, C.T. Freeman, and P.L. Lewin. Using 2D systems theory to design output signal based iterative learning control laws with experimental veriﬁcation. In Proceedings of the 47th IEEE Conference on Decision and Control, pages 3026–3031, Cancun, Mexico, December 2008. [8] D.H. Owens, N. Amann, E. Rogers, and M. French. Analysis of linear iterative learning control schemes – a 2D systems/repetitive processes approach. Multidimensional Systems and Signal Processing, 11(1/2):125–177, 2000. [9] P.D. Roberts. Numerical investigations of a stability theorem arising from 2dimensional analysis of an iterative optimal control algorithm. Multidimensional Systems and Signal Processing, 11 (1/2):109–124, 2000. [10] R.P. Roesser. A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, AC-20:1–10, 1975. [11] E. Rogers, K. Galkowski, and D.H. Owens. Control Systems Theory and Applications for Linear Repetitive Processes, volume 349 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 2007. [12] E. Rogers and D.H. Owens. Stability Analysis for Linear Repetitive Processes, volume 175 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 1992. [13] E. Rogers and D.H. Owens. Stability theory and performance bounds for a class of two-dimensional linear systems with interpass smoothing eﬀects. IMA Journal of Mathematical Control and Information, 14:415–427, 1997. ˇ [14] D.D. Siljak and D.M. Stipanovi´c. Robust stabilisation of nonlinear systems: The LMI approach. Mathematical Problems in Engineering, 6:461–493, 2000. Bla˙zej Cichy and Krzysztof Galkowski Institute of Control and Computation Engineering University of Zielona Gora ul. Podg´ orna 50 65-246 Zielona G´ ora, Poland e-mail: [email protected] [email protected] Eric Rogers School of Electronics and Computer Science University of Southampton Southampton SO17 1BJ, UK e-mail: [email protected] Received: February 23, 2009 Accepted: July 10, 2009

Operator Theory: Advances and Applications, Vol. 203, 195–236 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Fourier Method for One-dimensional Schr¨ odinger Operators with Singular Periodic Potentials Plamen Djakov and Boris Mityagin Dedicated to Israel Gohberg on the occasion of his eightieth birthday

Abstract. By using quasi-derivatives, we develop a Fourier method for studying the spectral properties of one-dimensional Schr¨ odinger operators with periodic singular potentials. Mathematics Subject Classiﬁcation (2000). Primary 34L40; Secondary 47E05. Keywords. Schr¨ odinger operator, periodic singular potential, quasi-derivative, spectrum, Fourier method.

1. Introduction Our goal in this paper is to develop a Fourier method for studying the spectral properties (in particular, spectral gap asymptotics) of the Schr¨ odinger operator L(v)y = −y

+ v(x)y,

x ∈ R,

(1.1)

−1 (R). v ∈ Hloc

(1.2)

where v is a singular potential such that v(x) = v(x + π),

In the case where the potential v is a real L2 ([0, π])-function, it is well known by the Floquet–Lyapunov theory (see [12, 24, 26, 43]), that the spectrum of L is absolutely continuous and has a band-gap structure, i.e., it is a union of closed intervals separated by spectral gaps + − + − + (−∞, λ0 ), (λ− 1 , λ1 ), (λ2 , λ2 ), . . . , (λn , λn ), . . . .

Communicated by J.A. Ball.

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The points (λ± n ) are deﬁned by the spectra of (1.1) considered on the interval [0, π], respectively, with periodic (for even n) and anti-periodic (for odd n) boundary conditions (bc) : (a) periodic Per+ : (b) antiperiodic Per− :

y(π) = y(0), y (π) = y (0); y(π) = −y(0), y (π) = −y (0);

So, one may consider the appropriate bases in L2 ([0, π]), which leads to a transformation of the periodic or anti-periodic Hill–Schr¨ odinger operator into an operator acting in an 2 -sequence space. This makes possible to develop a Fourier method for investigation of spectra, and especially, spectral gap asymptotics (see [19, 20], where the method has been used to estimate the gap asymptotics in terms of potential smoothness). Our papers [7, 8] (see also the survey [9]) give further development of that approach and provide a detailed analysis of (and extensive bibliography on) the intimate relationship between the smoothness of the potential v and the decay rate of the corresponding spectral gaps (and deviations of Dirichlet eigenvalues) under the assumption v ∈ L2 ([0, π]). But now singular potentials v ∈ H −1 bring a lot of new technical problems even in the framework of the same basic scheme as in [9]. The ﬁrst of these problems is to give proper understanding of the boundary conditions (a) and (b) or their broader interpretation and careful deﬁnition of the corresponding operators and their domains. This is done by using quasi-derivatives. To a great extent we follow the approach suggested (in the context of second-order o.d.e.) and developed by A. Savchuk and A. Shkalikov [35, 37] (see also [36, 39, 40]) and R. Hryniv and Ya. Mykytyuk [15] (see also [16]–[18]). E. Korotyaev [22, 23] follows a diﬀerent approach but it works only in the case of a real potential v. In the context of physical applications, let us notice that the analysis of Hill or Sturm–Liouville operators, or their multi-dimensional analogues −Δ+v(x) with point (surface) interaction (δ-type) potentials has a long history. From the early 1960’s (F. Berezin, L. Faddeev, R. Minlos [5, 6, 28]) to around 2000 the topic has been studied in detail; see the books [1, 2] and the references there. For speciﬁc potentials see for example W.N. Everitt, A. Zettl [13, 14] and P. Kurasov [25]. A more general approach which would consider any singular potential (beyond δ-functions or Coulomb type) in negative Sobolev spaces has been initiated by A. Shkalikov and his coauthors [4, 30, 34, 36]. It led to the theory of Sturm– Liouville operators with distribution potentials developed in [33, 36], and in particular [37]. It is known (e.g., see [15], Remark 2.3, or Proposition 1 below) that every −1 π-periodic potential v ∈ Hloc (R) has the form v = C + Q ,

where C = const, Q is π-periodic,

Q ∈ L2loc (R).

Therefore, one may introduce the quasi-derivative u = y − Qy and replace the distribution equation −y

+ vy = 0 by the following system of two linear equations

Fourier Method

197

with coeﬃcients in L1loc (R) y = Qy + u,

u = (C − Q2 )y − Qu.

(1.3)

By the Existence-Uniqueness theorem for systems of linear o.d.e. with L1loc (R)coeﬃcients (e.g., see [3, 29]), the Cauchy initial value problem for the system (1.3) has, for each pair of numbers (a, b), a unique solution (y, u) such that y(0) = a, u(0) = b. Moreover, following A. Savchuk and A. Shkalikov [35, 37], one may consider various boundary value problems on the interval [0, π]). In particular, let us consider the periodic or anti-periodic boundary conditions Per± , where (a∗ ) Per+ : (b∗ ) Per− :

y(π) = y(0), (y − Qy) (π) = (y − Qy) (0). y(π) = −y(0), (y − Qy) (π) = − (y − Qy) (0).

R. Hryniv and Ya. Mykytyuk [15] used also the system (1.3) in order to give complete analysis of the spectra of the Schr¨ odinger operator with real-valued periodic H −1 -potentials. They showed, that as in the case of periodic L2loc (R)potentials, the Floquet theory for the system (1.3) could be used to explain that if v is real-valued, then L(v) is a self-adjoint operator having absolutely continuous spectrum with band-gap structure, and the spectral gaps are determined by the spectra of the corresponding Hill–Schr¨ odinger operators LPer± deﬁned in the appropriate domains of L2 ([0, π])-functions, and considered, respectively, with the boundary conditions (a∗ ) and (b∗ ). In Section 2 we use the same quasi-derivative approach to deﬁne the domains of the operators L(v) for complex-valued potentials v, and to explain how their spectra are described in terms of the corresponding Lyapunov function. From a technical point of view, our approach is diﬀerent from the approach of R. Hryniv and Ya. Mykytyuk [15]: they consider only the self-adjoint case and use a quadratic form to deﬁne the domain of L(v), while we consider the non-self-adjoint case as well and use the Floquet theory and the resolvent method (see Lemma 3 and Theorem 4). Sections 3 and 4 contains the core results of this paper. In Section 3 we deﬁne odinger and study the operators LPer± which arise when considering the Hill–Schr¨ operator L(v) with the adjusted boundary conditions (a∗ ) and (b∗ ). We meticulously explain what is the Fourier representation of these operators1 in Proposition 10 and Theorem 11. 1 Maybe

it is worth to mention that T. Kappeler and C. M¨ohr [21] analyze “periodic and Dirichlet eigenvalues of Schr¨ odinger operators with singular potential” but the paper [21] does not tell how these operators (or boundary conditions) are deﬁned on the interval, i.e., in a Hilbert space L2 ([0, π]). At some point, without any justiﬁcation or explanation, a transition into weighted 2 sequence spaces (an analog of Sobolev spaces H a ) is made and the same sequence space operators as in the regular case when v ∈ L2per (R) are considered. But without formulating which Sturm– Liouville problem is considered, what are the corresponding boundary conditions, what is the domain of the operator, etc., it is not possible to pass from a non-deﬁned diﬀerential operator to its Fourier representation.

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In Section 4 we use the same approach as in Section 3 to deﬁne and study the Hill–Schr¨ odinger operator LDir (v) with Dirichlet boundary conditions Dir : y(0) = y(π) = 0. Our main result there is Theorem 16 which gives the Fourier representation of the operator LDir (v). In Section 5 we use the Fourier representations of the operators LPer± and LDir to study the localization of their spectra (see Theorem 21). Of course, Theo− rem 21 gives also rough asymptotics of the eigenvalues λ+ n , λn , μn of these operators. But we are interested to ﬁnd the asymptotics of spectral gaps γn = − λ+ n − λn in the self-adjoint case, or the asymptotics of both γn and the deviations − μn − (λ+ n + λn )/2 in the non-self-adjoint case, etc. Our results in that direction are announced in [10]; all details of proofs are given in [11].

2. Preliminary results 1. The operator (1.1) has a second term vy with v satisfying (1.2). First of all, 1 let us specify the structure of periodic functions and distributions in Hloc (R) and −1 Hloc (R). 1 (R) is deﬁned as the space of functions f (x) ∈ L2loc (R) which The space Hloc are absolutely continuous and have their derivatives f (x) ∈ L2loc (R). It is a Fr´echet space if considered with the topology deﬁned by the countable system of seminorms f 21,T , T = Tn = n, where T . / |f (x)|2 + |f (x)|2 dx. (2.1) f 21,T = −T

(The subspace of periodic functions 1 (R) : f (x + π) = f (x)} {f ∈ Hloc

is a normed space in the induced topology, with the norm f 1,T , T > π, and ( . / 1 1 2

2 W2 (R) = f ∈ Hloc (R) : |f (x)| + |f (x)| dx < ∞ R

is a Hilbert space.) In the same way, W1,1 loc (R) is deﬁned as the space of functions g(x) ∈ L1 (R) which are absolutely continuous with derivatives g (x) ∈ L1loc (R), i.e., for every T >0 T (|g(x)| + |g (x)|) dx < ∞. −T

Let D(R) be the space of all C ∞ -functions on R with compact support, and let D([−T, T ]) be the subset of all ϕ ∈ D(R) with supp ϕ ⊂ [−T, T ]. −1 (R) is the space of distributions v on R such that By deﬁnition, Hloc ∀T > 0 ∃C(T ) :

|v, ϕ| ≤ C(T )ϕ1,T

∀ϕ ∈ D([−T, T ]).

(2.2)

Fourier Method Of course, since

T

|ϕ(x)| dx ≤ 2T 2

−T

T

2 −T

199

|ϕ (x)|2 dx,

the condition (2.2) is equivalent to ˜ )ϕ L2 ([−T,T ]) |v, ϕ| ≤ C(T

∀T > 0 ∃C(T ) :

∀ϕ ∈ D([−T, T ]).

(2.3)

D1 ([−T, T ]) = {ϕ : ϕ ∈ D([−T, T ])}

(2.4)

Set D1 (R) = {ϕ : ϕ ∈ D(R)};

and consider the linear functional q deﬁned by q(ϕ ) := −v, ϕ,

ϕ ∈ D1 (R).

(2.5)

In view of (2.3), for each T > 0, q(·) is a continuous linear functional deﬁned in the space D1 ([−T, T ]) ⊂ L2 ([−T, T ]). By the Riesz Representation Theorem there exists a function QT (x) ∈ L2 ([−T, T ]) such that T

q(ϕ ) = QT (x)ϕ (x)dx ∀ϕ ∈ D([−T, T ]). (2.6) −T

The function QT is uniquely determined up to an additive constant because in L2 ([−T, T ]) only constants are orthogonal to D1 ([−T, T ]). Therefore, one can readily see that there is a function Q(x) ∈ L2loc (R) such that ∞

Q(x)ϕ (x)dx ∀ϕ ∈ D(R), q(ϕ ) = −∞

where the function Q is uniquely determined up to an additive constant. Thus, we have v, ϕ = −q(ϕ ) = −Q, ϕ = Q , ϕ, i.e., v = Q . A distribution v ∈

−1 Hloc (R)

(2.7)

is called periodic of period π if

v, ϕ(x) = v, ϕ(x − π)

∀ϕ ∈ D(R).

(2.8)

L. Schwartz [41] gave an equivalent deﬁnition of a periodic of period π distribution in the following way: Let ω : R → S 1 = R/πZ,

mod π. . /

A distribution F ∈ D (R) is periodic if, for some f ∈ C ∞ (S 1 ) , we have F (x) = f (ω(x)), where Φ=

k∈Z

ω(x) = x

i.e., ϕ, F = Φ, f , ϕ(x − kπ).

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Now, if v is periodic and Q ∈ L2loc (R) is chosen so that (2.7) holds, we have by (2.8) ∞ ∞ ∞ Q(x + π)ϕ (x)dx = Q(x)ϕ (x − π) = Q(x)ϕ (x)dx, −∞

i.e.,

−∞

∞ −∞

−∞

[Q(x + π) − Q(x)]ϕ (x)dx = 0

∀ϕ ∈ D(R).

Thus, there exists a constant c such that Q(x + π) − Q(x) = c

a.e.

Consider the function

c ˜ Q(x) = Q(x) − x; π ˜ + π) = Q(x) ˜ ˜ is π-periodic, and then we have Q(x a.e., so Q ˜ + c . v=Q π Let ˜ q(m)eimx Q(x) =

(2.9)

m∈2Z

˜ ∈ L2 ([0, π]). Set be the Fourier series expansion of the function Q c V (0) = , V (m) = imq(m) for m = 0. π Then |V (m)|2 ˜ 22 Q , L ([0,π]) = m2

(2.10)

m=0

and we can consider −1 −1 Hπ−per (R) = {v ∈ Hloc (R) : v satisﬁes(2.8)}

as a Hilbert space with a norm ˜ 2. v2 = |V (0)|2 + Q Convergence of the series (2.9) in L2 ([0, π]) implies its convergence in L2loc (R) because for any T > 0 with kπ ≤ T ≤ (k + 1)π it follows that T π 2 2 ˜ ˜ |Q(x)| dx ≤ 2(k + 1) |Q(x)| dx. −T

0

All this leads to the following statement. −1 (R) has the form Proposition 1. Every π-periodic distribution v ∈ Hloc

v = C + Q , with

Q ∈ L2loc (R), 1 q(0) = π

(2.11)

π

Q(x)dx = 0, 0

a.e.

Q(x + π) = Q(x)

(2.12)

Fourier Method and its Fourier series v=

201

V (m)eimx

(2.13)

m∈2Z

where V (0) = C,

V (m) = imq(m)

for m = 0,

and q(m) are the Fourier coeﬃcients of Q, converges in |V (m)|2 Q2L2([0,π]) = m2

−1 (R). Hloc

(2.14) Of course, (2.15)

m=0

Remark. R. Hryniv and Ya. Mykytyuk [15], (see Theorem 3.1 and Remark 2.3) −1 (R)give a more general claim about the structure of uniformly bounded Hloc distributions. −1 2. In view of (2.2), each distribution v ∈ Hloc (R) could be considered as 1 1 (R) with compact a linear functional on the space Hoo (R) of functions in Hloc −1 1 support. Therefore, if v ∈ Hloc (R) and y ∈ Hloc (R), then the diﬀerential expression (y) = −y

+ v · y is well deﬁned by −y

+ v · y, ϕ = y , ϕ + v, y · ϕ −1 (R). This observation suggests to consider the Schr¨ odinger as a distribution in Hloc 2 2 operator −d /dx + v in the domain 1 (R) ∩ L2 (R) : −y

+ v · y ∈ L2 (R) . (2.16) D(L(v)) = y ∈ Hloc

Moreover, suppose v = C +Q , where C is a constant and Q is a π-periodic function such that 1 π 2 Q ∈ L ([0, π]), q(0) = Q(x)dx = 0. (2.17) π 0 Then the diﬀerential expression (y) = −y

+ vy can be written in the form

(y) = − (y − Qy) − Qy + Cy. Notice that

(2.18)

(y) = − (y − Qy) − Qy + Cy = f ∈ L2 (R) if and only if 1 (R) u = y − Qy ∈ W1,loc

and the pair (y, u) satisﬁes the system of diﬀerential equations

y = Qy + u,

u = (C − Q2 )y − Qu + f. Consider the corresponding homogeneous system

y = Qy + u,

u = (C − Q2 )y − Qu.

(2.19)

(2.20)

with initial data y(0) = a,

u(0) = b.

(2.21)

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Since the coeﬃcients 1, Q, C − Q2 of the system (2.20) are in L1loc (R), the standard existence-uniqueness theorem for linear systems of equations with L1loc (R)coeﬃcients (e.g., see M. Naimark [29], Sect.16, or F. Atkinson [3]) guarantees that for any pair of numbers (a, b) the system (2.20) has a unique solution (y, u) with 1 y, u ∈ W1,loc (R) such that (2.21) holds. On the other hand, the coeﬃcients of the system (2.20) are π-periodic, so one may apply the classical Floquet theory. Let (y1 , u1 ) and (y2 , u2 ) be the solutions of (2.20) which satisfy y1 (0) = 1, u1 (0) = 0 and y2 (0) = 0, u2 (0) = 1. By the Caley–Hamilton theorem the Wronskian y (x) y2 (x) ≡1 det 1 u1 (x) u2 (x) because the trace of the coeﬃcient matrix of the system (2.20) is zero. If (y(x), u(x)) is a solution of (2.20) with initial data (a, b), then (y(x + π), u(x + π)) is a solution also, correspondingly with initial data y1 (π) y2 (π) a y(π) , M= =M . b u(π) u1 (π) u2 (π) Consider the characteristic equation of the monodromy matrix M : ρ2 − Δρ + 1 = 0,

Δ = y1 (π) + u2 (π).

(2.22)

Each root ρ of the characteristic equation (2.22) gives rise to a special solution (ϕ(x), ψ(x)) of (2.20) such that ϕ(x + π) = ρ · ϕ(x),

ψ(x + π) = ρ · ψ(x).

(2.23)

Since the product of the roots of (2.22) equals 1, the roots have the form ρ± = e±τ π ,

τ = α + iβ,

(2.24)

where β ∈ [0, 2] and α = 0 if the roots are on the unit circle or α > 0 otherwise. In the case where the equation (2.22) has two distinct roots, let (ϕ± , ψ ± ) be special solutions of (2.20) that correspond to the roots (2.24), i.e., (ϕ± (x + π), ψ ± (x + π)) = ρ± · (ϕ± (x), ψ ± (x)). Then one can readily see that the functions ϕ˜± (x) = e∓τ xϕ± (x),

ψ˜± (x) = e∓τ x ψ ± (x)

are π-periodic, and we have ϕ± (x) = e±τ x ϕ˜± (x),

ψ± (x) = e±τ x ψ˜± (x).

(2.25)

Consider the case where (2.22) has a double root ρ = ±1. If its geometric multiplicity equals 2 (i.e., the matrix M has two linearly independent eigenvectors), then the equation (2.20) has, respectively, two linearly independent solutions (ϕ± , ψ ± ) which are periodic if ρ = 1 or anti-periodic if ρ = −1.

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203

Otherwise, is a Jordan matrix), there are two linearly independent + (ifM− a a and such that vectors b+ b− + + − − + a a a a a M + = ρ + , M − = ρ − + ρκ + , ρ = ±1, κ = 0. (2.26) b b b b b Let (ϕ± , ψ ± ) be the corresponding solutions of (2.20). Then we have + − − + + ϕ (x) ϕ (x + π) ϕ (x) ϕ (x) ϕ (x + π) =ρ , =ρ + ρκ . ψ + (x + π) ψ + (x) ψ − (x + π) ψ − (x) ψ + (x) (2.27) Now, one can easily see that the functions ϕ˜− and ψ˜− given by − − κx ϕ+ (x) ϕ˜ (x) ϕ (x) − = ψ − (x) ψ˜− (x) π ψ + (x) are π-periodic (if ρ = 1) or anti-periodic (if ρ = −1). Therefore, the solution ϕ− (x) can be written in the form ψ − (x) − − κx ϕ+ (x) ϕ˜ (x) ϕ (x) = ˜− , (2.28) + ψ − (x) ψ (x) π ψ + (x) i.e., it is a linear combination of periodic (if ρ = 1), or anti-periodic (if ρ = −1) functions with coeﬃcients 1 and κx/π. The following lemma shows how the properties of the solutions of (2.19) and (2.20) depend on the roots of the characteristic equation (2.22). Lemma 2. (a) The homogeneous system (2.20) has no nonzero solution (y, u) with y ∈ L2 (R). Moreover, if the roots of the characteristic equation (2.22) lie on the unit circle, i.e., α = 0 in the representation (2.24), then (2.20) has no nonzero solution (y, u) with y ∈ L2 ((−∞, 0]) or y ∈ L2 ([0, +∞)). (b) If α = 0 in the representation (2.24), then there are functions f ∈ L2 (R) such that the corresponding non-homogeneous system (2.19) has no solution (y, u) with y ∈ L2 (R). (c) If the roots of the characteristic equation (2.22) lie outside the unit circle, i.e., α > 0 in the representation (2.24), then the non-homogeneous system (2.19) has, for each f ∈ L2 (R), a unique solution (y, u) = (R1 (f ), R2 (f )) such that R1 is a linear continuous operator from L2 (R) into W21 (R), and R2 1 is a linear continuous operator in L2 (R) with a range in W1,loc (R). Proof. (a) In view of the above discussion (see the text from (2.22) to (2.28)), if the characteristic equation (2.22) has two distinct roots ρ = e±τ π , then each solution (y, u) of the homogeneous system (2.20) is a linear combination of the

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special solutions, so y(x) = C + eτ x ϕ˜+ (x) + C − e−τ x ϕ˜− (x), where ϕ˜+ and ϕ˜− are π-periodic functions in H 1 . In the case where the real part of τ is strictly positive, i.e., τ = α+iβ with α > 0, one can readily see that eτ x ϕ˜+ (x) ∈ L2 ([0, ∞)) but eτ x ϕ˜+ (x) ∈ L2 ((−∞, 0]), while e−τ x ϕ˜− (x) ∈ L2 ([0, ∞)) but e−τ x ϕ˜− (x) ∈ L2 ((−∞, 0])). Therefore, if y ≡ 0 we have y ∈ L2 (R). Next we consider the case where τ = iβ with β = 0, 1. The Fourier series of the functions ϕ˜+ (x) and ϕ˜− (x) ikx ikx ϕ˜+ , ϕ˜− ∼ ϕ˜− ϕ˜+ ∼ ke ke k∈2Z

k∈2Z

converge uniformly in R because ϕ˜+ , ϕ˜− ∈ H 1 . Therefore, we have i(k+β)x i(k−β)x ϕ˜+ + C− ϕ˜− , y(x) = C + ke ke k∈2Z

k∈2Z

where the series on the right converge uniformly on R. If β is a rational number, then y is a periodic function, so y ∈ L2 ((−∞, 0]) and y ∈ L2 ([0, ∞)). If β is an irrational number, then 1 T y(x)e−i(k±β)x dx = C ± ϕ˜± ∀k ∈ 2Z. (2.29) lim k T →∞ T 0 On the other hand, if y ∈ L2 ([0, ∞)), then the Cauchy inequality implies + ,1/2 1 T T yL2([0,∞)) 1 √ → 0. y(x)e−i(k±β)x dx ≤ √ |y(x)|2 dx ≤ T 0 T T 0 But, in view of (2.29), this is impossible if y = 0. Thus y ∈ L2 ([0, ∞)). In a similar way, one can see that y ∈ L2 ((−∞, 0]). Finally, if the characteristic equation (2.22) has a double root ρ = ±1, then either every solution (y, u) of (2.20) is periodic or anti-periodic, and so y ∈ L2 ([0, ∞) and y ∈ L2 ((−∞, 0]), or it is a linear combination of some special solutions (see (2.28), and the preceding discussion), so we have κx y(x) = C + ϕ+ (x) + C − ϕ˜− + C − ϕ+ (x), π where the functions ϕ+ and ϕ˜− are periodic or anti-periodic. Now one can easily see that y ∈ L2 ([0, ∞) and y ∈ L2 ((−∞, 0]), which completes the proof of (a). (b) Let (ϕ± , ψ ± ) be special solutions of (2.20) that correspond to the roots (2.24) as above. We may assume without loss of generality that the Wronskian of the solutions (ϕ+ , ψ + ) and (ϕ− , ψ − ) equals 1 because these solutions are determined up to constant multipliers.

Fourier Method

205

The standard method of variation of constants leads to the following solution (y, u) of the non-homogeneous system (2.19): y = v + (x)ϕ+ (x) + v − (x)ϕ− (x), where v

+

and v

−

u = v + (x)ψ + (x) + v − (x)ψ − (x),

satisfy

dv − − dv · ϕ+ + · ϕ = 0, dx dx

dv + + dv − − ·ψ + · ψ = f, dx dx

+

so

(2.30)

x

v (x) = − +

−

+

ϕ (t)f (t)dt + C ,

−

x

v (x) =

0

(2.31)

ϕ+ (t)f (t)dt + C − .

(2.32)

0

Assume that the characteristic equation (2.22) has roots of the form ρ = eiβπ , β ∈ [0, 2). Take any function f ∈ L2 (R) with compact support, say supp f ⊂ (0, T ). By (2.30) and (2.32), if (y, u) is a solution of the non-homogeneous system (2.19), then the restriction of (y, u) on the intervals (−∞, 0) and [T, ∞) is a solution of the homogeneous system (2.20). So, by (a), if y ∈ L2 (R) then y ≡ 0 on the intervals (−∞, 0) and [T, ∞). This may happen if only if the constants C ± in (2.32) are zero, and we have T T ϕ− (t)f (t)dt = 0, ϕ+ (t)f (t)dt = 0. 0

0

Hence, if f is not orthogonal to the functions ϕ± on the interval [0, T ], then the non-homogeneous system (2.19) has no solution (y, u) with y ∈ L2 (R). This completes the proof of (b). (c) Now we consider the case where the characteristic equation (2.22) has roots of the form (2.24) with α > 0. Let (ϕ± , ψ ± ) be the corresponding special solutions. By (2.30), for each f ∈ L2 (R), the non-homogeneous system (2.19) has a solution of the form (y, u) = (R1 (f ), R2 (f ), where R1 (f ) = v + (x)ϕ+ (x) + v − (x)ϕ− (x),

R2 (f ) = v + (x)ψ + (x) + v − (x)ψ − (x), (2.33) and (2.31) holds. In order to have a solution that vanishes at ±∞ we set (taking into account (2.25)) ∞ x + −τ t − − e ϕ˜ (t)f (t)dt, v (x) = eτ t ϕ˜+ (t)f (t)dt. (2.34) v (x) = −∞

x ±

Let C± = max{|ϕ˜ (x)| : x ∈ [0, π]}. By (2.25), we have |ϕ± (x)| ≤ C± · e±αx . Therefore, by the Cauchy inequality, we get ∞ 2 + 2 2 −αt 2 e |f (t)|dt ≤ C− |v (x)| ≤ C− x

so

∞

e

−αt

(2.35) dt ·

x

C2 |v (x)| ≤ − e−αx α +

2

∞ x

∞

e

−αt

|f (t)| dt , 2

x

e−αt |f (t)|2 dt.

(2.36)

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P. Djakov and B. Mityagin

Thus, by (2.35), ∞ ∞ 2 2 ∞ + 2 + 2 αx v (x) ϕ (x) dx ≤ C− C+ e e−αt |f (t)|2 dtdx α −∞ −∞ x t C2 C2 ∞ C2 C2 ≤ − + |f (t)|2 eα(x−t) dx dt = − 2 + f 2L2(R) . α α −∞ −∞ In an analogous way one may prove that ∞ 2 2 − 2 − 2 v (x) ϕ (x) dx ≤ C− C+ f 2 2 . L (R) α2 −∞ In view of (2.30), these estimates prove that R1 is a continuous operator in L2 (R). d R1 (f ). In view of (2.31), we Next we estimate the L2 (R)-norm of y = dx have dϕ + dϕ − y (x) = v + (x) · (x) + v− (x) · (x). dx dx By (2.25), v + (x) ·

dϕ + dϕ˜ + (x) = αv + (x)ϕ+ + v + (x)eαx . dx dx

Since the L2 (R)-norm of v + (x)ϕ+ has been estimated above, we need to estimate only the L2 (R)-norm of v + (x)eαx dϕ˜+ /dx. By (2.36), we have ∞ ∞ 2 ∞ + v (x)eαx dϕ˜+ /dx2 dx ≤ C− dϕ˜+ /dx2 eαx e−αt |f (t)|2 dtdx α −∞ −∞ x t 2 ∞ 2 α(x−t) C− 2 + = dϕ˜ /dx e |f (t)| dx dt. α −∞ −∞ Firstly, we estimate the integral in the parentheses. Notice that the function dϕ± /dx (and therefore, dϕ˜± /dx ) are in the space L2 ([0, π]) due to the ﬁrst equation in (2.20). Therefore, π ± 2 dϕ˜ 2 (2.37) K± = dx (x) dx < ∞. 0 We have

t −∞

n=0

2 ≤ K+ ·

∞ n=0

Thus,

+ 2 dϕ˜ αξ dx (ξ + t) e dξ −(n+1)π

∞ + dϕ˜ /dx2 eα(x−t) dx =

e−αnπ =

−nπ

2 K2 K+ < (1 + απ) + . 1 − exp(−απ) απ

2 2 + v (x)eαx dϕ˜+ /dx2 dx ≤ (1 + απ) C− K+ f 2 . α2 π −∞ ∞

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207

In an analogous way it follows that ∞ − 2 2 2 − v (x)eαx dϕ˜ dx ≤ (1 + απ) C+ K− f 2 , dx α2 π −∞ so the operator R1 acts continuously from L2 (R) into the space W21 (R). The proof of the fact that the operator R2 acts continuously from L2 (R) into 1 W1,loc (R) is omitted because essentially it is the same (we only replace ϕ± with ψ ± in the proof that R1 is a continuous operator from L2 (R)) into W21 (R)). We need also the following lemma. Lemma 3. Let H be a Hilbert space with product (·, ·), and let A : D(A) → H,

B : D(B) → H

be (unbounded) linear operators with domains D(A) and D(B), such that (Af, g) = (f, Bg)

for f ∈ D(A), g ∈ D(B).

(2.38)

If there is a λ ∈ C such that the operators A − λ and B − λ are surjective, then (i) D(A) and D(B) are dense in H; (ii) A∗ = B and B ∗ = A, where A∗ and B ∗ are, respectively, the adjoint operators of A and B. Proof. We need to explain only that D(A) is dense in H and A∗ = B because one can replace the roles of A and B. To prove that D(A) is dense in H, we need to show that if h is orthogonal to D(A) then h = 0. Let (f, h) = 0 ∀f ∈ D(A). Since the operator B − λ is surjective, there is g ∈ D(B) such that h = (B − λ)g. Therefore, by (2.38), we have 0 = (f, h) = (f, (B − λ)g) = ((A − λ)f, g)

∀f ∈ D(A),

which yields g = 0 because the range of A − λ is H. Thus, h = (B − λ)g = 0. Hence (i) holds. Next we prove (ii). In view of (2.38), we have D(B) ⊂ Dom(A∗ ) and A∗ f = Bf for f ∈ D(B). Conversely, if g ∗ ∈ Dom(A∗ ), then ((A − λ)f, g ∗ ) = (f, w) ∗

∀f ∈ D(A),

(2.39)

∗

where w = (A − λ)g . Since the operator B − λ is surjective, there is g ∈ D(B) such that w = (B − λ)g. Therefore, by (2.38) and (2.39), we have ((A − λ)f, g ∗ ) = (f, (B − λ)g) = ((A − λ)f, g)

∀f ∈ D(A),

∗

which implies that g = g (because the range of A − λ is equal to H) and (A∗ − λ)g ∗ = (B − λ)g ∗ , i.e., A∗ g ∗ = Bg ∗ . This completes the proof of (ii). Consider the Schr¨ odinger operator with a spectral parameter L(v) − λ = −d2 /dx2 + (v − λ),

λ ∈ C.

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P. Djakov and B. Mityagin

In view of the formula (2.11) in Proposition 1, we may assume without loss of generality that (2.40) C = 0, v = Q , because a change of C results in a shift of the spectral parameter λ. Replacing C by −λ in the homogeneous system (2.20), we get

y = Qy + u, (2.41)

u = (−λ − Q2 )y − Qu. Let (y1 (x; λ), u1 (x; λ)) and (y2 (x; λ), u2 (x; λ)) be the solutions of (2.41) which satisfy the initial conditions y1 (0; λ) = 1, u1 (0; λ) = 0 and y2 (0; λ) = 0, u2 (0; λ) = 1. Since these solutions depend analytically on λ ∈ C, the Lyapunov function, or Hill discriminant, Δ(Q, λ) = y1 (π; λ) + u2 (π; λ) (2.42) is an entire function. Taking the conjugates of the equation in (2.41), one can easily see that Δ(Q, λ) = Δ(Q, λ). (2.43) Remark. A. Savchuk and A. Shkalikov gave asymptotic analysis of the functions yj (π, λ) and uj (π, λ), j = 1, 2. In particular, it follows from Formula (1.5) of Lemma 1.4 in [37] that, with z 2 = λ, 1 y1 (π, λ) = cos(πz)+o(1), y2 (π, λ) = [sin(πz)+o(1)], u2 (π, λ) = cos πz +o(1), z (2.44) and therefore, Δ(Q, λ) = 2 cos πz + o(1), z 2 = λ, (2.45) inside any parabola Pa = {λ ∈ C : |Im z| ≤ a}. (2.46) 2 In the regular case v ∈ L ([0, π]) these asymptotics of the fundamental solutions and the Lyapunov function Δ of the Hill–Schr¨ odinger operator could be found in [27], p. 32, Formula (1.3.11), or pp. 252–253, Formulae (3.4.23 ), (3.4.26). Consider the operator L(v), in the domain 1 D(L(v)) = y ∈ H 1 (R) : y − Qy ∈ L2 (R) ∩ W1,loc (R), Q (y) ∈ L2 (R) , (2.47) deﬁned by (2.48) L(v)y = Q (y), with Q (y) = −(y − Qy) − Qy , where v and Q are as in Proposition 1. −1 Theorem 4. Let v ∈ Hloc (R)) be π-periodic. Then (a) the domain D(L(v)) is dense in L2 (R); (b) the operator L(v) is closed, and its conjugate operator is

(L(v))∗ = L(v);

(2.49)

(In particular, if v is real-valued, then the operator L(v) is self-adjoint.)

Fourier Method

209

(c) the spectrum Sp(L(v)) of the operator L(v) is continuous, and moreover, Sp(L(v)) = {λ ∈ C |

∃θ ∈ [0, 2π) : Δ(λ) = 2 cos θ}.

(2.50)

2

Remark. In the case of L -potential v this result is known (see M. Serov [38], F. Rofe–Beketov and A. Kholkin [31, 32], and V. Tkachenko [42]). Proof. Firstly, we show that the operators L(v) and L(v) are formally adjoint, i.e., if y ∈ D(L(v)), h ∈ D(L(v)).

(L(v)y, h) = (f, L(v)h)

(2.51)

Since y − Qy and h are continuous L2 (R)-functions, their product is a continuous L1 (R)-function, so we have lim inf (y − Qy)h (x) = 0. x→±∞

Therefore, there exist two sequences of real numbers cn → −∞ and dn → ∞ such that .

/ .

/ (y − Qy)h (cn ) → 0, (y − Qy)h (dn ) → 0 as n → ∞. Now, we have dn ∞ . / −(y − Qy) h − Qy h dx (L(v)y, h) = Q (y)hdx = lim n→∞ c −∞ n , +

|c

−(y − Qy)h

= lim

n→∞

∞

=0+ −∞

dn

dn

.

+ n

(y − Qy)h dx −

cn

dn

Qy hdx

cn

/

y h − Qyh − Qy h dx.

The same argument shows that ∞ / .

y h − Qyh − Qy h dx = (y, L(v)h) , −∞

which completes the proof of (2.51). If the roots of the characteristic equation ρ2 − Δ(Q, λ)ρ + 1 = 0 lie on the unit circle {eiθ , θ ∈ [0, 2π)}, then they are of the form e±iθ , so we have Δ(Q, λ) = eiθ + e−iθ = 2 cos θ.

(2.52)

Therefore, if Δ(Q, λ) ∈ [−2, 2], then the roots of the characteristic equation lie outside of the unit circle {eiθ , θ ∈ [0, 2π)}. If so, by part (c) of Lemma 2, the operator L(v) − λ maps bijectively D(L(v)) onto L2 (R), and its inverse operator (L(v)) − λ)−1 : L2 (R) → D(L(v)) is a continuous linear operator. Thus, Δ(Q, λ) ∈ [−2, 2] ⇒ (L(v) − λ)−1 : L2 (R) → D(L(v))

exists.

(2.53)

Next we apply Lemma 3 with A = L(v) and B = L(v). Choose λ ∈ C so thatΔ(Q, λ) ∈ [−2, 2] (in view of (2.45), see the remark before Theorem 4, Δ(Q, λ) is a non-constant entire function, so such a choice is possible). Then, in

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P. Djakov and B. Mityagin

view of (2.43), we have that Δ(Q, λ) ∈ [−2, 2] also. In view of the above discussion, this means that the operator L(v) − λ maps bijectively D(L(v)) onto L2 (R) and L(v) − λ maps bijectively D(L(v)) onto L2 (R). Thus, by Lemma 3, D(L(v)) is dense in L2 (R) and L(v)∗ = L(v), i.e., (a) and (b) hold. Finally, in view of (2.53), (c) follows readily from part (b) of Lemma 2. 3. Theorem 4 shows that the spectrum of the operator L(v) is described by the equation (2.50). As we are going to explain below, this fact implies that the spectrum Sp(L(v)) could be described in terms of the spectra of the operators Lθ = Lθ (v), θ ∈ [0, π], that arise from the same diﬀerential expression = Q when it is considered on the interval [0, π] with the following boundary conditions: y(π) = eiθ y(0),

(y − Qy)(π) = eiθ (y − Qy)(0).

The domains D(Lθ ) of the operators Lθ are given by D(Lθ ) = y ∈ H 1 : y − Qy ∈ W11 ([0, π]), (2.54) holds, (y) ∈ H 0 ,

(2.54)

(2.55)

where H 1 = H 1 ([0, π]),

H 0 = L2 ([0, π]).

We set Lθ (y) = (y),

y ∈ D(Lθ ).

(2.56)

Notice that if y ∈ H ([0, π]), then Q (y) = f ∈ L ([0, π]) if and only if u = y − Qy ∈ W11 ([0, π]) and the pair (y, u) is a solution of the non-homogeneous system (2.19). y2 y1 and be the solutions of the homogeneous system (2.20) Lemma 5. Let u1 u2 which satisfy y2 (0) 1 0 y1 (0) , . (2.57) = = u1 (0) u2 (0) 0 1 1

2

If Δ = y1 (π) + u2 (π) = 2 cos θ,

θ ∈ [0, π],

(2.58)

then the non-homogeneous system (2.19) has, for each f ∈ H , a unique solution (y, u) = (R1 (f ), R2 (f )) such that y(π) y(0) = eiθ . (2.59) u(π) u(0) 0

Moreover, R1 is a linear continuous operator from H 0 into H 1 , and R2 is a linear continuous operator in H 0 with a range in W11 ([0, π]). Proof. By the variation of parameters method, every solution of the system (2.19) has the form y(x) y1 (x) y2 (x) = v1 (x) + v2 (x) , (2.60) u(x) u1 (x) u2 (x)

Fourier Method where

x

v1 (x) = −

211

y2 (x)f (t)dt + C1 ,

x

v2 (x) =

0

y1 (x)f (t)dt + C2 .

(2.61)

0

We set for convenience

m1 (f ) = −

π

y2 (t)f (t)dt,

π

m2 (f ) =

0

y1 (t)f (t)dt.

(2.62)

0

By (2.60)–(2.62), the condition (2.59) is equivalent to C1 y (π) y (π) + (m2 (f ) + C2 ) 2 = eiθ . (m1 (f ) + C1 ) 1 C2 u1 (π) u2 (π)

(2.63)

This is a system of two linear equations in two unknowns C1 and C2 . The corresponding determinant is equal to y (π) − eiθ y2 (π) = 1 + e2iθ − Δ · eiθ = eiθ (2 cos θ − Δ). det 1 u1 (π) u2 (π) − eiθ C1 , Therefore, if (2.58) holds, then the system (2.63) has a unique solution C2 where C1 = C1 (f ) and C2 = C2 (f ) are linear combinations of m1 (f ) and m2 (f ). With these values of C1 (f ) and C2 (f ) we set R1 (f ) = v1 · y1 + v2 · y2 ,

R2 (f ) = v1 · u1 + v2 · u2 .

By (2.61) and (2.62), the Cauchy inequality implies x |v1 (x)| ≤ |y2 (t)f (t)|dt + |C1 (f )| ≤ A · f ,

|v2 (x)| ≤ B · f ,

0

where A and B are constants. From here it follows that R1 and R2 are continuous linear operators in H 0 . Since d dy1 dy2 R1 (f ) = v1 + v2 , dx dx dx

R2 (f ) = v1

du1 du2 + v2 + f, dx dx

it follows also that R1 acts continuously from H 0 into H 1 , and R2 has range in W11 ([0, π]), which completes the proof. −1 (R)) is π-periodic. Then, Theorem 6. Suppose v ∈ Hloc

(a) for each θ ∈ [0, π], the domain D(Lθ (v)) deﬁned in (2.55) is dense in H 0 ; (b) the operator Lθ (v) ∈ (2.56) is closed, and its conjugate operator is Lθ (v)∗ = Lθ (v).

(2.64)

In particular, if v is real-valued, then the operator Lθ (v) is self-adjoint. (c) the spectrum Sp(Lθ (v)) of the operator Lθ (v) is discrete, and moreover, Sp(Lθ (v)) = {λ ∈ C : Δ(λ) = 2 cos θ}.

(2.65)

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P. Djakov and B. Mityagin

Proof. Integration by parts shows that the operators Lθ (v) and Lθ (v) are formally adjoint, i.e., (Lθ (v)y, h) = (f, Lθ (v)h)

if y ∈ D(Lθ (v)), h ∈ D(Lθ (v)).

(2.66)

Now we apply Lemma 3 with A = Lθ (v) and B = Lθ (v). Choose λ ∈ C so that Δ(Q, λ) = 2 cos θ (as one can easily see from the remark before Theorem 4, Δ(Q, λ) is a non-constant entire function, so such a choice is possible). Then, in view of (2.43), we have that Δ(Q, λ) = 2 cos θ also. By Lemma 5, Lθ (v)−λ maps bijectively D(Lθ (v)) onto H 0 and Lθ (v) − λ maps bijectively D(Lθ (v)) onto H 0 . Thus, by Lemma 3, D(Lθ (v) is dense in H 0 and Lθ (v)∗ = Lθ (v), i.e., (a) and (b) hold. If Δ(Q, λ) = 2 cos θ, then eiθ is a root of the characteristic equation (2.22), so there is a special solution (ϕ, ψ) of the homogeneous system (2.20) (considered with C = −λ) such that (2.23) holds with ρ = eiθ . But then ϕ ∈ D(Lθ (v)) and Lθ (v)ϕ = λϕ, i.e., λ is an eigenvalue of Lθ (v). In view of Lemma 5, this means that (2.65) holds. Since Δ(Q, λ) is a non-constant entire function (as one can easily see from the remark before Theorem 4) the set on the right in (2.65) is discrete. This completes the proof of (c). Corollary 7. In view of Theorem 4 and Theorem 6, we have 9 Sp (L(v)) = Sp (Lθ (v)).

(2.67)

θ∈[0,π]

In the self-adjoint case (i.e., when v, and therefore, Q are real-valued) the spectrum Sp (L(v)) ⊂ R has a band-gap structure. This is a well-known result in the regular case where v is an L2loc (R)-function. Its generalization in the singular case was proved by R. Hryniv and Ya. Mykytiuk [15]. In order to formulate that result more precisely, let us consider the following boundary conditions (bc): Per+ : y(π) = y(0), (y − Qy) (π) = (y − Qy) (0); (a∗ ) periodic ∗ (b ) antiperiodic Per− : y(π) = −y(0), (y − Qy) (π) = − (y − Qy) (0); In the case where Q is a continuous function, Per+ and Per− coincide, respectively, with the classical periodic boundary condition y(π) = y(0), y (π) = y (0) or anti-periodic boundary condition y(π) = −y(0), y (π) = −y (0) (see the related discussion in Section 6.2). The boundary conditions Per± are particular cases of (2.59), considered, respectively, for θ = 0 or θ = π. Therefore, by Theorem 6, for each of these two boundary conditions, the diﬀerential expression (2.18) gives a rise of a closed (self adjoint for real v) operator LPer± in H 0 = L2 ([0, π]), respectively, with a domain D(LPer+ ) = {y ∈ H 1 : y − Qy ∈ W11 ([0, π]), (a∗ ) holds, l(y) ∈ H 0 },

(2.68)

D(LPer− ) = {y ∈ H 1 : y − Qy ∈ W11 ([0, π]), (b∗ ) holds, l(y) ∈ H 0 }.

(2.69)

or The spectra of the operators LPer± are discrete. Let us enlist their eigenvalues in increasing order, by using even indices for the eigenvalues of LPer+ and odd

Fourier Method

213

indices for the eigenvalues of LPer− (the convenience of such enumeration will be clear later): + − + − + Sp (LPer+ ) = {λ0 , λ− (2.70) 2 , λ2 , λ4 , λ4 , λ6 , λ6 , . . .}, − + − + − + Sp (LPer− ) = {λ1 , λ1 , λ3 , λ3 , λ5 , λ5 . . .}. (2.71) Proposition 8. Suppose v = C + Q , where Q ∈ L2loc (R)) is a π-periodic real-valued function. Then, in the above notations, we have + − + − + − + − + λ0 < λ− 1 ≤ λ1 < λ2 ≤ λ2 < λ3 ≤ λ3 < λ4 ≤ λ4 < λ5 ≤ λ5 < · · · .

(2.72)

Moreover, the spectrum of the operator L(v) is absolutely continuous and has a band-gap structure: it is a union of closed intervals separated by spectral gaps + − + − + (−∞, λ0 ), (λ− 1 , λ1 ), (λ2 , λ2 ), . . . , (λn , λn ), . . . .

Let us mention that A. Savchuk and A. Shkalikov [35] have studied the Sturm–Liouville operators that arise when the diﬀerential expression Q , Q ∈ L2 ([0, 1]), is considered with adjusted regular boundary conditions (see Theorems 1.5 and 1.6 in [37]).

3. Fourier representation of the operators LPer± Let L0bc denote the free operator L0 = −d2 /dx2 considered with boundary conditions bc as a self-adjoint operator in L2 ([0, π]). It is easy to describe the spectra and eigenfunctions of L0bc for bc = Per± , Dir: (a) Sp(L0Per+ ) = {n2 , n = 0, 2, 4, . . .}; its eigenspaces are En0 = Span{e±inx } for n > 0 and E00 = {const}, dim En0 = 2 for n > 0, and dim E00 = 1. (b) Sp(L0Per− ) = {n2 , n = 1, 3, 5, . . .}; its eigenspaces are En0 = Span{e±inx }, and dim En0 = 2. (c) Sp(L0Dir ) = {n2 , n ∈ N}; √each eigenvalue n2 is simple; a corresponding normalized eigenfunction is 2 sin nx. Depending on the boundary conditions, we consider as our canonical orthogonal normalized basis (o.n.b.) in L2 ([0, π]) the system uk (x), k ∈ Γbc , where if bc = Per+ −

if bc = Per if bc = Dir

uk = exp(ikx), k ∈ ΓPer+ = 2Z;

(3.1)

uk = exp(ikx), k ∈ ΓPer− = 1 + 2Z; √ uk = 2 sin kx, k ∈ ΓDir = N.

(3.2) (3.3)

Let us notice that {uk (x), k ∈ Γbc } is a complete system of unit eigenvectors of the operator L0bc . We set 1 1 HPer+ = f ∈ H 1 : f (π) = f (0) , HPer f ∈ H 1 : f (π) = −f (0) (3.4) − = and 1 HDir = f ∈ H1 :

f (π) = f (0) = 0 .

(3.5)

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P. Djakov and B. Mityagin

1 ikx , k∈ One can easily see that {eikx , k ∈ 2Z} is an orthonormal basis in HPer + , {e √ 1 1+2Z} is an orthonormal basis in HPer− , and { 2 sin kx, k ∈ N} is an orthonormal 1 . basis in HDir From here it follows that

& 1 2 2 2 fk uk (x) : f H 1 = (1 + k )|fk | < ∞ . (3.6) Hbc = f (x) = k∈Γbc

k∈Γbc

The following statement is well known. Lemma 9. " " ikx ikx (a) If f, g ∈ L1 ([0, π]) and f ∼ , g ∼ are their k∈2Z fk e k∈2Z gk e ikx Fourier series with respect to the system {e , k ∈ 2Z}, then the following conditions are equivalent: (i) f is absolutely continuous, f (π) = f (0) and f (x) = g(x) a.e.; (ii) gk = ikfk ∀k ∈ 2Z. " " (b) If f, g ∈ L1 ([0, π]) and f ∼ k∈1+2Z fk eikx , g ∼ k∈1+2Z gk eikx are their Fourier series with respect to the system {eikx , k ∈ 1+2Z}, then the following conditions are equivalent: (i∗ ) f is absolutely continuous, f (π) = −f (0) and f (x) = g(x) a.e.; (ii∗ ) gk = ikfk ∀k ∈ 1 + 2Z. Proof. An integration by parts gives the implication (i) ⇒ (ii) [or (i∗ ) ⇒ (ii∗ )]. *x To prove that (ii) ⇒ (i) we set G(x) = 0 g(t)dt. By (ii) for k = 0, we have *π G(π) = 0 g(t)dt = πg0 = 0. Therefore, integrating by parts we get 1 π 1 π −ikx gk = g(x)e−ikx dx = e dG(x) = ikGk , π 0 π 0 *π where Gk = π1 0 e−ikx G(x)dx is the kth Fourier coeﬃcient of G. Thus, by (ii), we have Gk = fk for k = 0, so by the Uniqueness Theorem for Fourier series f (x) = G(x) + const, i.e., (i) holds. Finally, the proof of the implication (ii∗ ) ⇒ (i∗ ) could be reduced to part " (a) by considering the functions f˜(x) = f (x)eix ∼ k∈2Z fk−1 eikx and g˜(x) = g(x)eix + if (x)eix . We omit the details. The next proposition gives the Fourier representations of the operators LPer± and their domains. 1 Proposition 10. In the above notations, if y ∈ HPer ± , then we have yk eikx ∈ D(LPer± ) and (y) = h = hk eikx ∈ H 0 y= ΓPer±

if and only if hk = hk (y) := k2 yk +

ΓPer±

m∈ΓPer ±

V (k − m)ym + Cyk ,

|hk |2 < ∞,

(3.7)

Fourier Method i.e.,

1 D(LPer± ) = y ∈ HPer ± :

and LPer± (y) =

215

(hk (y))k∈ΓPer± ∈ 2 (ΓPer± )

hk (y)eikx .

(3.8) (3.9)

k∈ΓPer±

Proof. Since the proof is the same in the periodic and anti-periodic cases, we consider only the case of periodic boundary conditions. By (2.68), if y ∈ D(LPer+ ), 1 then y ∈ HPer + and where Let y(x) =

(y) = −z − Qy + Cy = h ∈ L2 ([0, π]),

(3.10)

z := y − Qy ∈ W11 ([0, π]),

(3.11)

k∈2Z

yk eikx ,

z(x) =

k∈2Z

z(π) = z(0).

zk eikx ,

h(x) =

hk eikx

k∈2Z

be the Fourier series of y, z and h. Since z(π) = z(0), Lemma 9 says that the Fourier series of z may be obtained by diﬀerentiating term by term the Fourier 1 series of z, and the same property is shared by y as a function in HPer + . Thus, (3.10) implies q(k − m)imym + Cyk = hk . (3.12) −ikzk − m " On the other hand, by (3.11), we have zk = ikyk − m q(k − m)ym , so substituting that in (3.12) we get q(k − m)ym − q(k − m)imym + Cyk = hk , (3.13) −ik ikyk − m

m

which leads to (3.7) because V (m) = imq(m), m ∈ 2Z. Conversely, " if (3.7) holds, then we have (3.13). Therefore, (3.12) holds with zk = ikyk − m q(k − m)ym . " 1 Since y = yk eikx ∈ HPer + , the Fourier coeﬃcients of its derivative are ikyk , k ∈ 2Z. Thus, (zk ) is the sequence of Fourier coeﬃcients of the function z = y − Qy ∈ L1 ([0, π]). On the other hand, by (3.12), (ikzk ) is the sequence of Fourier coeﬃcients of an L1 ([0, π])-function. Therefore, by Lemma 9, the function z is absolutely continuous, z(π) = z(0), and (ikzk ) is the sequence of Fourier coeﬃcients of its derivative z . Thus, (3.10) and (3.11) hold, i.e., y ∈ D(LPer+ ) and LPer+ y = (y) = h. Now, we are ready to explain the Fourier method for studying the spectra of the operators LPer± . Let F : H 0 → 2 (ΓPer± ) be the Fourier isomorphisms deﬁned by corresponding to each function f ∈ H 0 the sequence (fk ) of its Fourier coeﬃcients fk = (f, uk ), where {uk , k ∈ ΓPer± } is, respectively, the basis (3.1) or (3.2). Let F −1 be the inverse Fourier isomorphism.

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P. Djakov and B. Mityagin

Consider the unbounded operators L+ and L− acting in 2 (ΓPer± ) as L± (z) = (hk (z))k∈Γ ± , hk (z) = k 2 zk + V (k − m)zm + Czk , (3.14) Per

m∈ΓPer±

respectively, in the domains D(L± ) = z ∈ 2 (|k|, ΓPer± ) : L± (z) ∈ 2 (ΓPer± ) ,

(3.15)

where 2 (|k|, ΓPer± ) is the weighted 2 -space

2

(|k|, ΓPer± ) =

z = (zk )k∈ΓPer±

& 2 2 : (1 + |k| )|zk | < ∞ . k

In view of (3.6) and Proposition 10, the following theorem holds. Theorem 11. In the above notations, we have D(LPer± ) = F −1 (D(L± ))

(3.16)

LPer± = F −1 ◦ L± ◦ F .

(3.17)

and If it does not lead to confusion, for convenience we will loosely use one and the same notation LPer± for the operators LPer± and L± .

4. Fourier representation for the Hill–Schr¨ odinger operator with Dirichlet boundary conditions In this section we study the Hill–Schr¨ odinger operator LDir (v), v = C + Q , generated by the diﬀerential expression Q (y) = −(y − Qy) − Qy considered on the interval [0, π] with Dirichlet boundary conditions Dir :

y(0) = y(π) = 0.

Its domain is

D(LDir (v)) = y ∈ H 1 : y − Qy ∈ W11 ([0, π]), y(0) = y(π) = 0, Q (y) ∈ H 0 , (4.1) and we set (4.2) LDir (v)y = Q (y). y2 y1 and be the solutions of the homogeneous system (2.20) Lemma 12. Let u1 u2 which satisfy y2 (0) y1 (0) 1 0 , . (4.3) = = 0 1 u1 (0) u2 (0) If y2 (π) = 0,

(4.4)

Fourier Method

217

then the non-homogeneous system (2.19) has, for each f ∈ H 0 , a unique solution (y, u) = (R1 (f ), R2 (f )) such that y(0) = 0,

y(π) = 0.

(4.5) 0

1

Moreover, R1 is a linear continuous operator from H into H , and R2 is a linear continuous operator in H 0 with a range in W11 ([0, π]). Proof. By the variation of parameters method, every solution of the system (2.19) has the form y1 (x) y2 (x) y(x) + v2 (x) , = v1 (x) u1 (x) u2 (x) u(x) where x x v1 (x) = − y2 (x)f (t)dt + C1 , v2 (x) = y1 (x)f (t)dt + C2 . (4.6) 0

0

By (4.3), the condition y(0) = 0 will be satisﬁed if and only if C1 = 0. If so, the second condition y(π) = 0 in (4.5) is equivalent to m1 (f )y1 (π) + (m2 (f ) + C2 )y2 (π) = 0, where

m1 (f ) = −

π

y2 (x)f (t)dt,

π

m2 (f ) =

0

y1 (x)f (t)dt. 0

Thus, if y2 (π) = 0, then we have unique solution (y, u) of (2.19) that satisﬁes (4.5), and it is given by (4.6) with C1 = 0 and C2 (f ) = −

y1 (π) m1 (f ) − m2 (f ). y2 (π)

R1 (f ) y(x) , where = Thus, we have R2 (f ) u(x) x R1 (f ) = − y2 (x)f (t)dt · y1 (x) +

0

and R2 (f ) =

− 0

x

(4.7)

x

y1 (x)f (t)dt + C2 (f ) · y2 (x)

x

y1 (x)f (t)dt + C2 (f ) · u2 (x).

0

y2 (x)f (t)dt · u1 (x) + 0

It is easy to see (compare with the proof of Lemma 5) that R1 is a linear continuous operator from H 0 into H 1 , and R2 is a linear continuous operator in H 0 with a range in W11 ([0, π]). We omit the details. Now, let us consider the systems(2.19) and parameter (2.20) with a spectral y2 (x, λ) y1 (x, λ) and be the solutions λ by setting C = −λ there, and let u1 (x, λ) u2 (x, λ) of the homogeneous system (2.20) that satisfy (4.3) for x = 0. Notice that y2 (v; x, λ) = y2 (v; x, λ).

(4.8)

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P. Djakov and B. Mityagin

−1 (R) is π-periodic. Then, Theorem 13. Suppose v ∈ Hloc

(a) the domain D(LDir (v)) ∈ (4.1) is dense in H 0 ; (b) the operator LDir (v) is closed, and its conjugate operator is (LDir (v))∗ = LDir (v).

(4.9)

In particular, if v is real-valued, then the operator LDir (v) is self-adjoint. (c) the spectrum Sp(LDir (v)) of the operator LDir (v) is discrete, and moreover, Sp(LDir (v)) = {λ ∈ C : y2 (π, λ) = 0}.

(4.10)

Proof. Integration by parts shows that the operators LDir (v) and LDir (v) are formally adjoint, i.e., (LDir (v)y, h) = (f, LDir (v)h)

if y ∈ D(LDir (v)), h ∈ D(LDir (v)).

(4.11)

Now we apply Lemma 3 with A = LDir (v) and B = LDir (v). Choose λ ∈ C so that y2 (v; π, λ) = 0 (in view of (2.44), see the remark before Theorem 4, y2 (v; π, λ) is a non-constant entire function, so such a choice is possible). Then, in view of (4.8), we have y2 (v; π, λ) = 0 also. By Lemma 12, LDir (v) − λ maps bijectively D(LDir (v)) onto H 0 and LDir (v) − λ maps bijectively D(LDir (v)) onto H 0 . Thus, by Lemma 3, D(LDir (v)) is dense in H 0 and (LDir (v))∗ = LDir (v), i.e., (a) and (b) hold. If y2 (v; π, λ) = 0, then λ is an eigenvalue of the operator LDir (v), and y2 (v; x, λ) is a corresponding eigenvector. In view of Lemma 12, this means that (4.10) holds. Since y2 (π, λ) is a non-constant entire function, the set on the right in (4.10) is discrete. This completes the proof of (c). Lemma 14. (a) If f, g ∈ L1 ([0, π]) and f∼

∞

√

fk 2 sin kx,

g ∼ g0 +

k=1

∞

√ gk 2 cos kx

k=1

are, respectively, their sine and cosine Fourier series, then the following conditions are equivalent: (i) f is absolutely continuous, f (0) = f (π) = 0 and g(x) = f (x) a.e.; (ii) g0 = 0, gk = kfk ∀k ∈ N. (b) If f, g ∈ L1 ([0, π]) and f ∼ f0 +

∞ k=1

√ fk 2 cos kx,

g∼

∞

√ gk 2 sin kx

k=1

are, respectively, their cosine and sine Fourier series, then the following conditions are equivalent: (i∗ ) f is absolutely continuous and g(x) = f (x) a.e.; (ii∗ ) gk = −kfk k ∈ N.

Fourier Method

219

*π Proof. (a) We have (i) ⇒ (ii) because g0 = π1 0 g(x)dx = π1 (f (π) − f (0)) = 0, and π √ √ √ 1 π 1 k π gk = g(x) 2 cos kxdx = f (x) 2 cos kx 0 + f (x) 2 sin kxdx = kfk π 0 π π 0 for every k ∈ N. *x To prove that (ii) ⇒ (i), we set G(x) = 0 g(t)dt; then G(π) = G(0) = 0 because g0 = 0. The same computation as above shows that gk = kGk ∀k ∈ N, so the sine Fourier coeﬃcients of two L1 -functions G and f coincide. Thus, G(x) = f (x), which completes the proof of (a). The proof of (b) is omitted because it is similar to the proof of (a).

|

Let Q∼

∞

√ q˜(k) 2 sin kx

(4.12)

k=1

be the sine Fourier expansion of Q. We set also V˜ (0) = 0, V˜ (k) = k q˜(k)

for k ∈ N.

(4.13)

1 Proposition 15. In the above notations, if y ∈ HDir , then we have

y=

∞

yk sin kx ∈ D(LDir )

and

(y) = h =

k=1

∞

√ hk 2 sin kx ∈ H 0

k=1

if and only if ∞ % 1 $˜ V (|k − m|) − V˜ (k + m) ym +Cyk , hk = hk (y) = k 2 yk + √ 2 m=1

|hk |2 < ∞, (4.14)

i.e., 1 : D(LDir ) = y ∈ HDir

2 (hk (y))∞ 1 ∈ (N) ,

LDir (y) =

∞

√ hk (y) 2 sin kx.

k=1

(4.15) Proof. By (4.1), if y ∈ D(LDir ), then y ∈

1 HDir

and

(y) = −z − Qy + Cy = h ∈ L2 ([0, π]), where

z := y − Qy ∈ W11 ([0, π]).

Let y∼

∞

√ yk 2 sin kx,

k=1

z∼

∞

√ zk 2 cos kx,

k=1

h∼

(4.16) ∞

√ hk 2 sin kx

k=1

be the sine series of y and h, and the cosine series of z. Lemma 14 yields ∞ ∞ √ √ (−kzk ) 2 sin kx, y ∼ kyk 2 cos kx. z ∼ k=1

k=1

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P. Djakov and B. Mityagin

Therefore, hk = kzk − (Qy )k + Cyk ,

k ∈ N,

(4.17)

where (Qy )k are the sine coeﬃcients of the function Qy ∈ L ([0, π]). By (4.16), we have 1

zk = kyk − (Qy)k , where (Qy)k is the kth cosine coeﬃcient of Qy. It can be found by the formula ∞ √ 1 π Q(x)y(x) 2 cos kxdx = am · y m , (Qy)k = π 0 m=1 with am = am (k) =

1 π

π

0

1 π

π

√ √ Q(x) 2 cos kx 2 sin mxdx =

0

⎧ ⎪q˜(m + k) + q˜(m − k), m > k 1 ⎨ Q(x)[sin(m + k)x + sin(m − k)x]dx = √ q˜(2k), m=k 2⎪ ⎩ q˜(m + k) − q˜(k − m) m < k.

Therefore, ∞ k−1 ∞ 1 1 1 (Qy)k = √ q˜(m+k)ym − √ q˜(k−m)ym + √ q˜(m−k)ym . (4.18) 2 m=1 2 m=1 2 m=k+1

In an analogous way we can ﬁnd the sine coeﬃcients of Qy by the formula ∞ √ 1 π

(Qy )k = Q(x)y (x) 2 sin kxdx = bm · mym , π 0 m=1 √ where bm are the cosine coeﬃcients of Q(x) 2 sin kx, i.e., √ √ 1 π Q(x) 2 sin kx 2 cos mx = bm = bm (k) = π 0 1 π

π

0

⎧ ⎪q˜(k + m) + q˜(k − m), m < k, 1 ⎨ Q(x)[sin(k + m)x + sin(k − m)x]dx = √ q˜(2k), m = k, 2⎪ ⎩ q˜(k + m) − q˜(m − k) m > k.

Thus we get ∞ k−1 ∞ 1 1 1 √ √ √ (Qy )k = q˜(m+k)mym + q˜(k −m)mym − q˜(m−k)mym . 2 m=1 2 m=1 2 m=k+1 (4.19)

Fourier Method

221

Finally, (4.18) and (4.19), imply that ∞ 1 (m + k)˜ q (m + k) k 2 yk − k(Qy)k − (Qy )k = k 2 yk − √ 2 m=1 ∞ k−1 1 1 (m − k)˜ q (m − k) + √ (k − m)˜ q (k − m). +√ 2 m=k+1 2 m=1

Hence, in view of (4.13), we have ∞ % 1 $˜ 2 √ hk = k yk + V (|k − m|) − V˜ (k + m) ym + Cyk , 2 m=1 i.e., (4.14) holds. Conversely, if (4.14) holds, then going back we can see, by (4.17), that z = y −Qy ∈ L2 ([0, π]) has the property that kzk , k ∈ N, are the sine coeﬃcients of an L1 ([0, π])-function. Therefore, by Lemma 14, z is absolutely continuous and those numbers are the sine coeﬃcients of its derivative z . Hence, z = y −Qy ∈ W11 ([0, π]) and (y) = h, i.e., y ∈ D(LDir ) and LDir (y) = h. Let F : H 0 → 2 (N ) be the Fourier isomorphisms that corresponds to √ each function f ∈ H 0 the sequence (fk )k∈N of its Fourier coeﬃcients fk = (f, 2 sin kx), and let F −1 be the inverse Fourier isomorphism. Consider the unbounded operator Ld and acting in 2 (N) as % 1 $˜ V (|k − m|) − V˜ (k + m) zm +Czk Ld (z) = (hk (z))k∈N , hk (z) = k 2 zk + √ 2 m∈N (4.20) in the domain D(Ld ) = z ∈ 2 (|k|, N) : Ld (z) ∈ 2 (N) , (4.21) where 2 (|k|, N) is the weighted 2 -space

(|k|, N) = 2

z = (zk )k∈N :

& |k| |zk | < ∞ . 2

2

k

In view of (3.6) and Proposition 15, the following theorem holds. Theorem 16. In the above notations, we have D(LDir ) = F −1 (D(Ld ))

(4.22)

LDir = F −1 ◦ Ld ◦ F .

(4.23)

and If it does not lead to confusion, for convenience we will loosely use one and the same notation LDir for the operators LDir and Ld .

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5. Localization of spectra Throughout this section we need the following lemmas. Lemma 17. For each n ∈ N

k=±n

1 2 log 6n < ; |n2 − k 2 | n

k=±n

(5.1)

4 1 < 2. |n2 − k 2 |2 n

(5.2)

The proof is elementary; just apply consistently the identity 1 1 1 1 . + = n2 − k 2 2n n − k n + k Therefore we omit it. Lemma 18. There exists an absolute constant C > 0 such that (a) if n ∈ N and b ≥ 2, then 1 log b ≤C √ ; |n2 − k 2 | + b b

(5.3)

k

(b) if n ≥ 0 and b > 0 then 1 C ≤ 2 . |n2 − k2 |2 + b2 (n + b2 )1/2 (n4 + b2 )1/4

(5.4)

k=±n

A proof of this lemma can be found in [9], see Appendix, Lemma 79. We study the localization of spectra of the operators LPer± and LDir by using their Fourier representations. By (3.14) and Theorem 11, each of the operators L = LPer± has the form (5.5) L = L0 + V, 0 where the operators L " and V are deﬁned by their action on the sequence of Fourier 1 coeﬃcients of any y = Γ ± yk exp ikx ∈ HPer ± : Per

L : (yk ) → (k 2 yk ), 0

and V : (ym ) → (zk ),

zk =

m 0

k ∈ ΓPer±

V (k − m)ym ,

k, m ∈ ΓPer± .

(5.6) (5.7)

(We suppress in the notations of L and V the dependence on the boundary conditions Per± .) In the case of Dirichlet boundary condition, by (4.20) and Theorem 16, the operator L = LDir has the form (5.5), where the operators L0 and V √ are deﬁned " by their action on the sequence of Fourier coeﬃcients of any y = N yk 2 sin kx ∈ 1 HDir : (5.8) L0 : (yk ) → (k 2 yk ), k ∈ N

Fourier Method and V : (ym ) → (zk ),

223

% 1 $˜ V (|k − m|) − V˜ (k + m) ym , zk = √ 2 m

k, m ∈ N.

(5.9) (We suppress in the notations of L0 and V the dependence on the boundary conditions Dir .) Of course, in the regular case where v ∈ L2 ([0, π]), the operators L0 and V are, respectively, the Fourier representations of −d2 /dx2 and the multiplication −1 operator y → v · y. But if v ∈ Hloc (R) is a singular periodic potential, then the situation is more complicated, so we are able to write (5.5) with (5.6) and (5.7), or (5.8) and (5.9), only after having the results from Section 3 and 4 (see Theorem 11 and Theorem 16). In view of (5.6) and (5.8) the operator L0 is diagonal, so, for λ = k 2 , k ∈ Γbc , we may consider (in the space 2 (Γbc )) its inverse operator zk Rλ0 : (zk ) → (5.10) , k ∈ Γbc . λ − k2 One of the technical diﬃculties that arises for singular potentials is connected with the standard perturbation type formulae for the resolvent Rλ = (λ − L0 − V )−1 . In the case where v ∈ L2 ([0, π]) one can represent the resolvent in the form (e.g., see [9], Section 1.2) Rλ = (1 − Rλ0 V )−1 Rλ0 =

∞

(Rλ0 V )k Rλ0 ,

(5.11)

Rλ0 (V Rλ0 )k .

(5.12)

k=0

or Rλ = Rλ0 (1 − V Rλ0 )−1 =

∞ k=0

The simplest conditions that guarantee the convergence of the series (5.11) or (5.12) in 2 are Rλ0 V < 1, respectively, V Rλ0 < 1. Each of these conditions can be easily veriﬁed for large enough n if Re λ ∈ [n − 1, n + 1] and |λ − n2 | ≥ C(v), which leads to a series of results on the spectra, zones of instability and spectral decompositions. The situation is more complicated if v is a singular potential. Then, in general, there are no good estimates for the norms of Rλ0 V and V Rλ0 . However, one can write (5.11) or (5.12) as Rλ = Rλ0 + Rλ0 V Rλ0 + Rλ0 V Rλ0 V Rλ0 + · · · = Kλ2 +

∞

Kλ (Kλ V Kλ )m Kλ , (5.13)

m=1

provided (Kλ )2 = Rλ0 .

(5.14)

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We deﬁne an operator K = Kλ with the property (5.14) by its matrix representation 1 Kjm = δjm , j, m ∈ Γbc , (5.15) (λ − j 2 )1/2 where z 1/2 =

√

reiϕ/2

if z = reiϕ , 0 ≤ ϕ < 2π.

Then Rλ is well deﬁned if KλV Kλ : 2 (Γbc ) → 2 (Γbc ) < 1.

(5.16)

In view of (2.14), (5.7) and (5.15), the matrix representation of KV K for periodic or anti-periodic boundary conditions bc = Per± is (KV K)jm =

V (j − m) i(j − m)q(j − m) = , 2 1/2 (λ − −m ) (λ − j 2 )1/2 (λ − m2 )1/2 j 2 )1/2 (λ

(5.17)

where j, m ∈ 2Z for bc = Per+ , and j, m ∈ 1 + 2Z for bc = Per− . Therefore, we have for its Hilbert–Schmidt norm (which majorizes its 2 -norm)

KV K2HS =

j,m∈ΓPer±

(j − m)2 |q(j − m)|2 . |λ − j 2 ||λ − m2 |

(5.18)

By (4.13), (5.9) and (5.15), the matrix representation of KV K for Dirichlet boundary conditions bc = Dir is V˜ (|j − m|) V˜ (j + m) 1 1 (KV K)jm = √ −√ 2 1/2 2 1/2 2 2 (λ − j ) (λ − m ) 2 (λ − j )1/2 (λ − m2 )1/2 (5.19) |j − m|˜ q (|j − m|) (j + m)˜ q (j + m) 1 1 = √ −√ . 2 (λ − j 2 )1/2 (λ − m2 )1/2 2 (λ − j 2 )1/2 (λ − m2 )1/2 where j, m ∈ N. Therefore, we have for its Hilbert–Schmidt norm (which majorizes its 2 -norm) KV K2HS ≤ 2

(j + m)2 |˜ (j − m)2 |˜ q (|j − m|)|2 q (j + m)|2 +2 . (5.20) 2 2 2 |λ − j ||λ − m | |λ − j ||λ − m2 |

j,m∈N

j,m∈N

We set for convenience q˜(0) = 0,

r˜(s) = q˜(|s|) for s = 0,

s ∈ Z.

(5.21)

In view of (5.20) and (5.21), we have KV K2HS ≤

(j − m)2 |˜ r (j − m)|2 . |λ − j 2 ||λ − m2 |

(5.22)

j,m∈Z

We divide the plane C into strips, correspondingly to the boundary conditions, as follows:

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225

– if bc = Per+ then C = H0 ∪ H2 ∪ H4 ∪ · · · , and – if bc = Per− then C = H1 ∪ H3 ∪ H5 ∪ · · · , where H0 = {λ ∈ C : Re λ ≤ 1},

H1 = {λ ∈ C : Re λ ≤ 4},

Hn = {λ ∈ C : (n − 1) ≤ Re λ ≤ (n + 1) }, 2

2

(5.23)

n ≥ 2;

(5.24)

– if bc = Dir, then C = G1 ∪ G2 ∪ G3 ∪ · · · , where G1 = {λ : Re λ ≤ 2},

Gn = {λ : (n − 1)n ≤ Re λ ≤ n(n + 1)},

n ≥ 2. (5.25)

Consider also the discs Dn = {λ ∈ C : |λ − n2 | < n/4}, Then, for n ≥ 3, 1 log n ≤ C1 , |λ − k 2 | n k∈n+2Z

and

k∈Z

1 log n ≤ C1 , |λ − k2 | n

k∈n+2Z

k∈Z

n ∈ N,

C1 1 ≤ 2, |λ − k 2 |2 n

C1 1 ≤ 2, |λ − k2 |2 n

∀λ ∈ Hn \ Dn ,

∀λ ∈ Gn \ Dn ,

(5.26)

(5.27)

(5.28)

where C1 is an absolute constant. Indeed, if λ ∈ Hn , then one can easily see that |λ − k 2 | ≥ |n2 − k 2 |/4 for k ∈ n + 2Z. Therefore, if λ ∈ Hn \ Dn , then (5.1) implies that 2 8 8 log 6n 1 4 log n ≤ + ≤ + ≤ C1 , 2 |λ − k | n/4 |n2 − k 2 | n n n k∈n+2Z

k=±n

which proves the ﬁrst inequality in (5.27). The second inequality in (5.27) and the inequalities in (5.28) follow from Lemma 17 by the same argument. Next we estimate the Hilbert–Schmidt norm of the operator Kλ V Kλ for bc = Per± or Dir, and correspondingly, λ ∈ Hn \ Dn or λ ∈ Gn \ Dn , n ∈ N. For each 2 -sequence x = (x(j))j∈Z and m ∈ N we set ⎛ ⎞1/2 |x(j)|2 ⎠ . (5.29) Em (x) = ⎝ |j|≥m

√ " "∞ Lemma 19. Let v = Q , where Q(x) = k∈2Z q(k)eikx = m=1 q˜(m) 2 sin mx is a π-periodic L2 ([0, π]) function, and let q = (q(k))k∈2Z ,

q˜ = (˜ q (m))m∈N

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be the sequences of its √ Fourier coeﬃcients with respect to the orthonormal bases {eikx , k ∈ 2Z} and { 2 sin mx, m ∈ N}. Then, for n ≥ 3, . √ / Kλ V Kλ HS ≤ C E√n (q) + q/ n , λ ∈ Hn \ Dn , bc = Per± , (5.30) and

. √ / Kλ V Kλ HS ≤ C E√n (˜ q ) + ˜ q/ n ,

λ ∈ Gn \ Dn , bc = Dir,

(5.31)

where C is an absolute constant. Proof. Fix n ∈ N. We prove only (5.30) because, in view of (5.21) and (5.22), the proof of (5.31) is practically the same (the only diﬀerence is that the summation indices will run in Z). By (5.18), + , s2 2 KV KHS ≤ |q(s)|2 = Σ1 + Σ2 + Σ3 , (5.32) 2 ||λ − (m + s)2 | |λ − m s m where s ∈ 2Z, m ∈ n + 2Z and Σ1 = · · · , Σ2 = √ |s|≤ n

√

··· ,

Σ3 =

m∈n+2Z

··· .

(5.33)

|s|>4n

n<|s|≤4n

The Cauchy inequality implies that 1 ≤ |λ − m2 ||λ − (m + s)2 |

m∈n+2Z

Thus, by (5.28) and (5.29), √ C1 C1 C1 |q(s)|2 s2 2 ≤ ( n)2 2 q2 = q2 , Σ1 ≤ n n n √

1 . |λ − m2 |2

λ ∈ Hn \ D n ,

(5.34)

(5.35)

|s|≤ n

Σ2 ≤ (4n)2

/2 . C1 |q(s)|2 = 16C1 E√n (q) , 2 n √

λ ∈ Hn \ Dn .

(5.36)

|s|> n

Next we estimate Σ3 for n ≥ 3. First we show that if |s| > 4n then s2 C1 log n ≤ 16 , λ ∈ Hn \ D n . 2 2 |λ − m ||λ − (m + s) | n m

(5.37)

Indeed, if |m| ≥ |s|/2, then (since |s|/4 > n ≥ 3) |λ − m2 | ≥ m2 − | Re λ| ≥ s2 /4 − (n + 1)2 > s2 /4 − (|s|/4 + 1)2 ≥ s2 /8. Thus, by (5.27), |m|≥|s|/2

s2 8 C1 log n ≤ ≤8 2| |λ − m2 ||λ − (m + s)2 | |λ − (m + s) n m

for λ ∈ Hn \ Dn . If |m| < |s|/2, then |m + s| > |s| − |s|/2 = |s|/2, and therefore, |λ − (m + s)2 | ≥ (m + s)2 − | Re λ| ≥ s2 /4 − (n + 1)2 ≥ s2 /8.

Fourier Method Therefore, by (5.27), |m|<|s|/2

|λ −

227

C1 log n s2 8 ≤ ≤8 2 2| − (m + s) | |λ − m n m

m2 ||λ

for λ ∈ Hn \ Dn , which proves (5.37). Now, by (5.37), C1 log n C1 log n 2 Σ3 ≤ 16 (E4n (q)) . |q(s)|2 = 16 n n

(5.38)

|s|≥4n

Finally, (5.32), (5.35), (5.36) and (5.38) imply (5.30).

Let H N denote the half-plane H N = {λ ∈ C : Re λ < N 2 + N },

N ∈ N,

(5.39)

and let RN be the rectangle RN = {λ ∈ C : −N < Re λ < N 2 + N,

|Imλ| < N }.

Lemma 20. In the above notations, for bc = Per± or Dir, we have (log N )1/2 N √ q + E4 N (q) , sup Kλ V Kλ HS , λ ∈ H \ RN ≤ C N 1/4

(5.40)

(5.41)

where C is an absolute constant, and q is replaced by q˜ if bc = Dir . Proof. Consider the sequence r = (r(s))s∈Z , deﬁned by

0 for odd s, r(s) = . max(|q(s)|, |q(−s)|) for even s, Then, in view of (5.42), we have r ∈ 2 (Z) and r ≤ 2q. If bc = Per± , then we have, by (5.18), (j − m)2 |r(j − m)|2 . KV K2HS ≤ |λ − j 2 ||λ − m2 |

(5.42)

(5.43)

j,m∈Z

On the other hand, if bc = Dir, then (5.22) gives the same estimate for KV K2HS but with r replaced by the sequence r˜ ∈ (5.21). So, to prove (5.41), it is enough to estimate the right side of (5.43) for λ ∈ H N \ RN }. If Re λ ≤ −N, then (5.43) implies that (j − m)2 |r(j − m)|2 KV K2HS ≤ . |N + j 2 ||N + m2 | j,m∈Z

On the other hand, for b ≥ 1, the following estimate holds: (j − m)2 |r(j − m)|2 1+π ≤ 4r2 . |b2 + j 2 ||b2 + m2 | b j,m∈Z

(5.44)

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P. Djakov and B. Mityagin

Indeed, the left-hand side of (5.44) does not exceed 2(j 2 + m2 )|r(j − m)|2 |b2 + j 2 ||b2 + m2 |

j,m∈Z

1 1 |r(j − m)|2 + 2 |r(j − m)|2 2 2 2 + m | |b + j | m m j j ∞ 1 1 π 1 1+π 2 2 ≤ 4r2 . ≤ 4r +2 dx = 4r + 2 + x2 2 b2 b b b b 0 √ Now, with b = N , (5.44) yields ≤2

|b2

r2 KV K2HS ≤ C √ N

if Re λ ≤ −N,

where C is an absolute constant. By (5.43) and the elementary inequality ) √ |λ − m2 | = (x − m2 )2 + y 2 ≥ (|x − m2 | + |y|)/ 2, we have Kλ V Kλ 2HS ≤

(5.45)

λ = x + iy,

σ(x, y; s)|r(s)|2 ,

(5.46)

s∈Z

where σ(x, y; s) =

m∈Z

2s2 . (|x − m2 | + |y|)(|x − (m + s)2 | + |y|)

(5.47)

Now, suppose that λ = x + iy ∈ H N \ RN and |y| ≥ N. By (5.25) and (5.39), 9 Gn , HN ⊂ 1≤n≤N

so λ ∈ Gn for some n ≤ N. Moreover, σ(x, y; s) ≤ 16σ(n2 , N ; s) if λ ∈ Gn , |y| ≥ N.

(5.48)

Indeed, then one can easily see that |x − m2 | + |y| ≥

1 2 (|n − m2 | + N ), 4

m ∈ Z,

which implies (5.48). By (5.47) and (5.48), if λ = x + iy ∈ Gn \ RN and |y| ≥ N, then Kλ V Kλ 2HS ≤ σ(n2 , N ; s)|r(s)|2 ≤ Σ1 + Σ2 + Σ3 ,

(5.49)

s

where Σ1 =

√ |s|≤4 N

σ(n2 , N ; s)|r(s)|2 ,

Σ2 =

√ 4 N <|s|≤4n

··· ,

Σ3 =

|s|>4n

··· .

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229

√ If |s| ≤ 4 N , then the Cauchy inequality and (5.4) imply that C 32C 1 σ(n2 , N ; s) ≤ 32N · ≤ 32N ≤ √ . 2 − m2 |2 + N 2 4 + N 2 )1/4 |n N (n N m Thus 32C Σ1 ≤ √ r2 . N

(5.50)

. /2 Σ2 ≤ 32C · E4√N (r) .

(5.51)

√ If 4 N < |s| ≤ 4n then the Cauchy inequality and (5.4) yield 1 C ≤ 32n2 ≤ 32C σ(n2 , N ; s) ≤ 32n2 · 2 2 2 2 4 |n − m | + N N (n + N 2 )1/4 m because n ≤ N. Thus

Let |s| > 4n. If |m| < |s|/2 then |m + s| ≥ |s|/2, and therefore, |n2 − (m + s)2 | ≥ |m + s|2 − n2 ≥ (|s|/2)2 − (|s|/4)2 ≥ s2 /8. Thus, by (5.3), |m|<|s|/2

log N s2 8 ≤ ≤ 8C √ . 2 − m2 | + N (|n2 − m2 | + N )(|n2 − (m + s)2 | + N ) |n N m

If |m| ≥ |s|/2, then we have the same estimate because m2 − n2 ≥ (|s|/2)2 − (|s|/4)2 ≥ s2 /8, and therefore, again by (5.3), |m|≥|s|/2

s2 (|n2 − m2 | + N )(|n2 − (m + s)2 | + N )

log N 8 ≤ 8C √ . 2 − (m + s)2 | + N |n N m √ Thus σ(n2 , N ; s) ≤ 32C(log N )/ N , so we have ≤

log N Σ3 ≤ 32Cr2 √ . N

(5.52)

Now, in view of (5.42) and (5.21), the estimates (5.45) and (5.50)–(5.52) yield (5.41), which completes the proof. −1 Theorem 21. For each periodic potential v ∈ Hloc (R), the spectrum of the operators ± Lbc (v) with bc = Per , Dir is discrete. Moreover, if bc = Per± then, respectively, for each large enough even number N + > 0 or odd number N − , we have 9 Dn , (5.53) Sp (LPer± ) ⊂ RN ± ∪ n∈N ± +2N

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where RN is the rectangle (5.40), Dn = {λ : |λ − n2 | < n/4}, and

2N + + 1 # (Sp (LPer± ) ∩ RN ± ) = , 2N − # (Sp (LPer± ) ∩ Dn ) = 2 for n ∈ N ± + 2N, where each eigenvalue is counted with its algebraic multiplicity. If bc = Dir then, for each large enough number N ∈ N, we have Sp (LDir ) ⊂ RN ∪

∞ 9

Dn

(5.54)

# (Sp (LDir )) ∩ Dn ) = 1 for n > N.

(5.55)

n=N +1

and # (Sp (LDir ) ∩ RN ) = N + 1,

Proof. In view of (5.13), the resolvent Rλ is well deﬁned if KV K < 1. Therefore, (5.53) and (5.54) follow from Lemmas 19 and 20. To prove (5.54) and (5.55) we use a standard method of continuous parametrization. Let us consider the one-parameter family of potentials vτ (x) = τ v(x), τ ∈ [0, 1]. Then, in the notation of Lemma 19, we have vτ = τ ·Q , and the assertions of Lemmas 19 and 20 hold with q and q˜ replaced, respectively, by τ · q and τ · q˜. Therefore, (5.53) and (5.54) hold, with Lbc = Lbc (v) replaced by Lbc (vτ ). Moreover, the corresponding resolvents Rλ (Lbc (vτ )) are analytic in λ and continuous in τ. Now, let us prove the ﬁrst formula in (5.54) in the case bc = Per+ . Fix an even N + ∈ N so that (5.53) holds, and consider the projection 1 P N (τ ) = (λ − LPer+ (vτ ))−1 dλ. (5.56) 2πi λ∈∂RN / . The dimension dim P N (τ ) gives the number of eigenvalues inside the rectangle RN . Being an integer, it is a constant, so, by the relation (a) at the begging of Section 3, we have dim P N (1) = dim P N (0) = 2N + + 1. In view of the relations (a)–(c) at the begging of Section 3, the same argument shows that (5.54) and (5.55) hold in all cases. Remark. It is possible to choose the disks Dn = {λ : |λ−n2 | < rn } in Lemma√19 so that rn /n → 0. Indeed, if we take rn = n/ϕ(n), where ϕ(n) → ∞ but ϕ(n)/ n → 0 and ϕ(n)E√n (W ) → 0, then, modifying the proof of Lemma 19, one can get that Kλ V Kλ HS → 0 as n → ∞. Therefore, Theorem 21 could be sharpen: for large enough N ± and N , (5.53)–(5.55) hold with Dn = {λ : |λ − n2 | < rn } for some sequence {rn } such that rn /n → 0.

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231

6. Conclusion 1. The main goal of our paper was to bring into the framework of Fourier method −1 the analysis of Hill–Schr¨odinger operators with periodic Hloc (R) potential, considered with periodic, antiperiodic and Dirichlet boundary conditions. As soon as this is done we can apply the methodology developed in [20, 7, 8] (see a detailed exposition in [9]) to study the relationship between smoothness of a potential v − and rates of decay of spectral gaps γn = λ+ n − λn and deviations δn under a weak a −1 priori assumption v ∈ H . (In [20, 7, 8, 9] the basic assumption is v ∈ L2 ([0, π]).) Still, there is a lot of technical problems. All the details and proofs are given in [11]. Now we remind the results themselves. odinger operator with a real-valued (A) Let L = L0 + v(x) be a Hill–Schr¨ −1 π-periodic potential v ∈ Hloc (R), and let γ = (γn ) be its gap sequence. If ω = (ω(n))n∈Z is a sub-multiplicative weight such that log ω(n) 0 n

as

n → ∞,

(6.1)

then, with Ω = (Ω(n)),

Ω(n) =

ω(n) , n

we have γ ∈ 2 (N, Ω) ⇒ v ∈ H(Ω). If Ω is a sub-multiplicative weight of exponential type, i.e., log Ω(n) lim > 0, n→∞ n then there exists ε > 0 such that γ ∈ 2 (N, Ω) ⇒ v ∈ H(eε|n| ).

(6.2)

(6.3)

(6.4)

The statement (A) is a stronger version of Theorem 54 in [9]; see Section 6 and Theorem 28 in [11]. (B) Let L = L0 + v(x) be the Hill–Schr¨ odinger operator with a π-periodic −1 potential v ∈ Hloc (R). Then, for large enough n > N (v) the operator L has, in a disc of center n2 and radius rn = n/4, exactly two (counted with their algebraic multiplicity) − periodic (for even n), or antiperiodic (for odd n) eigenvalues λ+ n and λn , and one Dirichlet eigenvalue μn . Let − + Δn = |λ+ n > N (v); (6.5) n − λn | + |λn − μn |, then, for each sub-multiplicative weight ω and ω(n) Ω = (Ω(n)), Ω(n) = , n we have v ∈ H(Ω) ⇒ (Δn ) ∈ 2 (Ω). (6.6)

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Conversely, in the above notations, if ω = (ω(n))n∈Z is a sub-multiplicative weight such that log ω(n) 0 as n → ∞, (6.7) n then (Δn ) ∈ 2 (Ω) ⇒ v ∈ H(Ω). (6.8) If ω is a sub-multiplicative weight of exponential type, i.e., lim

n→∞

log ω(n) >0 n

(6.9)

then (Δn ) ∈ 2 (Ω) ⇒ ∃ε > 0 : v ∈ H(eε|n| ).

(6.10)

Statement (B) is a stronger version of Theorem 67 in [9]; see Section 7 and Theorem 29 in [11]. 2. Throughout the paper and in Statements (A) and (B) we consider three types of boundary conditions: Per± and Dir in the form (a∗ ) , (b∗ ) and (c∗ ≡ c) adjusted to the diﬀerential operators (1.1) with singular potentials v ∈ H −1 . It is worth to observe that if v happens to be a regular potential, i.e., v ∈ L2 ([0, π]) (or even v ∈ H α , α > −1/2) the boundary conditions (a∗ ) and (b∗ ) automatically become equivalent to the boundary conditions (a) and (b) as we used to write them in the regular case. Indeed (see the paragraph after (2.67)), we have (a∗ ) Per+ :

y(π) = y(0), (y − Qy) (π) = (y − Qy) (0).

Therefore, with v ∈ L2 , both the L2 -function Q and the quasi-derivative u = y −Qy are continuous functions, so the two terms y and Qy can be considered separately. Then the second condition in (a∗ ) can be rewritten as y (π) − y (0) = Q(π)y(π) − Q(0)y(0).

(6.11)

But, since Q is π-periodic (see Proposition 1), Q(π) = Q(0),

(6.12)

∗

and with the ﬁrst condition in (a ) the right side of (6.11) is Q(0)(y(π)−y(0)) = 0. Therefore, (a∗ ) comes to the form (a) y(π) = y(0),

y (π) = y (0).

Of course, in the same way the condition (b∗ ) automatically becomes equivalent to (b) if v ∈ H α , α > −1/2. A. Savchuk and A. Shkalikov checked ([37]), Theorem 1.5) which boundary conditions in terms of a function y and its quasi-derivative u = y − Qy are regular by Birkhoﬀ–Tamarkin. Not all of them are reduced to some canonical boundary conditions in the case of L2 -potentials; the result could depend on the value of Q(0). For example, Dirichlet–Neumann bc y(0) = 0,

(y − Qy)(π) = 0

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233

would became y(0) = 0,

y (π) = Q(π) · y(π).

Of course, one can adjust Q in advance by choosing (as it is done in [39]) π Q(x) = − v(t)dt if v ∈ L2 . x

But this choice is not good if Dirichlet–Neumann bc is written with changed roles of the end points, i.e., (y − Qy)(0) = 0,

y(π) = 0.

We want to restrict ourselves to such boundary conditions with v ∈ H −1 that if by chance v ∈ L2 then the reduced boundary conditions do not depend on Q(0). We consider as good self-adjoint bc only the following ones: Dir :

y(0) = 0,

y(π) = 0

and y(π) = eiθ y(0) (y − Qy)(π) = eiθ (y − Qy)(0) + Beiθ y(0), where θ ∈ [0, 2π) and B is real. Observations of this subsection are quite elementary but they would be important if we would try to extend statements like Statement (B) by ﬁnding other troikas of boundary conditions (and corresponding troikas of eigenvalues like {λ+ , λ− , μ}) and using these spectral triangles and the decay rates of their diameters to characterize a smoothness of potentials v with a priori assumption v ∈ H −1 (or even v ∈ L2 ([0, π]). Acknowledgment The authors thank Professors Rostyslav Hryniv, Andrei Shkalikov and Vadim Tkachenko for very useful discussions of many questions of spectral analysis of diﬀerential operators, both related and unrelated to the main topics of this paper.

References [1] S. Albeverio, F. Gesztesy, R. Hegh-Krohn, H. Holden, Solvable models in quantum mechanics. Texts and Monographs in Physics. Springer-Verlag, New York, 1988. [2] S. Albeverio and P. Kurasov, Singular perturbations of diﬀerential operators. Solvable Schr¨ odinger type operators. London Mathematical Society Lecture Note Series, 271. Cambridge University Press, Cambridge, 2000. [3] F.V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964. [4] Dzh.-G. Bak and A.A. Shkalikov, Multipliers in dual Sobolev spaces and Schr¨odinger operators with distribution potentials. (Russian) Mat. Zametki 71 (2002), no. 5, 643– 651; translation in Math. Notes 71 (2002), no. 5-6, 587–594.

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[5] F.A. Berezin and L.D. Faddeev, Remark on the Schr¨odinger equation with singular potential (Russian), Dokl. Akad. Nauk SSSR 137 (1961), 1011–1014; English transl., Soviet Math. Dokl. 2 (1961), 372–375. [6] F.A. Berezin, On the Lee model. (Russian) Mat. Sb. 60 (1963) 425–446; English transl., Amer. Math. Soc. Transl. (2) 56 (1966), 249–272. [7] P. Djakov and B. Mityagin, Smoothness of Schr¨odinger operator potential in the case of Gevrey type asymptotics of the gaps, J. Funct. Anal. 195 (2002), 89–128. [8] P. Djakov and B. Mityagin, Spectral triangles of Schr¨ odinger operators with complex potentials. Selecta Math. (N.S.) 9 (2003), 495–528. [9] P. Djakov and B. Mityagin, Instability zones of periodic 1D Schr¨odinger and Dirac operators (Russian), Uspehi Mat. Nauk 61 (2006), no 4, 77–182 (English: Russian Math. Surveys 61 (2006), no 4, 663–766). [10] P. Djakov and B. Mityagin, Spectral gap asymptotics of one dimensional Schr¨odinger operators with singular periodic potentials, Integral Transforms and Special Functions 20 (2009), 265–273. [11] P. Djakov and B. Mityagin, Spectral gaps of Schr¨ odinger operators with periodic singular potentials, Dyn. Partial Diﬀer. Equ. 6, no. 2 (2009), 95–165. [12] M.S.P. Eastham, The spectral theory of periodic diﬀerential operators, Hafner, New York 1974. [13] W.N. Everitt and A. Zettl, Generalized symmetric ordinary diﬀerential expressions. I. The general theory. Nieuw Arch. Wisk. (3) 27 (1979), 363–397. [14] W.N. Everitt and A. Zettl, A. Sturm–Liouville diﬀerential operators in direct sum spaces. Rocky Mountain J. Math. 16 (1986), 497–516. [15] R.O. Hryniv and Ya.V. Mykytyuk, 1-D Schr¨ odinger operators with periodic singular potentials. Methods Funct. Anal. Topology 7 (2001), 31–42. [16] R.O. Hryniv and Ya.V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Problems 19 (2003), 665–684. [17] R.O. Hryniv and Ya.V. Mykytyuk, Transformation operators for Sturm–Liouville operators with singular potentials. Math. Phys. Anal. Geom. 7 (2004), 119–149. [18] R.O. Hryniv and Ya.V. Mykytyuk, Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials, J. Funct. Anal. 238 (2006), 27–57. [19] T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill’s Equation and Action Variables for KdV, Trans. AMS 351 (1999), 619–646. [20] T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schr¨ odinger operator, SIAM J. Math. Anal. 33 (2001), 113–152. [21] T. Kappeler and C. M¨ ohr, Estimates for periodic and Dirichlet eigenvalues of the Schr¨ odinger operator with singular potential, J. Funct. Anal. 186 (2001), 69–91. [22] E. Korotyaev, Characterization of the spectrum of Schr¨ odinger operators with periodic distributions, Int. Math. Res. Not. 37 (2003), 2019–2031. [23] E. Korotyaev, A priori estimates for the Hill and Dirac operators, Russ. J. Math. Phys. 15 (2008), 320–331. [24] P. Kuchment, Floquet theory for partial diﬀerential equations, Basel-Boston, Birkh¨ auser Verlag, 1993.

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[25] P. Kurasov, On the Coulomb potential in dimension one J. Phys. A: Math. Gen., 29 (1996), 1767–1771. [26] W. Magnus and S. Winkler, “Hill’s equation”, Interscience Publishers, John Wiley, 1969. [27] V.A. Marchenko, “Sturm-Liouville operators and applications”, Oper. Theory Adv. Appl., Vol. 22, Birkh¨ auser, 1986. [28] R.A. Minlos and L.D. Faddeev, On the point interaction for a three-particle system in quantum mechanics, Dokl. Akad. Nauk SSSR 141 (1961), 1335–1338 (Russian); translated as Soviet Physics Dokl. 6 (1962), 1072–1074. [29] M.A. Naimark, Linear diﬀerential operators, Moscow, 1969. [30] M.I. Ne˘ıman-zade and A.A. Shkalikov, Schr¨odinger operators with singular potentials from spaces of multipliers. (Russian) Mat. Zametki 66 (1999), no. 5, 723–733; translation in Math. Notes 66 (1999), no. 5-6, 599–607 [31] F.S. Rofe–Beketov, On the spectrum of non-selfadjoint diﬀerential operators with periodic coeﬃcients. (Russian) Dokl. Akad. Nauk SSSR 152 1963 1312–1315; translation in Soviet Math. Dokl. 4 (1963), 1563–1566. [32] F.S. Rofe–Beketov and A.M. Kholkin, Spectral analysis of diﬀerential operators. Interplay between spectral and oscillatory properties. Translated from the Russian by Ognjen Milatovic and revised by the authors. With a foreword by Vladimir A. Marchenko. World Scientiﬁc Monograph Series in Mathematics, 7. World Scientiﬁc Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. [33] A.M. Savchuk, On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential (Russian) Mat. Zametki 69 (2001), no. 2, 277–285; translation in Math. Notes 69 (2001), no. 1-2, 245–252 [34] A.M. Savchuk and A.A. Shkalikov, Sturm-Liouville operators with singular potentials. (Russian) Mat. Zametki 66 (1999), no. 6, 897–912; translation in Math. Notes 66 (1999), no. 5-6, 741–753. [35] A.M. Savchuk and A.A. Shkalikov, Sturm-Liouville operators with singular potentials. (Russian) Mat. Zametki 66 (1999), 897–912; translation in Math. Notes 66 (1999), 741–753 (2000). [36] A.M. Savchuk and A.A. Shkalikov, The trace formula for Sturm-Liouville operators with singular potentials. (Russian) Mat. Zametki 69 (2001), 427–442; translation in Math. Notes 69 (2001), 387–400 [37] A.M. Savchuk and A.A. Shkalikov, Sturm–Liouville operators with distribution potentials. (Russian) Tr. Mosk. Mat. Obs. 64 (2003), 159–212; translation in Trans. Moscow Math. Soc. 2003, 143–192. [38] M.I. Serov, Certain properties of the spectrum of a non-selfadjoint diﬀerential operator of the second order, Soviet Math. Dokl. 1 (1960), pp. 190–192, [39] A.M. Savchuk and A.A. Shkalikov, Inverse problem for Sturm-Liouville operators with distribution potentials: reconstruction from two spectra. Russ. J. Math. Phys. 12 (2005), 507–514. [40] A.M. Savchuk and A.A. Shkalikov, On the eigenvalues of the Sturm–Liouville operator with potentials in Sobolev spaces, (Russian), Mat. Zametki 80 (2006), 864–884.

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[41] Laurent Schwartz, Th´eorie des distributions. (Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, nos. 9 and 10; Actualit´es Scientiﬁques et Industrielles, nos. 1091 and 1122.) Vol. I, 1950, 148 pp. Vol. II, 1951, 169 pp. [42] V. Tkachenko, On the spectral analysis of the one-dimensional Schr¨odinger operator with a periodic complex-valued potential. (Russian) Dokl. Akad. Nauk SSSR 155 (1964), 289–291. [43] J. Weidmann, Spectral theory of ordinary diﬀerential operators, Lect. Notes in Math. 1258, Springer, Berlin, 1987 Plamen Djakov Sabanci University Orhanli, 34956 Tuzla Istanbul, Turkey e-mail: [email protected] Boris Mityagin Department of Mathematics The Ohio State University 231 West 18th Ave Columbus, OH 43210, USA e-mail: [email protected] Received: February 19, 2009. Accepted: June 9, 2009.

Operator Theory: Advances and Applications, Vol. 203, 237–245 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Additive Invariants on Quantum Channels and Regularized Minimum Entropy Shmuel Friedland Abstract. We introduce two additive invariants of output quantum channels. If the value of one these invariants is less than 1 then the logarithm of the inverse of its value is a positive lower bound for the regularized minimum entropy of an output quantum channel. We give a few examples in which one of these invariants is less than 1. We also study the special cases where the above both invariants are equal to 1. Mathematics Subject Classiﬁcation (2000). 81P68, 94A17, 94A40, 15A42. Keywords. Quantum information theory, quantum channel, minimum entropy output, regularized minimum entropy output, additivity conjecture, additive invariants.

1. Introduction Denote by Sn (C) the Hilbert space of n × n hermitian matrices, where X, Y = tr XY . Denote by Sn,+,1 (C) ⊂ Sn,+ (C) ⊂ Sn (C) the convex set of positive hermitian matrices of trace one, and the cone of positive hermitian matrices respectively. A quantum channel is a completely positive linear transformation τ : Sn (C) → Sm (C): l Ai XA∗i , A1 , . . . , Al ∈ Cm×n , X ∈ Sn (C), (1.1) τ (X) = i=1

which is trace preserving: l

A∗i Ai = In ,

(1.2)

i=1

This research started during author’s participation in AIM workshop “Geometry and representation theory of tensors for computer science, statistics and other areas”, July 21–25, 2008. Communicated by L. Rodman.

238

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Denote by τ ∗ : Sm (C) → Sn (C) the adjoint linear transformation. The minimum entropy output of a quantum channel τ is deﬁned H(τ ) =

min X∈Sn,+,1 (C)

− tr τ (X) log τ (X).

(1.3)

If η : Sn (C) → Sm (C) is another quantum channel, then it is well known τ ⊗ η is a quantum channel, and H(τ ⊗ η) ≤ H(τ ) + H(η).

(1.4)

p

Hence the sequence H(⊗ τ ), p = 1, . . . , is subadditive. Thus the following limit exists: H(⊗p τ ) Hr (τ ) = lim , (1.5) p→∞ p and is called the regularized minimum entropy of quantum channel. Clearly, Hr (τ ) ≤ H(τ ). One of the major open problem of quantum information theory is the additivity conjecture, which claims that equality holds in (1.4). This additivity conjecture has several equivalent forms [8]. If the additivity conjecture holds then Hr (τ ) = H(τ ), and the computation of Hr (τ ) is relatively simple. There are known cases where the additivity conjecture is known, see references in [7]. It is also known that the p analog of the additivity conjecture is wrong [7]. It was shown in [2] that the additivity of the entanglement of subspaces fails over the real numbers. It was recently shown by Hastings [6] that the additivity conjecture is false. Hence the computation of Hr (τ ) is hard. This is the standard situation in computing the entropy of Potts models in statistical physics, e.g., [5]. Let l A(τ ) := Ai A∗i ∈ Sm,+ (C). (1.6) i=1

Then log λ1 (A(τ )) = log A(τ ), where λ1 (A) is the maximal eigenvalue of A(τ ), is the ﬁrst additive invariant of quantum channels, with respect to tensor products. Let σ1 (τ ) = τ ≥ σ2 (τ ) ≥ · · · ≥ 0 be the ﬁrst and the second singular value of the linear transformation given by τ . Then log σ1 (τ ) is the second additive invariant. (These two invariants are incomparable in general, see Section 3.) The ﬁrst result of this paper is Theorem 1.1. Let τ : Sn (C) → Sm (C) be a quantum channel. Assume that min(λ1 (A(τ )), τ ) < 1. Then Hr (τ ) ≥ max(− log λ1 (A(τ )), − log τ ).

(1.7)

In Section 3 we give examples where min(λ1 (A(τ )), σ1 (τ )) < 1. τ is called a unitary quantum channel if in (1.1) we assume Ai = ti Qi , Qi Q∗i = Q∗i Qi = In , i = 1, . . . , l, t = (t1 , . . . , tl ) ∈ Rl , t t = 1. (1.8) In that case λ1 (A(τ )) = σ1 (τ ) = 1. Note the counter example to the additivity conjecture in [6] is of this form. A quantum channel τ : Sn (C) → Sm (C) is called a

Additive Invariants on Quantum Channels

239

bi-quantum channel if m = n and τ ∗ : Sn (C) → Sn (C) is also a quantum channel. That is A(τ ) = In and it follows that σ1 (τ ) = 1. Note that a unitary quantum channel is a bi-quantum channel. The second major result of this paper is Theorem 1.2. Let τ : Sn (C) → Sn (C) be a bi-quantum channel. Then σ1 (τ ) = 1. Assume that n ≥ 2 and σ2 (τ ) < 1. Then 1 1 − σ2 (τ )2 H(τ ) ≥ − log(σ2 (τ )2 + ). 2 n

(1.9)

Note that (1.9) is nontrivial if σ2 (τ ) < 1. We show that the condition σ2 (τ ) < 1 holds for a generic unitary channel with l ≥ 3.

2. Proof of Theorem 1.1 Denote by Πn ⊂ Rn+ the convex set of probability vectors. For p = (p1 , . . . , pn ) ∈ Πn we have n 1 1 ≥( pi ) min log = − log max pj . j=1,...,n j=1,...,n pi pj i=1 i=1 i=1 (2.1) For X ∈ Sn (C) denote by λ(A) = (λ1 (X), . . . , λn (X)) the eigenvalue set of X, where λ1 (A) ≥ · · · ≥ λn (X). Then u1 , . . . , un is the corresponding orthonormal basis of Cn consisting of eigenvectors of X Xui = λi (X)ui where u∗i uj = δij for i, j = 1, . . . , n. Ky-Fan maximal characterization is, e.g., [3],

H(p) = −

n

k

pi log pi =

λj (X) =

j=1

n

pi log

k

max

x1 ,...,xk ∈Cn ,x∗ p xq =δpq

x∗j Xxj

j=1

=

k

tr(X(xj x∗j )).

(2.2)

j=1

Hence for x ∈ Cn , x∗ x = 1 we have k

λj (τ (xx∗ )) =

j=1

=

=

max m ∗

k

y1 ,...,yk ∈C ,yp yq =δpq l,k

max

∗ y =δ y1 ,...,yk ∈Cm ,yp q pq

max

∗ y =δ y1 ,...,yk ∈Cm ,yp q pq

(2.3)

j=1

|yj∗ Ai x|2

≤

i,j=1 k

tr(τ (xx∗ )(yj yj∗ ))

yj∗ A(τ )yj =

j=1

max

∗ y =δ y1 ,...,yk ∈Cm ,yp q pq

k

l,k

yj∗ Ai A∗i yj

i,j=1

λj (A(τ )).

(2.4)

j=1

"k Recall that j=1 λj (X) is a convex function on Sn (C). As the extreme points of Sn,+,1 are xx∗ , x ∈ Cn , x∗ x = 1 we obtain max

X∈Sn,+,1

k j=1

λj (τ (X)) ≤

k j=1

λj (A(τ )),

k = 1, . . . , m.

(2.5)

240

S. Friedland

X ∈ Sn,+,1 (C) iﬀ λ(X) ∈ Πn . Hence H(X) := H(λ(X)) ≥ − log λ1 (X) for X ∈ Sn,+,1 (C). (2.5) for k = 1 yields that H(τ ) ≥ − log λ1 (A(τ )). For C ∈ Rm×n let C = V ΣU be the singular value decomposition, (SVD), of C. So U = [u1 . . . un ] ∈ Rn×n , V = [v1 . . . vm ] ∈ Rm×m be orthogonal, and Σ = diag(σ1 (A), . . . , ) ∈ Rm×n , be a diagonal matrix with nonnegative diagonal entries + which form a nonincreasing sequence. The positive singular values of C are the √ √ positive eigenvalues of CC or C C. Let σ(C) = (σ) 1 (C), σ2 (C), ). . . , σl (C)) where σi (C) = 0 if i > r = rank C. Recall that CF := C, C = tr(CC ) = "rank C σi (C)2 . and σ1 (C) = C = maxu=v=1 |v (Cu)|. Thus, for x ∈ i=1 Cn , x∗ x = 1, we have the inequality λ1 (τ (xx∗ )) = max tr((yy∗ )τ (xx∗ )) = y=1

max

τ (xx∗ ), yy∗ ≤ σ1 (τ ).

yy∗ ,yy∗ =1

Hence max

X∈Sn,+,1

λ1 (τ (X)) ≤ σ1 (τ ).

(2.6)

Combine the above inequalities to deduce H(τ ) ≥ max(−logλ1 (A(τ )),−logσ1 (τ )). The properties of tensor products imply H(⊗p τ ) ≥ − log λ1 (A(⊗p τ )) = − log λ1 (⊗p A(τ )) = −p log λ1 (A(τ )), H(⊗p τ ) ≥ − log σ1 (⊗p τ ) = −p log σ1 (τ ) = −p log τ Hence (1.7) holds. If λ1 (A(τ )) < 1 then the inequality Hr (τ ) ≥ − log λ1 (A(τ )) can be improved [4, §4].

3. Examples Proposition 3.1. Let τ be a quantum channel given by (1.1). Then √ n n λ1 (A(τ )) ≥ , σ1 (τ ) ≥ √ . m m

(3.1)

Hence, λ1 (A(τ )), σ1 (τ ) ≥ 1 for m ≤ n. In particular, if m ≤ n then the condition either λ1 (A(τ )) = 1 or σ1 (τ ) = 1 holds if and only if m = n and τ ∗ is a quantum channel. Proof. Clearly, mλ1 (A(τ )) ≥

m

λj (A(τ )) = tr A(τ ) =

j=1

l i=1

tr Ai A∗i =

l

tr A∗i Ai = tr In = n.

i=1

Hence λ1 (A(τ )) ≥ Clearly, if m = n and A(τ ) = In then λ1 (A(τ )) = 1 and τ ∗ is a quantum channel. Vice versa if m ≤ n and λ1 (A(τ )) = 1 then m = n. Furthermore, all eigenvalues of A(τ ) have to be equal to 1, i.e., A(τ ) = In . n m.

Additive Invariants on Quantum Channels

241

Observe that the condition that τ of the form (1.1) is a quantum channel is equivalent to the condition τ ∗ (Im ) = In . As √ n 1 ∗ ∗ σ1 (τ ) = σ1 (τ ) ≥ τ ( √ Im ) = √ m m we deduce that second inequality in (3.1). Suppose that m ≤ n and σ1 (τ ) = 1. Hence m = n and σ1 (τ ∗ ) = τ ∗ ( √1n In ) = 1. So √1n In must be the left and the right singular vector of τ corresponding to the τ . I.e. τ (In ) = In , which is equivalent to the condition that τ ∗ is a quantum channel. Example 1. A quantum channel τ : S1 (C) → Sm (C) is of the form τ (x) =

l

ai xa∗i ,

ai ∈ C , i = 1, . . . , l, m

i=1

l

a∗i ai

= 1,

A(τ ) =

l

i=1

ai a∗i . (3.2)

i=1

Note that tr A(τ ) = 1. Hence λ1 (A(τ )) < 1, unless a1 , . . . , al are colinear. (This happens always if m = 1.) We claim that ) (3.3) σ1 (τ ) = tr A(τ )2 . Indeed max

|x|=1,Y ∈Sm (C),tr(Y 2 )=1

| tr τ (x)Y | =

max

Y ∈Sm (C),tr(Y 2 )=1

| tr A(τ )Y | =

) tr A(τ )2 .

Hence λ1 (A(τ )) < σ1 (τ ) < 1 iﬀ a1 , . . . , al are not colinear.

(3.4)

If a1 , . . . , al are co-linear then λ1 (A) = σ1 (A) = 1. Note that in this example H(τ ) = H(A(τ )). Example 2. A quantum channel τ : Sn (C) → S1 (C) is of the form τ (X) =

l

a∗i Xai ,

ai ∈ Cn , i = 1, . . . , l,

i=1

l

ai a∗i = In ,

A(τ ) =

i=1

l

a∗i ai = n.

i=1

(3.5) So λ1 (A(τ )) = n ≥ 1. On the other hand σ1 (τ ) =

max2

X∈Sn (C),tr X =1,|y|=1

| tr(τ (X)y)| =

X∈Sn

max

(C),tr X 2 =1

| tr X| =

√

n.

(3.6)

So for n > 1 λ1 (A(τ )) > σ1 (τ ). Example 3. A quantum channel of the form (1.1), where m = n and (1.2) holds, is called a strongly self-adjoint if there exists a permutation π on {1, . . . , l} such that A∗i = Aπ(i) for i = 1, . . . , l. So A(τ ) = In and λ1 (A(τ )) = 1. Note that τ is self-adjoint and τ (In ) = In . Since In is an interior point of Sn,+ it follows that σ1 (τ ) = 1.

242

S. Friedland

Example 4. Assume τj : Snj (C) → Smj (C), j = 1, 2 are two quantum channels. Consider the quantum channel τ = τ1 ⊗ τ2 . Then log λ1 (A(τ )) = log λ1 (A(τ1 )) + log λ1 (A(τ2 )), log σ1 (τ ) = log σ1 (τ1 ) + log σ1 (τ2 ). Thus, it is possible to have λ1 (A(τ )) < 1 without the assumption that both τ1 and τ2 satisfy the same condition. Combine Example 1 and Example 3 to obtain examples of quantum channels τ : Sn (C) → Smn (C), where n, m > 1 where λ1 (A(τ )) < 1. Similar arguments apply for σ1 (τ ). Example 5. Recall that if B ∈ Cm×n and C ∈ Cp×q then B 0m×q B⊕C = ∈ C(m+p)×(n+q) . 0p×n C Assume τj : Snj (C) → Smj (C), j = 1, 2 are two quantum channels given by "lj τj (Xj ) = i=1 Ai,j Xj A∗i,j , where Ai,j ∈ Cmj ×nj , i = 1, . . . , lj , j = 1, 2. Then τ1 ⊕ τ2 : Sn1 +n2 (C) :→ Sm1 +m2 (C) is deﬁned as follows.

l1 ,l2

(τ1 ⊕ τ2 )(X) =

(Ai1 ,1 ⊕ Ai2 ,2 )X(A∗i1 ⊕ A∗i2 ,2 ).

i1 =i2 =1

Clearly, τ1 ⊕ τ2 is a quantum channel. Furthermore, A(τ1 ⊕ τ2 ) = A(τ1 ) ⊕ A(τ2 ). Hence λ1 (A(τ1 ⊕ τ2 )) = max(λ1 (A(τ1 )), λ1 (A(τ2 ))).

(3.7)

This if λ1 (A(τi )) < 1 we get that λ1 (A(τ1 ⊕ τ2 ) < 1. The formula for σ1 (τ1 ⊕ τ2 ) does not seems to be as simple as (3.7). By viewing Sn1 (C) ⊕ Sn2 (C) as a subspace of Sn1 +n2 (C) we deduce the inequality σ1 (τ1 ⊕ τ2 ) ≥ max(σ1 (τ1 ), σ1 (τ2 )). Example 6. We ﬁrst show how to take a neighborhood of a given quantum channel given by (1.1). View A := (A1 , . . . , Al ) as a point in (Cm×n )l . Let O(A) ⊂ (Cm×n )l be an open neighborhood of A such that for any B := (B1 , . . . , Bl ) ∈ (Cm×n )l the "l matrix C(B) := i=1 Bi∗ Bi has positive eigenvalues. Deﬁne ˆl ) = (B1 C(B)− 12 , . . . , Bl C(B)− 12 ) ∈ (Cm×n )l . ˆ1 , . . . , B Bˆ = (B Then τB : Sn (C) → Sm (C) given by τB (X) =

l

ˆ i X(B ˆi )∗ B

i=1

is a quantum channel. So if O(A) is a small neighborhood A then τB is in the small neighborhood of τ . In particular of λ1 (A(τ )) < 1 then there exists a small neighborhood O(A) such that λ1 (A(τB )) < 1 for each B ∈ O(A). Similar claim holds if σ1 (τ ) < 1.

Additive Invariants on Quantum Channels

243

4. Bi-quantum channels Proof of Theorem 1.2. Observe ﬁrst that since τ and τ ∗ are quantum channels if follows that ω := τ ∗ τ is a self-adjoint quantum channel on Sn (C). As ω preserves the cone of positive hermitian matrices, ω(In ) = In and In is an interior point of Sn,+ (C), the Krein-Milman theorem, e.g., [1], it follows that 1 is the maximal eigenvalue of ω. Hence σ1 (τ ) = 1. Observe next n 1 λi (τ (xx∗ ))2 ) 2 = τ (xx∗ ). λ1 (τ (xx∗ )) ≤ ( i=1

We now estimate τ (xx∗ ) from above, assuming that x = 1. Consider the singular value decomposition of τ . Here m = n, and assume that U1 , . . . , Un , V1 , . . . , Vn ∈ Sn (C) are the right and left singular vectors of τ corresponding to σ1 (τ ), . . . , σn (τ ). Furthermore we assume that U1 = V1 = √1n In . Hence n

λi (τ (xx∗ ))2

=

rank τ

i=1

σi (τ )2 | tr Ui xx∗ |2

i=1

≤

σ1 (τ )2 | tr U1 xx∗ |2 +

rank τ

σ2 (τ )2 | tr Ui xx∗ |2 .

i=2 ∗

Since σ1 (τ ) = 1 and tr U1 xx = n

√1 n

∗

tr xx =

√1 , n

λi (τ (xx∗ ))2 ≤ σ2 (τ )2 +

i=1

we deduce that 1 − σ2 (τ )2 . n

(4.1)

:

So

1 − σ2 (τ )2 . n Use the arguments of the proof of Theorem 1.1 to deduce (1.9). ∗

λ1 (τ (xx ) ≤

σ2 (τ )2 +

Proposition 4.1. Let τi : Sni (C) → Sni (C) be a bi-quantum channel for i = 1, 2. Then τ1 ⊗ τ2 is a bi-channel. Furthermore σ2 (τ1 ⊗ τ2 ) = max(σ2 (τ1 ), σ2 (τ2 )).

(4.2)

In particular, if τ : Sn (C) → Sn (C) is a unitary channel and σ2 (τ ) < 1 then 1 1 − σ2 (τ )2 ). (4.3) H(⊗p τ ) ≥ − log(σ2 (τ )2 + 2 np Proof. Since (τ1 ⊗τ2 )∗ = τ1∗ ⊗τ2∗ it follows that a tensor product of two bi-quantum channels is a bi-quantum channel. Since the singular values of τ1 ⊗τ2 are all possible products of of singular values of τ1 and τ2 we deduce (4.2). Then (4.3) is implied by Theorem 1.2. Lemma 4.2. Consider a unitary channel of the form (1.1) and (1.8), where l ≥ 3, ti = 0, i = 1, . . . , l, Q1 = In , and Q2 , . . . , Ql do not have a common nontrivial invariant subspace. Then σ2 (τ ) < σ1 (τ ) = 1.

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Proof. Assume that X ∈ Sn,+ (C) has rank k ∈ [1, n − 1]. We claim that "k ∗ n rank τ (X) > rank X. Recall that X = j=1 xj xj , where x1 , . . . , xk ∈ C are 2 2 nonzero orthogonal vectors. As t1 , . . . , tk > 0 we deduce that τ (X) = t21 X +

k

t2j Qj XQ∗j ≥ t21 X.

j=2

So rank τ (X) ≥ k. Furthermore rank τ (X) = k if and only Qi xj ∈ U := span(x1 , . . . , xk ) for i = 2, . . . , l and j = 1, . . . , k. Since U is not invariant under Q2 , . . . , Ql we deduce that rank τ (X) > k. Clearly, if Y ≥ 0 and rank Y = n then rank τ (Y ) = n. Observe next that Q∗2 , . . . , Q∗l do not have a nontrivial common invariant subspace. Indeed, if V ⊂ Cn was a nontrivial common invariant of Q∗2 , . . . , Q∗l , then the orthogonal complement of V will be a nontrivial invariant subspace of Q2 , . . . , Ql , which contradicts our assumption. Hence τ ∗ (X) > rank X. Let η = τ ∗ τ . Thus, rank η n (Z) = n for any Z 0, i.e., ηn maps Sn,+ (C)\{0} to the interior of Sn,+ (C). By Krein-Milman theorem, i.e., [1], 1 = λ1 (η n ) > λ2 (η n ) = σ2 (τ )2n . Corollary 4.3. Let τ : Sn (C) → Sn (C) be a generic unitary quantum channel. I.e. τ of the form (1.1) and (1.8), where l ≥ 3, (t21 , . . . , t2l ) is a random probability vector, and Q1 , . . . , Ql are random unitary matrices. Then σ2 (τ ) < σ1 (τ ) = 1. Proof. Let τ1 (X) := τ (Q∗1 XQ1 ). Clearly, the l−1 unitary matrices Q2 Q∗1 , . . . , Ql Q∗1 are l − 1 random unitary matrices. Since l − 1 ≥ 2 these l − 1 matrices do not have a nontrivial common invariant subspace. Lemma 4.2 yields that σ2 (τ1 ) < 1. Clearly, σ2 (τ1 ) = σ2 (τ ). Acknowledgement I thank Gilad Gour for useful remarks.

References [1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press 1979. [2] H. Derksen, S. Friedland, G. Gour, D. Gross, L. Gurvits, A. Roy, and J. Yard, On minimum entropy output and the additivity conjecture, Notes of Quantum Information Group, American Mathematical Institute workshop “Geometry and representation theory of tensors for computer science, statistics and other areas”, July 21–25, 2008. [3] S. Friedland, Convex spectral functions, Linear Multilin. Algebra 9 (1981), 299–316. [4] S. Friedland, Additive invariants on quantum channels and applications to regularized minimum entropy, arXiv:0809.0078.

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[5] S. Friedland and U.N. Peled, Theory of Computation of Multidimensional Entropy with an Application to the Monomer-Dimer Problem, Advances of Applied Math. 34(2005), 486–522. [6] M.B. Hastings, A counterexample to additivity of minimum output entropy, arXiv:0809.3972v2 [quant-ph]. [7] P. Hayden and A. Winter, Counterexamples to maximal p-norm multiplicativity conjecture, arXiv: 0807.4753v1 [quant-ph] 30 July, 2008. [8] P.W. Shor, Equivalence of additivity questions in quantum information theory, Comm. Math. Phys. 246 (2004), 453–472, arXiv:quant-ph/030503v4, 3 July 2003. Shmuel Friedland Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, Illinois 60607-7045, USA e-mail: [email protected] Received: February 12, 2009 Accepted: May 5, 2009

Operator Theory: Advances and Applications, Vol. 203, 247–287 c 2010 Birkh¨ auser Verlag Basel/Switzerland

A Functional Model, Eigenvalues, and Finite Singular Critical Points for Indeﬁnite Sturm-Liouville Operators I.M. Karabash Dedicated to Israel Gohberg on the occasion of his eightieth birthday

Abstract. Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indeﬁnite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with ﬁnite singular critical points are considered. Mathematics Subject Classiﬁcation (2000). Primary 47E05, 34B24, 34B09; Secondary 34L10, 47B50. Keywords. Essential spectrum, discrete spectrum, eigenvalue, algebraic and geometric multiplicity, J-self-adjoint operator, indeﬁnite weight function, nonself-adjoint diﬀerential operator, singular critical point.

1. Introduction Let J be a signature operator in a complex Hilbert space H (i.e., J = J ∗ = J −1 ). Then J = P+ − P− and H = H+ ⊕ H− , where P± are the orthogonal projections onto H± := ker(J ∓ I). Recall that a closed symmetric operator S (in a Hilbert space H) is said to be simple if there is no nontrivial reducing subspace in which S is self-adjoint. This paper is concerned mainly with J-self-adjoint operators T such that Tmin := T ∩ T ∗ is a simple densely deﬁned symmetric operator in H with the This work was partly supported by the PIMS Postdoctoral Fellowship at the University of Calgary. Communicated by J.A. Ball.

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deﬁciency indices n+ (Tmin ) = n− (Tmin ) = 2. This class includes weighted SturmLiouville operators 1 d d 1 A= − p + q , r, , q ∈ L1loc (a, b), −∞ ≤ a < 0 0 a.e. on R, p > 0 a.e. on R, q is real-valued, (1.2) equipped with separated self-adjoint boundary conditions at a and b. This statement is a consequence of the fact that the weight function r has one turning point (i.e., the point where r changes sign), see, e.g., [47] and Section 2.3. (1.2) ﬁxes the turning point of r at 0, and therefore A is J-self-adjoint in the weighted space L2 ((a, b), |r(x)|dx) with the operator J deﬁned by (Jf )(x) := (sgn x)f (x). Note that the case of one turning point of r is principal for applications in kinetic theory (see [6, 5, 32] and a short review in [40, Section 1]). The eigenvalue problem for a regular indeﬁnite Sturm-Liouville operator was studied in a number of papers starting from Hilbert [33] (see, e.g., [61, 5, 4, 2, 16, 24, 67] and references therein). Till 2005, the spectral properties of singular diﬀerential operators with an indeﬁnite weight were studied mainly under the assumption of quasi J-nonnegativity, for A this means that σ(JA) ∩ (−∞, 0) is ﬁnite, for the deﬁnition and basic results see [16, 24]. In last decades, big attention have been attracted by the problem of similarity of A to a self-adjoint operator and the close problem of regularity of critical points (see a short review in [45]). In this paper, the problem under consideration is a detailed description of the spectrum σ(T ) of the operator T , of the set of eigenvalues (the point spectrum) σp (T ), and of algebraic and geometric multiplicities of eigenvalues. In Section 2.2, after some analysis of the more general case n+ (Tmin ) = n− (Tmin ) < ∞, we assume the above conditions on Tmin and construct a functional model of T based on that of symmetric operators [22, 29, 58]. It occurs that, for the operator A, the main objects of this model are the spectral measures dΣ+ and dΣ− of the classical Titchmarsh-Weyl m-coeﬃcients associated with A on (a, 0) and (0, b) (see Section 2.3 for details). In the abstract case, dΣ± are the spectral measures of two abstract Weyl functions M± (see [21, 22] for basic facts) naturally associated with T and the signature operator J. In Section 3, the model is used to ﬁnd all eigenvalues of T and their algebraic multiplicities in terms of M± and dΣ± (all geometric multiplicities are equal 1, the latter is obvious for the operator A). In turn, we obtain a description of the discrete and essential spectra and of the exceptional case when the resolvent set ρ(T ) is empty. For the operator A, these abstract results on the spectra of T reduce the eigenvalue problem to the problem of description of M± and dΣ± (or some of their properties) in terms of coeﬃcients p, q, r. The latter problem is diﬃcult, but, fortunately, for some classes of coeﬃcients is important for mathematical physics and is studied enough to get results on spectral properties of A (see Sections 3.3 and 4). Non-emptiness of ρ(A) is nontrivial and essential for the spectral analysis of A (see [4, 62] and [43, Problem 3.3]). In Section 3.3, the author generalizes slightly non-emptiness results noticed in [39, 47, 43].

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A part of this paper was obtained in the author’s candidate thesis [38], announced in the short communication [39], and used essentially in [47, 43]. Some of these applications, as well as connections with [45, 16, 48] and with the similarity problem, are discussed in Section 4. Section 5 provides an alternative approach to the examples of J-self-adjoint Sturm-Liouville operators with the singular critical point 0 given in [43, Sections 5 and 6] and [45, Section 5]. A class of operators with the singular critical point 0 is constructed. Relationships of the paper [16] with the example of [43, Sections 6.1] and with Theorem 3.1 are discussed in Section 6. The main advance of the method of the present paper is that it provides description of real eigenvalues and their algebraic multiplicities. The answer is especially nontrivial and has a rich structure in the case of embedded eigenvalues. The interest to the latter problem is partially motivated by the theory of kinetic equations of the Fokker-Plank (Kolmogorov) type (see references in Section 4.2). Also we drop completely the conditions of quasi-J-positivity and deﬁnitizability. The method of the paper is essentially based on the abstract approach to the theory of extensions of symmetric operators via boundary triplets, e.g., [50, 21, 20]. Some results on eigenvalues of non-self-adjoint extensions of symmetric operators were obtained in [21, 22, 18, 19] with the use of this abstract approach. Relationships of these results with the results of the present paper are indicated in Remarks 2.5 and 3.4. There is a kindred approach to the eigenvalue problem through characteristic functions, we refer the reader to the references in [23]. The characteristic function for the operator A was calculated in [47, Proposition 3.9], but the analysis of [47] shows that it is diﬃcult to apply this method to the eigenvalue problem for the operator A. Connections with deﬁnitizability and local deﬁnitizability of A and T (see, e.g., [55, 35] for basic facts and deﬁnitions) are given in Remarks 3.9 and 3.12. A preliminary version of this paper was published as a preprint [41]. Notation. Let H and H be Hilbert spaces with the scalar products (·, ·)H and (·, ·)H , respectively. The domain, kernel, and range of a (linear) operator S in H is denoted by dom(S), ker(S), and ran(S), respectively. If D is a subset of H, then SD is the image of D, SD := {Sh : h ∈ D}, and D is the closure of D. The discrete spectrum σdisc (S) of S is the set of isolated eigenvalues of ﬁnite algebraic multiplicity. The essential spectrum is deﬁned by σess (S) := σ(S) \ σdisc (S). The continuous spectrum is understood in the sense σc (S) := {λ ∈ C \ σp (S) : ran(S − λ) = ran(S − λ) = H }; −1

RS (λ) := (S − λI) , λ ∈ ρ(S), is the resolvent of S. Recall that an eigenvalue λ of S is called semi-simple if ker(S − λ)2 = ker(S − λ), and simple if it is semi-simple and dim ker(S − λ) = 1. By Sλ (S) we denote the root subspace (the algebraic eigensubspace) of S corresponding to the point λ. That is, Sλ (S) is the closed linear hull of the subspaces ker(S − λ)k , k ∈ N.

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If S is a symmetric operator, n± (S) denote the deﬁciency indices of S (see the Appendix). The topological support supp dΣ of a Borel measure dΣ on R is the smallest closed set S such that dΣ(R \ S) = 0; dΣ({λ}) denotes the measure of point λ (i.e., dΣ({λ}) := Σ(λ + 0) − Σ(λ − 0)) if the measure dΣ is determined by a function of bounded variation Σ. We denote the indicator function of a set S by χS (·). We write f ∈ L1loc (a, b) (f ∈ ACloc (a, b)) if the function f is Lebesgue integrable (absolutely continuous) on every closed bounded interval [a , b ] ⊂ (a, b).

2. The functional model for indeﬁnite Sturm-Liouville operators with one turning point 2.1. Preliminaries; the functional model of a symmetric operator Recall a functional model of symmetric operator following [22, Section 5.2], [58, Section 7] (a close version of a functional model can be found in [29]). In this paper, we need only the case of deﬁciency indices (1, 1). Let Σ(t) be a nondecreasing scalar function satisfying the conditions R

1 dΣ(t) < ∞, 1 + t2

R

dΣ(t) = ∞ , Σ(t) =

1 (Σ(t − 0) + Σ(t + 0)), 2

Σ(0) = 0. (2.1)

The operator of multiplication QΣ : f (t) → tf (t) is self-adjoint in L2 (R, dΣ(t)). It is assumed that QΣ is deﬁned on its natural domain 2 |tf (t)|2 dΣ(t) < ∞}. dom(QΣ ) = {f ∈ L (R, dΣ(t)) : R

Consider the following restriction of QΣ : TΣ = QΣ dom(TΣ ),

dom(TΣ ) = {f ∈ dom(QΣ ) :

f (t)dΣ(t) = 0}. R

Then TΣ is a simple densely deﬁned symmetric operator in L2 (R, dΣ(t)) with deﬁciency indices (1,1). The adjoint operator TΣ∗ has the form dom(TΣ∗ ) = {f = fQ + c

t : fQ ∈ dom(QΣ ), c ∈ C}, t2 + 1 TΣ∗ f = tfQ − c

1 , (2.2) t2 + 1

where the constant c is uniquely determined by the inclusion f − ct(t2 + 1)−1 ∈ dom(QΣ ) due to the second condition in (2.1).

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251

Σ,C from dom(TΣ∗ ) Let C be a ﬁxed real number. Deﬁne linear mappings ΓΣ 0 , Γ1 onto C by Σ,C Σ Γ0 f = c, Γ1 f = c C + fQ (t)dΣ(t), (2.3) R

where

t f = fQ + c 2 ∈ dom(TΣ∗ ), t +1

fQ ∈ dom(QΣ ),

c ∈ C.

Σ,C Then {C, ΓΣ } is a boundary triplet for TΣ∗ (see [22, Proposition 5.2 (3)], basic 0 , Γ1 facts on boundary triplets and abstract Weyl functions are given in the Appendix). The function 1 t dΣ(t), λ ∈ C \ supp dΣ, (2.4) − MΣ,C (λ) := C + t − λ 1 + t2 R

is the corresponding Weyl function of TΣ . Another way to describe the operator TΣ∗ is the following (see [22]). Note that the domain dom (TΣ∗ ) consists of the functions f ∈ L2 (R, dΣ(t)) such that for some constant c ∈ C the function f(t) := tf (t) − c belongs to L2 (R, dΣ(t)). It follows from (2.1) that the constant c is uniquely determined and coincides with the constant c introduced in (2.2). Therefore, and TΣ∗ f = f.

c = ΓΣ 0f

(2.5)

2.2. The functional model for J-self-adjoint extensions of symmetric operators Let J be a signature operator in a Hilbert space H, i.e., J = J ∗ = J −1 . Then J = P+ − P− and H = H+ ⊕ H− , where P± are the orthogonal projections onto H± := ker(J ∓ I). Let T be a J-self-adjoint operator in H, i.e., the operator B = JT is selfadjoint. The domains of T and B coincide, we denote them by D := dom(T ) (= dom(B)). Put Tmin := T ∩ T ∗ , Dmin := dom(Tmin ). By the deﬁnition, the operator Tmin is a symmetric operator and so is Bmin := B Dmin

= JTmin .

(2.6)

Let Σ+ and Σ− be nondecreasing scalar functions satisfying (2.1). Let C+ = A {Σ+ , C+ , Σ− , C− } in and C− be real constants. Consider the operator A 2 2 L (R, dΣ+ ) ⊕ L (R, dΣ− ) deﬁned by {Σ+ , C+ , Σ− , C− } = T ∗ ⊕ T ∗ dom(A), A Σ+ Σ−

(2.7)

= { f = f+ + f− : f± ∈ dom(TΣ∗ ), dom(A) ± Σ

Σ

Σ ,C+

Γ0 + f+ = Γ0 − f− , Γ1 + where

TΣ∗±

are the operators deﬁned in Subsection 2.1.

Σ ,C−

f + = Γ1 −

f− },

252

I.M. Karabash One of the main results of this paper is the following theorem.

Theorem 2.1. Let J be a signature operator in a separable Hilbert space H and let T be a J-self-adjoint operator such that Tmin := T ∩ T ∗ is a simple densely deﬁned symmetric operator in H with deﬁciency indices (2, 2). Then there exist nondecreasing scalar functions Σ+ , Σ− satisfying (2.1) and real constants C+ , C− {Σ+ , C+ , Σ− , C− }. such that T is unitarily equivalent to the operator A First, we prove several propositions that describe the structure of T as an extension of the symmetric operator Tmin , and then prove Theorem 2.1 at the end of this subsection. ± be deProposition 2.2. Let T be a J-self-adjoint operator. Let the operators Tmin ﬁned by ± ± ± ± Tmin := T Dmin , Dmin = dom(Tmin ) := Dmin ∩ H± . (2.8)

Then: ± is a symmetric operator in the Hilbert space H± and (i) Tmin + − Tmin = Tmin ⊕ Tmin ,

+ − Bmin = Bmin ⊕ Bmin ,

where

± ± Bmin := ±Tmin .

(2.9)

(ii) If any of the following two conditions (a) ρ(T ) = ∅ , (b) n+ (Tmin ) = n− (Tmin ), + + − − is satisﬁed, then n+ (Tmin ) = n− (Tmin ) and n+ (Tmin ) = n− (Tmin ). In particular, (a) implies (b). Proof. (i) Since B = B ∗ and D = dom(T ) = dom(B), we have T ∗ = BJ and Dmin = {f ∈ D ∩ JD : JBf = BJf }. So if f ∈ Dmin and g = Jf , then g ∈ D ∩ JD and JBg = JBJf = JJBf = Bf = BJg. This implies JDmin ⊂ Dmin (and in turn JDmin = Dmin since J is a unitary operator). Hence, for f ∈ Dmin we have P+ f +P− f ∈ Dmin and P+ f −P− f ∈ Dmin . So P± f ∈ Dmin and Dmin ⊂ (Dmin ∩ H+ ) ⊕ (Dmin ∩ H− ). The inverse inclusion is obvious, and we see that Dmin = (Dmin ∩ H+ ) ⊕ (Dmin ∩ H− ). Now note that Tmin (Dmin ∩ H± ) ⊂ H± . Indeed, let f± ∈ Dmin ∩ H± . Since Jf± = ±f± and T f± = T ∗ f± , we see that JBf± = BJf± = ±Bf± .

(2.10)

Note that g ∈ H± is equivalent to Jg = ±g. So (2.10) implies Bf± ∈ H± , and therefore the vector Tmin f± = T f± = JBf± belongs to H± . The ﬁrst part of (2.9) is proved. Since Tmin is a symmetric operator in H, the ± operators Tmin are symmetric too. Finally, the second part of (2.9) follows from (2.6) and (2.10).

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(ii) Since B = B ∗ , it easy to see that + − + − n+ (Bmin ) + n+ (Bmin ) = n+ (Bmin ) = n− (Bmin ) = n− (Bmin ) + n− (Bmin ) =: m. (2.11) + + − − The equalities n± (Tmin ) = n± (Bmin ) and n± (Tmin ) = n∓ (Bmin ) imply + − + − ) + n± (Tmin ) = n± (Bmin ) + n∓ (Bmin ). n± (Tmin ) = n± (Tmin

(2.12)

It follows from (2.11) and (2.12) that n+ (Tmin ) > m yields n− (Tmin) < m. In this case, C− ⊂ σp (T ) and H = (T − λI) dom(T ) for λ ∈ C+ . Hence, ρ(T ) = ∅. The case n+ (Tmin ) < m, n− (Tmin ) > m is similar. Thus, if ρ(T ) = ∅ or n+ (Tmin ) = n− (Tmin ), then + − + − ) + n− (Bmin ) = n− (Bmin ) + n+ (Bmin ) = m. n+ (Bmin ± ± ± Using (2.11), we get n+ (Bmin ) = n− (Bmin ) and, therefore, n+ (Tmin ) = ± n− (Tmin ).

Assume now that the operator Tmin is densely deﬁned in H. Put, for convenience’ sake, ± ∗ ± ± ± = (Tmin ) , Bmax = (Bmin )∗ . Tmax Clearly, ± ± Tmax = ±Bmax

and

± ± ± dom(Tmax ) = dom(Bmax ) =: Dmax .

(2.13)

Proposition 2.3. Let T be a J-self-adjoint operator. Assume that Tmin is densely deﬁned in H and − − n+ (Tmin ) = n− (Tmin ) =: N − < ∞.

+ + ) = n− (Tmin ) =: N + < ∞, n+ (Tmin

Then: + + − − (i) n+ (Tmin ) = n− (Tmin ) = n+ (Tmin ) = n− (Tmin ), that is, N + = N − =: N ; (ii) the mappings P± := P± D/Dmin are well deﬁned and are linear isomor± ± phisms from the quotient space D/Dmin onto the quotient space Dmax /Dmin . Proof. Note that + − ⊕ Dmax Dmax := Dmax

(2.14)

and Bmax :=

∗ Bmin .

D = dom(B) ⊂ dom(Bmax ) and P± Dmin =

± Dmin ,

is a domain of both the operators Tmax :=

∗ Tmin

Since

± ± /Dmin are well-deﬁned linear mappings. we see that P± : D/Dmin → Dmax Let us show that

the mappings

P±

are injective.

(2.15)

Indeed, if ker P+ = {0}, then there exists h ∈ D such that h ∈ Dmin and P+ h ∈ + + − Dmin . Recall that Dmin ⊂ D, so P− h = h − P+ h ∈ D ∩ H− = Dmin . By the ﬁrst + − equality in (2.9), Dmin = Dmin ⊕Dmin and this implies that h = P+ h+P− h belongs to Dmin , a contradiction.

254

I.M. Karabash ± ± /Dmin ), it follows from (2.15) that Since 2N ± = dim(Dmax

2N + ≥ m,

2N − ≥ m,

where

m := dim(D/Dmin )

(2.16)

(this deﬁnition of m coincides this that of (2.11)). Indeed, B is a self-adjoint extension of Bmin , therefore, + − + − dim (D/Dmin ) = n+ (Bmin ) + n+ (Bmin ) = n− (Bmin ) + n− (Bmin ).

We see that m = N + + N − . From this and (2.16), we get N + = N − = m/2. Thus, statement (i) holds true. Besides, taking (2.15) and N ± < ∞ into account, one obtains that P± are surjective. This complete the proof of (ii). Recall that existence of a boundary triplet for S ∗ , where S is a symmetric operator in a separable Hilbert space H, is equivalent to n+ (S) = n− (S) (see [50, 21]). Theorem 2.4 (cf. Theorem 6.4 of [20]). Let T be a J-self-adjoint operator. Assume that Tmin is densely deﬁned in H and n+ (Tmin ) = n− (Tmin ) =: m < ∞. Then: (i) m is an even number and + + − − n+ (Tmin ) = n− (Tmin ) = n+ (Tmin ) = n− (Tmin ) = m/2. + + (ii) Let {Cm/2 , Γ+ 0 , Γ1 } be a boundary triple for Tmax . Then there exist a boundary − − − triple {Cm/2 , Γ0 , Γ1 } for Tmax such that

D = {h ∈ Dmax : (Note that P± h ∈

− Γ+ 0 P+ h = Γ0 P− h,

± Dmax

− Γ+ 1 P+ h = Γ1 P− h}.

(2.17)

due to (2.14).)

Theorem 2.4 shows that the operator T admits the representation + − ⊕ Tmax D, T = Tmax

and its domain D has the representation (2.17). (2.18)

Proof. (i) follows from Proposition 2.3 (i). + + (ii) Let {Cm/2 , Γ+ 0 , Γ1 } be a boundary triple for the operator Tmax (actually, statement (i) implies that such a boundary triple exists, for the case when the + space H is separable see, e.g., [50]). It follows from Deﬁnition A.1, that Γ+ 0 Dmin = + + + + Γ1 Dmin = {0}. So one can consider the mappings Γ+ : h+ → {Γ0 h+ , Γ1 h+ } as a + + /Dmin onto Cm/2 ⊕ Cm/2 . Introducing the mappings linear isomorphism from Dmax + −1 Γ− j := Γj P+ P− ,

j = 0, 1,

(2.19)

− one can get from Proposition 2.3 (ii) the fact that Γ− : h− → {Γ− 0 h− , Γ1 h− } is a − − − m/2 m/2 linear isomorphism from Dmax /Dmin onto C ⊕C . Putting Γ0 h− = Γ− 1 h− = − − − − 0 for all h− ∈ Dmin , we get natural linear extensions of Γ− , Γ , and Γ on Dmax . 0 1 Let h ∈ Dmax /Dmin and h± = P± h, where P± are mappings from Dmax /Dmin ± ± to Dmax /Dmin induced by P± . By Proposition 2.3 (ii), h ∈ D/Dmin if and only if −1 h+ = P+ P− h− . From this and (2.19), one can obtain easily that

D = {h = h+ + h− :

± h± ∈ Dmax ,

− Γ+ j h+ = Γj h− , j = 0, 1 }.

(2.20)

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255

− − Let us show that {Cm/2 , Γ− 0 , Γ1 } is a boundary triple for Tmax . The property (ii) of Deﬁnition A.1 follows from the same property for the boundary triple + {Cm/2 , Γ+ 0 , Γ1 } and from Proposition 2.3 (ii). Now we have to prove property (i) of Deﬁnition A.1. Since B = B ∗ , for all f, g ∈ D = dom(B) we have

0 = (Bf,g)H − (f,Bg)H

(2.21)

= ( Bmax (P+ f + P− f ) , P+ g + P− g )H − ( P+ f + P− f , Bmax (P+ g + P− g) )H + − + − = (Bmax P+ f,P+ g)H + (Bmax P− f,P− g)H − (P+ f,Bmax P+ g)H − (P− f,Bmax P− g)H + ± + + Since P± f, P± g ∈ Dmax and {Cm/2 , Γ+ 0 , Γ1 } is a boundary triple for Tmax = Bmax , Deﬁnition A.1 yields / . / . + + P+ g H Bmax P+ f, P+ g H − P+ f, Bmax + + + = (Γ+ 1 P+ f, Γ0 g)Cm/2 − (Γ0 P+ f, Γ1 P+ g)Cm/2 . (2.22)

From (2.20) and f, g ∈ D, we get + + + (Γ+ 1 P+ f, Γ0 P+ g)Cm/2 − (Γ0 P+ f, Γ1 P+ g)Cm/2 − − − = (Γ− 1 P− f, Γ0 P− g)Cm/2 − (Γ0 P− f, Γ1 P− g)Cm/2 . (2.23)

It follows from (2.21), (2.22), and (2.23) that − − − 0 = (Γ− 1 P− f, Γ0 P− g)Cm/2 − (Γ0 P− f, Γ1 P− g)Cm/2 . − / . / − + Bmax P− f, P− g H − P− f, Bmax P− g H .

or, equivalently, . − / . / − Tmax P− f, P− g H − P− f, Tmax P− g H − − − = (Γ− 1 P− f, Γ0 P− g)Cm/2 − (Γ0 P− f, Γ1 P− g)Cm/2

(2.24)

for all f, g ∈ D. It follows easily from Proposition 2.3 (ii) that the mapping P− : − D → Dmax is surjective. Therefore (2.24) implies that property (ii) of Deﬁnition − − A.1 is fulﬁlled for {Cm/2 , Γ− 0 , Γ1 } and so this triple is a boundary triple for Tmax . Finally, note that (2.20) coincides with (2.17). ± ) = 1 and there exist boundary Proof of Theorem 2.1. By Theorem 2.4 (i), n± (Tmin ± ± ± ± triplets Π = {C, Γ0 , Γ1 } for Tmax such that (2.17) holds. Let M± be the Weyl ± ± corresponding to the boundary triplets Π± . Since Tmin are densely functions of Tmin deﬁned operators, M± have the form (2.4) with certain constants C± ∈ R± and positive measures dΣ± (t) satisfying (2.1). This fact follows from Corollary 2 in [21, Section 1.2] as well as from the remark after [22, Theorem 1.1] and [22, Remark 5.1] (note that, in the case of deﬁciency indices (1,1), condition (3) of Corollary 2 in [21, Section 1.2] is equivalent to the second condition in (2.1)). By Corollary 1 in

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I.M. Karabash

[21, Section 1.2] (see also [21, Corollary 7.1]), the simplicity of both the operators ± and TΣ± implies that Tmin ± −1 U± Tmin U± = TΣ± ,

(2.25)

where TΣ± are the operators deﬁned in Subsection 2.1, and U± are certain unitary operators from H± onto L2 (R, dΣ± (t)). Moreover, the unitary operators U± can be chosen such that ± Γ± 0 = Γ0 U± ,

Σ

± Γ± 1 = Γ1

Σ ,C±

U± .

(2.26)

The last statement follows from the description of all possible boundary triples in terms of chosen one (see, e.g., [50] and [22, Proposition 1.7]). Indeed, since the ± deﬁciency indices of Tmin are (1,1), [22, formulae (1.12) and (1.13)] imply that Σ Σ ,C ± ± ± iα± Γ0 U± and Γ1 = eiα± Γ1 ± ± U± with α ∈ [0, 2π). Now changing U± Γ0 = e to eiα± U± we save (2.25) and get (2.26). Formulae (2.18) and (2.17) complete the proof. Remark 2.5. (1) Self-adjoint couplings of symmetric operators were studied in [20, 26] (see also references therein). Theorem 2.4 (ii) can be considered as a modiﬁcation of [20, Theorem 6.4] for J-self-adjoint operators. (2) Note that in Proposition 2.2 we do not assume that the domain Dmin of Tmin is dense in H. However, for convenience’ sake, the operator Tmin is assumed to be densely deﬁned in the other statements of this subsection. The assumption dom(Tmin ) = H can be removed from Proposition 2.3 and Theorem 2.4 with the use of the linear relation notion in the way similar to [20, Section 6]. (3) Theorems 2.4 (ii) and 2.1 show that the operator T admits an inﬁnite family of functional models, which corresponds to the inﬁnite family of boundary triples. All the functional models can be derived from a chosen one due to [22, Proposition 1.7]. 2.3. The Sturm-Liouville case Consider the diﬀerential expressions l[y] =

1 (−(py ) + qy) |r|

and

a[y] =

1 (−(py ) + qy) , r

(2.27)

assuming that 1/p, q, r ∈ L1loc (a, b) are real-valued coeﬃcients, that p(x) > 0 and xr(x) > 0 for almost all x ∈ (a, b), and that −∞ ≤ a < 0 < b ≤ +∞. So the weight function r has the only turning point at 0 and the diﬀerential expressions a and l are regular at all points of the interval (a, b) (but may be singular at the endpoints a and b). The diﬀerential expressions are understood in the sense of M.G. Krein’s quasi-derivatives (see, e.g., [16]). If the endpoint a (the endpoint b) is regular or is in the limit circle case for l[·], we equip the expressions l[·] and a[·] with a separated self-adjoint boundary condition (see, e.g., [66] or [55]) at a (resp., b), and get in this way the self-adjoint

Functional Model for Indeﬁnite Sturm-Liouville Operators

257

operator L and the J-self-adjoint operator A in the Hilbert space L2 (R, |r(x)|dx). Indeed, A = JL with J deﬁned by (Jf )(x) = (sgn x)f (x).

(2.28)

Obviously, J ∗ = J −1 = J in L2 ( (a, b), |r(x)|dx). So J is a signature operator and A is a J-self-adjoint operator. In the case when l[·] is in the limit point case at a and/or b, we do not need boundary conditions at a and/or b. It is not diﬃcult to see that the operator Amin := A ∩ A∗ is a closed densely deﬁned symmetric operator with the deﬁciency indices (2,2) and that Amin admits − + − an orthogonal decomposition Amin = A+ min ⊕ Amin , where Amin (Amin ) is a part of Amin in L2 ((0, b), |r(x)|dx) (resp., L2 ((a, 0), |r(x)|dx)), see, e.g., [47, Section 2.1], and (2.31) below for a particular case (note that Amin is not a minimal operator associated with a[·] in the usual sense). The operators A± min are simple. This fact considered known by specialists, it was proved in [30], formally, under some additional conditions on the coeﬃcients. A modiﬁcation of the same proof is brieﬂy indicated in Remark 2.7 below. So Amin is a simple symmetric operator. Applying Theorem 2.1, one obtains a functional model for A. However, we will show that a model for A can be obtained directly from the classical spectral theory of Sturm-Liouville operators and that dΣ± are spectral measures associated with Titchmarsh-Weyl m-coeﬃcients of A. To avoid superﬂuous notation and consideration of several diﬀerent cases, we argue for the case when (a, b) = R,

p ≡ 1,

r(x) ≡ sgn x,

and the diﬀerential expression l[·] is limit-point at + ∞ and − ∞.

(2.29) (2.30)

That is we assume that the operator d2 d2 A = (sgn x) − 2 + q(x) L = − 2 + q(x) dx dx is deﬁned on the maximal domain and is self-adjoint (resp., J-self-adjoint). Under these assumptions, dom(L) = dom(A) = {y ∈ L2 (R) : y, y ∈ ACloc (R), y

+ qy ∈ L2 (R)}. The operator Amin = A ∩ A∗ has the form Amin = A dom(Amin ), dom(Amin ) = {y ∈ dom(A) : y(0) = y (0) = 0}.

(2.31)

2 By A± min we deﬁne the restrictions of Amin on dom(Amin ) ∩ L (R± ). Let us deﬁne the Titchmarsh-Weyl m-coeﬃcients MN+ (λ) and MN− (λ) for the Neumann problem associated with the diﬀerential expression a[·] on R+ and R− , respectively. Facts mentioned below can be found, e.g., in [57, 64], where they

258

I.M. Karabash

are given for spectral problems on R+ , but the modiﬁcation for R− is straightforward. Let s(x, λ), c(x, λ) be the solutions of the equation −y

(x) + q(x)y(x) = λy(x) subject to boundary conditions s(0, λ) =

d s(0, λ) = c(0, λ) = 1. dx

d c(0, λ) = 0, dx

Then MN± (λ) are well deﬁned by the inclusions ψ± (·, λ) = −s(·, ±λ) + MN± (λ) c(·, ±λ) ∈ L2 (R± ),

(2.32)

for all λ ∈ C \ R. The functions MN± (λ) are (R)-functions (belong to the class (R)) ; i.e., MN± (λ) are holomorphic in C \ R, MN±(λ) = MN± (λ) and Im λ Im MN± (λ) ≥ 0, λ ∈ C \ R (see, e.g., [37]). Moreover, MN± (λ) admit the following representation dΣN± (t) , (2.33) MN± (λ) = t−λ R where ΣN± are nondecreasing scalar function such that conditions (2.1) are fulﬁlled and R

(1 + |t|)−1 dΣN± (t) < ∞;

the functions MN± (λ) have the asymptotic formula 1 i , (λ → ∞, 0 < δ < arg λ < π − δ) . (2.34) MN± (λ) = ± √ +O λ ±λ √ Here and below z is the branch of the multifunction √ on the complex plane C with the cut along R , singled out by the condition −1 = i. We assume that + √ λ ≥ 0 for λ ∈ [0, +∞). Let A± 0 be the self-adjoint operators associated with the Neumann problem

y (±0) = 0 for the diﬀerential expression a[·] on R± . The measures dΣN± (t) are called the spectral measures of the operators A± 0 since −1 QΣN± = F± A± 0 F±

where QΣN± are the operators of multiplication by t in the space L2 (R, dΣN± (t)) and F± are the (generalized) Fourier transformations deﬁned by x1 (F± f )(t) := l.i.m. ± f (x)c(x, ±t)dx. (2.35) x1 →±∞

0

Here l.i.m. denotes the strong limit in L (R, dΣN± ). Recall that F± are unitary operators from L2 (R± ) onto L2 (R, dΣN± ). 2

Functional Model for Indeﬁnite Sturm-Liouville Operators

259

Note that supp dΣN± = σ(QΣN± ) = σ(A± 0 ), that (2.33) gives a holomorphic continuation of MN± (λ) to C \ supp dΣN± , and that, in this domain, MN± (λ) = MΣN± ,CN± (λ), where t dΣN± (2.36) CN± := 1 + t2 R and MΣN± ,CN± (λ) are deﬁned by (2.4). Theorem 2.6. Assume that conditions (2.29) and (2.30) are fulﬁlled and the Jself-adjoint operator A = (sgn x)(−d2 /dx2 + q(x)) is deﬁned as above. Then A is = A{Σ N+ , CN+ , ΣN− , CN− }. More precisely, unitarily equivalent to the operator A −1 −1 . ⊕ F− )=A (F+ ⊕ F− )A(F+

(2.37)

Proof. The proof is based on two following representations of the resolvent RA± 0 (see [57, 65]): ±x ±∞ (RA± (λ)f± )(x) = ∓ψ± (x, λ) c(s, ±λ)f (s)ds ∓ c(x, ±λ) ψ± (s, λ)f (s)ds, 0

±x

0

(RA± (λ)f± )(x) = 0

R

c(x, ±t) (F± f± )(t) dΣ± (t) , t−λ

x ∈ R± .

(2.38) (2.39)

It is not diﬃcult to see (e.g., [47, Section 2.1]) that / . − ∗/ . ∗ ⊕ (Amin ) : y(+0) = y(−0), y (+0) = y (−0) . dom(A) := y ∈ dom (A+ min ) (2.40) ± := F± A± F −1 and recall that A ± := F± A± F −1 is the operator Put A 0 0 ± min min ± ± = QΣ . of multiplication by t in the space L2 (R, dΣN± (t)), i.e., A 0 N± Let functions f ∈ L2 (R) and f± ∈ L2 (R± ) be such that f = f+ + f− . Denote g ± (t) := (F± f± )(t). From (2.39) we get ± g (t)dΣN± (t) (RA± (λ)f± )(±0) = . (2.41) 0 t−λ R Since RA ± (λ)g ± (t) = g ± (t)(t − λ)−1 , we see that 0 y± (0) = (F± y± )(t)dΣN± (t) for all y± ∈ dom(A± 0 ), R

± = QΣ dom(A ± ) , A min min ± ± ) = { dom(A y± ∈ dom(QΣN± ) : y± (t)dΣN± (t) = 0}. min and

R

± = TΣ . That is, A min N±

(2.42)

260

I.M. Karabash % $ * ±∞ It follows from (2.38) that RA± (λ)f± (±0) = ± 0 ψ± (x, λ)f± (x)dx for

λ∈ / R. From this and (2.41), we get

0

(F± ψ± (·, λ)) (t) =

1 t−λ

∈ L2 (R, dΣN± ) .

(2.43)

∗ Let y± (x) ∈ dom((A± min ) ). Then, by the von Neumann formula,

y± (t) = y0± (t) + c1 ψ± (t, i) + c2 ψ± (t, −i) ,

(2.44)

where y0± (t) ∈ dom(A± min ) and c1 , c2 ∈ C are certain constants. Therefore (2.32) yields 1 1 y± (0) = c1 MN± (i) + c2 MN±(−i) = c1 dΣN± (t) + c2 dΣN± (t). t − i t + i R R ∗ This, (2.43), and (2.42) implies that (2.42) holds for all y± (x) ∈ dom((A± min ) ). Taking (2.36) and (2.3) into account, we get Σ ,C (2.45) y± (0) = (F± y± )(t) dΣ± (t) = Γ1 N± N± F± y± . R

± = (0) = −c1 − c2 . On the other hand, it follows from A Further, by (2.44), y± min ΣN± ΣN± −1 TΣN± and (2.3) that Γ0 Fy0± = 0 and Γ0 (t − λ) = 1. Hence, Σ

Σ

Γ0 N± Fy± = c1 Γ0 N±

1 1 Σ

+ c2 Γ0 N± = c1 + c2 = −y± (0). t−i t+i

Combining (2.40), (2.45), and (2.46), we get (2.37).

(2.46)

Remark 2.7. Since the operators TΣN± are simple (see [58, Proposition 7.9]), in passing it is proved that so are the operators A± min and Amin . This proof of simplicity works in general case of Sturm-Liouville operator with one turning point described in the beginning of this section. Formally, it removes extra smoothness assumptions on the coeﬃcient p imposed in [30]. But actually it is just another version of the proof of [30, Theorem 3] since the essence of both the proofs is based on Kreins criterion for simplicity [54, Section 1.3].

and of 3. Point and essential spectra of the model operator A indeﬁnite Sturm-Liouville operators 3.1. Point spectrum of the model operator The main result of this section and of the paper is a description of the point + , C+ , Σ− , C− }. spectrum and algebraic multiplicities of eigenvalues of A{Σ

Functional Model for Indeﬁnite Sturm-Liouville Operators

261

First, to classify eigenvalues of the operator TΣ∗ deﬁned in Subsection 2.1, we introduce the following mutually disjoint sets: ( −2 A0 (Σ) = λ ∈ σc (QΣ ) : |t − λ| dΣ(t) = ∞ , R ( −2 |t − λ| dΣ(t) < ∞ , Ar (Σ) = λ ∈ σp (QΣ ) : Ap (Σ) = σp (QΣ ).

(3.1)

R

Observe that C = A0 (Σ) ∪ Ar (Σ) ∪ Ap (Σ) and A0 (Σ) = {λ ∈ C : ker(TΣ∗ − λI) = {0} } , Ar (Σ) = λ ∈ C : ker(TΣ∗ − λI) = {c(t − λ)−1 , c ∈ C} , Ap (Σ) = λ ∈ C : ker(TΣ∗ − λI) = {cχ{λ} (t), c ∈ C} .

(3.2) (3.3)

± ± ± ± , where Γ0 ± , In this section we denote for brevity Γ± 0 := Γ0 , Γ1 := Γ1 are linear mappings from dom(TΣ∗± ) to C deﬁned by (2.3).

Σ

Σ ,C

Σ

Σ ,C Γ1 ± ±

In this paper, for ﬁxed λ ∈ R, the notation is equal to 0 at t = λ and

1 (t−λ)j

χR\{λ} (t) (t−λ)j

means the function that

for t = λ. If λ ∈ R, then

χR\{λ} (t) (t−λ)j

means just

χR\{λ} (t) (t−λ)j

1 (t−λ)j .

In what follows the functions and jump discontinuities of Σ play an essential role. Note that the set of jump discontinuities of Σ coincides with χR\{λ} (t) 1 Ap (Σ) = σp (QΣ ). If λ ∈ R \ Ap (Σ), then (t−λ) and (t−λ) j j belong to the same 2 class of L (R, dΣ) and any of these two notations can be used. We also use notation dΣ({λ}) := Σ(λ + 0) − Σ(λ − 0). For the sake of simplicity, we start from the case when (1 + |t|)−1 dΣ± < ∞ and C± = t(1 + t2 )−1 dΣ± , R

(3.4)

R

(which arises, in particular, in Section 2.3) and then consider the general case. Theorem 3.1. Let Σ± be nondecreasing scalar functions satisfying (2.1) and let C± be real constants. Assume also that conditions (3.4) are fulﬁlled. Then the following = A{Σ + , C+ , Σ− , C− }. statements describe the point spectrum of the operator A 1) If λ ∈ A0 (Σ+ ) ∪ A0 (Σ− ), then λ ∈ σp (A). 2) If λ ∈ Ap (Σ+ ) ∩ Ap (Σ− ), then the geometric multiplicity of λ equals 1; (i) λ is an eigenvalue of A; (ii) the eigenvalue λ is simple (i.e., the algebraic and geometric multiplicities are equal to 1) if an only if at least one of the following conditions is

262

I.M. Karabash not fulﬁlled:

dΣ− ({λ}) = dΣ+ ({λ}),

R\{λ}

R\{λ}

(3.5)

1 dΣ+ (t) < ∞, |t − λ|2 1 dΣ− (t) < ∞; |t − λ|2

(3.6) (3.7)

(iii) if conditions (3.5), (3.6) and (3.7) hold true, then the algebraic multiplicity of λ equals the greatest number k (k ∈ {2, 3, 4, . . . } ∪ {+∞}) such that the conditions 1 1 dΣ− (t) < ∞, dΣ+ (t) < ∞, (3.8) 2j |t − λ| |t − λ|2j R\{λ} R\{λ} 1 1 dΣ− (t) = dΣ+ (t), (3.9) j−1 (t − λ) (t − λ)j−1 R\{λ} R\{λ} are fulﬁlled for all natural j such that 2 ≤ j ≤ k −1 (in particular, k = 2 if at least one of conditions (3.8), (3.9) is not fulﬁlled for j = 2). if and only if 3) Assume that λ ∈ Ar (Σ+ ) ∩ Ar (Σ− ). Then λ ∈ σp (A) 1 1 (3.10) dΣ+ (t) = dΣ− (t) . t − λ t − λ R R If (3.10) holds true, then the geometric multiplicity of λ is 1, and the algebraic multiplicity is the greatest number k (k ∈ {1, 2, 3, . . . } ∪ {+∞}) such that the conditions 1 1 dΣ (t) < ∞, dΣ+ (t) < ∞, (3.11) − 2j |t − λ| |t − λ|2j R R 1 1 dΣ (t) = dΣ+ (t) (3.12) − j j R (t − λ) R (t − λ) are fulﬁlled for all j ∈ N such that 1 ≤ j ≤ k. 4) If λ ∈ Ap (Σ+ ) ∩ Ar (Σ− ) or λ ∈ Ap (Σ− ) ∩ Ar (Σ+ ), then λ ∈ σp (A). y− ∈ L2 (R, dΣ+ ) ⊕ L2 (R, dΣ− ) is a solution of the Proof. A vector y = y+ = λy if and only if equation Ay y ∈ ker(TΣ∗+ − λI) ⊕ ker(TΣ∗− − λI) and y ∈ dom(A). h− if and only if ∈ dom(TΣ∗+ )⊕dom(TΣ∗− ) belongs to dom(A) Recall that h = h+ + Γ− 0 h− = Γ0 h+ ,

+ Γ− 1 h− = Γ1 h+ .

It follows from (2.3) that tΓ± ± ± 0 h± dΣ± (t), h± (t) − 2 Γ1 h± = C± Γ0 h± + t +1 R

h± ∈ dom(TΣ∗± ).

(3.13)

(3.14)

Functional Model for Indeﬁnite Sturm-Liouville Operators

263

(3.4) and (2.3) yield h± (t) ∈ L1 (R, dΣ± ) for arbitrary h± (t) ∈ dom(TΣ∗± ), and using (3.14), we obtain ± Γ 1 y± = y± (t)dΣ± (t) . (3.15) R

∈ L (R, dΣ± ) and (2.3) (or even simpler (2.5)) yields If λ ∈ Ar , then 1 ∗ ∈ dom(T ) and Σ± t−λ 1 t−λ

that

2

Γ± 0

1 =1. t−λ

(3.16)

The function χ{λ} (t), λ ∈ R, is a nonzero vector in L2 (R, dΣ± ) exactly when λ ∈ Ap ; in this case, ± χ = 0, Γ χ = dΣ± (t) = dΣ± ({λ}) . (3.17) Γ± 0 {λ} 1 {λ} {λ}

= λy and consider the case ker(T ∗ − λ) = {0} (the case 1) Suppose Ay Σ− ker(TΣ∗+ − λ) = {0} is analogous). Then y− = 0 and, by (3.13), we get Γ+ 0 y+ = 0, y = 0. Hence y ∈ dom(Q ) (see (2.3)), and Q y = λy . This implies Γ+ + Σ+ Σ+ + 1 + *+ + y+ (t) = c1 χ{λ} (t), c1 ∈ C. On the other hand, 0 = Γ1 y+ (t) = R y+ (t)dΣ+ (t). Thus c1 = 0 and y+ = 0 a.e. with respect to the measure dΣ+ . 2) Let λ ∈ Ap (Σ+ ) ∩ Ap (Σ− ). By (3.3), we have − c1 χ{λ} (t) , c± y(t) = 1 ∈ C. c+ 1 χ{λ} (t) Since λ ∈ Ap (Σ± ), we see that λ ∈ R and dΣ± ({λ}) = 0. Taking into account + (3.17), we see that system (3.13) is equivalent to c− 1 dΣ− ({λ}) = c1 dΣ+ ({λ}). Therefore the geometric multiplicity of λ equals 1 and , + 1 χ (t) {λ} dΣ− ({λ}) (3.18) is one of corresponding eigenvectors of A. y0 = 1 χ (t) dΣ+ ({λ}) {λ} − y1 1 − λy1 = y0 . By (2.5), we have Let y1 = and Ay y1+ − − − − Γ0 y1 y1 (t) ty1 (t) − = y0 . − λ + Γ+ ty1+ (t) y1+ (t) 0 y1 Thus, (t − λ)y1± (t) =

1 ± χ{λ} (t) + Γ± 0 y1 . dΣ± ({λ})

Choosing t = λ, we obtain ± Γ± 0 y1 = −

1 = 0. dΣ± ({λ})

(3.19)

264

I.M. Karabash

Therefore, y1± = −

χR\{λ} (t) 1 + c± 2 χ{λ} (t) , dΣ± ({λ}) t − λ

(3.20)

+ − 2 2 where c± 2 ∈ C. The conditions y1 ∈ L (R, dΣ+ ) and y1 ∈ L (R, dΣ− ) are equivalent to (3.6) and (3.7), respectively. Assume that (3.6) and (3.7) are fulﬁlled. By (3.15), we have 1 1 ± ± dΣ± (t) + c± Γ1 y1 = − 2 dΣ± ({λ}). dΣ± ({λ}) R\{λ} t − λ

if and only if the conditions (3.5) The latter and (3.19) implies that y1 ∈ dom(A) and 1 1 − dΣ− (t) + c− 2 dΣ− ({λ}) dΣ− ({λ}) R\{λ} t − λ 1 1 (3.21) =− dΣ+ (t) + c+ 2 dΣ+ ({λ}) dΣ+ ({λ}) R\{λ} t − λ 2 are fulﬁlled. Thus, the quotient space ker(A−λ) / ker(A−λ) = {0} if and only if the conditions (3.5), (3.6), and (3.7) are satisﬁed. In this case, generalized eigenvectors of ﬁrst order y1 have the form (3.20) with constants c± 2 such that (3.21) holds. − Assume that all condition mentioned above are satisﬁed. Then dim ker(A 2 λ) / ker(A − λ) = 1 and one of generalized eigenvectors of ﬁrst order is given by the constants 1 ± 2 dΣ∓ (t), c2 = −α1 t − λ R\{λ}

α1 := dΣ− 1({λ}) = dΣ+ 1({λ}) . − y2 2 − λy2 = y1 , then and Ay If y2 = y2+

where

± (t − λ)y2± (t) = y1± (t) + Γ± 0 y2

= −α1

χR\{λ} (t) ± ± + c± 2 χ{λ} (t) + Γ0 y2 . t−λ

(3.22)

For t = λ we have ± ± Γ± 0 y2 = −c2 .

(3.23)

Consequently, y2± = −α1

χR\{λ} (t) χR\{λ} (t) + c± − c± 2 3 χ{λ} (t), (t − λ)2 t−λ

c± 3 ∈ C.

(3.24)

Functional Model for Indeﬁnite Sturm-Liouville Operators

265

By (3.23), conditions (3.13) for y2 has the form + c− 2 = c2 , − α1

(3.25)

1 1 −1 dΣ− (t) − c− dΣ− (t) + c− 2 3 α1 2 (t − λ) t − λ R\{λ} R\{λ} 1 1 + −1 dΣ+ (t) + c+ = −α1 dΣ (t) − c + 2 3 α1 . 2 (t − λ) t − λ R\{λ} R\{λ}

χR\{λ} (t) ∈ L2 (R, dΣ± ) and (3.25) is fulﬁlled. This (t − λ)2 is equivalent (3.8) and (3.9) for j = 2. Continuing this line of reasoning, we obtain part 2) of the theorem. 3) The idea of the proof for part 3) is similar to that of part calcula −2),1 but c1 t−λ . Hence tions are simpler. Let λ ∈ Ar (Σ+ ) ∩ Ar (Σ+ ). Then y(t) = 1 c+ 1 t−λ (3.13) has the form 1 1 − + − + c1 c1 = c1 , dΣ− (t) = c1 dΣ+ (t) . t − λ t − λ R R Thus y2 exists if and only if

if and only if (3.10) holds true; in this case Consequently λ is an eigenvalue of A 1 t−λ the geometric multiplicity is 1 and y0 = is a corresponding eigenvector 1 t−λ

of A. 1 − λy1 = y0 where y1 = Let Ay (t − λ)y1± (t) =

y1− y1+

. Then

1 + c± 2 , t−λ

± ± c± 2 = Γ0 y 1 .

c±

± 1 2 2 Therefore y1± = (t−λ) 2 + t−λ . The case y1 ∈ L (R, dΣ± ) is characterized by (3.11) with j = 2. Conditions (3.13) become 1 1 c− c+ + 2 2 dΣ dΣ+ (t), c− + (t) = + − 2 = c2 . (t − λ)2 t−λ (t − λ)2 t−λ R R

Taking into account (3.10), we see that the generalized eigenvector y1 exists if and only if conditions (3.11), (3.12) are satisﬁed for j = 2. Continuing this line of reasoning, we obtain part 3) of the theorem. ∩ Ar (Σ− ) (the case λ ∈ Ap (Σ− ) ∩ Ar (Σ+ ) is similar). 4) Supposeλ ∈ Ap (Σ+ ) 1 c− 1 t−λ Then y(t) = and (3.13) has the form c+ 1 χ{λ} (t) 1 − − dΣ− (t) = c+ c1 c1 = 0, 1 dΣ+ ({λ}) . R t−λ + Thus c− 1 = c1 = 0 and λ ∈ σp (A).

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Now we consider the general case when the functions Σ± satisfy (2.1) and C± are arbitrary real constants. Lemma 3.2. Let k ∈ N and let one of the following two assumptions be fulﬁlled: (a) λ ∈ C \ R or (b) λ ∈ R, dΣ+ ({λ}) = dΣ− ({λ}), and

χR\{λ} (t) (t−λ)k

χ

χR\{λ} (t) (t−λ)k

∈ L2 (R, dΣ+ ),

∈ L2 (R, dΣ− ).

(t)

R\{λ} ∈ dom(TΣ∗+ ), Then (t−λ) k ments are equivalent:

χR\{λ} (t) (t−λ)k

∈ dom(TΣ∗− ), and the following two state-

(i) Γ− 1 (ii)

χR\{λ} (t) χR\{λ} (t) = Γ+ 1 (t−λ)k ; (t−λ)k lim Φ(k−1) (λ + iε) = 0, where ε→0 ε∈R

the function Φ is deﬁned by

Φ := MΣ+ ,C+ − MΣ− ,C− and Φ(j) is its jth derivative (Φ(0) = Φ). If, additionally, λ ∈ σess (QΣ+ ) ∪ σess (QΣ− ), then statements (i) and (ii) are equivalent to (iii) the function Φ is analytic in a certain neighborhood of λ and Φ(k−1) (λ) = 0. (If MΣ+ ,C+ − MΣ− ,C− is deﬁned in a punctured neighborhood of λ and has a removable singularity at λ, then we assume that Φ is analytically extended over λ.) Proof. We assume here and below that j ∈ N. χR\{λ} 2 First note that if λ ∈ σess (QΣ± ), then (t−λ) j ∈ L (R, dΣ± ) for any j ∈ N, χ R\{λ} ∗ and using the deﬁnition of dom(TΣ∗± ), we see that (t−λ) j ∈ dom(TΣ± ) for any j. Generally, the last statement is not true for λ ∈ σess (QΣ± ). But under assumptions of the lemma, we have 1 dΣ± (t) < ∞ (3.26) |t − λ|2j R\{λ} for j = k. Taking into account the ﬁrst assumption in (2.1), we see that (3.26) χR\{λ} (t) is valid for all j ≤ k. The latter implies that (t−λ) ∈ dom(TΣ∗± ) for all j ≤ k. j Moreover,

χR\{λ} (t) (t−λ)j

∈ dom(QΣ± ) if 2 ≤ j ≤ k (assuming k ≥ 2). Therefore, Γ± 0

χR\{λ} (t) = 0, (t − λ)j

2≤j≤k .

The last statement does not hold in the case j = 1. Using (2.5), one has Γ± 0

χR\{λ} (t) = 1. t−λ

(3.27)

Functional Model for Indeﬁnite Sturm-Liouville Operators Eqs. (2.3) (see also (3.14)) allow us to conclude that χR\{λ} (t) χR\{λ} (t) t + = C − dΣ± (t), Γ± ± 1 t−λ t−λ t2 + 1 R χR\{λ} (t) ± χR\{λ} (t) Γ1 = dΣ± (t) if 2 ≤ j ≤ k . j (t − λ)j R (t − λ)

267

(3.28) (3.29)

If λ ∈ σ(QΣ± ) (in particular, if λ ∈ R), then (2.4) shows that Γ± 1

χR\{λ} (t) 1 (j−1) = Γ± = (j − 1)! MΣ± ,C± (λ). 1 (t − λ)j (t − λ)j

This proves the equivalence of (i), (ii), and (iii) for the case when λ ∈ σ(QΣ+ ) ∪ σ(QΣ− ) (this simplest case explains the crux of the lemma). Consider the case λ ∈ σ(QΣ+ ) ∪ σ(QΣ− ) and λ ∈ σess (QΣ+ ) ∪ σess (QΣ− ). The assumptions of the lemma state that dΣ+ ({λ}) = dΣ− ({λ}). So λ is an isolated eigenvalue of both the operators QΣ+ and QΣ− and is an isolated jump discontinuity of Σ+ and Σ− . This and dΣ+ ({λ}) = dΣ− ({λ}) imply that Φ has a removable singularity at λ and can be considered as an analytic function in a certain neighborhood of λ. Moreover, χR\{λ} χR\{λ} (k − 1)! Φ(k−1) (λ) = Γ+ − Γ− , 1 1 (t − λ)k (t − λ)k and (i) ⇔ (ii) ⇔ (iii) is shown again. Now let assumption (b) be satisﬁed and let λ ∈ σess (QΣ+ ) ∪ σess (QΣ− ). Then the function Φ is not analytic in λ, but the limit in statement (ii) exists and (k − 1)! lim Φ(k−1) (λ + iε) = Γ+ 1 ε→0 ε∈R

χR\{λ} (t) χR\{λ} (t) − Γ− . 1 (t − λ)k (t − λ)k

(3.30)

Indeed, taking dΣ+ ({λ}) = dΣ− ({λ}) into account, we get Φ(z) = C+ − C− + I∞ (z) + Iλ (z), where t 1 − 2 (dΣ+ (t) − dΣ− (t)) , I∞ (z) := t−λ t +1 R\[λ−δ,λ+δ]

Iλ (z) :=

t 1 − t − λ t2 + 1

(dΣ+ (t) − dΣ− (t)) ,

[λ−δ,λ)∪(λ,λ+δ]

and δ is any ﬁxed positive number. The function I∞ (z) is analytic at λ. Formula (3.26) is valid for j ≤ k and allows us to apply Lebesgue’s dominated convergence (j−1) theorem to the limit lim Iλ (λ + iε). As a result, we see that (3.28) implies ε→0 ε∈R

(3.30) for k = 1 and (3.29) implies (3.30) for k ≥ 2.

= A{Σ + , C+ , Σ− , C− }, where the functions Σ± satisfy (2.1) Theorem 3.3. Let A and C± are certain real constants. Then the following statements hold:

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I.M. Karabash

1) If λ ∈ A0 (Σ+ ) ∪ A0 (Σ− ), then λ ∈ σp (A). 2) If λ ∈ Ap (Σ+ ) ∩ Ap (Σ− ), then the geometric multiplicity of λ equals 1; (i) λ is an eigenvalue of A; (ii) the eigenvalue λ is simple if an only if at least one of conditions (3.5), (3.6), (3.7) is not fulﬁlled; (iii) if conditions (3.5), (3.6) and (3.7) hold true, then the algebraic multiplicity of λ equals the greatest number k (k ∈ {2, 3, 4, . . . } ∪ {+∞}) such that conditions (3.8) and lim Φ(j−2) (λ + iε) = 0

ε→0 ε∈R

(the function Φ is deﬁned in Lemma 3.2),

(3.31)

are fulﬁlled for all j ∈ N such that 2 ≤ j ≤ k − 1 (in particular, k = 2 if at least one of conditions (3.8), (3.31) is not fulﬁlled for j = 2). if and only if 3) Assume that λ ∈ Ar (Σ+ ) ∩ Ar (Σ− ). Then λ ∈ σp (A) lim Φ(λ + iε) = 0.

ε→0 ε∈R

(3.32)

If (3.32) holds true, then the geometric multiplicity of λ is 1, and the algebraic multiplicity is the greatest number k (1 ≤ k ≤ ∞) such that the conditions (3.11) and lim Φ(j−1) (λ + iε) = 0

ε→0 ε∈R

(3.33)

are fulﬁlled for all j ∈ N such that 1 ≤ j ≤ k. 4) If λ ∈ Ap (Σ+ ) ∩ Ar (Σ− ) or λ ∈ Ap (Σ− ) ∩ Ar (Σ+ ), then λ ∈ σp (A). Proof. The proof is similar to that of Theorem 3.1, but some technical complications appear. Namely, (3.15) is not valid whenever any of conditions in (3.4) is not satisﬁed. We have to use (3.14), which is valid in the general case. Note that (3.17) holds true. In the case λ ∈ Ar (Σ± ), (3.16) holds also. When λ ∈ σp (QΣ± ), Eq. (3.16) should be changed to (3.27). The proof of statements 1) and 4) remains the same. 2) Let λ ∈ Ap (Σ+ ) ∩ Ap (Σ− ). As before, we see that λ is an eigenvalue of A with geometric multiplicity 1 and one of corresponding eigenvectors has the form (3.18). − y1 1 − λy1 = y0 . In the same way, we get (3.19), (3.20) and Ay Let y1 = y1+ as well as the fact that the conditions y1+ ∈ L2 (R, dΣ+ ) and y1− ∈ L2 (R, dΣ− ) are equivalent to (3.6) and (3.7), respectively. If (3.6) and (3.7) are fulﬁlled, we obtain ± Γ± 1 y1 = −

χR\{λ} (t) 1 Γ± + c± 2 dΣ± ({λ}). dΣ± ({λ}) 1 t − λ

Functional Model for Indeﬁnite Sturm-Liouville Operators

269

if and only if conditions (3.5) and The latter and (3.19) implies that y1 ∈ dom(A) − dΣ−1({λ}) Γ− 1

χR\{λ} (t) t−λ

+ 1 + c− 2 dΣ− ({λ}) = − dΣ+ ({λ}) Γ1

χR\{λ} (t) t−λ

+ c+ 2 dΣ+ ({λ}) (3.34)

are fulﬁlled. Thus, generalized eigenvectors of ﬁrst order exist if and only if conditions (3.5), (3.6), and (3.7) are satisﬁed. In this case, y1 has the form (3.20) with constants c± 2 such that (3.34) holds. In particular, the constants 2 ∓ c± 2 = −α1 Γ1

χR\{λ} (t) , t−λ

(3.35)

give a generalized eigenvector (as before, α1 = dΣ−1({λ}) = dΣ+ 1({λ}) ). − y2 2 − λy2 = y1 . Then (3.22), (3.23), and (3.24) have Let y2 = and Ay y2+ ± 2 to be fulﬁlled with c± 2 given by (3.35). So y2 belong to L (R, dΣ± ) if and only if + (3.8) is satisﬁed for j = 2. Conditions (3.13) are equivalent to c− 2 = c2 and χR\{λ} − χR\{λ} −1 + c− − c− −α1 Γ− 1 2 Γ1 3 α1 = 2 (t − λ) t−λ χR\{λ} + χR\{λ} −1 + c+ = −α1 Γ+ − c+ 1 2 Γ1 3 α1 . 2 (t − λ) t−λ Thus y2 exists if and only if, for j = 2, conditions (3.8) and χR\{λ} χR\{λ} = Γ+ Γ− 1 1 j−1 (t − λ) (t − λ)j−1

(3.36)

are fulﬁlled. By Lemma 3.2, (3.36) is equivalent to (3.31) with j = 2. Continuing this line of reasoning, we obtain parts 2) and 3) of the theorem. Remark 3.4. (1) In the last theorem, the conditions that determine the algebraic multiplicities are given in the terms of the function Φ = MΣ+ ,C+ − MΣ− ,C− , so in the terms of abstract Weyl functions MΣ± ,C± . Using Lemma 3.2 and (3.28), (3.29), Theorem 3.3 can be easily rewritten in terms of the spectral measures Σ± , χR\{λ} (t) but this makes the answer longer due to the diﬀerent forms of Γ± for the 1 (t−λ)j cases j = 1 and j ≥ 2, see (3.28) and (3.29). In the case when assumptions (3.4) are fulﬁlled, (3.28) can be written in the form of (3.29) and we get Theorem 3.1. that belong to ρ(QΣ+ ⊕ QΣ+ ) can be found in (2) Note that eigenvalues of A the terms of MΣ± ,C± using [21] (and, perhaps, [18]), see the next section. Algebraic multiplicities of eigenvalues in ρ(QΣ+ ⊕ QΣ+ ) can be found using Krein’s resolvent formula (see [21, 22] for a convenient abstract form), root subspaces for eigenvalues in ρ(QΣ+ ⊕ QΣ+ ) were found in [19]. Theorem 3.3 has some common points with [12], where the abstract Weyl function was used to ﬁnd eigenvalues of a self-adjoint operator. But the approach of the present paper goes in the backward direction: we use the spectral measures dΣ± and the functional model to ﬁnd eigenvalues and root subspaces and then, using Lemma 3.2, return to the answer in the terms of the abstract Weyl functions given in Theorem 3.3.

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(3) Various generalizations of (R)-functions and their functional models were considered in [25, 34]. These results were applied to certain classes of regular Sturm-Liouviile problems in [26, 10, 27]. 3.2. Essential and discrete spectra of the model operator and of indeﬁnite Sturm-Liouville operators Besides the symmetry condition σ(T ) = σ(T )∗ the spectrum of a J-self-adjoint operator can be fairly arbitrary (see [55]). An example of a diﬀerential operator with a “wild” spectrum was given in [4, 2]. Example 3.5. Consider the operator A in L2 [−1, 1] associated with the diﬀerential expression

(sgn x) ((sgn x)y ) and boundary conditions y(−1) = 0 = y(1). More precisely, Ay = −y

, dom(A) = {y ∈ W22 (−1, 0) ⊕ W22 (0, 1) : y(−0) = y(+0), y (−0) = −y (+0) and y(−1) = 0 = y(1)}. The operator A is J-self-adjoint with J given by Jf (x) = (sgn x)f (x). It was observed in [4, 2] that every λ ∈ C is an eigenvalue of A and, moreover, every λ ∈ R is a nonsimple eigenvalue. Theorem 3.1 shows that every λ ∈ C is an eigenvalue of inﬁnite algebraic multiplicity (the geometric multiplicity of λ equals 1). Indeed, introducing as in Theorem 2.4 the operator Amin := A ∩ A∗ , we see that − + − dom(A) = {y ∈ A∗min : Γ+ 0 y+ = Γ0 y− , Γ1 y+ = Γ1 y− }, where y+ (y− ) is the orthoprojection of y on L2 [0, 1] (resp., L2 [−1, 0]),

Γ+ 0 y+ := −y (+0),

Γ− 0 y− := y (−0),

and Γ± 1 y± := y(±0).

+ + 2 On the other hand, {C, Γ− 0 , Γ1 } is a boundary triple for Amin = Amin L [0, 1] and − − − 2 {C, Γ0 , Γ1 } is a boundary triple for Amin = Amin L [−1, 0]. It is easy to see that d2 the diﬀerential expression − dx 2 is associated with both the symmetric operators . These operators and their boundary triples are unitarily equivalent. This A± min means that the corresponding Weyl functions M± coincide. Now Theorem 3.3 implies that any λ ∈ C \ R is an eigenvalue of inﬁnite algebraic multiplicity and therefore σ(A) = C. (Actually in this case conditions (3.4) hold, so Theorem 3.1 can also be applied.) Finally, note that the functions M± are meromorphic and therefore Theorem 3.3 (2)–(3) and Lemma 3.2 (ii)⇔(iii) imply that each point λ ∈ C is an eigenvalue of inﬁnite algebraic multiplicity.

Remark 3.6. In [62], a characterization of the case σ(A) = C was given in terms 1 d d of coeﬃcients for regular operators A = r(x) dx p(x) dx with Dirichlet boundary conditions. Both coeﬃcients r and p were allowed to change sign, modiﬁcations of arguments for general regular problems were suggested also. = C is exceptional in the Arguments of Example 3.5 show that the case σ(A) sense of the next proposition.

Functional Model for Indeﬁnite Sturm-Liouville Operators

271

Proposition 3.7. The following statements are equivalent: (i) MΣ+ ,C+ (λ) = MΣ− ,C− (λ) for all λ ∈ C \ (supp dΣ+ ∪ supp dΣ− ); (ii) the measures dΣ+ and dΣ− coincide, and C+ = C− ; = C. (iii) σ(A) Moreover, if statements (i)–(iii) hold true, then every point in the set . / C \ σess (QΣ+ ) ∪ σess (QΣ− ) of inﬁnite algebraic multiplicity. is an eigenvalue of A If MΣ+ ,C+ (·) ≡ MΣ− ,C− (·), then the nonreal spectrum is the set of zeros of analytic function Φ deﬁned in Lemma 3.2. More precisely, Theorem 3.3 shows that ∩ ρ(QΣ ⊕ QΣ ) σ(A) + − = {λ ∈ ρ(QΣ+ ) ∩ ρ(QΣ− ) : MΣ+ ,C+ (λ) = MΣ− ,C− (λ)} ⊂ σp (A)

(3.37)

(this statement also can be obtained from [21, Proposition 2.1]). It is easy to see that (3.37) and Theorem 3.3 yield the following description of the discrete and essential spectra (cf. [1, p. 106, Theorem 1]). Proposition 3.8. Assume that MΣ+ ,C+ (λ0 ) = MΣ− ,C− (λ0 ) for certain λ0 in the set ρ(QΣ+ ) ∩ ρ(QΣ− ). Then: = σess (QΣ+ ) ∪ σess (QΣ− ) ⊂ R; (i) σess (A) . / = σdisc (QΣ+ ) ∩ σdisc (QΣ− ) (ii) σdisc (A) ∪ {λ ∈ ρ(QΣ+ ) ∩ ρ(QΣ− ) : MΣ+ ,C+ (λ) = MΣ− ,C− (λ)}; (iii) the geometric multiplicity equals 1/ for all eigenvalues of A; . (iv) if λ0 ∈ σdisc (QΣ+ ) ∩ σdisc (QΣ− ) , then the algebraic multiplicity of λ0 is equal to the multiplicity of λ0 as a zero of the holomorphic function 1 MΣ+ ,C+ (λ)

−

1 ; MΣ− ,C− (λ)

(v) if λ0 ∈ ρ(QΣ+ ) ∩ ρ(QΣ− ), then the algebraic multiplicity of λ0 is equal to the multiplicity of λ0 as zero of the holomorphic function MΣ+ ,C+ (λ) − MΣ− ,C− (λ). is deﬁnitizable if and only if the sets supp dΣ+ and Remark 3.9. The operator A supp dΣ− are separated by a ﬁnite number of points (in the sense of [48, Deﬁnition 3.4]). This criterion was obtained for operators A = (sgn x)(−d2 /dx2 + q(x)) in [38, 39] (see also [47, Section 2.3]) using the result of [36] and the fact that ρ(A) = ∅; the detailed proof was published in [48, Theorem 3.6]. The same proof is valid for if we note that ρ(A) = ∅ whenever supp dΣ+ and supp dΣ− are the operator A separated by a ﬁnite number of points. Indeed, in this case supp dΣ+ = supp dΣ− since supp dΣ± are unbounded due to the second assumption in (2.1).

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3.3. Non-emptiness of resolvent set for Sturm-Liouville operators To apply Proposition 3.8 to the J-self-adjoint Sturm-Liouville operator d d sgn x − p(x) + q(x) A= |r(x)| dx dx introduced in Section 2.3, one has to insure that ρ(A) = ∅. Here we discuss brieﬂy + , C+ , Σ− , C− } is one of model operaresults of this type. We assume that A{Σ tors unitarily equivalent to the operator A and that M± (·) = MΣ± ,C± (·) are the associated Weyl functions. Sometimes it is known that the asymptotic formulae of M+ and M− at ∞ are diﬀerent. This argument was used in [47, Proposition 2.5 (iv)] to show that ρ(A) = ∅ for the operator (sgn x)(−d2 /dx2 + q(x)). Indeed, (2.34) shows that MN+ (·) ≡ MN− (·). One can extend this result using [3, Theorem 4] in the following way: if p ≡ 1 and there exist constants r± > 0 such that x (±r(t) − r± )dt = o(x) as x → ±0, 0

then ρ(A) = ∅. *x If p ≡ 1, one may use the standard change of variable s = 0 to the form with p ≡ 1:

dτ p(τ )

to get back

Proposition 3.10. Assume that there exist positive constants r± such that x x dt r(t)dt = (r+ + o(1)) as x → +0, (3.38) p(t) 0 0 0 0 dt as x → −0. (3.39) |r(t)|dt = (r− + o(1)) p(t) x x Then ρ(A) = ∅. Another simple way to prove ρ(A) = ∅ uses information on the supports of spectral measures dΣ± . In this way, it was obtained in [43, Proposition 3.1] that 1 d d (− dx p dx + q) is semi-bounded from below (the proof [43, p. ρ(A) = ∅ if L = |r| 811] given for p ≡ 1 is valid in the general case). Moreover, modifying slightly the same arguments, we get the next result. − Proposition 3.11. Assume that at least one of the symmetric operators A+ min , Amin (deﬁned in Section 2.3) is semi-bounded. Then ρ(A) = ∅.

Remark 3.12. (1) Proposition 3.11 has the following application to the theory of locally deﬁnitizable operators (see [35] for basic deﬁnitions): the operator A = sgn(x) d d |r| (− dx p dx + q) introduced in Section 2.3 is locally deﬁnitizable in some open 1 d d neighborhood of ∞ if and only if corresponding operator L = |r| (− dx p dx + q) is semi-bounded from below. This is a natural generalization of [48, Theorem 3.10], where the above criterion for r(x) = sgn x and p ≡ 1 was obtained. The proof of

Functional Model for Indeﬁnite Sturm-Liouville Operators

273

[48, Theorem 3.10] (based on [7]) remains valid in general case if Proposition 3.11 is used instead of [47, Proposition 2.5 (iv)]. Local deﬁnitizability of Sturm-Liouville operators with the weight function r having more than one turning point was considered in [8]. (2) And vice versa, it was noticed in [48, Proposition 4.1] that local deﬁnitizability results could be used to get additional information on non-real spectrum. Namely, the above criterion of local deﬁnitizability implies that the non-real specd d (− dx p dx + q) is bounded if the operator trum σ(A) \ R of the operator A = sgn(x) |r| 1 d d (− dx p dx + q) is semi-bounded from below (the proof is immediate from the L = |r| deﬁnition of the local deﬁnitizability). (3) Under the assumption that a[y] = (sgn x)(−y

+ qy) is in the limit point case in ±∞, the fact that ρ(A) = ∅ was noticed by M.M. Malamud and the author of this paper during the work on [46], and was published in [39, 47].

4. The absence of embedded eigenvalues and other applications 4.1. The absence of embedded eigenvalues for the case of inﬁnite-zone potentials Theorems 3.1 and 3.3 can be applied to prove that the Sturm-Liouville operator A has no embedded eigenvalues in the essential spectrum if some information on the spectral measures dΣ± is known. We illustrate the use of this idea on operators A = (sgn x)L, where L = −d2 /dx2 + q(x) is an operator in L2 (R) with inﬁnite-zone potentials q (in the sense of [56], the deﬁnition is given below). First recall that the operator L = −d2 /dx2 + q(x) with inﬁnite zone potentials q is deﬁned on the maximal natural domain and is self-adjoint in L2 (R) (i.e., the diﬀerential expression is in the limit point case both at ±∞). The spectrum of L is absolutely continuous and has the zone structure, i.e., σ(L) = σac (L) = [μr0 , μl1 ] ∪ [μr1 , μl2 ] ∪ · · · ,

(4.1)

l ∞ where {μrj }∞ 0 and {μj }j=1 are sequences of real numbers such that

μr0 < μl1 < μr1 < · · · < μrj−1 < μlj < μrj < · · ·

,

(4.2)

and lim μrj = lim μlj = +∞.

j→∞

μlj

j→∞

(μrj )

is the left (right, resp.) endpoint of the jth gap in the spectrum σ(L), the “zeroth” gap is (−∞, μr0 ). Following [56], we brieﬂy recall the deﬁnition of inﬁnite-zone potential under the additional assumptions that ∞ j=1

μrj (μrj

−

μlj )

< ∞,

∞ 1 < ∞. l μ j=1 j

(4.3)

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I.M. Karabash

∞ l r Consider inﬁnite sequences {ξj }∞ 1 and {j }1 such that ξj ∈ [μj , μj ], j ∈ {−1, +1} for all j ≥ 1. For every N ∈ N, put ; ;N λ−μl λ−μr ξj −λ gN = N fN = (λ − μr0 ) j=1 μl j μl j , (4.4) j=1 μlj , j j √ " j −fN (ξj ) f (λ)+k2 (λ) kN (λ) = gN (λ) N hN (λ) = N gN (λ)N . (4.5) j=1 g (ξj )(λ−ξj ) , N

It is easy to see from (4.3) that gN and fN converge uniformly on every compact subset of C. Denote lim gN (λ) =: g(λ),

N →∞

lim fN (λ) =: f (λ).

N →∞

[56, Theorem 9.1.1] states that there exist limits lim kN (λ) =: k(λ) for all λ ∈ C.

lim hN (λ) =: h(λ),

N →∞

N →∞

Moreover, the functions g, f , h, and k are holomorphic in C. It follows from [56, Subsection 9.1.2] that the functions mN± (λ) := ±

g(λ) ) k(λ) ∓ i f (λ)

(4.6)

are the Titchmarsh-Weyl m-coeﬃcients on R± (corresponding to the Neumann 2 2 boundary conditions) for some Sturm-Liouville ) operator L = −d /dx + q(x) with a real bounded potential q(·). The branch f (·) of the multifunction is chosen such that both m± belong to the class (R) (see Section 2.3 for the deﬁnition). Deﬁnition 4.1 ([56]). A real potential q is called an inﬁnite-zone potential if the Titchmarsh-Weyl m-coeﬃcients mN± associated with −d2 /dx2 +q(x) on R± admit representations (4.6). Let q be an inﬁnite-zone potential deﬁned as above. B. Levitan proved that under the additional condition inf(μlj+1 − μlj ) > 0, the potential q is almostperiodical (see [56, Chapter 11]). The following theorem describes the structure of the spectrum of the J-selfadjoint operator A = (sgn x)L. Note that the Titchmarsh-Weyl m-coeﬃcients MN± for A introduced in Section 2.3 are connected with m-coeﬃcients for L through MN± (λ) = ±mN± (±λ),

λ ∈ C \ supp dΣ±

(see, e.g., [47, Section 2.2]). (4.7)

Theorem 4.2. Let L = −d /dx + q(x) be a Sturm-Liouville operator with an inﬁnite-zone potential q and let A = (sgn x)L. Assume also that assumptions (4.3) are satisﬁed for the zones of the spectrum σ(L). Then: (i) σp (A) = σdisc (A), that is all the eigenvalues of A are isolated and have ﬁnite algebraic multiplicity. Besides, all the eigenvalues and their geometric and algebraic multiplicities are given by statements (ii)–(v) of Proposition 3.8. (ii) The nonreal of a ﬁnite number of eigenvalues. $<spectrum σ(A) % \ R $consists % <∞ ∞ r l l r (iii) σess (A) = [μ , μ ] ∪ [−μ , −μ ] . j+1 j+1 j j=0 j j=0 2

2

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275

The functions g, f , k, and h deﬁned above are holomorphic in C. Moreover, g and f admit the following representations g(λ) =

∞ = ξj − λ , μlj j=1

f (λ) = (λ − μr0 )

N = λ − μlj λ − μrj j=1

μlj

μlj

,

where the inﬁnite products converge uniformly on all compact subsets of C due to assumptions (4.3) (see [56, Section 9]). It follows from (4.5) that 2 (λ) = fN (λ) hN (λ)gN (λ) − kN

and h(λ)g(λ) − k 2 (λ) = f (λ).

(4.8)

From this and (4.7) we get M± (λ) =

) k(±λ) ± i f (±λ) g(±λ) ) = . h(±λ) k(±λ) ∓ i f (±λ)

(4.9)

It from (4.8) that the zeros of the function h belongs to (−∞, μr0 ] ∪ <∞follows l r j=1 [μj , μj ]). Besides, all the zeroes of h have multiplicity 1 (otherwise one of the functions MN± does not belongs to the class (R)). This implies that the spectra of the operators A± 0 deﬁned in Section 2.3 have the following structure: ± σess (A± 0 ) = σac (A0 ),

− σac (A+ 0 ) = −σac (A0 ) = σ(L) =

∞ 9

[μrj , μlj+1 ],

(4.10)

j=0

) ± σp (A± 0 ) = σdisc (A0 ) = {λ ∈ R \ σ(±L) : h(±λ) = 0, k(±λ) ± i f (±λ) = 0}. (4.11) Proof of Theorem 4.2. Since σ(A) = C (see Section 3.3), Proposition 3.8 (i) proves (iii). (i) We have to show that are no eigenvalues in σess (A). This statement < there r follows from the fact that ± ∞ [μ , μlj+1 ] ⊂ A0 (ΣN± ) (see (3.1) for the deﬁnition) j j=0 <∞ and Theorem 3.1 (1). Indeed, let λ0 be in j=0 [μrj , μlj+1 ]. Note that (4.10) implies that λ0 ∈ σac (A+ 0 ) = σac (QΣN+ ). It follows from (4.9), (2.33) and the Stieltjes inversion formula (see, e.g., [43, formula (2.9)]) that ) ∞ 9 f (±t)

for a.a. t ∈ ± [μrj , μlj+1 ] (4.12) ΣN± (t) = πh(±t) j=0

<∞

In particular, if λ0 ∈ j=0 (μrj , μlj+1 ), then Σ N+ (t) ≥ C1 for |t − λ0 | small enough and a certain constant C1 > 0. So |t − λ0 |−2 dΣN+ (t) = ∞. (4.13) R

(or λ0 = μlj ), (4.13) also holds since Σ + (t) ≥ C2 |t−λ0 |1/2 , In the case when λ0 = C2 > 0, for t in a certain left (resp., right) neighborhood of λ0 . Arguments for σ(−L) ⊂ A0 (Σ− ) are the same. This concludes the proof of (i). μrj

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(ii) We have to prove only that σ(A) \ R is ﬁnite. [48, Proposition 4.1] (see also Remark 3.12 (2)) implies that σ(A) \ R

is a bounded set.

(4.14)

Proposition 3.8 (ii) states that points of σ(A)\R are zeros of the holomorphic in C \ (supp dΣN+ ∪ supp dΣN− ) function MN+ (λ) − MN− (λ), and therefore, are roots of each of the following equations (each subsequent equation is a modiﬁcation of the previous one) g(λ) g(−λ) ) ) = , k(λ) − i f (λ) k(−λ) + i f (−λ) (g(λ)k(−λ) − g(−λ)k(λ))2

) ) = −g 2 (λ)f (−λ) − g 2 (−λ)f (λ) − 2g(λ)g(−λ) f (−λ) f (λ),

$

%2 [g(λ)k(−λ) − g(−λ)k(λ)]2 + g 2 (λ)f (−λ) + g 2 (−λ)f (λ) − 4g 2 (λ)g 2 (−λ)f (−λ)f (λ) = 0.

The entire function in the left side of the last equation is not identically zero since it is positive in the points of the set {λ ∈ (σ(L) \ σ(−L)) ∪ (σ(−L) \ σ(L)) : g(λ)g(−λ) = 0} (this set is nonempty due to (4.1), (4.6), and the fact that MN± (·) ≡ 0). Therefore, MN+ − MN−

has a ﬁnite number of zeros

in any bounded subset of C \ (supp dΣN+ ∪ supp dΣN− ) (⊃ C \ R). Combining this and (4.14), we see that σ(A) \ R is ﬁnite.

(4.15)

Note that (4.15) implies that each gap of the essential spectrum σess (A) has at most ﬁnite number of eigenvalues. 4.2. Other applications A part of this paper was obtained in author’s candidate thesis [38], was announced together with some applications in the short communication [39], and was a base of several author’s conference talks in 2004–2005. Namely, in [38, 39], the oper + , C+ , Σ− , C− } was introduced as a functional model for the operator ator A{Σ (sgn x)(−d2 /dx2 + q) (see Theorem 2.6) and the description of eigenvalues under conditions (3.4) was given (see Theorem 3.1). These results were used in [47, 43]. The idea of the functional model originated from [42, 46], where a representation of the operator (sgn x)(−d2 /dx2 + q) as an extension of a direct sum of two symmetric Sturm-Liouville operators was used essentially (in a less explicit form the same idea appeared earlier in [17, 28]). The absence of embedded eigenvalues in the essential spectrum of the operator (sgn x)(−d2 /dx2 + q) with a ﬁnite-zone potential q was proved in [47, Theorem

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277

7.1 (2)] via Theorem 3.1. This proof is adopted in part (i) of Theorem 4.2 for the inﬁnite-zone case. Theorem 3.1 helps to ﬁnd algebraic multiplicity of embedded eigenvalues. This was used in a paper of A. Kostenko and the author (see [43], Proposition 2.2, Theorems 6.1 (ii) and 6.4 (ii)) to prove simplicity of the eigenvalue λ = 0 for two operators of type (sgn x)(−d2 /dx2 + q). This fact and necessary conditions for regularity of critical points (see [43, Theorem 3.9]) allowed us to show that 0 is a singular critical point for the considered operators (see [43, Remark 6.3, Theorem 6.4 (iii)] and also Section 5 of the present paper). The fact that 0 may be a non-semi-simple eigenvalue (i.e., ker A2 = ker A) is essential for the theory of “two-way” diﬀusion equations. In the simplest case, such equations lead to spectral analysis of J-nonnegative operators that take the form of the operator A introduced in Section 2.3. If 0 and ∞ are not singular critical points of A than the algebraic multiplicity of 0 aﬀects proper settings of boundary value problems for the corresponding diﬀusion equation (see [6, 5, 32]). If 0 is a singular critical point of A, as in the examples constructed in [43, 45] and Section 5, the existence and uniqueness theory for corresponding diﬀusion equations is not well understood (see [49, Section 1], [14, 60, 40]). Remark 4.3. For periodic potentials and certain classes of decaying potentials, asymptotic behavior of solutions of −y

+ qy = λy is well known and yields the absence of eigenvalues in some parts of the spectrum of the operator A = (sgn x)L (where L = −d2 /dx2 + q is a self-adjoint operator in L2 (R)). * +∞ For example, the assumption −∞ (1 + |x|)|q(x)|dx < ∞ yields σp (A) ∩ R = ∅ (see, e.g., [59, Lemma 3.1.1 and formula (3.2.4)]). This fact was used essentially in [45, Section 4] to prove that, for this class of potentials, A is similar to self-adjoint operator exactly when L ≥ 0. For periodic potentials, σess (A) ∩ σp (A) = ∅ follows from [64, Section 21]. For q ∈ L1 (R), the fact that the only possible real eigenvalue is 0 (i.e., σp (A) ∩ R ⊂ {0}) follows immediately from [64, Section 5.7] or from [13, Problem IX.4].

5. Remarks on indeﬁnite Sturm-Liouville operators with the singular critical point 0 In this section, we provide an alternative approach to examples of J-self-adjoint Sturm-Liouville operators with the singular critical point 0 constructed in [43, Sections 5 and 6] and [45, Section 5]. As before, let H be a Hilbert space with a scalar product (·, ·)H . Suppose that H = H+ ⊕ H− , where H+ and H− are (closed) subspaces of H. Denote by P± the orthogonal projections from H onto H± . Let J = P+ − P− and [·, ·] := (J·, ·)H . Then the pair K = (H, [·, ·]) is called a Krein space. The operator J is called a fundamental symmetry (or a signature operator) in the Krein space K. Basic facts on the theory of Krein spaces and the theory of J-self-adjoint deﬁnitizable operators can be found, e.g., in [55]. An account on (non-diﬀerential) operators

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with a ﬁnite singular critical point can be found in [15]. The following proposition is a simple consequence of [55, Theorem II.5.7]. Proposition 5.1. Assume that a J-self-adjoint deﬁnitizable operator B has a simple eigenvalue λ0 ∈ R and that a corresponding eigenvector f0 ∈ ker(B − λ0 ) \ {0} is neutral (i.e., [f0 , f0 ] = 0). Then λ0 is a singular critical point of B. Proof. Assume that λ0 is not a singular critical point of B. Then there exist a J˙ 1 orthogonal decomposition into a direct sum of two closed subspaces H = H0 [+]H that reduces the operator B and such that H0 is a root subspace corresponding to λ0 and λ0 ∈ σp (B H1 ) (this follows from [55], Proposition II.5.1, Proposition II.5.6, Theorem II.5.7, and the decomposition (II.2.10)). Since the eigenvalue λ0 is simple, we have H0 = {cf0 , c ∈ C}. Since f0 is neutral, we see that [f0 , f ] = 0 for all f in H. So {H, [·, ·]} is not a Krein space, a contradiction. It is not diﬃcult to see that the operators considered in [43, Sections 5 and 6] and [45, Section 5] satisfy conditions of Proposition 5.1, and so Proposition 5.1 can be used in these two papers instead of [43, Theorem 3.4]. This does not makes the proofs much simpler, but the results become clearer from the Krein space point of view. Remark 5.2. The necessary similarity condition given in [43, Theorem 3.4] is of independent interest since it provides a criterion of similarity to a self-adjoint operator for operators (sgn x)(−d2 /dx2 + q) with ﬁnite-zone potentials (see [43, Remark 3.7]). And it is unknown whether the condition of [43, Theorem 3.4] provides a criterion of similarity to a self-adjoint operator for the general operator sgn x d d |r| (− dx p dx + q) with one turning point introduced in Section 2.3. Using Proposition 5.1, a large class of operators with the singular critical point 0 similar to that of [43, 45] can be constructed. In the next theorem, we characterize the case described in Proposition 5.1 among the operators Ar := sgn x d2 − |r(x)| that have the limit point case both at ±∞ (and act in L2 (R; |r(x)|dx)). dx2 So we assume that r ∈ L1loc (R),

r = 0 a.e., r(x) = (sgn x)|r(x)|, x2 |r(x)|dx = ∞,

(5.1) (5.2)

R±

and the operator Ar is deﬁned on its maximal domain. Condition (5.2) is equivalent to J-self-adjointness of Ar with J : f (x) → (sgn x)f (x). Obviously, Ar is J-nonnegative and deﬁnitizable (see [16] and also Proposition 3.11). Theorem 5.3. Let assumptions (5.1), (5.2) be satisﬁed. Then: (1) Two following statements are equivalent: (r1) Ar has a simple eigenvalue at 0 and [f0 , f0 ] = 0 for f0 ∈ ker Ar \ {0}.

Functional Model for Indeﬁnite Sturm-Liouville Operators * (r2) r ∈ L1 (R), R r(x)dx = 0, and 2 (y1 (x)) |r(x)|dx = +∞, where y1 (x) := R

x

+∞

r(t) dtds. 0

279

(5.3)

s

(2) If (r2) is satisﬁed, then 0 is a singular critical point of Ar . Proof. We need to prove only (1). Let us note that r ∈ L1 (R) is equivalent to 0 ∈ σp (Ar ). If the latter holds, then f0 (x) ≡ 1 is an eigenfunction (unique up to * multiplication by a constant), and [f0 , f0 ] = 0 is equivalent to R r(x)dx = 0. Assume that the eigenvalue 0 is not simple. Then there is a generalized eigenfunction of ﬁrst order y ∈ L2 (R, |r(x)|dx), which is a solution of Ar y = f0 . It is easy to see that the derivative of y has the form y = y1 + C1 , where C1 ∈ C is a constant and y1 is deﬁned by (5.3). Condition y ∈ L2 (R, |r(x)|dx) implies that C1 = 0 (otherwise y (x) → C1 = 0 as x → ±∞ and, therefore, y ∈ L2 (R; |r(x)|dx) due to (5.2)). This shows that y1 is the only possible generalized eigenvector, and (5.3) ensures that y1 ∈ L2 (R, |r(x)|dx). Thus, (5.3) is equivalent to the fact that * 0 is a simple eigenvalue (under the assumptions r ∈ L1 (R), R r(x)dx = 0). In the following corollary f (x) g(x) as x → +∞ (x → −∞) means that for X > 0 large enough both fg and fg are bounded on (X, +∞) (resp., (−∞, −X)). Corollary 5.4. Assume that condition (5.1) is satisﬁed and r(x) ±|x|α as x → * ±∞. If α ∈ [−5/3, −1) and R r(x)dx = 0, then the operator Ar has a singular critical point at 0. This includes [43, Theorem 5.1 (ii)–(iv)], also it is interesting to compare this result with the suﬃcient condition on regularity of 0 given in [52],[45, Theorem 1.3], and the discussions in [45, Section 5.2] and [44, Section 5]. Remark 5.5. Theorem 5.3 can be easily generalized to weight functions r with many turning points under the assumption that r(x)

is of constant sign a.e. on (−∞, −X) and (X, +∞)

(5.4)

r = for certain X > 0 large enough. Indeed, assume additionally r ∈ 0 a.e. on R, and (5.2). [16, Proposition 2.5] implies that the maximal operator d2 Ar := − r1 dx 2 is deﬁnitizable. Now it is easy to see that statements (1) and (2) of d2 Theorem 5.3 are valid with the same proof for Ar = − r1 dx 2. L1loc (R),

6. Discussion From another point of view algebraic multiplicities of eigenvalues of deﬁnitizable operators was considered in [55, Proposition II.2.1] and [16, Section 1.3] in terms sgn x d d (− dx p dx + q), Theorem 3.3 solves of deﬁnitizing polynomials. For operators |r(x)| the same problem in terms of Titchmarsh-Weyl m-coeﬃcients. Combining both approaches, it is possible to get quite precise results both on eigenvalues and on deﬁnitizing polynomials. Such analysis was done in [45, Section 4.3] for operators

280

I.M. Karabash

(sgn x)(−d2 /dx2 + q) with potentials q ∈ L1 (R; (1 + |x|)dx); in particular, the minimal deﬁnitizing polynomial was described in terms of Titchmarsh-Weyl mcoeﬃcients (recently, [53] was used in [9] to extended some results of [45, Section 4.3] on a slightly more general class of potentials). ´ A. Kostenko and later B. Curgus informed the author that Theorem 3.1 and [43, Section 6.1] are in disagreement with one of the statements of [16, p. 39, 1st paragraph]. Namely, [16, Section 1.3] is concerned with J-self-adjoint operators A (in a Krein space K = (H, [·, ·])) such that the form [A·, ·] has a ﬁnite number kA of negative squares. Such operators are sometimes called quasi-J-nonnegative. It is assumed also that ρ(A) = ∅. Here, as before, H is a Hilbert space, J is a fundamental symmetry, and [·, ·] := (J·, ·)H . According to [16, p. 39, 1st paragraph] (the author changes slightly the appearance): (p1) an operator A of the type mentioned above has a deﬁnitizing polynomial pA of the form pA (z) = zqA (z)qA (z), where the polynomial qA can be chosen monic and of minimal degree. Under these assumptions, qA is unique and its degree is less than or equal to kA . (p2) A real number λ = 0 is a zero of qA if and only if it is an eigenvalue of A such that λ[f, f ] ≤ 0 for some corresponding eigenvector f . (p3) qA (0) = 0 implies that 0 is an eigenvalue of A and one of corresponding Jordan chains is of length ≥ 2. Note also that in the settings of [16, p. 39, 1st paragraph], qA is of minimal degree, but it easy to see the deﬁnitizing polynomial pA may be not a deﬁnitizing polynomial of minimal degree. From the author’s point of view, two following statements in [16, Section 1.3] 0 are incorrect: assertion (p3) and the equality dim L0 = kA + kA in [16, Proposition 1.5]. Statement (p3) was given as a simple consequence of considerations in [55]. The proof of [16, Proposition 1.5] has an unclear point, which is discussed below. As far as the author understand, all other results of [16] (as well as results obtained in [45, Section 4.3]) do not depend of these two statements. Let us explain points of contradiction in more details. In [43, Section 6.1], the operator A, deﬁned by A := JL, (Jf )(x) = (sgn x)f (x), x4 − 6|x| (Ly)(x) = −y (x) + 6 y(x), (|x|3 + 3)2

(6.1)

is considered in L2 (R). The operators L and A are deﬁned on their maximal domains, L is self-adjoint, and A is J-self-adjoint. We will use here notations of Section 2.3. In particular, l[·] is the diﬀerential expression of the operator L.

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281

We will need the following properties of the operator A. Proposition 6.1 (cf. Section 6.1 in [43]). Let A be the J-self-adjoint operator deﬁned by (6.1). Then: (i) A is a quasi-J-nonnegative operator, with kA = 1, i.e., the sesquilinear form [A·, ·] has one negative square; (ii) σ(A) ⊂ R; (iii) σp (A) = {0}; (iv) 0 is a simple eigenvalue of A. Statements (ii) and (iv) of Proposition 6.1 were proved in [43, Theorem 6.1]. Statement (i) was given in [43, Remark 6.3] with a very shortened proof. For the sake of completeness, we give below the proofs of statements (i) and (iii). Proof. Statement (i) follows from [16, Remark 1.2] and the fact that the negative part of the spectrum of the operator L consists of one simple eigenvalue at λ0 = −1. Let us prove that σ(L) ∩ (−∞, 0) = {−1}. Consider the operators L± 0 associated with the diﬀerential expression l[·] and the Neumann problems y (±0) = 0 on R± , and let mN± be the corresponding Titchmarsh-Weyl m-coeﬃcients, see, e.g., [45, formula (2.7)]. It follows from [43, Lemma 6.2] that both the Titchmarsh-Weyl m-coeﬃcient mN± are equal to the function m0 deﬁned by λ m0 (λ) = , λ ∈ C \ ({−1} ∪ [0, +∞)) , (6.2) 1 + λ(−λ)1/2 where z 1/2 denotes the branch of the complex root with a cut along the negative semi-axis R− such that (−1 + i0)1/2 = i. λ0 = −1 is a pole of m0 , and therefore, an eigenvalue of both the operators − L+ , L 0 0 , and in turn an eigenvalue of L. The support of the spectral measure of m0 is equal to {−1} ∪ [0, +∞), see part (ii) of the proof of [43, Theorem 6.1]. It is easy to see from the standard deﬁnition of Titchmarsh-Weyl m-coeﬃcients (or from [21, Proposition 2.1]) that λ ∈ (−∞, −1) ∪ (−1, 0) belongs to the spectrum of L if and only if mN+ (λ) + mN− (λ) = 0. But [43, formula (6.2)] implies that mN+ (λ)+mN− (λ) = 2m0 (λ) = 0 for all λ ∈ (−∞, −1) ∪ (−1, 0). Thus, the eigenvalue λ0 = −1 is the only point of the spectrum of L in (−∞, 0). λ0 = −1 is a simple eigenvalue since l[·] is in the limit point case at ±∞ (see [66, Theorem 5.3]). (iii) It is proved in [43, Theorem 6.1] that 0 is an eigenvalue of A and that σ(A) ⊂ R. Here we have to show that A has no eigenvalues in R \ {0}. The proof of [43, Theorem 6.1 (ii)] states that the Titchmarsh-Weyl mcoeﬃcient MN+ is equal to m0 , that its spectral measure ΣN+ is absolutely continuous on intervals [0, X], X > 0, and that for t > 0 we have Σ N+ (t) :=

t5/2 . π(1 + t3 )

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I.M. Karabash

Combining this with Theorem 3.1 (1), we see that (0, +∞) ⊂ A0 (ΣN+ ), and therefore σp (A) ∩ (0, +∞) = ∅. Since the potential of L is even, we see that σp (A) ∩ (−∞, 0) = ∅. This concludes the proof. Remark 6.2. Actually, σ(A) = R. This follows from Proposition 3.8 (i) and the fact that MN± (·) = ±mN± (±·). The fact that A has no eigenvalues in R \ {0} can also be easily obtained from [64, Section 5.7] or from [13, Problem IX.4]. Combining Proposition 6.1 with (p1) and (p2), we will show that qA (z) = z, and that qA (z) = z contradicts (p3). Indeed, since L is not nonnegative, the polynomial z is not a deﬁnitizing polynomial of the operator A. So pA (z) ≡ z, and therefore qA is nontrivial. qA has the degree equal to kA = 1 due to (p1). Since the polynomial qA is of minimal degree, Proposition 6.1 (ii) implies that qA has no nonreal zeros, see [55, p. 11, the second paragraph] or [16, p. 38, the last paragraph]. (Note also that in our case pA is a deﬁnitizing polynomial of minimal degree since 0 is a critical point of A.) By Proposition 6.1 (iii), A has no eigenvalues in R \ {0}. Therefore, statement (p2) implies that qA has no zeros in R \ {0}. Summarizing, we see that qA (z) = z and pA (z) = z 3 . Proposition 6.1 (iv) states that 0 is a simple eigenvalue. This fact contradicts (p3). 0 The equality dim L0 = kA + kA from [16, Proposition 1.5] is not valid for the operator A deﬁned by (6.1). Namely, [16, Proposition 1.5] states that there exists an invariant under A 0 0 subspace L0 of dimension dim L0 = kA + kA , where kA is the dimension of the isotropic part of the root subspace S0 (A) with respect to the sesquilinear form [A·, ·]. For the operator A deﬁned by (6.1), statements (i) and (iv) of Proposition 0 = 1, respectively. So L0 is a two-dimensional 6.1 imply that kA = 1 and kA invariant subspace of A. All the root subspaces of the restriction A L0 are root subspaces of A, and therefore Proposition 6.1 (iii)–(iv) implies dim L0 ≤ 1. This 0 contradicts dim L0 = kA + kA = 2. Remark 6.3. From the author’s point of view, the statement “the inner product

[A·, ·] has kA negative squares on S0 (A)” in the proof of [16, Proposition 1.5] is not valid for the operator A deﬁned by (6.1), since in this case S0 (A) = ker A,

but kA = 1.

Appendix A. Boundary triplets for symmetric operators In this section we recall necessary deﬁnitions and facts from the theory of boundary triplets and abstract Weyl functions following [50, 31, 21, 22]. Let H, H, H1 , and H2 be complex Hilbert spaces. By [H1 , H2 ] we denote the set of bounded linear operators acting from the space H1 to the space H2 and deﬁned on all the space H1 . If H1 = H2 , we write [H1 ] instead of [H1 , H1 ]. Let S be a closed densely deﬁned symmetric operator in H with equal deﬁciency indices n+ (S) = n− (S) = n (by deﬁnition, n± (S) := dim N±i (S), where Nλ (S) := ker(S ∗ − λI)).

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283

Deﬁnition A.1. A triplet Π = {H, Γ0 , Γ1 } consisting of an auxiliary Hilbert space H and linear mappings Γj : dom(S ∗ ) −→ H, (j = 0, 1), is called a boundary triplet for S ∗ if the following two conditions are satisﬁed: f, g ∈ dom(S ∗ ); (i) (S ∗ f, g)H − (f, S ∗ g)H = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H , ∗ (ii) the linear mapping Γ = {Γ0 f, Γ1 f } : dom(S ) −→ H ⊕ H is surjective. In the rest of this section we assume that the Hilbert space H is separable. Then the existence of a boundary triplet for S ∗ is equivalent to n+ (S) = n− (S). The mappings Γ0 and Γ1 naturally induce two extensions S0 and S1 of S given by Sj := S ∗ dom(Sj ), dom(Sj ) = ker Γj , j = 0, 1. It turns out that S0 and S1 are self-adjoint operators in H, Sj∗ = Sj , j = 0, 1. The γ-ﬁeld of the operator S corresponding to the boundary triplet Π is the operator function γ(·) : ρ(S0 ) → [H, Nλ (S)] deﬁned by γ(λ) := (Γ0 Nλ (S))−1 . The function γ is well deﬁned and holomorphic on ρ(S0 ). Deﬁnition A.2 ([21, 22]). Let Π = {H, Γ0 , Γ1 } be a boundary triplet for the operator S ∗ . The operator-valued function M (·) : ρ(S0 ) → [H] deﬁned by M (λ) := Γ1 γ(λ),

λ ∈ ρ(S0 ),

is called the Weyl function of S corresponding to the boundary triplet Π. Note that the Weyl function M is holomorphic on ρ(S0 ) and is an (operatorvalued) (R)-function obeying 0 ∈ ρ(Im(M (i))). Acknowledgment ´ The author expresses his gratitude to Paul Binding, Branko Curgus, Aleksey Kostenko, and Cornelis van der Mee for useful discussions. The author would like to thank the anonymous referees for careful reading of the paper and for numerous suggestions on improving it. The author would like to thank the organizers of the conference IWOTA 2008 for the hospitality of the College of William and Mary.

References [1] N.I. Achieser, I.M. Glasmann, Theory of linear operators in Hilbert space. V. II. Visha skola, Kharkov, 1978 (Russian). [2] W. Allegretto, A.B. Mingarelli, Boundary problems of the second order with an indeﬁnite weight-function. J. reine angew. Math. 398 (1989), 1–24. [3] F.V. Atkinson, On the location of the Weyl circles. Proc. Royal Soc. Edinburgh Sect.A 88 (1981), 345–356. [4] F.V. Atkinson, A.B. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted Sturm-Liouville problems. J. reine angew. Math. 375/376 (1987), 380–393. [5] R. Beals, Indeﬁnite Sturm-Liouville problems and half-range completeness. J. Diﬀerential Equations 56 (1985), 391–407.

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[6] R. Beals, V. Protopopescu, Half-range completeness for the Fokker-Plank equation. J. Stat. Phys. 32 (1983), 565–584. [7] J. Behrndt, Finite rank perturbations of locally deﬁnitizable self-adjoint operators in Krein spaces. J. Operator Theory 58 (2007), 101–126. [8] J. Behrndt, On the spectral theory of singular indeﬁnite Sturm-Liouville operators. J. Math. Anal. Appl. 334 (2007), 1439–1449. [9] J. Behrndt, Q. Katatbeh, C. Trunk, Non-real eigenvalues of singular indeﬁnite Sturm-Liouville operators, (to appear in Proc. Amer. Math. Soc.). [10] J. Behrndt, C. Trunk, Sturm-Liouville operators with indeﬁnite weight functions and eigenvalue depending boundary conditions. J. Diﬀerential Equations 222 (2006), no. 2, 297–324. [11] P. Binding, H. Volkmer, Eigencurves for two-parameter Sturm-Liouville equations. SIAM Review 38 (1996), no. 1, 27–48. [12] J.F. Brasche, M.M. Malamud, H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions. Integral Equations and Operator Theory 43 (2002), no. 3, 264–289. [13] E.A. Coddington, N. Levinson, Theory of ordinary diﬀerential equations. McGrawHill Book Company, New York-Toronto-London, 1955; Russian translation: Inostrannaya Literatura, Moscow, 1958. ´ [14] B. Curgus, Boundary value problems in Kre˘ın spaces. Glas. Mat. Ser. III 35(55) (2000), no. 1, 45–58. ´ [15] B. Curgus, A. Gheondea, H. Langer, On singular critical points of positive operators in Krein spaces. Proc. Amer. Math. Soc. 128 (2000), no. 9, 2621–2626. ´ [16] B Curgus, H. Langer, A Krein space approach to symmetric ordinary diﬀerential operators with an indeﬁnite weight function J. Diﬀerential Equations 79 (1989), 31– 61. 2 ´ [17] B. Curgus, B. Najman, The operator (sgn x) d 2 is similar to a selfadjoint operator dx

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[42] I.M. Karabash, A.S. Kostenko, On the similarity of operators of the type d2 sgn x(− dx 2 + cδ) to a normal and a selfadjoint operator. Math. Notes 74 (2003), no. 1-2, 127–131. [43] I.M. Karabash, A.S. Kostenko, Indeﬁnite Sturm-Liouville operators with the singular critical point zero. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 801–820. [44] I.M. Karabash, A.S. Kostenko, On similarity of a J-nonnegative Sturm-Liouville operator to a self-adjoint operator. Funct. Anal. Appl. 43 (2009), no. 1, 65–68. [45] I.M. Karabash, A.S. Kostenko, M.M. Malamud, The similarity problem for Jnonnegative Sturm-Liouville operators. J. Diﬀerential Equations 246 (2009), 964– 997. [46] I.M. Karabash, M.M. Malamud, On similarity of J-selfadjoint Sturm-Liouville operators with ﬁnite-gap potential to selfadjoint ones. Dokl. Akad. Nauk 394 (2004), no. 4, 17–21 (Russian); translation in Doklady Mathematics 69 (2004), no. 2, 195–199. [47] I.M. Karabash, M.M. Malamud, Indeﬁnite Sturm-Liouville operators d2 (sgn x)(− dx 2 + q) with ﬁnite-zone potentials. Operators and Matrices 1 (2007), no. 3, 301–368. [48] I.M. Karabash, C. Trunk, Spectral properties of singular Sturm-Liouville operators with indeﬁnite weight sgn x. Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 483–503. [49] M. Klaus, C.V.M. van der Mee, V. Protopopescu, Half-range solutions of indeﬁnite Sturm-Liouville problems. J. Funct. Anal. 70 (1987), no. 2, 254–288. [50] A.N. Kochubei, On extensions of symmetric operators and symmetric binary relations. Mat. Zametki 17 (1975), no. 1, 41–48; Engl. transl: Math. Notes 17 (1975). [51] A.N. Kochubei, On characteristic functions of symmetric operators and their extensions. Sov. Y. Contemporary Math. Anal. 15 (1980). [52] A.S. Kostenko, The similarity of some J-nonnegative operators to a selfadjoint operator. Mat. Zametki 80 (2006), no. 1, 135–138 (Russian); translation in Math. Notes 80 (2006), no. 1, 131–135. [53] I. Knowles, On the location of eigenvalues of second order linear diﬀerential operators. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 15–22. [54] M.G. Krein, Basic propositions of the theory of representation of Hermitian operators with deﬁciency index (m, m). Ukrain. Mat. Z. 1 (1949), 3–66. [55] H. Langer, Spectral functions of deﬁnitizable operators in Krein space. Lecture Notes in Mathematics, Vol. 948, 1982, 1–46. [56] B.M. Levitan, Inverse Sturm-Liouville problems. Nauka, Moscow, 1984 (Russian); English translation: VNU Science Press, Utrecht, 1987. [57] B.M. Levitan, I.S. Sargsjan, Sturm-Liouville and Dirac operators. Nauka, Moscow, 1988 (Russian); Engl. translation: Kluwer, Dordrecht 1990. [58] M.M. Malamud, S.M. Malamud, Spectral theory of operator measures in Hilbert spaces. Algebra i Analiz 15 (2003), no. 3, 1–77 (Russian); translation in St. Petersburg Math. J. 15 (2003), no. 3, 1–53. [59] V.A. Marchenko, Sturm-Liouville operators and applications. Kiev, “Naukova Dumka”, 1977 (Russian); translation in: Oper. Theory Adv. Appl., Vol. 22, Birkh¨ auser, Basel, 1986.

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[60] C.V.M. van der Mee, Exponentially dichotomous operators and applications. Oper. Theory Adv. Appl., Vol. 182, Birkh¨ auser, 2008. [61] A.B. Mingarelli, Volterra–Stieltjes integral equations and generalized ordinary diﬀerential expressions. Lecture Notes in Mathematics, Vol. 989, Springer-Verlag, Berlin, 1983. [62] A.B. Mingarelli, Characterizing degenerate Sturm-Liouville problems. Electron. J. Diﬀerential Equations (2004), no. 130, 8 pp. [63] R.G.D. Richardson, Contributions to the study of oscillation properties of the solutions of linear diﬀerential equations of the second order. Amer. J. Math. 40 (1918), 283–316. [64] E.C. Titchmarsh, Eigenfunction expansions associated with second-order diﬀerential equations. Vol. II. Clarendon Press, Oxford, 1958. [65] E.C. Titchmarsh, Eigenfunction expansions associated with second-order diﬀerential equations. Vol. I, 2nd Edition. Clarendon Press, Oxford, 1962. [66] J. Weidmann, Spectral theory of ordinary diﬀerential operators. Lecture Notes in Math., Vol. 1258, Springer-Verlag, Berlin, 1987. [67] A. Zettl, Sturm-Liouville Theory. AMS, 2005. I.M. Karabash Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary T2N 1N4 Alberta, Canada and Department of PDE Institute of Applied Mathematics and Mechanics R. Luxemburg str. 74 Donetsk 83114, Ukraine e-mail: [email protected] [email protected] Received: February 28, 2009 Accepted: July 12, 2009

Operator Theory: Advances and Applications, Vol. 203, 289–323 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On the Eigenvalues of the Lax Operator for the Matrix-valued AKNS System Martin Klaus Dedicated to Israel Gohberg on the occasion of his eightieth birthday

Abstract. We consider the eigenvalues of the matrix AKNS system and establish bounds on the location of eigenvalues and criteria for the nonexistence of eigenvalues. We also identify properties of the system which guarantee that eigenvalues cannot lie on the imaginary axis or can only lie on the imaginary axis. Moreover, we study the deﬁciency indices of the underlying non-selfadjoint diﬀerential operator. Mathematics Subject Classiﬁcation (2000). Primary: 34L40, 47E05; Secondary: 34A30, 47B50. Keywords. Matrix-valued AKNS system, focusing nonlinear Schr¨odinger equation, Zakharov-Shabat system, J-self-adjoint operator, Krein space, non-selfadjoint eigenvalue problem.

1. Introduction In this paper we are concerned with the eigenvalues of diﬀerential systems of the form [1, 2, 3, 14, 44] −iξIn Q

v = v, (1.1) R iξIm where v is an (n + m)-component vector function of the real variable x, Q and R are n × m and m × n complex-valued matrix functions of x, and In , Im , are the n × n and m × m identity matrices, respectively; ξ is a complex-valued eigenvalue parameter. The precise assumptions on Q and R will vary and be stated when they are pertinent to the discussion. We call (1.1) the matrix-valued AKNS system because the system with n = m = 1 was ﬁrst introduced in [1] to solve certain nonlinear evolution equations by the inverse scattering transform technique. Communicated by J.A. Ball.

290

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Our motivation for studying these systems stems from the fact that they are at the heart of the inverse scattering transform method by which certain nonlinear evolution equations can be linearized. As is well known, in this method one has two linear problems (a Lax pair [30]), one of which is the eigenvalue problem (1.1). The corresponding nonlinear matrix PDE is iQx = Qtt − 2QRQ −iRx = Rtt − 2RQR,

(1.2)

also known as the matrix nonlinear Schr¨ odinger equation. Special cases of this system are obtained by demanding that R = ±Q∗ (where the asterisk denotes the complex conjugate transpose). Then (1.1) reduces to the single matrix PDE iQx = Qtt ∓ 2QQ∗ Q, which for n = m = 1 is the standard nonlinear Schr¨ odinger equation (NLS), where the + (−) sign corresponds to the focusing (defocusing) case, respectively. The associated linear system (1.1) is the Zakharov-Shabat system [44]. When n = 1, m = 2, the PDEs in (1.2) represent two coupled NLS equations which have ﬁrst been studied by Manakov [32]. In this paper we will concentrate on the complex eigenvalues of the linear system (1.1) associated with (1.2). We are especially interested in the nonreal eigenvalues because they are responsible for the soliton solutions of the associated evolution equation. Therefore we limit ourselves to systems of the form (1.1) that do not correspond to a self-adjoint (matrix) operator. Our results provide new and detailed information on the eigenvalues of (1.1) which may be relevant for the inverse scattering theory of (1.1), a topic of current interest [2, 13, 14]. We also believe that there are issues regarding the spectral properties of (1.1) that are of interest in their own right and warrant further study. The paper is organized as follows. In Section 2 we deﬁne the diﬀerential operator that underlies (1.1) and whose eigenvalues we will study. We will work under minimal assumptions, requiring only that Q and R be locally integrable. Similar assumptions have been made in [9] in work on Darboux transformations for the focusing nonlinear Schr¨ odinger equation. It turns out that there is only one feasible choice of an operator to be associated with (1.1), namely the closure of the minimal operator. Our method for arriving at this result is via Theorem 2.1 and is diﬀerent from that used in [9] for the Zakharov-Shabat system. In Section 3 we look at symmetries whereby we have to distinguish between symmetries with respect to an indeﬁnite inner product (Krein space), symmetries with respect to a conjugation, and symmetries induced by simple similarity transformations. Such symmetries have immediate implications for the location of the eigenvalues and provide useful information for the inverse scattering procedure and for the numerical computation of eigenvalues. In fact, we show that as a consequence of certain symmetries we can ﬁnd the eigenvalues located on the imaginary axis by seeking the roots of a real-valued function. In Section 4 with study the deﬁciency indices of the diﬀerential operator associated with (1.1). Among other things we

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show that if m = n, then for every ξ ∈ C there can never be n + m solutions that are in L2 (R+ )n+m , resp. L2 (R− )n+m . This is a consequence of the trace of system (1.1) being equal to zero. From this we obtain the existence of solutions that are L2 towards +∞ or −∞, respectively. At present we do not know if all the possible deﬁciency indices can actually occur; there seem to be no results in this direction for system (1.1). Then we use the information about the deﬁciency indices to construct the Green’s function. In Section 5 we introduce the Jost solutions assuming Q and R belong to L1 . We characterize the eigenvalues in terms of the zeros of a determinant, which in inverse scattering theory represents the inverse of a transmission coeﬃcient, and we establish the connection with the Green’s function constructed in Section 4. In Section 6 we determine a bound on the location of eigenvalues in terms of explicit constants and quantities related to Q and R. The bound conﬁnes the eigenvalues to a disk in the ξ-plane. We remark that such a bound cannot possibly be of the form |ξ| < F (Q1 , R1), where F * is a suitable function, Q1 = R Q(x)dx, and throughout the paper denotes the Euclidean vector norm and the associated operator matrix norm. This is so because the substitution Q(x) → eicx Q(x), R(x) → e−icx R(x), c ∈ R, preserves the L1 norms but causes a comprehensive shift ξ → ξ − c/2 of all eigenvalues. In Section 7 we prove some results that ensure that eigenvalues do not exist. Some of these results are shown to be best possible. A result that has been known for some time is of the form: if Q1 R1 < c, then there are no eigenvalues. We show that c = 1 is the best possible (i.e., the largest) constant for this statement to be true. Up until now, c = 0.817 had been the best value [3]. For the Zakharov-Shabat system (R = −Q∗ , n = m = 1) it was shown in [23] that if Q1 ≤ π/2, then (1.1) has no eigenvalues. We generalize this result to the AKNS system in Theorem 7.4. In Section 8 we identify some criteria which ensure that there are no eigenvalues on the imaginary axis. For example, this is always so if Q and R are odd functions, a case which has been discussed in the applied literature [19]. In Section 9 we present two theorems that guarantee that all eigenvalues are situated on the imaginary axis. This continues our previous study of imaginary eigenvalues that began in [21] with the observation that in the case of the Zakharov-Shabat system a positive real symmetric potential may very well support nonimaginary eigenvalues. This led to the discovery of conditions which guarantee that all eigenvalues must be purely imaginary [22]. In particular, Theorem 9.3 is a matrix generalization of the principal result of [22], which says that single lobe potentials can only produce imaginary eigenvalues. However, we have reduced the assumptions of this theorem to the extent that the term “single lobe” only faintly captures the essential features of Q and R, especially since the condition R = −Q∗ has been completely removed. Also, while developing Theorem 9.3, we came across a system of the form (1.1) for which the spectrum covers an entire half-plane (Theorem 9.2). Finally, in Theorem 9.4, we generalize Theorem 9.3 to certain multi-hump coeﬃcients, but we only do this for the standard Zakharov-Shabat case. The main condition (see (9.5)) again captures certain shape features of the graph of Q(x).

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2. The AKNS diﬀerential operator We ﬁrst introduce some notation and discuss the diﬀerential operator associated with (1.1). We write τ (Q, R)v = ξv, τ (Q, R) = iJ with

J=

In 0

0 , −Im

d + V, dx 0 −iQ . iR 0

V =

(2.1)

Here and in the sequel, the zero sub-matrices are always understood to match up in size with the other blocks in the matrix. The underlying Hilbert space is L2 (R)n+m for which we will write H. The inner product in H will be denoted by ( · , · )H . We start under minimal assumptions on Q and R and require only that Q ∈ L1loc (R)n×m ,

R ∈ L1loc (R)m×n .

(2.2)

As usual we deﬁne the maximal operator Hmax (Q, R) : D[Hmax (Q, R)] → H by Hmax (Q, R)v = τ (Q, R)v for every v ∈ D[Hmax (Q, R)] = {v ∈ H : v ∈ ACloc (R)n+m , τ (Q, R)v ∈ H}. Similarly, we deﬁne Hmin (Q, R) : D[Hmin (Q, R)] → H by Hmin (Q, R)v = τ (Q, R)v for every v ∈ D[Hmin (Q, R)] = {v ∈ D[Hmax (Q, R)] : v has compact support}. The (formal) adjoint of τ (Q, R) is the diﬀerential expression τ (R∗ , Q∗ ) with associated minimal and maximal operators Hmin (R∗ , Q∗ ) and Hmax (R∗ , Q∗ ), respectively. As in the self-adjoint case one proves that D[Hmin (Q, R)] is dense in H and that Hmin (Q, R)∗ = Hmax (R∗ , Q∗ ),

Hmax (Q, R)∗ = H min (R∗ , Q∗ ),

(2.3)

where A denotes the closure of an operator A. The reader may verify (2.3) by stepping through the proofs of Theorems 3.1–3.9 in [43]. From the theory of symmetric Dirac operators [43, p. 99, p. 253], [31, p. 240–241] and certain Hamiltonian systems [28, p. 117], we know that the limit-point case always prevails at inﬁnity. Hence the closure of the minimal operator is self-adjoint and equal to the maximal operator. The analog in our situation is the following theorem, which is probably familiar but nevertheless proves to be quite useful.

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Theorem 2.1. Let Q and R obey (2.2). Then H min (Q, R) = Hmax (Q, R). Furthermore, lim v ∗ (x)Jw(x) = 0 (2.4) x→±∞

for every v ∈ D[Hmax (Q, R)], w ∈ D[Hmax (R∗ , Q∗ )]. Proof. Let T1 = Hmax (Q, R), T2 = Hmax (R∗ , Q∗ ), and pick any v ∈ D(T1 ), w ∈ D(T2 ). Then Green’s formula gives b [(T1 v)∗ w − v ∗ (T2 w)]dx = −iv(b)∗ Jw(b) + iv(a)∗ Jw(a). a

Since v, T1 v, w, and T2 w all belong to H, we conclude that the limits a → −∞ and b → +∞ exist independently of each other on either side of the equation. Since v(x)∗ Jw(x) is integrable (as a sum of products of L2 functions), these limits must be zero separately, and so (2.4) holds. Hence (T1 v, w)H = (v, T2 w)H , which implies T1 ⊂ T2∗ and T2 ⊂ T1∗ . Thus, Hmax (Q, R) ⊂ Hmax (R∗ , Q∗ )∗ = Hmin (Q, R)∗∗ = H min (Q, R), where we have used the ﬁrst of (2.3). The reverse inclusion is obvious since, by (2.3), Hmax (Q, R) = Hmin (R∗ , Q∗ )∗ is closed. We remark that H min (Q, R) and H min (R∗ , Q∗ ) form an adjoint pair [15, p. 103]. For such pairs there exists a deﬁciency index theory and a theory of closed extensions [15, 41]. In view of Theorem 2.1 we do not have to worry about how to select the physically most appropriate closed extension; it is clear that we have to choose Hmax (Q, R) as the diﬀerential operator associated with (1.1). For simplicity we will from now on abbreviate Hmax (Q, R) as H(Q, R).

3. Operators with symmetries There are two main types of symmetries that have interesting implications for the spectral theory of H(Q, R). The ﬁrst one is J-symmetry, where J deﬁnes a Krein space indeﬁnite inner product [v, w] = (v, Jw)H , for all v, w ∈ H, and J satisﬁes J = J ∗ , J 2 = I. Symmetry of a (densely deﬁned) operator A with respect to the J ⊂ A∗ . Moreover, indeﬁnite inner product is equivalent to the statement that JA J = A∗ . The second symmetry is J-symmetry, A is J-self-adjoint if and only if JA where J is a conjugation, that is, J satisﬁes (v, Jw)H = (w, J v)H for all v, w ∈ H and J-self-adjointness are and J 2 = I. Hence J is conjugate linear. J-symmetry deﬁned in analogy to the Krein space case. In our applications to the operator H(Q, R) it will always be easy to verify min (Q, R)J = Hmin (R∗ , Q∗ ) and this implies, owing to Theorem directly that JH The 2.1, that JH(Q, R)J = H(R∗ , Q∗ ) = H(Q, R)∗ ; the same holds true for J. reason why Hmin (Q, R) has the stated property is that the Js (or Js) used in the examples given below leave the support of a function invariant. The requirement must map D[Hmin (R∗ , Q∗ )] to D[Hmin (Q, R)] is then fulﬁlled due to that J (or J) the restrictions on Q(x) and R(x), which diﬀer from case to case. We found that

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there is such a large variety of diﬀerent operators that exhibit symmetries that it is impossible to list them all here. Therefore we will discuss only a few examples, some of which have been selected because they will play a role later in the paper. More details about the case n = 1, m = 2, can be found in [24]. We will need to use the following matrices if m = n : J± =

0 ±In

In . 0

Also, some of the symmetries will involve a reﬂection about the origin which we represent by the operator P, (P v)(x) = v(−x); moreover, C denotes complex conjugation. The veriﬁcation that the symmetries listed below are as stated is straightforward and details are omitted. Example 3.1. Krein space J-self-adjoint operators: (a) If R = Q∗ , then H(Q, R) is self-adjoint. As we have mentioned earlier, this case will not be pursued further here. (b) If R = −Q∗ then H(Q, R) is J-self-adjoint with J = J given in (2.1). ∗ (c) Let m = n and suppose R(x) = R(−x) and Q(x) = Q(−x)∗ . Then H(Q, R) is J-selfadjoint with J = iJ− P. (d) Let m = n and suppose R(x) = −R(−x)∗ and Q(x) = −Q(−x)∗ . Then H(Q, R) is J-selfadjoint with J = J+ P. Example 3.2. Conjugation-type J-self-adjoint operators: (a) Suppose that n = m and that Q(x) = Q(x)T and R(x) = R(x)T . Set J = J+ C. Then H(Q, R) is J-selfadjoint. (b) Here n, m ≥ 1 are arbitrary. If Q(x) = R(−x)T , then set J = JP C; if Q(x) = −R(−x)T , then set J = P C. Again, H(Q, R) is J-selfadjoint. We add some further comments on the examples listed above. Examples 3.2 (a) and (b) are just special cases of larger families of operators. For example, if m = n, one can make the ansatz 0 A1 J= C A2 0 and seek the associated J-self-adjoint operators. One ﬁnds the conditions A1 = AT1 , −1 A2 = (A1 ) (here the bar denotes the complex conjugate), and Q(x) = A1 Q(x)T A1 ,

R(x) = A1 R(x)T A1 .

Example 3.2 (a) is the special case A1 = A2 = In . Similarly, we can start with B1 0 J= PC 0 B2

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where B1 , B2 are constant n × n, resp. m × m, matrices. We ﬁnd that B1 = B1T , B2 = B2T , B1 B 1 = In , and B2 B 2 = Im must be true, together with Q(x) = −B1 R(−x)T B 2 ,

R(x) = −B2 Q(−x)T B 1 .

Example 3.2 (b) is the special case with B1 = In , B2 = −Im , resp. B1 = In , B2 = Im . It may also be useful to know when H(Q, R) is J-skew-selfadjoint. This happens, for example, when Q(−x) = R(x)∗ , with J = JP (n, m are arbitrary). Then JH(Q, R)J = −H(R∗ , Q∗ ). Assuming suitable forms of J one can ﬁnd more cases. Under the additional hypothesis R = −Q∗ , Example 3.2 (a) was studied in [10] and the J-self-adjointness was established by employing a result by Race [37]. For those H(Q, R) that are J-self-adjoint we have that if ξ is an eigenvalue, ∗ then ξ is also an eigenvalue for H(Q, R) . Hence ξ belongs to the spectrum of H(Q, R); it is an eigenvalue of H(Q, R) if ξ is an isolated eigenvalue of H(Q, R) (cf. [18, p. 184]). Moreover, there is no real residual spectrum [5, p. 97]. If H(Q, R) is J-self-adjoint, then the residual spectrum is empty [16]. Moreover, ξ is an eigenvalue for H(Q, R) if and only if ξ is an eigenvalue of H(R∗ , Q∗ ). For more information, see [15, 16]. The next theorem states conditions under which ξ, −ξ, and −ξ are also eigenvalues, provided ξ is an eigenvalue for H(Q, R). Theorem 3.3. Suppose ξ is an isolated eigenvalue (EV) of (1.1). Then the following are true. (i) If Q and R are both even (odd), then −ξ is an EV. If Q and R are also real, then ξ and −ξ are also EVs. (ii) If m = n, Q = ±Q∗ , and R = ±R∗ , then −ξ is an EV. If, in addition, R = −Q, then ξ and −ξ are also EVs. (iii) If Q = −R∗ , then ξ is an EV. If, in addition, Q is even (odd) or Q is real, then −ξ and −ξ are EVs. Proof. It suﬃces to write down the relevant symmetry relations in each case. In (i) we have (JP )H(Q, R)(JP ) = −H(Q, R) (even case), P H(Q, R)P = −H(Q, R) (odd case), and CH(Q, R)C = −H(Q, R) if Q and R are also real. In (ii) we have J± H(Q, R)J± = ∓H(R∗ , Q∗ ) and, in addition, J− H(Q, R)J− = H(Q, R) if −1 = −J− . In (iii) we have that Jv is an eigenfunction for the R = −Q; note that J− ∗ ∗ EV ξ of H(R , Q ) = H(Q, R)∗ . Thus ξ is an EV of H(Q, R). For the additional claims we use the similarity transformations of part (i). The assumption that ξ be an isolated eigenvalue was used in (iii) to ensure that ξ is an eigenvalue, not just a point in the spectrum, of H(Q, R). Symmetries are also important for the numerical computation of eigenvalues, especially of eigenvalues on the imaginary axis. One method is to use the shooting method on a ﬁnite interval [−d, d] which should be so large that Q(x) and R(x) are negligible for |x| > d. In applications to optical ﬁbres, Q(x) and R(x) typically decay exponentially so that cutting oﬀ these functions is numerically expedient. In

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order to calculate eigenvalues we choose a fundamental matrix Φ(x, ξ) of the system (1.1) with Φ(−d, ξ) = I. Although the fundamental matrix will be studied in more detail in the subsequent sections, we can immediately see from the forms of the solutions to (1.1) as x → ±∞ that ξ ∈ C+ is an eigenvalue of (1.1) provided there is a vector α ∈ Cn+m such that Φ(−d, ξ)α = α = (α1 , 0)T and Φ(d, ξ)α = (0, ∗)T . Here α1 is an n-component vector and ∗ denotes a nonzero m-component vector. To put this into a diﬀerent but equivalent form, we partition Φ(x, ξ) as Φ11 (x, ξ) Φ12 (x, ξ) Φ(x, ξ) = , Φ21 (x, ξ) Φ22 (x, ξ) where Φ11 and Φ22 are n × n and m × m blocks, respectively. Then ξ ∈ C+ is an eigenvalue if and only if det[Φ11 (d, ξ)] = 0. Thus, in order to ﬁnd purely imaginary eigenvalues computationally, it is of great help to know whether det[Φ11 (d, is)] is real-valued for s > 0. By the Schwarz reﬂection principle this also tells us that complex eigenvalues must lie symmetrically with respect to the imaginary axis. The following theorem tells us which symmetry properties lead to a real-valued function det[Φ11 (d, is)]. Theorem 3.4. Suppose Q(x) and R(x) have compact support [−d, d]. Then, if either (a) Q = ±Q, R = ±R, or (b) m = n and R(x) = ±Q(−x), or (c) Q = ±Q∗ , R = ±R∗ , or (d) Q(x) = ±R∗ (−x), then det[Φ11 (d, −ξ)] = det[Φ11 (d, ξ)] and so det[Φ11 (d, is)], s > 0, is real valued. Proof. (a) In case of (+) we have Φ(x, −ξ) = Φ(x, ξ) for all x ∈ [−d, d]. Hence Φ11 (x, −ξ) = Φ11 (x, ξ) follows immediately and Φ11 (x, is) is real valued. In case of (−) we have that Φ(x, −ξ) = JΦ(x, ξ)J for all x ∈ [−d, d]. In view of the block structure of J the assertion also holds in this case. (b) One veriﬁes that Φ(x, −ξ) = ∓J∓ Φ(−x, ξ)Φ(d, ξ)−1 J∓ . Now let χ11 (x, ξ) χ12 (x, ξ) χ(x, ξ) = Φ(x, ξ)−1 = . χ21 (x, ξ) χ22 (x, ξ) Then, by Jacobi’s theorem on minors of matrices and their inverses [34, Theorem 1.5.3], det[χ22 (x, ξ)] = det[Φ11 (x, ξ)]. (3.1) We include a quick argument adapted to our special situation. Since Φ11 Φ12 Φ11 Φ12 In 0 = , χ21 χ22 Φ21 Φ22 0 In then taking determinants on both sides gives (3.1); note that det[Φ(x, ξ)] = 1 for all x, since m = n. Since χ22 (d, ξ) ∓χ21 (d, ξ) Φ(d, −ξ) = , ∓χ12 (d, ξ) χ11 (d, ξ)

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we conclude that (for both signs) Φ11 (d, −ξ) = χ22 (d, ξ). Thus, by (3.1), det[Φ11 (d, −ξ)] = det[Φ11 (d, ξ)], and the assertion follows. (c) This case is similar to the previous one, except that here we start from the identity Φ(x, −ξ) = ∓J∓ [Φ(x, ξ)−1 ]∗ J∓ ; further details are omitted. (d) In this case we have Φ(x, −ξ) = [Φ(−x, ξ)−1 ]∗ Φ(d, ξ)∗ if (+) holds, resp. Φ(x,−ξ) = J[Φ(−x,ξ)−1 ]∗ Φ(d,ξ)∗ J if (−) holds. This gives Φ11 (d,−ξ) = Φ11 (d,ξ)∗ , and the result follows. It is easy to check that all situations described in Theorem 3.3 in which −ξ is an eigenvalue are covered by Theorem 3.4.

4. Square-integrable solutions on a half-line The question of how many solutions are square-integrable near either +∞ or −∞ at a given ξ ∈ C is important for the construction of the Green’s function and the implementation of Darboux transformations; we refer to the extensive study of these questions in [9], where the case n = m = 1 with R = −Q∗ was completely answered. Here we only wish to establish some basic results that may be useful for a later, more in-depth treatment of the spectral and inverse scattering theory of (1.1). Part of our motivation was also to simply see what can be salvaged of the extensive body of work that exists for symmetric Dirac operators and Hamiltonian diﬀerential systems. We begin with the simple question: For a given ξ ∈ C, can it be that all solutions of (τ (P, Q) − ξI)v = 0 lie in L2 (R+ )n+m ? Certainly, one expects the answer to depend on ξ. Somewhat surprisingly maybe, the answer does not depend on ξ at all, at least if m = n; if m = n our answer depends only on the sign of Im ξ. Theorem 4.1. Suppose Q and R obey (2.2). If m = n, then for any ξ ∈ C, the number of linearly independent solutions of (τ (Q, R) − ξI)v = 0 that lie in L2 (R+ )n+m is strictly less than n + m. The same conclusion holds if n > m (n < m) and Im ξ ≥ 0 (Im ξ ≤ 0). An analogous result holds for solutions lying in L2 (R− )n+m , provided we choose Im ξ ≤ 0 (Im ξ ≥ 0) if n > m (n < m), respectively. Proof. Consider the case of R+ with m = n; the argument for R− is similar. Let Φ(x) (we suppress ξ) be a fundamental matrix of solutions of (τ (Q, R) − ξI)v = 0 on x ≥ 0 such that Φ(0) = I, and assume all its columns are square-integrable. It follows that Φ(x)∗ Φ(x) has all its entries in L1 (R+ ); hence tr[Φ(x)∗ Φ(x)] ∈ L1 (R+ ). Expressing this trace in terms of the singular values of Φ(x) and applying the arithmetic-geometric mean inequality gives tr[Φ(x)∗ Φ(x)] ≥ 2n |det[Φ(x)]|1/n .

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This inequality is known, see, e.g., [29, Ex.1, p. 231]. But the system (1.1) has trace zero, so det [Φ(x)] is constant and equal to 1 for all x; here n = m is essential. So tr[Φ(x)∗ Φ(x)] ≥ 2n, which is a contradiction, since the left-hand side is in L1 (R+ ). The remaining assertions follow from the formula det[Φ(x)] = ei(m−n)ξx . The argument using the trace has been used in [36] in the context of limitcircle criteria for (self-adjoint) Hamiltonian systems. For such systems one has other powerful methods to estimate the number of square-integrable solutions, see [4, p. 295] for a general result. For non-self-adjoint systems, under certain technical assumptions on the coeﬃcients, results on the number of square-integrable solutions have been obtained in [8]. For self-adjoint systems of odd order, see [7]. Theorem 4.2. Suppose (2.2) holds and ξ ∈ ρ(H(Q, R)). If m = n, then (τ (Q, R) − ξI)v = 0 has at least one (nontrivial ) solution in L2 (R+ )n+m and at least one (nontrivial ) solution in L2 (R− )n+m . If n > m, then for Im ξ ≤ 0 (Im ξ ≥ 0), (τ (Q, R) − ξI)v = 0 has a solution in L2 (R+ )n+m (L2 (R− )n+m ), while if n < m, then for Im ξ ≥ 0 (Im ξ ≤ 0), (τ (Q, R) − ξI)v = 0 has a solution in L2 (R+ )n+m (L2 (R− )n+m ). The assumption ξ ∈ ρ(H(Q, R)) is crucial. Technically one needs ξ ∈ Π(H min (Q, R; +∞)) and ξ ∈ Π(H min (R∗ , Q∗ ; +∞)) (these minimal operators are deﬁned below) for the results pertaining to R+ and analogous conditions for R− . Here Π(A) = {λ ∈ C : (A − λI)u ≥ k(λ)u}, k(λ) > 0, denotes the ﬁeld of regularity of a closed operator A. The assumption ξ ∈ ρ(H(Q, R)) implies both of them. Further, if n = 1, m = 2, and Q and R are in L1 , then we know there are two solutions in L2 (R+ )3 for Im ξ > 0 and the theorem guarantees one; similarly, if Im ξ < 0, it gives one L2 -solution on R− when there are actually two. If m = n = 1, we get a unique L2 -solution at both ends; when R = −Q∗ this is known ([9]) and has been proved by diﬀerent methods. Proof. Consider solutions on R+ . Let Hmin (Q, R; +∞) and Hmax (Q, R; +∞) denote the minimal and maximal operators associated with τ (Q, R) on R+ . The domains of these operators are denoted by D[Hmin (Q, R; +∞)] and D[Hmax (Q, R; +∞)]. Then D[Hmin (Q, R; +∞)] consists of all v ∈ AC(R+ )n+m that satisfy τ (Q, R)v ∈ L2 (R+ )n+m , v(0) = 0, and have compact support. Then dim D[Hmax (Q, R; +∞)]/D[H min (Q, R; +∞)] = n + m. This follows from (the proof of) [15, Theorem 10.13], since D[H min (Q, R; +∞)] is the restriction of D[Hmax (Q, R; +∞)] to those functions that vanish at 0 and satisfy (2.4) for x → +∞, which also holds for v ∈ D[Hmax (Q, R; +∞)],

w ∈ D[Hmax (R∗ , Q∗ ; +∞)].

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Since H min (Q, R; +∞) and H min (R∗ , Q∗ ; +∞) form an adjoint pair, it follows from [15, Corollary 10.21] that nul[Hmax (Q, R; +∞) − ξI] + nul[Hmax (R∗ , Q∗ ; +∞) − ξI] = n + m,

(4.1)

where nul[A] = dim[N (A)]. If m = n, then by Theorem 4.1, none of the terms on the left-hand side can be zero and the assertion is proved. If n > m (n < m) and Im ξ ≤ 0 (Im ξ ≥ 0), then the second term on the left-hand side of (4.1) is < n + m; this follows from Theorem 4.1 applied to τ (R∗ , Q∗ ) − ξI. So the ﬁrst term on the left-hand side must be nonzero, proving the claim. The proof for R− is the same. For self-adjoint Hamiltonian systems there is a direct argument ([17, p. Lemma 1.1]) showing that the sum on the left-hand side of the analog of (4.1) cannot be > n+m. This argument can be adapted to our system. Let Φ(x, ξ) be a fundamental matrix for (1.1) and let Ψ(x, ξ) be a fundamental matrix for the adjoint system, and assume that Φ(0, ξ) = Ψ(0, ξ) = I. Then Ψ(x, ξ) = J[Φ(x, ξ)∗ ]−1 J as a calculation shows. So JΨ(x, ξ)∗ JΦ(x, ξ) = I. Now, if the left-hand side of (4.1) were greater than n + m, then for dimensionality reasons, there would exist, since J is bijective, a vector α ∈ N (Hmax (R∗ , Q∗ ; +∞) − ξI) ∩ JN (Hmax (Q, R; +∞) − ξI) with α = 1. Then Ψ(x, ξ)α and Φ(x, ξ)Jα are both in L2 (R+ )n+m . But then 1 = (Jα)∗ JΨ(x, ξ)∗ JΦ(x, ξ)Jα = [Ψ(x, ξ)α]∗ J[Φ(x, ξ)Jα]. Now the vectors in brackets are both in L2 , which gives a contradiction. There is also a link between the number of solutions that are in L2 towards +∞ and the number of solutions that are in L2 towards −∞ at a point ξ ∈ ρ(H(Q, R)). Clearly, these two numbers cannot add up to more than n + m, for otherwise we could construct an eigenfunction of H(Q, R). More precisely, we have the following results. Theorem 4.3. Assume (2.2) and let ξ ∈ ρ(H(Q, R)). Then nul[Hmax (Q, R; −∞) − ξI] + nul[Hmax (Q, R; +∞) − ξI] = n + m.

(4.2)

nul[Hmax (R∗ , Q∗ ; −∞) − ξI] + nul[Hmax (R∗ , Q∗ ; +∞) − ξI] = n + m.

(4.3)

Proof. Let Hmin,0 (Q, R) = Hmin (Q, R; −∞) ⊕ Hmin (Q, R; +∞). Then dim D[H(Q, R)]/D[H min,0 (Q, R)] = n + m. By [15, Theorem 3.1, Theorem 10.20] n + m = nul[H(Q, R) − ξI] + nul[H min,0 (Q, R)∗ − ξI] − nul[H(Q, R)∗ − ξI]. The ﬁrst and last term on the right-hand side are zero by the assumptions, and since H(Q, R)∗ = H(R∗ , Q∗ ) and ξ ∈ ρ(H(R∗ , Q∗ )). For the second term we have

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(cf. [15, p. 156]) nul[H min,0 (Q, R)∗ − ξI] = nul[Hmax (R∗ , Q∗ ; +∞) − ξI] + nul[Hmax (R∗ , Q∗ ; −∞) − ξI]. Hence (4.3) and then (4.2) follows.

If there exists a conjugation operator J satisfying max (Q, R; +∞)J = Hmax (R∗ , Q∗ ; +∞), JH

(4.4)

then the two nullities in (4.1) are equal. Note that this is not the same as J-self∗ ∗ ∗ ∗ ∗ adjointness, since Hmax (Q, R; +∞) = H min (R , Q ; +∞) = Hmax (R , Q ; +∞). On taking adjoints we get max (Q, R; +∞)J) ∗ H min (Q, R; +∞) = Hmax (R∗ , Q∗ ; +∞)∗ = (JH max (Q, R; +∞)∗ J = JH min (R∗ , Q∗ ; +∞)J, = JH ∗ J for a densely deﬁned linear J) ∗ = JA where we have used the fact that (JA operator A. Hence J H min (Q, R; +∞) J = H min (R∗ , Q∗ ; +∞), which is equivalent to (4.4). It is obvious that if n + m is odd, then we cannot have a conjugation J satisfying (4.4). Looking back to Example 3.2 (b), we see that the two conjugations J = JP C and J = P C, which are deﬁned for arbitrary m and n, do not leave L2 (R± )n+m invariant owing to the presence of P. Hence (4.4) does not hold. We can summarize these ﬁndings as follows. Theorem 4.4. Let m + n = 2k and suppose there is a conjugation J such that (4.4) holds. Let ξ ∈ ρ(H(Q, R)). Then nul[Hmax (Q, R; ±∞) − ξI] = k. Proof. Combine (4.1) with (4.2) and (4.3).

Instead of relating N (Hmax (Q, R; ±∞) − ξI) to N (Hmax (R∗ , Q∗ ; ±∞) − ξI) by a conjugation one can try to relate N (Hmax (Q, R; +∞) − ξI) to N (Hmax (Q, R; −∞) − ξI) by a similarity transformation. This works, for example, if m = n and Q(x) = ±R(−x). Then, for every v ∈ N (Hmax (Q, R; +∞) − ξI) we get that z = J∓ P v ∈ N (Hmax (Q, R; −∞) − ξI). Therefore the conclusion of Theorem 4.4 also holds in this case. If Q and R belong to L1 , then the unperturbed operator with Q = R = 0 determines the number of L2 solutions at ±∞. Thus N (Hmax (Q, R; +∞)−ξI) = m (n) if Im ξ > 0 (Im ξ < 0), and N (Hmax (Q, R; −∞) − ξI) = n (m) if Im ξ > 0 (Im ξ < 0).

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As an application of the topics discussed in this section we construct the resolvent kernel or Green’s function in terms of solutions to (1.1). For any ξ ∈ ρ(H(Q, R)), we put α = nul[Hmax (Q, R; +∞) − ξI],

β = n + m − α.

Then we form two (n + m) × (n + m) matrices F = ( F1 | F2 ), >?@A α

F = ( F1 | F2 ), >?@A

(4.5)

β

where F1 is a submatrix consisting of α column vectors that form a basis for N (Hmax (Q, R; +∞) − ξI) and F2 consists of β linearly independent columns that form a basis for N (Hmax (Q, R; −∞) − ξI). In F the submatrix F2 has as its columns a basis of N (Hmax (R∗ , Q∗ ; +∞)−ξI) and the columns of F1 are a basis for N (Hmax (R∗ , Q∗ ; −∞)− ξI). Since ξ ∈ ρ(H(Q, R)), F and F are both nonsingular. Now deﬁne a matrix K(x, ξ) = F (x, ξ)∗ JF (x, ξ), (4.6) and note that it is constant in x by a straightforward calculation. Moreover, as a consequence of (2.4), using both limits there, K(x, ξ) = K(ξ) is seen to have the special form K1 (ξ) 0 K(ξ) = , (4.7) 0 K2 (ξ) where K1 , K2 have sizes α × α, β × β, respectively, and are both invertible. It follows from (4.6) that JF (x, ξ)K(ξ)−1 F (x, ξ)∗ = I.

(4.8)

In the context of scattering theory, K1 (ξ) and K2 (ξ) have (up to multiplicative constants) physical interpretations as transmission coeﬃcients [2], or of inverses of transmission coeﬃcients [13], as there are diﬀerent deﬁnitions in use. We now partition F into blocks as follows: , + F11 F12 , (4.9) F = F21 F22 where F11 has dimensions α × α. Theorem 4.5. Suppose Q and R obey (2.2) and ξ ∈ ρ(H(Q, R)). Then the resolvent (H(Q, R) − ξI)−1 has integral kernel

y < x, −iF1 (x, ξ)K1 (ξ)−1 (F11 (y, ξ)∗ | F21 (y, ξ)∗ ) −1 (H(Q, R) − ξI) [x, y] = −1 ∗ ∗ iF2 (x, ξ)K2 (ξ) ( F12 (y, ξ) | F22 (y, ξ) ) y > x. (4.10) If α = 0, which could happen if n > m and Im ξ > 0, or if n < m and Im ξ < 0, then the part with y < x is absent in (4.10). Similarly, if α = n + m, which could happen if n > m and Im ξ < 0, or if n < m and Im ξ > 0, then the

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part with y > x is absent. However, we emphasize that we don’t know if any of these cases actually occurs. We have derived (4.10) by proceeding as in the case of a self-adjoint Hamiltonian system with equal deﬁciency indices (cf. [27]). Proof. The formal veriﬁcation is a somewhat tedious calculation, wending the way through the various deﬁnitions and block matrices. One also uses (4.8) in view of the order in which the matrices appear in (4.10). For a rigorous proof it suﬃces to verify that the resolvent acts correctly on functions of compact support; this is straightforward. Then we note that a typical matrix element of the kernel is of the form g(x)θ(x − y)f (y) (or g(x)θ(y − x)f (y)). Using the fact that, since ξ ∈ ρ(H(Q, R)), the kernel in (4.10) represents a bounded operator when acting on functions of compact support, together with a boundedness criterion for such kernels [11], we conclude that the kernel in (4.10) represents the resolvent on all of H. For details we refer the reader to [9], where this argument was used in the Zakharov-Shabat case. In the special case n = m = 1 and R = −Q∗ we know from Example 3.2 (a) that H(Q, R) is J-self-adjoint. If we choose the matrix F = (F1 | F2 ) with det F = −1 and put F = (J+ F 2 | J+ F 1 ). Then K(ξ) = −J and the resolvent kernel agrees with that given in [9]. If the operator has symmetries, then these will be reﬂected in the resolvent and may lead to a simpler expression.

5. Jost solutions In this section we assume that Q ∈ L1 (R)n×m ,

R ∈ L1 (R)m×n .

(5.1)

Under (5.1), variation of parameters allows us to deﬁne matrix-valued solutions that are asymptotic to solutions of the unperturbed problem. In particular, for ξ ∈ R, we have the scattering solutions deﬁned by I 0 φ(x, ξ) ∼ e−iξx n x → −∞, ψ(x, ξ) ∼ eiξx x → +∞. 0 Im We partition φ and ψ into blocks as follows. The top n × n (bottom m × n) block of φ will be denoted by φ1 (φ2 ) and the top n × m (bottom m × m) block of ψ will be denoted by ψ1 (ψ2 ), respectively. We recall that φ and ψ are the unique solutions of the integral equations x −iξx −iξx φ1 (x, ξ) = e In + e eiξt Q(t)φ2 (t, ξ) dt (5.2) −∞ x φ2 (x, ξ) = eiξx e−iξt R(t)φ1 (t, ξ) dt (5.3) −∞ ∞ eiξt Q(t)ψ2 (t, ξ) dt (5.4) ψ1 (x, ξ) = −e−iξx x

Lax Operator for AKNS System ψ2 (x, ξ) = e

iξx

Im − e

iξx

∞

303

e−iξt R(t)ψ1 (t, ξ) dt.

(5.5)

x

These integral equations can be solved by iteration in a standard way. Moreover, + we can allow ξ ∈ C and then φ(x, ·) and ψ(x, ·) are analytic in C+ and continuous on the real axis. The bounds that are obtained from the iteration process are based on Gronwall’s inequality which typically gives rise to exponential factors that are very large and considerably overestimate the actual solution. Since for later use we need upper bounds that are as realistic as possible, we derive them by another method. The meaning of the symbols and 1 is as deﬁned in the Introduction. For example, in the following lemma, φ1 (x, ξ) is the norm of the linear operator φ1 (x, ξ) : Cn → Cn . In particular, each column of φ1 obeys the estimate given in the lemma; analogous statements hold for φ2 , ψ1 , and ψ2 . Lemma 5.1. Let β = Im ξ ≥ 0 and let σ(x) = (4β 2 + Q(x) + R(x)∗ 2 )1/2 . Then φ1 (x, ξ) ≤ eβx e(1/2) φ2 (x, ξ) ≤ eβx e(1/2)

*x

−∞

*x

−∞

ψ1 (x, ξ) ≤ e−βx e(1/2) ψ2 (x, ξ) ≤ e−βx e(1/2)

* *

(σ(t)−2β)dt

,

(5.6)

x

(σ(t)−2β)dt −∞ ∞

R(t) dt,

(5.7)

Q(t) dt,

(5.8)

∞ (σ(t)−2β)dt x

x ∞ (σ(t)−2β)dt x

.

(5.9)

If R = −Q∗ , the exponential factors go away and in that case the estimates were derived earlier in [25]. The method of proof is well known and may go back to [42]. Note that as x → −∞ (x → +∞) the bound in (5.6)–(5.9) approaches the correct asymptotic form. Proof. From (1.1) we have (arguments are suppressed if not needed) Q + R∗ 2β In ∗

∗ φ. (φ φ) = φ Q∗ + R −2β Im

(5.10)

Put B = Q + R∗ . The square of the matrix on the right-hand side is a diagonal block operator with entries 4β 2 In + BB ∗ and 4β 2 Im + B ∗ B. Since the nonzero eigenvalues of BB ∗ and B ∗ B coincide, we see that the maximum eigenvalue of the matrix in (5.10) is σ(x). From (5.10), it follows that for any α ∈ Cn , (φ α2 ) ≤ σ(x)φ α2 ,

(5.11)

so that on integrating (5.11) from y to x (y < x) we get φ(x, ξ) α2 ≤ φ(y, ξ) α2 e

*

x y

σ(t) dt

= e2βx (φ(y, ξ) α2 e−2βy )e

*

x (σ(t)−2β)dt y

. (5.12)

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Since, as y → −∞, φ(y, ξ) α2 ∼ e2βy , taking y → −∞ in (5.12) gives φ(x, ξ) α2 ≤ e2βx e

*x

−∞

(σ(t)−2β)dt

.

Since α is an arbitrary unit vector, (5.6) follows. Then (5.7) follows from (5.3), and (5.8), (5.9) are proved similarly using (5.4), (5.5). In the next lemma we estimate the diﬀerence between solutions φ(1) (x, ξ) and φ (x, ξ) that belong to diﬀerent pairs of coeﬃcients Qk (x), Rk (x), k = 1, 2. We use a subscript to refer to either pair, e.g., we write σ1 (x) and σ2 (x), except for the solutions themselves, where we use superscripts because in that case we need subscripts for their components. (2)

Lemma 5.2. Let Qk (x), Rk (x) (k = 1, 2) satisfy (5.1) and let β = Im ξ ≥ 0. Then x *x (2) (1) βx (1/2) −∞ [τ1 (t)+τ2 (t)]dt M (t) dt, (5.13) φ (x, ξ) − φ (x, ξ) ≤ e e −∞

∗

where τk (t) = Qk (t) + Rk (t) for k = 1, 2, and M (x) = min{R1 1 , R2 1 } Q1(x) − Q2 (x) + R1 (x) − R2 (x).

(5.14)

Proof. Set Δφ = φ(2) − φ(1) (suppressing x and ξ if not needed). A calculation gives 2β In Q1 + R1∗ ∗

∗ Δφ (Δφ Δφ) = Δφ Q∗1 + R1 −2β Im 0 Q1 − Q2 ∗ (2) + 2Re Δφ φ . R1 − R2 0 Let ΔQ(x) = Q1 (x) − Q2 (x), ΔR(x) = R1 (x) − R2 (x). Then, for any unit vector α ∈ Cn , (Δφα2 ) ≤ σ1 Δφα2 + 2Δφα(ΔQ(x)2 φ2 α2 + ΔR(x)2 φ1 α2 )1/2 . (2)

(2)

Since, by (5.6) and (5.7), (2)

φ2 α ≤ eβx e(1/2)

*x

−∞

(σ2 (t)−2β)dt

R2 1 ,

(2)

φ1 α ≤ eβx e(1/2)

*x

−∞

(σ2 (t)−2β)dt

,

we obtain Δφ α ≤

*x 1 σ1 Δφ α + eβx e(1/2) −∞ (σ2 (t)−2β)dt (R2 1 ΔQ(x) + ΔR(x)). 2

Integrating the inequality gives (5.13) with (5.14) if we note that we may interchange Q1 , R1 , and Q2 , R2 ; we also used σk (t) − 2β ≤ τk (t) to simplify some exponents.

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The factor min{R1 1 , R2 1 } in (5.14) accounts for the special situation R1 = R2 = 0, which implies M (x) = 0, and which conforms with (5.3), since, by (1) (2) (1) (2) (5.13), φ2 (x, ξ) = φ2 (x, ξ) = 0 and thus φ1 (x, ξ) = φ1 (x, ξ) = e−iξx In . It follows from basic asymptotic theory [12, p. 92], in view of the fact that Q(x) and R(x) are both integrable, that for Im ξ > 0, (1.1) has two fundamental matrices satisfying the asymptotic estimates −iξx e [In + o(1)] ψ1 (x, ξ) Φ+ (x, ξ) = x → +∞, (5.15) ψ2 (x, ξ) o(e−iξx ) o(eiξx ) φ1 (x, ξ) Φ− (x, ξ) = x → −∞, (5.16) φ2 (x, ξ) eiξx [Im + o(1)] where o(1) is a term approaching 0. We immediately see from (5.15) see that ξ ∈ C+ is an eigenvalue for H(Q, R) if and only if there are two vectors ζ ∈ C n and η ∈ C m such that φ(x, ξ)ζ = ψ(x, ξ)η (5.17) T T for all x ∈ R; then Φ+ (x, ξ)(0, η) = Φ− (x, ξ)(ζ, 0) is the corresponding eigenfunction. Also note that there are exactly n linearly independent solutions that are in L2 near −∞ and m linearly independent solutions that are in L2 toward ∞. So, in the notation of Theorem 4.5, we have α = m and β = n. Using Lemma 5.1 and (5.2) and (5.5) we see that for ξ ∈ C+ , eiξx φ1 (x, ξ) → A(ξ) e

−iξx

x → +∞,

ψ2 (x, ξ) → B(ξ)

where

A(ξ) = In + B(ξ) = Im −

∞

−∞ ∞

x → −∞,

eiξt Q(t)φ2 (t, ξ) dt,

(5.18)

e−iξt R(t)ψ1 (t, ξ) dt.

(5.19)

−∞

It follows from (5.7) and (5.8) that A(ξ) and B(ξ) represent analytic functions in C+ which are continuous down to the real axis. Taking x → +∞ in (5.17) we get lim eiξx φ1 (x, ξ)ζ = A(ξ)ζ = lim eiξx ψ1 (x, ξ)η = 0.

x→+∞

x→+∞

Hence ζ ∈ N (A(ξ)). Similarly, on letting x → −∞ in (5.17), we ﬁnd that η ∈ N (B(ξ)). Furthermore lim e−iξx φ2 (x, ξ)ζ = lim e−iξx ψ2 (x, ξ)η = η,

x→+∞

x→+∞

(5.20)

which suggests, in view of (5.3), that we deﬁne a mapping S : N (A(ξ)) → N (B(ξ)) by Sζ =

∞

−∞

e−iξt R(t)φ1 (t, ξ)ζ dt,

then Sζ = η by (5.20). Of course, S depends on ξ.

ζ ∈ N (A(ξ));

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Theorem 5.3. Suppose Q and R obey (4.2). Then (i) ξ ∈ C+ is an eigenvalue of H(Q, R) if and only if det[A(ξ)] = 0, and this is true if and only if det[B(ξ)] = 0. (ii) Suppose ξ ∈ C+ is an eigenvalue. Then S is a bijection between N (A(ξ)) and N (B(ξ)). (iii) The geometric multiplicity of an eigenvalue is not larger than min{m, n}; the same conclusion holds if ξ ∈ C − . Part (iii) was already proved in [13, Corollary 3.17] as a consequence of properties of the scattering matrix. Proof. (i) We already know that if ξ is an eigenvalue then ζ ∈ N (A(ξ)) and η ∈ N (B(ξ)), hence det[A(ξ)] = det[B(ξ)] = 0. Conversely, if det[A(ξ)] = 0, we pick a nonzero ζ ∈ N (A(ξ)). Then, by (5.2), φ1 (x, ξ)ζ = o(e−iξx ) as x → +∞. Let γ = (γ 1 , γ 2 )T ∈ Cn+m (γ 1 ∈ Cn , γ 2 ∈ Cm ) be a vector such that Φ+ (x, ξ)γ = φ(x, ξ)ζ. Then γ 1 = 0 and hence ψ(x, ξ)γ 2 = φ(x, ξ)ζ. This vector is in L2 towards both ±∞, so ξ is an eigenvalue. The proof when det[B(ξ)] = 0 is similar. (ii) We ﬁrst show that S is injective. Suppose ζ ∈ N (A(ξ)) and Sζ = 0. We know from the ﬁrst part that φ(x, ξ)ζ is an eigenfunction for ξ and, from (5.20), that Sζ = η. Hence η = 0, which forces ζ = 0. To show that S is onto, pick any vector γ ∈ N (B(ξ)). It follows from (5.5) and (5.19) that ψ2 (x, ξ)γ = o(eiξx ) as x → −∞. Since ψ(x, ξ)γ = Φ− (x, ξ)ω for some ω ∈ Cn+m (ω 1 ∈ Cn , ω 2 ∈ Cm ), we conclude that ω2 = 0. This implies that ψ(x, ξ)γ = φ(x, ξ)ω1 is an eigenfunction. Taking x → +∞ shows that ω1 ∈ N (A(ξ)); hence S is onto. (iii) The geometric multiplicity of ξ is equal to nul[N (A(ξ))] = nul[N (B(ξ))] which are ≤ n and ≤ m, respectively. The bound for the multiplicity also holds when Im ξ < 0 by a similar proof using the appropriate solutions for Im ξ < 0. Alternatively, we can use the fact that nul[H(Q, R) − ξI] = nul[H(R∗ , Q∗ ) − ξI], since, as we will see below, all eigenvalues are isolated eigenvalues. In the subsequent sections we will occasionally want to replace Q and R by smooth approximations in order to avoid unnecessary technicalities. The following lemma is helpful in this respect. Its proof is easy using Lemma 5.1 and Lemma 5.2. Lemma 5.4. Let A(ξ; Q1 , R1 ) and A(ξ; Q2 , R2 ) belong to two pairs Q1 , R1 and Q2 , R2 , respectively, which both satisfy (5.1). Then ∗

∗

A(ξ; Q1 , R1 ) − A(ξ; Q2 , R2 ) ≤ C e(1/2)(Q1 +R1 1 +Q2 +R2 1 ) , where C = min{Q1 − Q2 1 R2 1 + Q1 1 M 1 , Q1 − Q2 1 R1 1 + Q2 1 M 1} and M (x) is deﬁned in (5.14). Clearly, in view of Rouch´e’s theorem and the analyticity of A(ξ), this implies that isolated eigenvalues in C+ and C− depend continuously on Q and R in the L1 norm.

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So far we have only looked at eigenvalues and not mentioned the other parts of the spectrum. We ﬁll in some of those details now. In fact, we can reap the fruits of our work in Section 4 and write down the Green’s function. We only consider the case ξ ∈ C+ , since for ξ ∈ C− the calculations are completely analogous. To do this, we also need the solutions of the adjoint equation (τ (R∗ , Q∗ ) − ξI)v = 0 deﬁned by x −iξx φ1 (x, ξ) = e eiξt R(t)∗ φ2 (t, ξ) dt −∞ x iξx iξx φ2 (x, ξ) = e Im + e e−iξt Q(t)∗ φ1 (t, ξ) dt −∞ ∞ −iξx −iξx In − e eiξt R(t)∗ ψ2 (t, ξ) dt ψ1 (x, ξ) = e x ∞ iξx ψ2 (x, ξ) = −e e−iξt Q(t)∗ ψ1 (t, ξ) dt. x

and B(ξ) by We deﬁne matrices A(ξ) + o(1)] φ2 (x, ξ) = eiξx [A(ξ)

x → +∞

ψ1 (x, ξ) = e−iξx [B(ξ) + o(1)]

x → −∞.

is m × m and that of B(ξ) is n × n. The size of A(ξ) Let (cf. (4.5)) + , 1 (x, ξ) ψ1 (x, ξ) φ ψ1 (x, ξ) φ1 (x, ξ) , F (x, ξ) = . F (x, ξ) = ψ2 (x, ξ) φ2 (x, ξ) φ2 (x, ξ) ψ2 (x, ξ) Also, in the notation of Theorem 4.5, we have α = m, β = n. It is straightforward to compute ∗, K1 (ξ) = −A(ξ) K2 (ξ) = A(ξ), where we have used the x → +∞ asymptotics. Alternatively, using the x → −∞ asymptotics yields ∗, K1 (ξ) = −B(ξ), K2 (ξ) = B(ξ) where B(ξ) is deﬁned in (5.19). This gives us all the pieces needed for the Green’s function (see Theorem 4.5). Since the matrix elements of F (x, ξ) and F(x, ξ)∗ are analytic in C+ , it is clear that the singularities in C+ of the Green’s function are precisely the zeros of det[A(ξ)]. These zeros are isolated points in C± and have ﬁnite multiplicities, since det[A(ξ)] does not vanish identically, in fact det[A(ξ)] → 1 as |ξ| → ∞. Hence the poles of the resolvent correspond to eigenvalues of ﬁnite algebraic multiplicities. The real axis belongs to the spectrum because for real ξ there are no solutions of (1.1) or its adjoint that are in L2 towards either +∞ or −∞. Thus in view of (4.1), ξ must belong to the spectrum. It also follows that R(H(Q, R)−ξI) is dense, so the real axis belongs to the continuous spectrum.

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Alternatively, one can also appeal to perturbation theory to prove that the essential spectrum is the real line. First, since the matrix multiplication operator in (2.1) viewed as a perturbation of iJd/dx is not relatively bounded (D(V ) does not contain D(iJd/dx) in general) there is a problem. However, under (2.2), V is what can be considered to be the equivalent of a form-compact perturbation in the self-adjoint case [38, p. 369, prob. 39]. This technique has been extended to the self-adjoint Dirac case long ago (see, e.g., [20], [35]) and has been applied to the matrix AKNS system also [13]. One shows by using the resolvent expansion that the diﬀerence of the resolvents of H(Q, R) and iJd/dx is compact for |ξ| suﬃciently large. It may happen that det[A(ξ)] = 0 for a real ξ. Then ξ is not an eigenvalue but is often referred to as a spectral singularity. For the 2 × 2 Zakharov-Shabat system there is detailed information available on the location of spectral singularities [26]. We do not study them here but they will play a role in the proof of Theorem 9.4.

6. Bounds on the location of eigenvalues +

Clearly, a number ξ ∈ C cannot be an eigenvalue or a spectral singularity if A(ξ) − In < 1. This observation allows us to determine subsets of C+ where eigenvalues (spectral singularities) cannot occur. To estimate this quantity we proceed as in [25]. Theorem 6.1. Suppose (5.1) holds. Then: (i) There is a radius r0 such that all eigenvalues ξ in C+ satisfy |ξ| ≤ r0 . (ii) Suppose that, in addition to (5.1), at least one of Q(x) and R(x) has an L1 -derivative. Then the eigenvalues in C+ satisfy |ξ| ≤ r0 , where

1 ∗ Q 1 R1 e(1/2)R +Q1 , if Q (x) ∈ L1 , 2 r0 = 1

(1/2)R∗ +Q1 , if R (x) ∈ L1 . 2 R 1 Q1 e The ﬁrst statement is essentially known from inverse scattering theory and comes from the fact that A(ξ) → 0 as |x| → ∞ in C+ . However, it would be false to believe that this approach to zero is uniform in the coeﬃcients Q and R if these are conﬁned to bounded sets in the L1 norm; the reason was given in the Introduction. This is the reason why the derivatives enter in part (ii). Of course, if both Q and R are diﬀerentiable, we have the choice of picking the smaller radius. As in [25] one could also use the total variation instead of the L1 norms for the derivatives of Q and R. Then Q or R need be only piece-wise continuous. For the two-dimensional Zakharov-Shabat system several other bounds are known [6], [25] and it seems to us that these could also be extended to the matrix case. However, the bounds in [6] do not conﬁne the eigenvalues to a bounded region. Theorem 6.1 is needed for the proof Theorem 9.4.

Lax Operator for AKNS System Proof. Inserting (5.3) in (5.18) and integrating by parts yields t ∞ e2iξt Q(t) e−iξτ R(τ )φ1 (τ ) dτ dt A(ξ) − In = −∞ −∞ ∞ ∞ 2iξs e Q(s) ds e−2iξt R(t)[eiξt φ1 (t)] dt. =− −∞

(6.1)

t

To prove (i), use the fact that (β = Im ξ ≥ 0) # # ∞ # # 2βt # 2iξs sup e # e Q(s)ds# # → 0, t

309

|ξ| → ∞,

t

which follows from the Riemann-Lebesgue lemma, and also use (5.6). Now ∞ 2iξs ∞ e2iξt e − Q (s) ds e2iξs Q(s) ds = −Q(t) 2iξ 2iξ t t and consequently # # ∞ # # −2iξt ∞ 2iξs 1

# #e Q(t) + e Q(s) ds# ≤ Q (s) ds . # 2|ξ| t

t

The term in parenthesis is easily seen to be decreasing in t. Therefore # # ∞ # −2iξt ∞ 2iξs # #e #≤ 1 e Q(s) ds Q (s) ds. # # 2|ξ| t

−∞

Inserting this bound in (6.1) and using (5.6) gives ∗ 1 Q 1 R1 e(1/2)R +Q1 , A(ξ) − In ≤ 2|ξ| where we have also used the simpliﬁcation σ(t) − 2β ≤ Q(t) + R(t)∗ . This proves the ﬁrst inequality of (ii). The second follows by estimating B(ξ) − Im < 1; this leads to an interchange of Q and R.

7. Nonexistence of eigenvalues In this section we consider criteria that guarantee the absence of eigenvalues in the upper half-plane. It is immediately obvious from (5.18) that there can be no eigenvalues if the potentials have small enough L1 norms. By estimating the iterated Neumann series associated with (5.2)–(5.3) (or (5.4)–(5.5)) one sees that there are no eigenvalues when [3] ) I0 (2 Q1 R1 ) < 2, (7.1) where I0 is the modiﬁed Bessel function of order 0 (note that I0 (x) ≥ 1 for x ≥ 0), or when [40] ) Q1R1 I0 (2 Q1 R1 ) < 1. (7.2) From (7.1) we see that there are no eigenvalues if Q1 R1 ≤ 0.817 and from (7.2) if Q1 R1 ≤ 0.592; so the ﬁrst bound beats the second. There are many

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M. Klaus

ways to come up with such bounds from the Neumann series. Here are two more; the reader should have no diﬃculty deriving them. First, there are no eigenvalues if ∞ x exp Q(x) R(t)dt dx ≤ 2, (7.3) −∞

−∞

and this is true provided Q1 R1 ≤ ln 2 = 0.693, which lies between the previous two values. However, this bound correctly reﬂects another feature of system (1.1). It tells us that, if there is a point x0 such that supp Q ⊂ (−∞, x0 ) and supp R ⊂ (x0 , +∞), then the double integral in the exponent in (7.3) is zero; hence there can be no eigenvalues in the upper half-plane. That this is indeed correct can also immediately be seen from (1.1). The second bound follows from (5.6), setting β = 0, so that ∗ φ1 (x, ξ) ≤ e(1/2)Q+R 1 . Then ∞ x ∗

A(ξ) − In ≤ e(1/2)Q+R

1

−∞

Q(x)

−∞ ∗

R(t)dt dx

and this is less than 2 if Q1 R1 < .901. If R = −Q we get ∞ 2 1 1 A(ξ) − In ≤ Q(x)dx = Q21 , 2 2 −∞ √ which gives Q1 ≤ 2; a result that was found by a diﬀerent method in [25, p. 33]. Now, if R = −Q∗ and n = m = 1 (see [23]), or n = 1, m = 2 (see [24]), it is known that there are no eigenvalues provided Q1 ≤ π/2. These π/2 bounds are optimal in the sense that the constant π/2 cannot be replaced by a larger number. Before we continue with our discussion of the general matrix case we present a new direct proof of the π/2 result under the condition R = −Q∗ , but for arbitrary m and n. Note that the spectrum is symmetric about the real axis, which means that our result will automatically also hold with respect to the lower half-plane. Theorem 7.1. Suppose (5.1) holds and R(x) = −Q(x)∗ . Then, if Q1 ≤ π/2, H(Q, R) has no eigenvalues (in C+ and C− ). Proof. Pick a ξ with β = Im ξ > 0 and pick any α ∈ Cn , α = 1. Then (φ1 (x, ξ)α2 ) = α∗ (φ1 (x, ξ)∗ φ1 (x, ξ)) α = 2βφ1 (x, ξ)α2 + 2Re (α∗ φ1 (x, ξ)∗ Q(x)φ2 (x, ξ)α) ≥ 2βφ1 (x, ξ)α2 − 2Q(x) φ1 (x, ξ)α φ2 (x, ξ)α. Hence φ1 (x, ξ)α ≥ βφ1 (x, ξ)α − Q(x) φ2 (x, ξ)α (7.4) −βx provided that φ1 (x, ξ)α = 0. Since e φ1 (x, ξ)α → α = 1 as x → −∞, (7.4) is certainly valid on some largest interval (−∞, x0 ), where x0 = +∞ is allowed. From (5.11) we obtain, since σ1 (x) = 2β, the bound φ1 (x, ξ)2 + φ2 (x, ξ)2 ≤ e2βx .

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311

Thus φ1 (x, ξ)α ≥ βφ1 (x, ξ)α − Q(x) (e2βx − φ1 (x, ξ)α2 )1/2 on (−∞, x0 ). In terms of z(x, ξ) = e−βx φ1 (x, ξ)α, this diﬀerential inequality becomes z (x, ξ) ≥ −Q(x)(1 − z(x, ξ)2 )1/2 ,

x < x0 .

Integrating from −∞ to x0 yields, since z(x, ξ) → 1 as x → −∞, x0 π sin−1 [z(x0 , ξ)] − ≥ − Q(x) dx. 2 −∞ Now, if x0 is ﬁnite, then z(x0 , ξ) = 0, and we get the inequality π/2 ≤ Q1 , which is necessary for an eigenvalue to exist. If x0 = +∞, then lim z(x, ξ) = 0 x→+∞

and the same inequality results. Finally, if Q1 = π/2 and an eigenvalue exists, then, in view of Lemma 5.4 and the fact that A(ξ) is analytic, we conclude that an eigenvalue still exists if we replace Q by (1 − )Q, with > 0 suﬃciently small. This gives a contradiction. Next we return to the problem with Q and R unrelated. The criteria in (7.1) and (7.2) are of the form Q1 R1 ≤ c

=⇒

no eigenvalues exist.

This leads to the question: What is the largest c such that this implication holds, for any n, m ≥ 1 and any Q, R satisfying (5.1). The best constant so far is that obtained from (7.1). However, as we will show now, it is not optimal. Theorem 7.2. Suppose (5.1) is satisﬁed. Then the largest c = 1. Proof. We ﬁrst show that if c > 1, then there are always functions Q(x) and R(x) such that Q1 R1 = c and H(Q, R) has an eigenvalue. It suﬃces to give an example with n = m = 1. Let R(x) = −μ on [−1, 0], R(x) = 0 otherwise, and Q(x) = μ on [0, 1], Q(x) = 0 otherwise, where μ > 0 is a parameter. Then a calculation gives sin2 ξ A(ξ) = 1 − μ2 e2iξ ξ2 from which we see that there is a purely imaginary eigenvalue located at approximately ξ = i(μ − 1) + O((μ − 1)2 ) as μ → 1. Now we show that there are no eigenvalues when c ≤ 1. Note that the diﬃculty is the following. If we insert (5.3) in (5.2) and iterate, we obtain x t e2iξt Q(t) e−2iξs R(s)ds dt + · · · . eiξx φ1 (x, ξ) = In + −∞

−∞

We immediately see that the integral term on the right-hand side has norm less than Q1 R1 . So, if we could simply ignore the remainder terms (indicated

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M. Klaus

by · · · ) we would be done. Estimating the remainder terms leads to the familiar bounds like (7.1), (7.2), and (7.3). So we cannot go this route. Instead, we employ the Birman-Schwinger principle which is well known from the study of eigenvalue problems for the Schr¨ odinger equation but has also seen applications to the Dirac equation and the Zakharov-Shabat system. To this end we write V (x) = A B C, where

+ A=

,

Q(x)1/2 In

0

0

R(x)1/2 Im + R(x)1/2 In

C=

+ ,

B= 0

0

Q(x) −i Q(x)

R(x) i R(x) ,

0

Q(x)1/2 Im

0

, ,

.

If Q(x) = 0, resp. R(x) = 0, we set (arbitrarily, but without loss of generality) Q(x)/Q(x) = 0, resp. R(x)/R(x) = 0. Alternatively, we could simply restrict the operator to supp R ⊕ supp Q. Now assume v is an eigenfunction for the eigenvalue ξ of H(Q, R) and deﬁne f = Cv. Then Kξ = C (H0 − ξI)−1 A B f = −f,

H0 = iJ

d . dx

The integral kernel of C (H0 − ξI)−1 A (without B) is D1 (x, y)In 0 −1 [C (H0 − ξI) A](x, y) = , 0 D2 (x, y)Im

(7.5)

(7.6)

where D1 (x, y) = iR(x)1/2 eiξ(y−x) θ(y − x)Q(y)1/2 ,

D2 (x, y) = D1 (y, x).

Since B ≤ 1, and the kernels Dk (x, y) are Hilbert-Schmidt (HS), it follows that ∞ 1/2 ∞ Kξ ≤ D1 B ≤ D1 HS = R(x) Q(y) dy dx . (7.7) −∞

x

Note that the norm of the matrix operator (7.6) is equal to the norm of D1 as a scalar operator on L1 (R). So, if c < 1, the right-hand side of (7.7) is less than one and (7.5) cannot have a solution. If c = 1 we argue as at the end of the proof of Theorem 7.1. We know that if we put additional restrictions on Q and R, then c = 1 is not necessarily the best constant. For example, if we demand that R = −Q∗ , then we know that c = π 2 /4 is the best constant (as a bound for the product Q1 R1 ). By pushing this method a bit further we can obtain another π/2-type result which applies to arbitrary functions Q(x) and R(x) satisfying (5.1). We ﬁrst need a lemma.

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Lemma 7.3. Let q ∈ L1 (R) be real and nonnegative. Then the integral operator B with kernel B(x, y) = q(x)1/2 θ(x − y)q(y)1/2 has norm B = (2/π)q1 . Proof. This is a consequence of the results obtained in [25]. The operator norm of the 2 × 2 matrix kernel 0 B(x, y) B(y, x) 0 was determined there (see the proof of Theorem 4.3) and found to be (2/π)q1 . We remark that this matrix kernel is the Birman-Schwinger kernel at ξ = 0 for the standard 2 × 2 Zakharov-Shabat system with potential q. Since the system at ξ = 0 can be solved explicitly, we can also determine the spectrum of the operator; in [25] all the eigenvalues are given. The above result may very well be known but we are not aware of a reference. We would like to point out though that the above kernel is a special case of kernels studied in [11],[33] (we learned this from [9], where further references are given). According to these references the norm would be at most x ∞ 1/2 2q1 2 sup q(t)dt q(t)dt = q1 > . π −∞ x The estimates contained in [11],[33] are optimal for the entire class of operators studied there, but not for the special class considered here. Let q(x) = max{Q(x), R(x)}. Theorem 7.4. Suppose q1 ≤ π/2. Then H(Q, R) has no eigenvalues. Proof. Suppose Im ξ > 0. Write Kξ as Kξ = (Cq −1/2 )q 1/2 (H0 − ξI)−1 q 1/2 (q −1/2 A)B. and note that the multiplication operators Cq −1/2 and q −1/2 A have norms less than 1. Hence Kξ ≤ q 1/2 (H0 − ξI)−1 q 1/2 . But q 1/2 (H0 − ξI)−1 q 1/2 is a diagonal matrix operator whose entries are kernels that are bounded in absolute value by the kernel B given in Lemma 7.3 or its adjoint. Hence Kξ ≤ (2/π)q1 . If Im ξ < 0, use (H0 − ξI)−1 = [(H0 − ξI)−1 ]∗ .

8. Nonexistence of purely imaginary eigenvalues Purely imaginary eigenvalues play a special role in the spectral theory of (1.1). Theorem 8.1. (i) Suppose n = 1, but m is arbitrary. Suppose Q(x) and R(x) have real and nonnegative entries. Then there are no imaginary eigenvalues.

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(ii) Suppose m = n and Q(x) and R(x) are both positive (negative) self-adjoint matrices. Then there are no imaginary eigenvalues. (iii) Suppose Q(x) = R(−x)∗ (n and m are not restricted). Then there are no imaginary eigenvalues. If, in case (iii), we also impose the condition R(x) = −Q(x)∗ , then we see that Q(x) = −Q(−x) is odd. If n = m = 1 and if n = 1, m = 2, then this case was already proved in [22] and [24]. Odd functions Q are of interest in ﬁber optics [19]. One can show by an example that the conclusion of (i) need not be true if n > 1. Proof. (i) This is obvious if we set ξ = iβ, β > 0, α = (1, 0, . . . ) in (1.1) and solve for φ(x, iβ)α, and hence φ1 (x, iβ)α, by iteration. All terms in the resulting series are nonnegative, in particular the ﬁrst component of φ1 (x, iβ)α is always greater than 1. Hence (1.1) cannot hold. (ii) Let v1 (v2 ) denote the upper (lower) n components of an eigenfunction v for an eigenvalue ξ ∈ C. Then a straightforward computation yields v2∗ v1 + v1∗ v2 = iξ(v1∗ v2 − v2∗ v1 ) + v2∗ Q(x)v2 + v1∗ R(x)v1 . Integrating both sides over and using an integration by parts on the left-hand side, we obtain ∞ (−(v2∗ ) v1 − (v1∗ ) v2 )dx −∞ (8.1) ∞ ∞ ∗ ∗ ∗ ∗ = iξ (v1 v2 − v2 v1 )dx + [v2 Q(x)v2 + v1 R(x)v1 ] dx. −∞

−∞

The left-hand side of (8.1) is imaginary. For ξ imaginary, the ﬁrst integral on the right-hand side is also purely imaginary whereas the second integral is real and nonnegative; it is zero only if Q(x)v2 (x) = R(x)v1 (x) = 0 (a.e.) and this cannot happen for an eigenfunction v. Since the two sides do not match up, ξ cannot be purely imaginary. (iii) Let ξ = −iβ, β > 0, be an eigenvalue. First, a calculation gives (φ(−x, iβ)∗ φ(x, iβ)) = 0. Now let α ∈ Cn be a vector such that φ(x, iβ)α is an eigenfunction. Then φ(0, iβ)α2 = α∗ φ(−x, iβ)∗ φ(x, iβ)α ≤ φ(x, iβ)αφ(−x, iβ)α. Since both factors on the right-hand side go to zero as x → ∞, we have reached a contradiction.

9. Purely imaginary eigenvalues For the standard 2 × 2 Zakharov-Shabat system (R = −Q∗ ) it was proved in [22] that if Q has the “single lobe” property, then the eigenvalues are all conﬁned to the imaginary axis. We recall that according to the deﬁnition given in [22], Q is

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single lobe provided it is non-negative, piece-wise smooth, bounded, in L1 , nondecreasing for x < 0, and non-increasing for x > 0. Of course, the point where Q(x) has its maximum can be shifted from 0 to any point x0 on the real line. In this section we ﬁrst generalize the single lobe result to systems of the form (1.1). As it turns out, the condition R = −Q∗ can be dropped entirely and some of the other technical assumptions can be relaxed as well. In the last part of this section we consider the standard Zakharov-Shabat and show that for a certain class of multi-hump functions Q(x) the eigenvalues must be purely imaginary. The main feature of this last generalization is that it is not a perturbative criterion but rather a “shape-related” one, whereby we refer to the shape of the graph of Q(x). As we will show by an example, these generalizations of the single lobe theorem require m = n. We begin with an observation about the spectrum of (1.1) under very special assumptions, which, however, will be relevant for the subsequent developments. Lemma 9.1. Let n = m ≥ 1. Suppose on x ≥ 0, Q(x) = 0 and R(x) ∈ L∞ (R+ )m×n and on x < 0, R(x) = 0 and Q(x) ∈ L∞ (R+ )n×m . Then C+ ⊂ ρ(H(Q, R)). Note that if Q and R were in L1 , then the result would follow from (7.3), since the support of Q lies entirely to the left of the support of R. Then there could be no eigenvalues in the upper half-plane. We only need the result for C+ , hence we do not fully discuss the spectrum of this operator here. Proof. For every ξ ∈ C+ we can write down the resolvent operator. We only state the result, the veriﬁcation is just a calculation. Of course, (4.10) could also be employed. Let z 0 −2iξt R(z) = e R(t) dt, Q(z) = e2iξt Q(t) dt. 0

z

The kernel of (H(Q, R) − ξI)−1 is given by , + −iξ(x+t) 0 −ie Q(x) , [(H(Q, R) − ξI)−1 ](x, t) = ieiξ(x−t) In ieiξ(x+t) R(t) + , −ie−iξ(x+t) Q(t) ieiξ(t−x) In −1 [(H(Q, R) − ξI) ](x, t) = , 0 ieiξ(x+t) R(x)

x > t, (9.1)

x < t. (9.2)

The resolvent represents a bounded operator, because for β > 0 the kernels e−β(x−t) θ(x − t) and e−β(t−x) θ(t − x) each have norm 1/β. Moreover, Q(x) ≤ (2β)−1 e−2βx Q∞,

x ≤ 0,

R(x) ≤ (2β)−1 e2βx R∞ ,

x ≥ 0,

where Q∞ = ess supQ(x). Therefore we have [e−iξ(x+t) Q(t)θ(t − x)]op ≤ (2β 2 )−1 Q∞, − x)]op ≤ (2β 2 )−1 R∞ , [eiξ(x+t) R(x)θ(t

(9.3)

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and similar estimates for the oﬀ-diagonal terms in (9.1). The subscript “op” says that we mean the operator norm of the given integral kernel. We can derive an explicit estimate for the norm of the resolvent. If we let the resolvent act on a vector (f1 , f2 )T , then the contribution from the (12)-entry gives x ∞ −iξ(x+t) C12 (x) = −iQ(x) e f2 (t)dt − i e−iξ(x+t) Q(t)f 2 (t)dt. −∞

x

Using the fact that Q(x) = 0 for x > 0 and using (9.3), we see that 0 C12 (x) ≤ Q∞ (2β)−1 e−β|x−t|f2 (t)dt, x < 0, −∞

and C12 (x) = 0 if x > 0. Hence, since (2β)−1 e−β|x−t] is the integral kernel of (−d2 /dx2 + β 2 )−1 , we have that C12 2 ≤ Q∞ β

−2

1/2

0

f2 (t) dt 2

−∞

.

We get an analogous estimate for the contribution from the (2, 1)-entry. The diagonal terms have norms equal to 1/β. It follows that # # # β −1 Q∞ β −2 # #, Im ξ > 0, [(H(Q, R) − ξI)−1 ]op ≤ # # # R∞ β −2 β −1 where on the right-hand side we have the uniform matrix norm, for which we could write down a lengthy expression. If Q∞ = R∞ , this norm is (β + Q∞ )β −2 . The O(β −2 ) behavior as β → 0 is due to the fact that H(Q, R) is not self-adjoint. If we relax the assumptions in the above theorem and do not require R to be bounded, then something dramatic happens. Theorem 9.2. Let m = n = 1. Suppose on x ≥ 0, Q(x) = 0 and R(x) is ﬁnite but diverging to +∞ or −∞ as x → +∞, and that on x < 0, R(x) = 0 and Q(x) satisﬁes (only) (2.2). Then ρ(H(Q, R)) ∩ C+ = ∅. We remark that the divergence of R(x) is only suﬃcient, not necessary, for the conclusion of this theorem to be true; further details may appear elsewhere. Proof. Pick ξ = α + iβ ∈ C+ and put gδ (x) = (gδ;1 (x), 0)T , where gδ;1 (x) = √ iαx−δx 2δe for x > 0, gδ;1 (x) = 0 for x < 0, and 0 < δ < β; we will let δ → 0 at the end. Then gδ 2 = 1 for every δ > 0. It is easy to see that H(Q, R) − ξI is injective, i.e., ξ is not an eigenvalue. We will show that gδ ∈ R(H(Q, R) − ξI) but that (H(Q, R) − ξI)−1 gδ is unbounded as δ → 0. This implies ξ ∈ / ρ(H(Q, R)).

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First, fδ (x, ξ) given by

√ i 2δ e(iα−δ)x * fδ (x, ξ) = x β + δ − 2iα eiξx 0 R(t)e(β−δ)t dt √ (−iα+β)x i 2δ e fδ (x, ξ) = 0 β + δ − 2iα

x > 0, x < 0,

satisﬁes (H(Q, R) − ξI)fδ = gδ and fδ ∈ H. Now, to estimate fδ = (H(Q, R) − ξI)−1 gδ , it suﬃces to consider the case when R(t) diverges to +∞ as t → +∞. For a given M > 0, pick N so that R(t) > M for t > N. Put xN ;δ = N + (ln 2)/(β − δ), so that e−(β−δ)(x−N ) ≤ 1/2 if x ≥ xN ;δ . Then, for the second component fδ;2 of fδ and x > 0, we have 2 x ∞ 2δM 2 2 −2βx (β−δ)t fδ;2 (·, ξ)2 ≥ e e dt dx (β + δ)2 + 4α2 xN ;δ N ≥

M 2 e−2δxN ;δ . 4(β − δ)2 ((β + δ)2 + 4α2 )

So lim inf fδ;2 (·, ξ)22 ≥ δ→0

M2 . + 4α2 )

4β 2 (β 2

Since M is arbitrary, the theorem is proved.

That we can also get the whole plane as spectrum is easy to see if we put 2 R(x) = ex for x ≥ 0. The purpose of Lemma 9.1 and the subsequent estimates on the resolvent is that they provide us with explicit control of the resolvent, so that if we perturb the operator we can get information about the change in the spectrum. For example, it is now easy to see that if we add L1 perturbations to Q and R, then the essential spectrum of H(Q, R) does not change. Note that since R and Q are bounded, we have that D[H(Q, R)] = D(H0 ) and the perturbation methods mentioned in Section 5 apply. Theorem 9.3. Suppose m = n and that on x < 0, Q ∈ L∞ (R− )n×n and on x > 0, R ∈ L∞ (R+ )n×n . Moreover, suppose that for all x ∈ R, Q(x) and R(x) are selfadjoint matrices and also satisfy the following conditions (i) on x ≥ 0, Q(x) is positive semi-deﬁnite, decreasing, and approaching 0 as x → +∞; (ii) on x ≤ 0, R(x) is negative semi-deﬁnite, decreasing, and approaching 0 as x → −∞. Then H(Q, R) has only purely imaginary eigenvalues in C+ . The same conclusion holds if we replace Q and R by −Q and −R, respectively, so that R(x) is positive increasing on x < 0 and Q(x) is negative increasing on x > 0. By switching Q and R in the assumptions we get analogous results for C− .

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The operator introduced in Lemma 9.1 serves as the background operator, the perturbation is given by Q(x) on x > 0 and R(x) on x < 0. Theorem 9.3 is our extension of the single lobe theorem of [23]. If we also impose the condition R(x) = −Q(x)∗ = −Q(x), then the above assumptions are met precisely when Q(x) has the typical single lobe shape (generalized to matrices). Proof. We know from Theorem 3.3(ii) that −ξ is an eigenvalue of H(Q, P ). Hence −ξ is an eigenvalue of H(R∗ , Q∗ ). In fact, there is a simple link between the two eigenspaces, since J+ H(Q, R)J+ = −H(R∗ , Q∗ ). So, if v is the eigenfunction of the eigenvalue ξ of H(Q, R), then Sv is the eigenfunction of the eigenvalue −ξ of H(R∗ , Q∗ ). Now suppose Re ξ = 0. Since −ξ = ξ, the vectors v and Sv are orthogonal, that is ∞ (v1∗ v2 + v2∗ v1 ) dx = 0. (9.4) −∞

This relation goes back to [39] in the context of the Zakharov-Shabat system. It was then exploited in [22] to prove the single lobe theorem. To prepare for the essential part of the proof we replace, if necessary, Q(x) by a matrix function that is close in L1 norm, is diﬀerentiable, is decreasing, is zero when x > a for some a > 0, and is strictly positive deﬁnite on [0, a]. This can be accomplished by choosing a positive, inﬁnitely diﬀerentiable function with support [−1, 1] so *1 that −1 j(x)dx = 1, and which is also symmetric (j(x) = j(−x)) and increasing (decreasing) for x < 0 (x > 0). Deﬁne j (x) = −1 j(x/), > 0, and let, for x > 0, ∞ j (x − t)Q(t)dt. Q (x) = −∞

Since Q(x) is assumed to be decreasing, which means that given any vector γ ∈ C n , γ ∗ Q(x)γ is decreasing, it follows that γ ∗ Q (x)γ is also decreasing, at least for x > . A little care is needed near zero when 0 ≤ x < , since the convolution picks up a contribution from x < 0 where Q(x) is not necessarily decreasing. To ﬁx this, we can ﬁrst modify Q(x) on [−, ] by setting Q(x) = Q(); this amounts to a small L1 perturbation, since Q(x) may have a (one-sided) singularity at 0. Applying the convolution to this modiﬁed Q(x) makes Q (x) diﬀerentiable and decreasing for all x ≥ 0. Then we cut oﬀ Q(x) at some large distance a > 0 and add In to Q (if necessary) so that then Q (x) ≥ In on [0, a]. All these modiﬁcations amount to small L1 perturbations or small L∞ perturbations and therefore leave the eigenvalue close to where it was initially. In particular, the real part of the perturbed eigenvalue is still nonzero. Then we modify R(x) in a similar manner, if necessary, and cut it oﬀ at some b < 0. Hence the eigenfunction v = (v1 , v2 )T for the (perturbed) eigenvalue ξ satisﬁes v2 (b, ξ) = 0 and v1 (a, ξ) = 0. Hence the integration in (9.4) only goes from b to a. Now from (1.1) we infer that Q−1 v1 = −iξQ−1 v1 + v2 ,

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and therefore v1∗ v2 + v2∗ v1 = −2βv1∗ Q−1 v1 + v1∗ Q−1 v1 + v1∗ Q−1 v1 . Using v1∗ Q−1 v1 + v1∗ Q−1v1 = (v1∗ Q−1 v1 ) + v1∗ Q−1 Q Q−1 v1 leads to a ∗ ∗ (v1 v2 + v2 v1 ) dx = −2β 0

a

v1∗ Q−1 v1 dx

+

0

a

v1∗ Q−1Q Q−1 v1 dx + v1∗ Q−1 v1 |a0 ,

0

where β = Im ξ > 0. Since v1 (a, ξ) = 0, we see that the right-hand side is strictly negative. A similar calculation, but this time only using the second of (1.1), gives 0 0 0 ∗ ∗ ∗ −1 (v1 v2 + v2 v1 ) dx = 2β v2 R v2 dx + v2∗ R−1 R R−1 v2 dx + v2∗ R−1 v2 |0b . b

b

b

Since R ≤ 0, R ≤ 0, and v2 (b, ξ) = 0, we conclude that the right-hand side is again negative. Hence we get that the left-hand side of (9.2) is negative, a contradiction. The reason why we have to assume m = n in Theorem 9.3 is that otherwise we do not even know if there is a theorem of this kind for a reasonably large class of functions Q and R. The following example illustrates the problem. Consider the Manakov case (n = 1, m = 2, R = −Q∗ ) with Q = (q1 , q2 ), q1 (x) = h on [−1, 1] and zero otherwise, and q2 (x) = 1 on [−2, 2] and zero otherwise. Thus the individual entries of Q are single lobe and one might expect that the conclusion of Theorem 9.3 would apply in this case. However, this is not so because if h = 1.9, then there exists a complex pair of eigenvalues near, but not on, the imaginary axis. For approximately h = 1.96, the two eigenvalues collide on the imaginary axis at approximately ξ = 0.3i and then split into a pair of purely imaginary eigenvalues if h is increased further. Finally, we extend Theorem 9.3 to multi-hump functions, but, as already mentioned above, only for the 2 × 2 Zakharov-Shabat case. So we assume R(x) = −Q(x)∗ and set Q(x) = q(x). The assumptions regarding diﬀerentiability and compact support made below could be weakened by using the approximation technique described in the proof of the previous theorem. In particular, q need only be monotone and piece-wise continuous between consecutive maxima and minima. Theorem 9.4. Suppose q(x) > 0 on an interval [d1 , d2 ] and q(x) = 0 for x < d1 and x > d2 . Suppose q is absolutely continuous with N strict local maxima a1 , . . . , aN and N − 1 strict local minima b1 , . . . , bN −1 such that d1 < a1 < b1 < a2 < · · · < bN −1 < aN < d2 . Furthermore, suppose N −1 k=1

1 1 ≤ . q(bk ) q(ak ) N

k=1

Then all eigenvalues (in C+ ∪ C− ) of (1.1) are purely imaginary.

(9.5)

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Note that the endpoints d1 and d2 of the support of q are not included in the list of maxima and minima. This is because q(x) is increasing on [d1 , a1 ) and decreasing on (aN , d2 ]. However, if we move the left cut-oﬀ point to d1 = a1 , then the theorem remains valid with the term q(a1 ) = q(d1 ) included on the right-hand side of (9.5). A similar statement holds at the other endpoint. Also, if the reader has seen the proof he will have no diﬃculty applying the theorem in situations where q is piecewise constant. If H(x1 , x2 , . . . , xn ) is the harmonic mean of x1 , . . . , xn , then the condition (9.5) can be stated as H(q(a1 ), . . . , q(aN )) N ≤ . H(q(b1 ), . . . , q(bN −1 )) N −1 Proof. Suppose there exists a solution v = (v1 , v2 )T of (1.1) with ξ ∈ R \ {0} satisfying v1 (d1 , ξ) = 1 (normalization), v2 (d1 , ξ) = 0, and v1 (d2 , ξ) = 0. In other words, ξ is a spectral singularity; (9.4) also applies to spectral singularities since Q and R have compact support. We think of the integral in (9.4) as a sum of integrals over the subintervals [d1 , a1 ), (a1 , b1 ), (b1 , a2 ), . . . , (aN , d2 ]. Set b0 = d1 , bN = d2 . First we consider the intervals of the form (bk , ak+1 ) (k = 0, . . . , N − 1). On each such interval, q(x) increases. From the equation for v2 we get, since ξ ∈ R, that v1 v 2 = −v2 v2 /q + (iξ/q)|v2 |2 and hence

ak+1

(v1 v 2 + v2 v 1 ) dx = −

bk

ak+1

bk

(|v2 (x)|2 )

dx. q(x)

Integrating by parts on the right-hand side yields ak+1 |v2 (ak+1 )|2 |v2 (bk )|2 |v2 (x)|2 q (x) − + − dx. q(ak+1 ) q(bk ) q(x)2 bk If bk = b0 = d1 , then the second term is zero since v2 (d1 , ξ) = 0. Turning to the intervals (ak , bk ) on which q(x) is decreasing, we use the ﬁrst of (1.1) and proceed as above to obtain bk bk |v1 (ak )|2 |v1 (bk )|2 |v1 (x)|2 q (x) − + (v1 v 2 + v2 v 1 ) dx = dx. q(bk ) q(ak ) q(x)2 ak ak If bk = bN = d2 , then v1 (bk , ξ) = 0. Adding up all the contributions, we obtain

∞ −∞

N −1 |v1 (ak )|2 |v2 (a1 )|2 |v1 (bk )|2 + − q(a1 ) q(bk ) q(ak ) k=1 (9.6) N −1 |v2 (bk )|2 |v1 (aN )|2 |v2 (ak+1 )|2 + + − . − q(ak+1 ) q(bk ) q(aN )

(v1 v2 + v2 v 1 ) dx ≤ −

k=1

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321

Now we also know from (5.11) that |v1 (x, ξ)|2 +|v2 (x, ξ)|2 = 1 for every x ∈ [d1 , d2 ], since ξ is real (and R = −Q∗ ). Using this and combining the terms in (9.6) gives ∞ N N −1 1 1 (v1 v 2 + v2 v 1 ) dx ≤ − + . (9.7) q(ak ) q(bk ) −∞ k=1

k=1

By assumption (9.5), the right-hand side of (9.7) is nonpositive. It suﬃces now to proceed with the case when it is strictly negative; otherwise we use a perturbation argument as at the end of the proof of Theorem 7.1. It follows that there are no spectral singularities on the real axis except possibly at ξ = 0; remember that (9.4) was derived under the assumption that Re ξ = 0. Now we claim that this implies that all eigenvalues must lie on the imaginary axis. To prove this we consider the problem where q is replaced by μq, with 0 ≤ μ ≤ 1. We have to show that as we increase μ from 0 to 1, no nonimaginary eigenvalues can be created, for example through a collision of two initially purely imaginary eigenvalues. This is exactly a problem that has been studied in [26] and the details needed to complete this proof can be found in the proof of Theorem 4.1(i) in [26]. This part of the proof involves a compactness argument for which Theorem 6.1 is needed. The reason why this proof takes a circuitous path via spectral singularities is that the equation |v1 (x, ξ)|2 + |v2 (x, ξ)|2 = 1 only holds if ξ is real. If N = 2 and q is symmetric then q(a1 ) = q(a2 ) = max q, b1 = 0, and q(b1 ) = min q. Then the condition (9.5) reads 2 qmin ≥ qmax , which already appeared in [26, Theorem 2.4]. That this condition for the nonexistence of nonimaginary eigenvalues is sharp, in the sense that the factor 2 cannot be replaced by a larger number, can be seen from an example given in [26, p. 15].

References [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249–315. [2] M.J. Ablowitz, B. Prinari, and A.D. Trubatch, Discrete and continuous nonlinear Schr¨ odinger systems, Cambridge Univ. Press, Cambridge, 2004. [3] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981. [4] F.V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964. [5] T.Y. Azizov and I.S. Iokhvidov, Linear operators in spaces with an indeﬁnite metric, John Wiley & Sons, New York, 1989. [6] J. Bronski, Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem, Physica 97 (1996), 376–397. [7] H. Behncke and D.B. Hinton, Eigenfunctions, deﬁciency indices and spectra of oddorder diﬀerential operators, Proc. London Math. Soc. 97 (2008), 425–449.

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[8] B.M. Brown, W.D. Evans, and M. Plum, Titchmarsh-Sims-Weyl theory for complex Hamiltonian systems, Proc. London Math. Soc. 87 (2003), 419–450. [9] R.C. Cascaval, F. Gesztesy, H. Holden, and Y. Latushkin, Spectral analysis of Darboux transformations for the focusing NLS hierarchy, J. Anal. Math. 93 (2004), 139– 197. [10] R.C. Cascaval and F. Gesztesy, J -self-adjointness of a class of Dirac-type operators, J. Math. Anal. Appl. 294 (2004), 113–121. [11] R.S. Chisholm and W.N. Everitt, On bounded integral operators in the space of integrable-square functions, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/71), 199– 204. [12] W.A. Coppel, Stability and asymptotic behavior of diﬀerential equations, Heath, Boston, 1965. [13] F. Demontis, Direct and inverse scattering of the matrix Zakharov-Shabat system, Ph.D. thesis, University of Cagliari, Italy, 2007. [14] F. Demontis and C. van der Mee, Marchenko equations and norming constants of the matrix Zakharov-Shabat system, Operators and Matrices 2 (2008), 79–113. [15] D.E. Edmunds and W.E. Evans, Spectral theory and diﬀerential operators, Clarendon Press, Oxford, 1987. [16] I.M. Glazman, Direct methods of qualitative spectral analysis of singular diﬀerential operators, Moscow, 1963. English Translation by Israel Program for Scientiﬁc Translations, 1965. [17] D. Hinton and K. Shaw, Titchmarsh-Weyl theory for Hamiltonian systems, in Spectral theory of diﬀerential operators, I.W. Knowles and R.T. Lewis, Eds., North Holland, New York, 1981. [18] T. Kato, Perturbation theory for linear operators, Springer, New York, 1976. [19] D.J. Kaup and L.R. Scacca, Generation of 0π pulses from a zero-area pulse in coherent pulse propagation, J. Opt. Soc. Am. 70 (1980), 224–230. [20] M. Klaus, Dirac operators with several Coulomb singularities, Helv. Phys. Acta 53 (1980), 463–482. [21] M. Klaus and J.K. Shaw, Inﬂuence of pulse shape and frequency chirp on stability of optical solitons, Optics Commun. 197 (2001), 491–500. [22] M. Klaus and J.K. Shaw, Purely imaginary eigenvalues of Zakharov-Shabat systems, Phys. Rev. E. (3) 65, (2002), article 036607. [23] M. Klaus and J.K. Shaw, On the eigenvalues of Zakharov-Shabat systems, SIAM J. Math. Anal. 34 (2003), 759–773. [24] M. Klaus, Remarks on the eigenvalues of the Manakov system, Mathematics and computers in simulation 69 (2005), 356–367. [25] M. Klaus, On the Zakharov-Shabat eigenvalue problem, in Contemporary Mathematics 379, 21–45, Amer. Math. Soc., Providence, RI, 2005. [26] M. Klaus and B. Mityagin, Coupling constant behavior of eigenvalues of ZakharovShabat systems, J. Math. Phys. 48 (2007), article 123502. [27] A. Krall, M(λ) theory for singular Hamiltonian systems with two singular endpoints, 20 (1989), 701–715.

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[28] A. Krall, A limit-point criterion for linear Hamiltonian systems, Applicable Analysis 61 (1996), 115–119. [29] P. Lancaster, Theory of Matrices, Academic Press, New York, 1969. [30] P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure and Appl. Math. 21 (1968), 467–490. [31] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac operators, Kluwer Acad. Publ., Dordrecht, 1991. [32] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38 (1974), 248–253. [33] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. [34] L. Mirsky, An introduction to linear algebra, Clarendon Press, Oxford, 1955. [35] G. Nenciu, Self-adjointness and invariance of the essential spectrum for Dirac operators deﬁned as quadratic forms, Comm. Math. Phys. 48 (1976), 235–247. [36] J. Qi, Non-limit-circle criteria for singular Hamiltonian diﬀerential systems, J. Math. Anal. Appl. 305 (2005), 599–616. [37] D. Race, The theory of J-self-adjoint extensions of J-symmetric operators, J. Diﬀerential Equations 57 (1985), 258–274. [38] M. Reed and B. Simon, Methods of modern mathematical physics, Academic Press, 1978. [39] J. Satsuma and N. Yajima, Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Suppl. Prog. Theor. Phys. 55 (1974), 284–306. [40] J. Villarroel, M.J. Ablowitz, and B. Prinari, Solvability of the direct and inverse problems for the nonlinear Schr¨ odinger equation, Acta Applicandae Mathematicae 87 (2005), 245–280. [41] M.I. Vishik, On general boundary problems for elliptic diﬀerential equations, Amer. Math. Soc. Transl.(2) 24 (1963), 107–172. [42] T. Wa˙zewski, Sur la limitation des int´ egrales des syst`emes d’´equations diﬀ´erentielles lin´eaires ordinaires, Studia Math. 10 (1948), 48–59. [43] J. Weidmann, Spectral theory of ordinary diﬀerential operators, Lect. Notes in Math. 1258, Springer, New York, 1987. [44] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. J. Exp. Theor. Phys. 34 (1972), 62–69. Martin Klaus Department of Mathematics Virginia Tech Blacksburg, VA 24061, USA e-mail: [email protected] Received: May 18, 2009 Accepted: July 17, 2009

Operator Theory: Advances and Applications, Vol. 203, 325–345 c 2010 Birkh¨ auser Verlag Basel/Switzerland

An Extension Theorem for Bounded Forms Deﬁned in Relaxed Discrete Algebraic Scattering Systems and the Relaxed Commutant Lifting Theorem S.A.M. Marcantognini and M.D. Mor´an Abstract. The concept of relaxed discrete algebraic scattering system is introduced. For a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ) and a set {B1 , B2 , B0 } of sesquilinear forms deﬁned in the relaxed discrete algebraic scattering system such that B1 : G1 × G1 → C and B2 : G2 × G2 → C are nonnegative, B1 (Γg 1 , Γg 1 ) ≤ B1 (g 1 , g 1 ) for all g 1 ∈ G1 , B2 (Γg, Γg) = B2 (g, g) 1 1 for all g ∈ G, and |B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2 for all g 1 ∈ G1 and 2 2 1 1 g ∈ G , a map Φ : ΓG → G interpolating the system and the forms is considered. An extension theorem for a set {B1 , B2 , B0 } of sesquilinear forms deﬁned in a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ) with interpolant map Φ : ΓG1 → G1 is established. It is shown that the result encompasses the Cotlar-Sadosky extension theorem for bounded forms deﬁned in discrete algebraic scattering systems as well as the Relaxed Commutant Lifting Theorem. Furthermore, the interpolants D in the relaxed lifting problem are obtained in correspondence with the extension forms B in a related extension problem so that D and B determine each other uniquely. Mathematics Subject Classiﬁcation (2000). Primary: 47A20; Secondary: 47A07, 47A40. Keywords. Bounded forms, scattering systems, Relaxed Commutant Lifting Theorem.

1. Introduction The algebraic scattering systems are abstract structures which were introduced by M. Cotlar and C. Sadosky [5] to provide theoretical schemes when dealing with translation-like invariant forms. It has indeed proved to be the case that Communicated by J.A. Ball.

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the algebraic scattering systems yield a common general framework for a large collection of disparate problems where, as joint feature, invariant forms appear explicitly or lurk beneath (cf. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). The celebrated Commutant Lifting Theorem by D. Sarason, B. Sz.-Nagy and C. Foias [26, 27] is amongst the results that can be treated from the viewpoint of algebraic scattering systems and forms deﬁned in them (cf. [14], see also [17, Section VII.8] and [16, Section 5]). In fact, the lifting problem can be translated into an extension problem for forms so that each solution of the latter uniquely determines a solution of the former, and vice-versa. The algebraic scattering system arisen in this situation is rather simple and the related extension problem refers to whether there exists a Toeplitz-type extension of a Hankel-type form deﬁned in the system. An extension theorem established by M. Cotlar and C. Sadosky gives for granted that the form admits such an extension if it is subordinated to a pair of nonnegative Toeplitz-type forms. The boundedness condition is satisﬁed by the form underlying the data set of the lifting problem and the Commutant Lifting Theorem readily follows. It seems to be natural to treat the relaxed version of the Commutant Lifting Theorem by C. Foias, A.E. Frazho and M.A. Kaashoek [18] in the same fashion. It happens, however, that none of the Cotlar-Sadosky scattering systems ﬁts in with the relaxed lifting problem. We present a kind of algebraic scattering system and a set of forms that, together with an interpolant function for the system and the forms, suit the purpose of translating the lifting problem into an extension problem for forms. We also establish the analog of the Cotlar-Sadosky extension theorem for the new setting of systems and forms and we get a proof of the Relaxed Commutant Lifting Theorem from it. The new concepts and the generalized extension theorem we bring in may be of interest by themselves within the scattering system theory. Descriptions of the interpolants in the Relaxed Commutant Lifting Theorem are provided by A.E. Frazho, S. ter Host and M.A. Kaashoek in [19] and [20], and by W.S. Li and D. Timotin in [22]. The coupling method is used in [19] in combination with system theory techniques and in [22] in conjunction with a choice sequence approach. An elementary harmonic majorant argument is employed in [20]. Other results in the same direction are proved in [23] by merging the coupling method with a functional model due to D.Z. Arov and L.Z. Grossman [2, 3]. Both techniques, the coupling method and the model, serve as tools also in this note but here they rely on the scattering system approach. The same methods and approach were adopted in [24] to treat weighted versions of Nehari’s problem for matrix-valued Wiener functions. Algebraic scattering systems were also explored in [4] to deal with the problem of describing the symbols of a given ` a la Pt´ ak-Vrbov´ a abstract Hankel operator. The coupling method and the Arov-Grossman model resulted in a parameterization of the interpolants in the classical Commutant Lifting Theorem in [25]. The paper is organized in ﬁve sections. Section 1, this section, serves as an introduction. In Section 2 we ﬁx the notation and state some known results. The concepts of relaxed discrete algebraic scattering system, on one hand, and inter-

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polant map, on the other, are discussed, along with some examples, in Sections 3 and 4, respectively. Section 5 comprises two subsections. First we present our main result in the central part of Section 5. Then we show that the result includes as particular cases the Cotlar-Sadosky extension theorem for bounded forms (Subsection 5.1) and the Relaxed Commutant Lifting Theorem (Subsection 5.2).

2. Preliminaries We follow the standard notation, so N, Z and C are, respectively, the set of natural, integer and complex numbers; D denotes the open unit disk in the complex plane and T its boundary. Throughout this note, all Hilbert spaces are assumed to be complex and separable. If {Gι }ι∈I is a collection of linear subspaces of a Hilbert space H then B G is the least closed subspace of H containing {Gι }ι∈I . If H is just a linear ι∈I ι B space, we keep the notation ι∈I Gι for the linear span of {Gι }ι∈I . When Gι ⊥ Gκ for ι = κ, we write ⊕ι∈I Gι instead. As usual, L(H, K) denotes the space of all everywhere deﬁned bounded linear operators on the Hilbert space H to the Hilbert space K, and L(H) is used instead of L(H, H). By 1 we indicate either the scalar unit or the identity operator, depending on the context. The null space of A ∈ L(H, K) is denoted by ker A. If G is a closed linear subspace of a Hilbert space H, then PG stands for the orthogonal projection from H onto G. 1 If T ∈ L(H, K) is a contraction operator then DT := (1 − T ∗ T ) 2 and DT := DT H are the defect operator and the defect space of T , respectively. If V ∈ L(D, R) is a unitary operator and D, R are closed subspaces of a Hilbert space H, we call V an isometry on H with domain D and range R. Its defect subspaces are N := H D and M := H R. By a minimal unitary extension ⊇ H such that U |D = V of V we mean a unitary operator U on a Hilbert space H B ∞ n = and H n=−∞ U H. Two minimal unitary extensions of V , say U ∈ L(H)

and U ∈ L(H ), are regarded as identical if there exists a unitary isomorphism such that τ |H = 1 and τ U = U τ . We write U(V ) to denote the set of → H τ :H minimal unitary extensions of V . In the sequel, if N and M are two Hilbert spaces, then S(N , M) stands for the L(N , M)-Schur class, so that ϑ ∈ S(N , M) if and only if ϑ : D → L(N , M) is an analytic function such that sup ϑ(z) ≤ 1. z∈D

For any (separable) Hilbert space E we denote by L2 (E) the class of all the functions f : T → E which are Lebesgue-measurable (strongly or weakly, which comes to be the same due to the separability of E) and such that 2π 1 2 f := f (eit )2 dt < ∞. 2π 0

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With the pointwise linear operations and the scalar product 2π 1 f (eit ), g(eit )E dt (f, g ∈ L2 (E)) f, gL2 (E) := 2π 0 L2 (E) becomes a (separable) Hilbert space under the interpretation that two func2 tions in LC (E) are viewed as identical if they coincide almost everywhere. Moreover, 2 L (E) = ∞ n=−∞ Gn (E), where, for each integer number n, Gn (E) is the subspace of those functions f ∈ L2 (E) such that f (eit ) = eint x for some x ∈ E. elements of H 2 (E) are all the analytic functions u : D → E, u(z) = "∞ The n n=0 z un , z ∈ D and {un } ⊆ E, such that u := 2

∞

un 2 < ∞.

n=0

We recall that H 2 (E) is a Hilbert space with the pointwise linear operations and the scalar product , + ∞ ∞ ∞ u, vH 2 (E) := un , vn E z n un , v(z) = z nvn ∈ H 2 (E) . u(z) = n=0

n=0

n=0

As a consequence of Fatou’s Theorem, the radial limit limr↑1 u(reit ) exists almost everywhere. The application that maps each u(z) ∈ H 2 (E) into its radial limit provides an embedding of H 2 (E) into L2 (E) preserving the Hilbert space structures. Via the Poisson be shown that the application maps H 2 (E) C∞integral, it can 2 ontoC the subspace n=0 Gn (E) of L (E). Therefore we may consider that H 2 (E) ∞ and n=0 Gn (E) amount to the same Hilbert space. If ϑ ∈ S(N , M) then limr↑1 ϑ(reit ) exists almost everywhere as a strong limit of operators and determines a contraction operator in L(N , M). With each ϑ ∈ S(N , M) we associate a contraction operator from L2 (N ) into L2 (M) deﬁned by f (eit ) → ϑ(eit )f (eit ) (f (eit ) ∈ L2 (N )) and a contraction operator from H 2 (N ) into H 2 (M) deﬁned by / . u(z) → ϑ(z)u(z) u(z) ∈ H 2 (N ) and z ∈ D .

C Due to identiﬁcation of H 2 (N ) (and H 2 (M)) with the subspace ∞ n=0 Gn (N ) Cthe (and ∞ G (M), respectively) the latter operator may be consider as a restricn n=0 tion of the former one. We denote both of them by ϑ. When N = M = E and ϑ(z) ≡ z (z times the identity operator on E) the associated operator is the (forward) shift S. Given ϑ ∈ S(N , M) we can likewise consider the operator Δ(eit ) = Dϑ(eit ) almost everywhere. The basic reference for vector- and operator-valued analytic functions is [28]. We refer the reader to the detailed exposition given therein.

An Extension Theorem for Bounded Forms . . .

329

As a matter of notation, if C, D are Hilbert spaces and E = C ⊕ D, we will c . Both notations write the elements of E either as rows c d or as columns d are also adopted when C, D are just linear spaces and E = C × D. To conclude this section we state the result encompassing the Arov-Grossman model. The model gives a labeling of the minimal unitary extensions of a given Hilbert space isometry by means of operator-valued Schur functions. It is therefore a useful theoretical device for constructing unitary extensions of isometries and, likewise, as we will see shortly, Toeplitz-type extensions of Hankel-type forms. Theorem 2.1. (Arov-Grossman [2, 3]) Let V : D ⊆ H → H be an isometry with defect subspaces N and M. Given ϑ ∈ S(N , M), set Eϑ := H 2 (M) ⊕ ΔL2 (N ) ∩ { ϑχ Δχ : χ ∈ H 2 (N )}⊥ , where Δ(ζ) := Dϑ(ζ) ,

|ζ| = 1.

Deﬁne Fϑ := H ⊕ Eϑ and Uϑ : Fϑ → Fϑ by ⎡ ⎤ ⎡ ⎤ V PD h + ϑ(0)PN h + φ(0) h ⎦ S ∗ (φ + ϑPN h) Uϑ ⎣ φ ⎦ := ⎣ S ∗ (ψ + ΔPN h) ψ

φ ∈ Eϑ h ∈ H, ψ

where S is the shift on either H 2 (M) or L2 (N ), depending on the context. Then: (i) Uϑ ∈ L(Fϑ ) is a minimal unitary extension of V such that PM Uϑ (1 − zPEϑ Uϑ )−1 |N = ϑ(z) for all z ∈ D. the function (ii) For any minimal unitary extension U of V on H, z :→ PM U (1 − zPHH U )−1 |N

(z ∈ D)

belongs to S(N , M). ) are two minimal unitary extensions of V , and U ∈ L(H (iii) If U ∈ L(H) such that τ |H = 1 and → H then there exists a unitary isomorphism τ : H

τ U = U τ , if and only if U )−1 |N PM U (1 − zPH H U )−1 |N = PM U (1 − zPHH for all z ∈ D. Therefore, the map ϑ → Uϑ ∈ L(Fϑ ) establishes a bijective correspondence between S(N , M) and U(V ), up to unitary isomorphisms as far as U(V ) is concerned.

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3. Relaxed discrete algebraic scattering systems A relaxed discrete algebraic scattering system is a quadruple (G, G1 , G2 , Γ) where • G is a linear space, • G1 and G2 are linear subspaces of G, and • Γ is a linear transformation on G such that ΓG1 ⊆ G1 and G2 ⊆ ΓG2 . It is further assumed that • G is the least linear space containing G1 and {Γn G2 }n≥0 : ⎛ ⎞ D D ⎝ G = G1 Γn G2 ⎠ . n≥0

Example 3.1. If Γ is an algebraic isomorphism on G then (G, G1 , G2 , Γ) is a CotlarSadosky scattering system (cfr. [5]). If G is a Hilbert space, G1 and G2 are closed subspaces of G and Γ is a unitary operator on G, then (G, G1 , G2 , Γ) is a so-called Hilbert space scattering system. If, in addition, E E Γn G1 = {0} = Γ−n G2 n≥0

and

⎛ G=⎝

n≥0

⎞

D

Γ−n G1 ⎠

D

n≥0 1

⎛ ⎝

D

⎞ Γn G2 ⎠ ,

n≥0

2

then (G, G , G , Γ) turns out to be an Adamyan-Arov scattering system (cfr.[1]). A Lax-Phillips scattering system (cfr. [21]) is an Adamyan-Arov scattering system such that G1 ⊥ G2 and G=

D

Γ−n G1 =

n≥0

D

Γn G2 .

n≥0

The next example exhibits a relaxed discrete algebraic scattering system which is not a Cotlar-Sadosky scattering system. Example 3.2. Let G be the linear space of all polynomials on T, so that

s & n s G := an ζ : r, s ∈ N ∪ {0}, {an}n=−r ⊆ C, ζ ∈ T . n=−r 1

2

Let G and G be the linear subspaces of G comprising the analytic and antianalytic polynomials, respectively. Deﬁne Γ on G by setting Γ

s n=−r

an ζ n :=

s n=−r, n=0

an ζ n .

An Extension Theorem for Bounded Forms . . .

331

4. Interpolants of relaxed discrete algebraic scattering systems and forms deﬁned on them Given a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ), two nonnegative sesquilinear forms B1 : G1 × G1 → C, B2 : G × G → C and a sesquilinear form B0 : G1 × G2 → C such that (i) B1 (Γg 1 , Γg 1 ) ≤ B1 (g 1 , g 1 ) for all g 1 ∈ G1 , (ii) B2 (Γg, Γg) = B2 (g, g) for all g ∈ G, and 1 1 (iii) |B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2 for all g 1 ∈ G1 and g 2 ∈ G2 , we say that a map Φ : ΓG1 → G1 interpolates the system and the forms if and only if (a) B1 (ΦΓg 1 , ΦΓg 1 ) ≤ B1 (Γg 1 , Γg 1 ) for all g 1 ∈ G1 , and (b) B0 (ΦΓg 1 , u) = B0 (Γg 1 , Γu) for all g 1 ∈ G1 and all u ∈ G2 such that Γu ∈ G2 . Example 4.1. Let (G, G1 , G2 , Γ) be a Cotlar-Sadosky scattering system. A sesquilinear form B : G×G → C is said to be Γ-Toeplitz if and only if B(Γg, Γg) = B(g, g) for all g ∈ G. In turn, a sesquilinear form B0 : G1 × G2 → C is said to be Γ-Hankel if and only if B0 (Γg 1 , g 2 ) = B0 (g 1 , Γ−1 g 2 ) for all g 1 ∈ G1 and g 2 ∈ G2 . If (G, G1 , G2 , Γ) is a Cotlar-Sadosky scattering system, B1 , B2 : G × G → C are two nonnegative Γ-Toeplitz forms and B0 : G1 × G2 → C is a Γ-Hankel form such that 1

1

|B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2

for all g 1 ∈ G1 and g 2 ∈ G2 ,

then the map Φ = Γ−1 interpolates the system and the forms. With the notation just introduced, notice that (ii) as in the above deﬁnition states that B2 is a Γ-Toeplitz form on G × G while (i) says that B1 is a sort of “weak” or “relaxed” Γ-Toeplitz form on G1 × G1 . Note also that the map Φ : ΓG1 → G1 interpolates the system and the forms whenever Φ turns B1 into a weak or relaxed Φ-Toeplitz form on ΓG1 × ΓG1 and B0 into a Φ-Hankel form on ΓG1 × {u ∈ G2 : Γu ∈ G2 } under the meaningless manipulation Φ−1 u = Γu for u ∈ G2 such that Γu ∈ G2 . Example 4.2. Let (G, G1 , G2 , Γ) be the relaxed discrete algebraic scattering system given in Example 3.2. For each n ∈ Z, let δn (m) be deﬁned as 1 if n = m and 0 otherwise. For nonnegative integers r, s, p, q and complex numbers {an }sn=−r , {bn }qn=−p , consider + s , q q s 1 1 n m = an ζ , bm ζ an bm δn (m), B1 : G × G → C, B1 + B2 : G × G → C,

B2

n=0 s

an ζ n ,

n=−r

B0 : G1 × G2 → C,

m=0

+ B0

n=0 m=0

,

q

bm ζ m

=

m=−p s n=0

an ζ n ,

−1 −1

an bm δn (m),

n=−r m=−p −1 m=−p

,

bm ζ m

= a0 b−1 .

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S.A.M. Marcantognini and M.D. Mor´ an

Then B1 , B2 are two nonnegative sesquilinear forms and B0 is a sesquilinear 1 1 form." They verify (i), "s (ii) and (iii) above and the map Φ : ΓG → G given s by Φ n=1 an en := n=2 an en interpolates the system and the forms. In the Cotlar-Sadosky treatment of invariant forms deﬁned in algebraic scattering systems the results playing a key role are those providing conditions for a Hankel-type form to be extended or lifted to a form constrained to be of Toeplitztype and to satisfy certain conditions. As for a Γ-Hankel form B0 : G1 × G2 → C deﬁned in a Cotlar-Sadosky scattering system (G, G1 , G2 , Γ) and subordinated to two nonnegative Γ-Toeplitz forms B1 , B2 : G × G → C as in Example 4.1, the extension property is granted by the following theorem. Theorem 4.3. (Cotlar-Sadosky [5]) Given a Cotlar-Sadosky scattering system (G, G1 , G2 , Γ), set D D Γ−n G1 and G 2 := Γn G2 . G 1 := n≥0

n≥0

Let B1 , B2 : G × G → C be a pair of nonnegative Γ-Toeplitz forms and let B0 : G1 × G2 → C be a Γ-Hankel form such that 1

1

|B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2

for all g 1 ∈ G1 and g 2 ∈ G2 .

Then there exists a Γ-Toeplitz form B : G 1 × G 2 → C such that (1) B(g 1 , g 2 ) = B0 (g 1 , g 2 ) for all g 1 ∈ G1 and g 2 ∈ G2 , and 1 1 (2) |B(v, u)| ≤ B1 (v, v) 2 B2 (u, u) 2 for all v ∈ G 1 and u ∈ G 2 . In the next section we deal with the analogue of Theorem 4.3 for bounded forms deﬁned in relaxed discrete algebraic scattering systems.

5. An extension theorem for bounded forms deﬁned in relaxed discrete algebraic scattering systems Theorem 5.1. Given a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ), set D G 2 := Γn G2 . n≥0

Consider two nonnegative sesquilinear forms B1 : G1 × G1 → C, B2 : G × G → C and a sesquilinear form B0 : G1 × G2 → C such that (i) B1 (Γg 1 , Γg 1 ) ≤ B1 (g 1 , g 1 ) for all g 1 ∈ G1 , (ii) B2 (Γg, Γg) = B2 (g, g) for all g ∈ G, and 1 1 (iii) |B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2 for all g 1 ∈ G1 and g 2 ∈ G2 . If Φ : ΓG1 → G1 interpolates the system and the forms then there exists a sesquilinear form B : G1 × G 2 → C such that (1) B(g 1 , g 2 ) = B0 (g 1 , g 2 ) for all g 1 ∈ G1 and g 2 ∈ G2 , (2) B(ΦΓg 1 , u) = B(Γg 1 , Γu) for all g 1 ∈ G1 and u ∈ G 2 , and

An Extension Theorem for Bounded Forms . . . 1

333

1

(3) |B(g 1 , u)| ≤ B1 (g 1 , g 1 ) 2 B2 (u, u) 2 for all g 1 ∈ G1 and u ∈ G 2 . Proof. Consider the product space G1 × G2 with the sesquilinear form F G u u u u

1 2

, ∈ G ×G . ,

:= B1 (u, u )+B0 (u, v )+B0 (u , v)+B2 (v, v ) v v v v The condition stated in (iii) guarantees that there exist a Hilbert space H and a linear map σ : G1 × G2 → H such that σ(G1 × G2 ) is dense in H and F G F G u u u u u u , ∈ G1 × G2 . , ,

= ,σ

σ v v v v v v H Since G2 ⊆ ΓG2 , we get that, for given n ∈ N and g ∈ G2 , there exists un ∈ G2 such that g = Γn un . Moreover, un is uniquely determined by n ∈ N and g ∈ G2 . In fact, if u, v ∈ G2 are such that Γn u = Γn v , then, for all g 1 ∈ G1 and g 2 ∈ G2 , G G F 1 F 1 g 0 0 g = B0 (g 1 , u − v) + B2 (g 2 , u − v) σ 2 ,σ , = g g2 u−v H u−v where B2 (g 2 , u − v) = B2 (Γn g 2 , Γn (u − v)) = 0 and |B0 (g 1 , u − v)|

1

1

≤ B1 (g 1 , g 1 ) 2 B2 (u − v, u − v) 2 1

1

= B1 (g 1 , g 1 ) 2 B2 (Γn (u − v), Γn (u − v)) 2 = 0. Since Φ interpolates the system and the forms and B2 is Γ-Toeplitz, it follows that, for g 1 ∈ G1 , g 2 ∈ G2 and u ∈ G2 such that Γu = g 2 , # 1 # # # # # ΦΓg 1 # # # ≤ #σ Γg2 # . #σ # # u # g # H

Therefore, by setting 1 ΦΓg 1 Γg := σ Xσ g2 u

H

(g 1 ∈ G1 , g 2 ∈ G2 , u ∈ G2 such that Γu = g 2 )

we get a contraction X : σ(ΓG1 × G2 ) → H. If S is the shift on H 2 (DX ) then the 2 × 2 block matrix X 0 V = DX S gives an isometry on the Hilbert space H ⊕ H 2 (DX ) with domain D := σ(ΓG1 × G2 ) ⊕ H 2 (DX ) and range R :=

Xh DX h + Sϕ : h

ϕ ∈D .

334

S.A.M. Marcantognini and M.D. Mor´ an The defect subspaces of V are N = H σ(ΓG1 × G2 )

and

M={ h

x : h ∈ H, x ∈ DX ⊆ H 2 (DX ), X ∗ h + DX x = 0}.

Given ϑ ∈ S(N , M), let Uϑ ∈ L(Fϑ ) be the corresponding minimal unitary extension of V as in Theorem 2.1. If n ∈ N and g ∈ G2 , take un ∈ G2 such that Γn un = g. Then 0 0 n =σ V σ . un g Therefore, for all g, g ∈ G2 , G F 0 0 ,σ

= B2 (Γn g, g ). Uϑ−n σ g g F ϑ

So, if {g(n)}n≥0 is a G2 -valued sequence with ﬁnite support, then # # ⎞ ⎛ # #2 # −n # 0 # # Uϑ σ = B2 ⎝ Γn g(n), Γn g(n)⎠ . # g(n) # #n≥0 # n≥0 n≥0 Fϑ B Γn G2 and deﬁne B : G1 × G 2 → C by Recall that G 2 := n≥0

⎛ B ⎝g , 1

n≥0

⎞

H I 1 g 0 −n Γ g(n)⎠ := σ Uϑ σ , g(n) 0 n

n≥0

Fϑ

for g ∈ G and {g(n)}n≥0 ⊆ G with ﬁnite support. We claim that B satisﬁes (1), (2) and (3) as required. Indeed: • For all g 1 ∈ G1 and g 2 ∈ G2 , F 1 G F 1 G g g 0 0 1 2 = σ = B0 (g 1 , g 2 ). ,σ 2 ,σ 2 B(g , g ) = σ g g 0 0 F H 1

1

2

ϑ

• For all g 1 ∈ G1 and g 2 ∈ G2 , F 1 G Γg 0 −(n+1) 1 n+1 2 g ) = σ σ 2 , Uϑ B(Γg , Γ g 0 Fϑ F G ΦΓg 1 Γg 1 0 −n = DX σ , Uϑ σ 2 g 0 0 Fϑ G F ΦΓg 1 0 , Uϑ−n σ 2 = = B(ΦΓg 1 , Γn g 2 ). 0 g F ϑ

Therefore, for all g 1 ∈ G1 and u ∈ G 2 , B(Γg 1 , Γu) = B(ΦΓg 1 , u).

An Extension Theorem for Bounded Forms . . .

335

• For all g 1 ∈ G1 and {g(n)}n≥0 ⊆ G2 with ﬁnite support, + , # 1 # # # # # g # # " n " −n 0 # # 1 # # σ Γ g(n) ≤ # U σ B g , # ϑ # # # g(n) 0 # n≥0 H n≥0 +

1

1

1 2

= B1 (g , g ) B2

Fϑ

"

n

Γ g(n),

n≥0

"

, 12 n

Γ g(n)

.

n≥0

That is, for all g 1 ∈ G1 and u ∈ G 2 , 1

1

|B(g 1 , u)| ≤ B1 (g 1 , g 1 ) 2 B2 (u, u) 2 .

The proof is complete.

In the above construction B is given in terms of ϑ. With the aid of Theorem 2.1, we can further analyze the direct connection between ϑ and B and, matterof-factly, we can show that all extensions are given in this way. To this end, let B : G1 × G 2 → C be a sesquilinear form such that (1), (2) and (3) hold. Repeat the construction that led on to the Hilbert space H and the map σ, replacing B0 by B and taking into account that B is also subordinated to B1 and B2 according with (3). In this way we get a Hilbert space HB and a map σB : G1 × G 2 → HB such that σB (G1 × G 2 ) is dense in HB and F G g g = B1 (g, g ) + B(g, u ) + B(g , u) + B2 (u, u ) σB , σB

u u HB g g , ∈ G1 × G 2 . for all u u From (1) it follows that the map densely deﬁned on H by 1 1 1 g g g 1 2 σ 2 → σB 2 × G ∈ G g g g2 gives rise to an isometry I from H into HB . Since Φ : ΓG1 → G1 interpolates the system and the forms, B2 is Γ-Toeplitz and B satisﬁes (2), it can be seen that the map 1 Γg ΦΓg 1 XB σB := σB (g 1 ∈ G1 , u ∈ G 2 , u ∈ G 2 such that Γu = u) u u

yields a contraction XB : σB (ΓG1 × G 2 ) → HB . Clearly, XB I|σ(ΓG1 ×G2 ) = IX.

(5.1)

For all g 1 ∈ G1 , g 2 ∈ G2 and u ∈ G 2 , # # # # 1 # 1 # 1 # 1 # # # # # # # # # #DXB σB Γg # = #DXB σB Γg # = #DX σ Γg # = #DX σ Γg2 # . # # # # # # # u 0 0 g #H F F H

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S.A.M. Marcantognini and M.D. Mor´ an

Therefore there exists a unitary operator δ : H 2 (DX ) → H 2 (DXB ) such that 1 1 Γg Γg δDX σ σ = D , g 1 ∈ G1 , g 2 ∈ G2 , XB B g2 g2 and δS = SB δ

(5.2)

2

where SB is the shift operator on H (DXB ). Thus ⎡ ⎤ ⎡ ⎤ H F I 0 ⎣ ⎦ ⊕ ⎦→⎣ : J= ⊕ 0 δ H 2 (DX ) H 2 (DXB ) is an isometry. The 2 × 2 block matrix

XB VB = DXB

0 SB

gives rise to an isometry on HB ⊕ H 2 (DXB ) with domain DB := σB (ΓG1 × G 2 ) ⊕ H 2 (DXB ) and range RB :=

XB f

DXB f + Sϕ : f

ϕ ∈ DB .

From (5.1) and (5.2) it follows that VB J|D = JV . Denote by NB and MB the defect subspaces of VB . Let U ∈ L(F ) be the minimal unitary extension of VB associated with the S(NB , MB )-function constantly equal to 0 in the corresponding version of Theorem 2.1. Deﬁne D U n J(H ⊕ H 2 (DX )) and U = U |F . F = n∈Z

Set ϑ(z) := J ∗ PJM U (1 − zPF J(H⊕H 2 (DX )) U )−1 J|N

(z ∈ D).

Then ϑ ∈ S(N , M). So, if Uϑ ∈ L(Fϑ ) is the associated minimal unitary extension of V in Theorem 2.1, then, by using that JV = VB J|D , we get that F 1 G g 0 B(g 1 , Γn g 2 ) = σ , Uϑ−n σ 2 g 0 F ϑ

for all g ∈ G , g ∈ G and n ≥ 0. Finally, note that B is completely determined by the sequence {B|G1 ×Γn G2 }n≥0 . From Theorem 2.1 it follows that, for all z ∈ D and all g1 g 2 ∈ G1 × G2 , G F 1 " " 0 g −(n+1) n 1 n+1 2 n g ) = σ 2 , Uϑ σ n≥0 z B(g , Γ n≥0 z g 0 Fϑ F 1 G 0 g ,σ 2 = Tϑ (z)(1 − zTϑ (z))−1 σ , g 0 H 1

1

2

2

An Extension Theorem for Bounded Forms . . .

337

where Tϑ (z) := V PD + ϑ(z)PN

(z ∈ D).

This gives the connection between ϑ and B. In the following, if (G, G1 , G2 , Γ) is a relaxed discrete algebraic scattering system, {B1 , B2 , B0 } is a set of forms constrained to satisfy (i), (ii) and (iii) in Theorem 5.1 and Φ : ΓG1 → G1 interpolates the system and the forms, we will ﬁnd it convenient to say that a sesquilinear form B : G1 × G 2 verifying (1), (2) and (3) in Theorem 5.1 is a solution of the extension problem with data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ}. The arguments in the proof of Theorem 5.1 along with the above discussion provide a proof of the following reﬁnement of Theorem 5.1. Theorem 5.2. Given a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ), set D Γn G2 . G 2 := n≥0

Consider two nonnegative sesquilinear forms B1 : G1 × G1 → C, B2 : G × G → C and a sesquilinear form B0 : G1 × G2 → C satisfying (i), (ii) and (iii) in Theorem 5.1. If Φ : ΓG1 → G1 interpolates the system and the forms, then there exists a Hilbert space isometry V , with domain D, range R and defect subspaces N and M, such that each ϑ ∈ S(N , M) determines a solution B : G1 × G 2 of the extension problem with data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ} via the relation F 1 G g 0 n 1 n+1 2 −1 z B(g , Γ g ) = Tϑ (z)(1 − zTϑ(z)) σ ,σ 2 g 0 H n≥0

where Tϑ (z) := V PD + ϑ(z)PN

(z ∈ D).

Moreover, all solutions are given in this way. Unfortunately it may happen that diﬀerent ϑ’s give rise to the same extension B. Hence, unless certain conditions are met, Theorem 5.2 does not yield a proper parameterization of the extension forms B in terms of the S(N , M)-functions ϑ. We will leave the discussion on this matter out of account. However, to illustrate the point, we present an example. Example 5.3. Consider G, G1 and G2 as in Example 3.2. Set Γ

s

n

an ζ :=

n=−r

Clearly, ΓG1 ⊆ G1 , G2 = ΓG2 and G = G1 discrete algebraic scattering system.

0

an ζ n .

n=−r

B

G2 . Thus (G, G1 , G2 , Γ) is a relaxed

338

S.A.M. Marcantognini and M.D. Mor´ an

Let B1 , B2 be the pair of nonnegative sesquilinear forms deﬁned in Example 4.2, to wit + s , q q s 1 1 n m B1 : G × G → C, B1 = an ζ , bm ζ an bm δn (m) n=0

and

+

B2 : G × G → C,

s

B2

m=0

n

an ζ ,

n=−r

q

n=0 m=0

, bm ζ

m

=

m=−p

−1 −1

an bm δn (m).

n=−r m=−p

Deﬁne B0 : G1 × G2 → C as B0 ≡ 0, so that {B1 , B2 , B0 } satisfy the required conditions (i), (ii) and (iii). If Φ : ΓG1 → G1 is set to be 0 everywhere on ΓG1 , then Φ interpolates the system and the forms. Since G 2 = G2 , it is clear that B ≡ B0 ≡ 0 is the only solution of the extension problem with the given data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ}. 2 2 the usual Hardy space, and by H− its orthogonal comNow, denote by H+ 2 2 plement in L . Deﬁne the L -function e0 by e0 (ζ) ≡ 1, for ζ ∈ T. Write K0 for the linear subspace of L2 comprising all constant functions: K0 := {λe0 : λ ∈ C}. It can be seen that the coupling isometry V underlying the data set acts on 2 2 H+ ⊕ H− ⊕ H 2 (K0 ). Indeed, straightforward computations yield 2 D = K0 ⊕ H− ⊕ H 2 (K0 ), 2 ⊕ H 2 (K0 ), R = H−

and V : D → R given by ⎡ ⎤ ⎡ ⎤ λe0 0 V ⎣ u− ⎦ = ⎣ u − ⎦ ϕ λe0 + Sϕ

2 , ϕ ∈ H 2 (K0 )). (λ ∈ C, u− ∈ H−

It follows that the defect subspaces of V are 2 N = SH+

2 and M = H+ .

2 2 Clearly, every ϑ ∈ S(SH+ , H+ ) produces the same B, the unique solution of the extension problem.

5.1. The extension theorem for bounded forms deﬁned in relaxed discrete algebraic scattering systems generalizes the Cotlar-Sadosky Extension Theorem Let (V, W 1 , W 2 , τ ) be a Cotlar-Sadosky scattering system. Consider a pair (B1 , B2 ) of nonnegative τ -Toeplitz forms on V × V and a τ -Hankel form B0 on W 1 × W 2 such that 1

1

|B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2 , Set G = W1

D

⎛ ⎝

D

n≥0

g1 ∈ W 1, g2 ∈ W 2.

⎞ τ n W 2 ⎠ , G1 = W 1 and G2 = W 2 .

An Extension Theorem for Bounded Forms . . .

339

Since τ W 1 ⊆ W 1 , we have that τ G ⊆ G. So Γ := τ |G is a linear transformation on G . Moreover ΓG1 = τ W 1 ⊆ W 1 and G2 = W 2 = τ τ −1 W 2 ⊆ τ W 2 = ΓG2 . Whence (G, G1 , G2 , Γ) is a relaxed discrete algebraic scattering system. Put B1 := B1 |G1 ×G1 , B2 := B2 |G×G , B0 := B0 |G1 ×G2 and Φ := τ −1 |ΓG1 . We get that B1 : G1 × G1 → C, B2 : G × G → C are two nonnegative sesquilinear forms and B0 : G1 × G2 → C is a sesquilinear form. Besides (i) B1 (Γg 1 , Γg 1 ) ≤ B1 (g 1 , g 1 ) for all g 1 ∈ G1 , (ii) B2 (Γg, Γg) = B2 (g, g) for all g ∈ G, 1 1 (iii) |B0 (g 1 , g 2 )| ≤ B1 (g 1 , g 1 ) 2 B2 (g 2 , g 2 ) 2 for all g 1 ∈ G1 and g 2 ∈ G2 , while Φ : ΓG1 → G1 interpolates the system and the forms. By Theorem 5.1 there exists a solution B of the extension problem with data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ}. Write D D τ −n W 1 and W2 := τ nW 2 . W1 := n≥0

n≥0

Deﬁne B : W × W → C by , , + k + k j j τ −n w1 (n), τ m w2 (m) = B

Γk−n w1 (n), Γm+k w2 (m) B 1

n=0

2

m=0

n=0

(m)}jm=0 1 2

m=0

⊆ W and {w ⊆W . for {w Next we show that B : W × W → C is a τ -Toeplitz form satisfying 1

(n)}kn=0

1

2

B(w1 , w2 ) = B0 (w1 , w2 ),

2

w 1 ∈ W 1 , w2 ∈ W 2

(5.3)

and 1

1

|B(v, u)| ≤ B1 (v, v) 2 B2 (u, u) 2 ,

v ∈ W 1, u ∈ W 2

(5.4)

(corresponding with (1) and (2) in Theorem 4.3, respectively). • Recall that τ W 1 ⊆ W 1 and take into account that Γ = τ |G . Given that B is a solution of the extension problem with the above detailed data set, recall also that, for all w1 ∈ W 1 , w2 ∈ W 2 and m ≥ 0, B (ΦΓw1 , Γm w2 ) = B (Γw1 , Γm+1 w2 ). Then, for all w1 ∈ W 1 , w2 ∈ W 2 , m ≥ 0 and s ≥ 1, B(τ w1 , τ τ m w2 ) = B(τ w1 , τ m+1 w2 ) = B (Γw1 , Γm+1 w2 ) = B (ΦΓw1 , Γm w2 ) = B (w1 , Γm w2 ) = B(w1 , τ m w2 )

340

S.A.M. Marcantognini and M.D. Mor´ an and B(τ τ −s w1 , τ τ m w2 ) = B(τ −s (τ w1 ), τ m+1 w2 ) = B (Γw1 , Γm+1+s w2 ) = B (ΦΓw1 , Γm+s w2 ) = B (w1 , Γm+s w2 ) = B(τ −s w1 , τ m w2 ).

Therefore B is τ -Toeplitz. • Since B is an extension of B0 , it follows that, for all w1 ∈ W 1 and w2 ∈ W 2 , B(w1 , w2 ) = B (w1 , w2 ) = B0 (w1 , w2 ) = B0 (w1 , w2 ). So (5.3) holds. • As for (5.4), take {w1 (n)}kn=0 ⊆ W 1 and {w2 (m)}jm=0 ⊆ W 2 and note that, since B is subordinated to (B1 , B2 ), j k " " −n 1 m 2 B τ w (n), τ w (m) n=0

m=0

k j " k−n 1 " Γ w (n), Γm+k w2 (m) = B

n=0

≤

B1

k "

m=0

k−n

Γ

1

w (n),

n=0

×B2

≤ B1

k−n

Γ

j "

Γ

m+k

τ

w (m)),

m=0

Γ

m+k

12 w (m) 2

m=0

−n

j "

j "

2

m=0 k "

12 w (n) 1

n=0

1

w (n),

n=0

×B2

k "

k "

τ

−n

12 w (n) 1

n=0

m

2

τ w (m)),

j "

12 τ w (m) . m

2

m=0

5.2. The extension theorem for bounded forms deﬁned in relaxed discrete algebraic scattering systems generalizes the Relaxed Commuting Lifting Theorem We recall that a minimal isometric dilation of a contraction T on a Hilbert space H is an isometryBV on a Hilbert space K ⊇ H such that T n = PH V n |H for all n n ∈ N and K = ∞ n=0 V H. A minimal isometric dilation V ∈ L(K) of a given contraction T ∈ L(H) is obtained by setting K := H ⊕ H 2 (DT ) and Th h (h ∈ H, ϕ ∈ H 2 (DT )) := V DT h + Sϕ ϕ with S the shift operator on H 2 (DT ). If K is another Hilbert space containing H as closed subspace, and if V := τ V τ −1 where τ : K → K is a unitary isomorphism such that τ |H = 1, then V ∈ L(K ) is also a minimal isometric dilation of T . Two

An Extension Theorem for Bounded Forms . . .

341

minimal isometric dilations of T related in this way are abstractly indistinguishable. It is known that any two minimal isometric dilations of a given contraction T ∈ L(H) are indistinguishable. More on contractions and their dilations can be seen in [28]. We consider the following version of the Relaxed Commutant Lifting Theorem [18]. Theorem 5.4. Let E0 , E and H be Hilbert spaces. Let {C, T, R, Q} be a set of four operators: a contraction C ∈ L(E, H), a coisometry T ∈ L(H) with minimal isometric dilation V ∈ L(K) and two bounded linear operators R, Q ∈ L(E0 , E). Assume that Q∗ Q − R∗ R ≥ 0

and

T CR = CQ.

Then there exists a contraction D ∈ L(E, K) such that PH D = C

and

V DR = DQ.

Proof. Since Q∗ Q−R∗ R ≥ 0 we get that ker Q ⊆ ker R. In what follows we further assume that ker Q = {0}. We write Q := QE0 for short. 1 2 We set G := E × K, G := E × {0}, G := {0} × H1 and deﬁne Γ on 1G as Γ e k := PQ e V k (e ∈ E, k ∈ K). On one hand, ΓG = Q × {0} ⊆ G . On the other hand, G2 ⊆ ΓG2 , since the assumption that T is a coisometry grants ∗ h = V T ∗ h. Also, given that K = that every h ∈ H can be written as h = T T B B n 1 n 2 1 2 n≥0 V H, it readily follows that G = G × n≥0 Γ G . Whence (G, G , G , Γ) is a relaxed discrete algebraic scattering system. Deﬁne B1 : G1 × G1 → C, B2 : G × G → C and B0 : G1 × G2 → C by the relations B1 ( e 0 , e 0 ) = e, e E (e, e ∈ E), B2 ( e k , e k ) = k, k K (e, e ∈ E, k, k ∈ K) and

B0 ( e

0 , 0

h ) = Ce, hH

(e ∈ E, h ∈ H).

It readily follows that B1 : G ×G → C, B2 : G×G → C are two nonnegative sesquilinear forms and B0 : G1 × G2 → C is a sesquilinear form satisfying the constraints (i), (ii) and (iii) in Theorem 5.1. Let Φ : ΓG1 → G1 be densely deﬁned by (e0 ∈ E0 ). Φ Qe0 0 := Re0 0 1

1

For all e0 ∈ E0 , B1 (Φ Qe0 0 , Φ Qe0 0 ) = Re0 E ≤ Qe0 E = B1 ( Qe0 0 , Qe0 0 ). Thus, for all g 1 ∈ G1 , B1 (ΦΓg 1 , ΦΓg 1 ) ≤ B1 (Γg 1 , Γg 1 ).

342

S.A.M. Marcantognini and M.D. Mor´ an

Note that u = 0 h ∈ G2 is such that Γu ∈ G2 whenever V h = T h, in which case h = T ∗ T h. Whence, for all e0 ∈ E0 and all h ∈ H such that V h = T h, B0 (ΦΓ Qe0 0 , 0 h ) = B0 (Φ Qe0 0 , 0 h ) = B0 ( Re0 0 , 0 h ) = CRe0 , hH = CRe0 , T ∗ T hH = T CRe0 , T hH = CQe0 , V hH = B0 ( Qe0 0 , Γ 0 h ) = B0 (Γ Qe0 0 , Γ 0 h ). So, for all g 1 ∈ G1 and all u ∈ G2 such that Γu ∈ G2 , B0 (ΦΓg 1 , u) = B0 (Γg 1 , Γu). We conclude that Φ interpolates the system and the forms. Now we apply Theorem 5.1 to get a sesquilinear form B : G1 × G 2 → C (note that G 2 = {0} × K) such that: (1) B(g 1 , g 2 ) = B0 (g 1 , g 2 ) for all g 1 ∈ G1 and g 2 ∈ G2 , (2) B(ΦΓg 1 , u) = B(Γg 1 , Γu) for all g 1 ∈ G1 and u ∈ G 2 , and 1 1 (3) |B(g 1 , u)| ≤ B1 (g 1 , g 1 ) 2 B2 (u, u) 2 for all g 1 ∈ G1 and u ∈ G 2 . We deﬁne D : E → K by De, kK := B( e 0 , 0 k ) (e ∈ E, k ∈ K). The proof is complete if we show that D is a contraction such that PH D = C and V DR = DQ. • According to (3), for all e ∈ E and k ∈ K,

≤

|B( e 0 , 0 k )| 1 1 B1 ( e 0 , e 0 ) 2 B2 ( 0 k , 0 k ) 2

=

eE kK .

|De, kK | =

Hence D is a contraction operator. • As (1) states it, for all e ∈ E and h ∈ H, De, hK = B( e Therefore PH D = C.

0 , 0 h ) = B0 ( e 0 , 0 h ) = Ce, hH .

An Extension Theorem for Bounded Forms . . .

343

• Since to be a coisometry, V is unitary. Therefore, for all k ∈ K, T is assumed Γ 0 V ∗ k = 0 k . From (2) it follows that, for all e0 ∈ E0 and k ∈ K, V DRe0 , kK = DRe0 , V ∗ kK = B( Re0 0 , 0 V ∗ k )) = B(Φ Qe0 0 , 0 V ∗ k ) = B(ΦΓ Qe0 0 , 0 V ∗ k ) = B(Γ Qe0 0 , Γ 0 V ∗ k ) = B( Qe0 0 , 0 k ) = DQe0 , kK . Hence V DR = DQ.

In the above proof, from the data set {C, T, R, Q} in the Relaxed Commutant Lifting Theorem we built up a relaxed discrete algebraic scattering system (G, G1 , G2 , Γ), three forms B1 , B2 , B0 and a function Φ : ΓG1 → G1 that interpolates the system and the forms so that a contraction D ∈ L(E, K) satisfying the constraints PH D = C and V DR = DQ is given in correspondence with a solution B for the extension problem with data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ}. It can be seen that, conversely, if D is an interpolant for the relaxed lifting problem with data set {C, T, R, Q}, meaning that D ∈ L(E, K) is a contraction operator such that PH D = C and V DR = DQ, then B : G1 × G 2 given by (5.5) B(g 1 , u) := De, kK (g 1 = e 0 ∈ G1 , u = 0 k ∈ G 2 ) is a solution for the extension problem with data set {(G, G1 , G2 , Γ); B1 , B2 , B0 ; Φ}. Furthermore, D and B in (5.5) determine each other uniquely. As a ﬁnal remark, we point out that what seems to be a more general version of the Relaxed Commutant Lifting Theorem assumes that T is a contraction, not necessarily a coisometry. However, it can be shown (see, for instance, [22]) that, for a suitable coisometric extension T ∈ L(H ) of T with minimal isometric dilation V ∈ L(K ), there is a bijection between the set of interpolants for {C, T, R, Q} and the set of interpolants for {C, T , R, Q} when C is viewed as a contraction from E into H ⊇ H. Therefore, in the Relaxed Commutant Lifting Theorem, we may always suppose that T is a coisometry.

References [1] V.M. Adamyan and D.Z. Arov, On unitary coupling of semiunitary operators, Amer. Math. Soc. Trans. Ser., 95 (1970), 75–169. [2] D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of dilations of isometric operators, Soviet Math. Dokl., 27 (1983), 518–522. , Scattering matrices in the theory of unitary extension of isometric operators, [3] Math. Nachr., 157 (1992), 105–123. [4] S. Bermudo, S.A.M. Marcantognini and M.D. Mor´ an, Operators of Hankel type, Czechoslovak Math. J., 56(131) (2006), No. 4, 1147–1163.

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[5] M. Cotlar and C. Sadosky, A lifting theorem for subordinated invariant kernels, J. Funct. Anal. 67 (1986), No. 3, 345–359. , Toeplitz liftings of Hankel forms, in Function spaces and applications (Lund, [6] 1986), Lecture Notes in Math., 1302, Springer Berlin, 1988, 22–43. [7] , Integral representations of bounded Hankel forms deﬁned in scattering systems with a multiparametric evolution group, in Contributions to operator theory and its applications (Mesa, AZ, 1987), Oper. Theory Adv. and Appl., 35, Birkh¨ auser Verlag Basel, 1988, 357–375. , Integral representations of bounded Hankel forms deﬁned in scattering sys[8] tems with a multiparametric evolution group, Operator Theory Adv. and Appl., 38 (1988), 357–375. [9] , The generalized Bochner theorem in algebraic scattering systems, in Analysis at Urbana, Vol. II (Urbana, IL 1986–1987), London Math. Soc. Lecture Notes, 138, Cambridge University Press, 1989, 144–169. , Nonlinear lifting theorems, integral representations and stationary processes [10] in algebraic scattering systems, in The Gohberg Anniversary Collection II, Operator Theory: Adv. and Appl., 41, Birkh¨ auser Verlag Basel, 1989, 97–123. , Two-parameter lifting theorems and double Hilbert transforms in commu[11] tative and non-commutative settings, J. Math. Anal. and Appl., 151 (1990), No. 2, 439–480. [12] , Toeplitz liftings of Hankel forms bounded by non-Toeplitz norms, Integral Equations Operator Theory, 14 (1991), No. 4, 501–532. , Weakly positive matrix measures, generalized Toeplitz forms, and their ap[13] plications to Hankel and Hilbert transform operators, in Continuous and discrete Fourier transforms, extension problems and Wiener-Hopf equations, Oper. Theory Adv. Appl., 58, Birkh¨ auser Verlag, Basel, 1992, 93–120. , Transference of metrics induced by unitary couplings, a Sarason theorem [14] for the bidimensional torus and a Sz.-Nagy-Foias theorem for two pairs of dilations, J. Funct. Anal., 111 (1993), 473–488. [15] , Liftings of Kernels Shift-Invariant in Scattering Systems, in Holomorphic Spaces (Berkeley, 1995), MSRI Publications, 33, Cambridge University Press, 1998, 303–336. [16] C. Foias, On the extension of intertwining operators, in: Harmonic Analysis and Operator Theory, A Conference in Honor of Mischa Cotlar, January 3–8, 1994, Caracas, Venezuela, Contemporary Mathematics, 189, American Mathematical Society, Providence, Rhode Island, 1995, 227–234. [17] C. Foias and A.E Frazho, The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Adv. and Appl., 44, Birkh¨ auser Verlag Basel, 1990. [18] C. Foias, A.E. Frazho and M.A. Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations Operator Theory, 42 (2002), 253–310. [19] A.E. Frazho, S. ter Horst and M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations Operator Theory, 54 (2006), 33–67. , All solutions to the relaxed commutant lifting problem, Acta Sci. Math. [20] (Szeged), 72 (2006), No. 1-2, 299–318.

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[21] P.D. Lax and R.S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26 Academic Press, New York-London 1967. [22] W.S. Li and D. Timotin, The relaxed intertwining lifting in the coupling approach, Integral Equations Operator Theory, 54 (2006), 97–111. [23] S.A.M. Marcantognini and M.D. Mor´ an, A Schur analysis of the minimal weak unitary dilations of a contraction operator and the Relaxed Commutant Lifting Theorem, Integral Equations Operator Theory, 64 (2009), 273–299. [24] S.A.M. Marcantognini, M.D. Mor´ an and A. Octavio, The weighted Nehari-DymGohberg problem, Integral Equations Operator Theory, 46 (2003), No. 3, 341–362. [25] M.D. Mor´ an, On intertwining dilations, J. Math. Anal. Appl., 141 (1989), No. 1, 219–234. [26] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. , 127 (1967), 179–203. [27] B. Sz.-Nagy and C. Foia¸s, Dilatation des commutants d’op´erateurs, C. R. Acad. Sci. Paris, S´erie A, 266 (1968), 493–495. [28] , Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London, 1970. S.A.M. Marcantognini Department of Mathematics Instituto Venezolano de Investigaciones Cient´ıﬁcas P.O. Box 21827 Caracas 1020A, Venezuela e-mail: [email protected] M.D. Mor´ an Escuela de Matem´ aticas Facultad de Ciencias Universidad Central de Venezuela Apartado Postal 20513 Caracas 1020A, Venezuela e-mail: [email protected] Received: February 27, 2009 Accepted: March 27, 2009

Operator Theory: Advances and Applications, Vol. 203, 347–362 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Deconstructing Dirac Operators. III: Dirac and Semi-Dirac Pairs Mircea Martin Abstract. The Dirac operator on the Euclidean space Rn , n ≥ 2, is a ﬁrstorder diﬀerential operator Deuc,n with coeﬃcients in the real Cliﬀord algebra Aeuc,n associated with Rn that has the deﬁning property D2euc,n = −Δeuc,n , where Δeuc,n stands for the standard Laplace operator on Rn . As generalizations of this class of operators, we investigate pairs (D, D† ) of ﬁrstorder homogeneous diﬀerential operators on Rn with coeﬃcients in a real Banach algebra A, such that DD† = μL Δeuc,n and D† D = μR Δeuc,n , or DD† + D† D = μΔeuc,n , where μL , μR , or μ are some elements of A. Every pair (D, D† ) that has the former property is called a Dirac pair of diﬀerential operators, and every pair (D, D† ) with the latter property is called a semi-Dirac pair. Our goal is to prove that for any Dirac, or semi-Dirac pair, (D, D† ), there are two interrelated Cauchy-Pompeiu type, and, respectively, two Bochner-Martinelli-Koppelman type integral representation formulas, one for D and another for D† . In addition, we show that the existence of such integral representation formulas characterizes the two classes of pairs of differential operators. Mathematics Subject Classiﬁcation (2000). 32A26, 35F05, 47B34, 47F05. Keywords. First-order partial diﬀerential operators, integral representation formulas, Dirac operators.

1. Introduction The study of Dirac operators in an Euclidean or Hermitian setting is nowadays regarded as part of Cliﬀord analysis, a far reaching extension of single variable complex analysis to several real or complex variables. Excellent accounts on the subject can be found in the monographs by Brackx, Delanghe, and Sommen [BDS], Delanghe, Sommen, and Souˇcek [DSS], Gilbert and Murray [GM], G¨ urlebeck and Spr¨ ossig [GS], Mitrea [Mi], and Rocha-Chavez, Shapiro, and Sommen [RSS4]. The Communicated by J.A. Ball.

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volumes edited by Ryan [R4] and Ryan and Spr¨ ossig [RS] also provide a good illustration of the work done in this area. For speciﬁc contributions related in part to some of the problems addressed in our article we refer to Bernstein [B], RochaChavez, Shapiro, and Sommen [RSS1–3], Ryan [R1–3, 5], Shapiro [Sh], Sommen [So1, 2], and Vasilevski and Shapiro [VS]. This article complements the investigations reported in Martin [M7, 8]. We will pursue the same general goal, namely, to detect the origin of some basic properties of Euclidean Dirac operators by studying ﬁrst-order diﬀerential operators with coeﬃcients in a Banach algebra. Though this class is limited, one expects that their study will lead to a better understanding of the class of Dirac operators, and perhaps connect Cliﬀord analysis with some new issues of harmonic and complex analysis in several variables, or of multi-dimensional operator theory. Some connections of this kind, that motivated us in developing such an approach, are pointed out in Martin [M1–6]. For basic facts concerning diﬀerential operators we refer to H¨ ormander [H¨ o] and Tarkhanov [T]. Throughout our article, we will let Aeuc,n denote the real Cliﬀord algebra associated with the Euclidean space Rn , n ≥ 2. We recall that Aeuc,n is a unital associative real algebra with identity e0 , equipped with a real linear embedding σeuc,n : Rn → Aeuc,n such that σeuc,n (ξ)2 = −|ξ|2 e0 ,

ξ ∈ Rn ,

(1.1)

where | · | is the Euclidian norm on Rn . Moreover, Aeuc,n is uniquely deﬁned by the next universal property: If (A, σ) is any pair consisting of a unital associative real algebra A with identity e, and a real linear mapping σ : Rn → A with the property σ(ξ)2 = −|ξ|2 e,

ξ ∈ Rn ,

then there exists a unique real algebra homomorphism α : Aeuc,n → A such that σ = α ◦ σeuc,n . Actually, Aeuc,n is generated by n elements that are customarily identiﬁed with the standard orthonormal basis {e1 , e2 , . . . , en } for Rn , subject to the relations, ei ej + ej ei = −2δij e0 , 1 ≤ i, j ≤ n, (1.2) where δij equals 1 or 0, according as i = j or i = j. The embedding σeuc,n : Rn → Aeuc,n is deﬁned by σeuc,n (ξ) = ξ1 e1 + · · · + ξn en , ξ = (ξ1 , . . . , ξn ) ∈ Rn , and property (1.1) is equivalent to (1.2). The set consisting of e0 and all products eI = ei1 · · · · · eip ,

I = (i1 , . . . , ip ),

1 ≤ i1 < · · · < ip ≤ n,

yields a basis for Aeuc,n as a real vector space, and Aeuc,n is equipped with an inner product ·, · such that the basis just deﬁned is orthonormal. Further, by regarding Aeuc,n as an algebra of left multiplication operators acting on the Hilbert space (Aeuc,n , ·, ·) we convert Aeuc,n into a real C ∗ -algebra. Let now H be a Hilbert left or right Aeuc,n -module, that is, a real inner product space upon which the algebra Aeuc,n acts on the left or right, such that each generator ei ∈ Aeuc,n , 1 ≤ i ≤ n, determines a skew-adjoint unitary operator.

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The space C ∞ (Rn , H) of smooth H-valued functions on Rn is a left or right Aeuc,n module under pointwise multiplication. Therefore, it makes sense to introduce the ﬁrst-order diﬀerential operator Deuc,n : C ∞ (Rn , H) → C ∞ (Rn , H), called the Euclidean Dirac operator on Rn , by setting Deuc,n = e1 D1 + e2 D2 + · · · + en Dn ,

(1.3)

with Di = ∂/∂xi , 1 ≤ i ≤ n. It is easy to check, based on (1.2), that Deuc,n is an elliptic self-adjoint operator. Moreover, the embedding σeuc,n : Rn → Aeuc,n equals the symbol mapping of Deuc,n , and equation (1.1) amounts to D2euc,n = −Δeuc,n ,

(1.4)

where Δeuc,n = D12 + D22 + · · · + Dn2 is the Laplace operator on Rn . As yet another important property, we recall that Deuc,n has a fundamental solution EC,n : Rn0 → Aeuc,n , called the Euclidean Cauchy kernel on Rn , and deﬁned by EC,n (ξ) =

1 n−1

|S

|

·

−σeuc,n (ξ) , |ξ|n

ξ ∈ Rn0 = Rn \ {0},

where |Sn−1 | is the surface area of the unit sphere Sn−1 in Rn . This kernel makes it possible to set up a Cauchy-Pompeiu formula for Deuc,n , that generalizes the classical formula for the Cauchy-Riemann operator in single variable complex analysis. As a matter of fact, if one takes the fundamental solution EL,n : Rn0 → R of the Laplace operator Δeuc,n for n ≥ 2, then from (1.4) one gets that the Euclidean Cauchy kernel is given by EC,n = −Deuc,n EL,n . The signiﬁcance of this property was fully explained in a quite interesting article due to Hile [Hi]. One of the main results of that article generalizes the Cauchy-Pompeiu formula alluded to above for ﬁrst-order constant matrix coeﬃcient diﬀerential operators D on Rn that satisfy an equation of the form (1.5), D† D = Δeuc,n , where D† is another constant matrix coeﬃcient diﬀerential operator. The kernel EH involved in the integral representation formula proved by Hile comes from equation (1.5) and is given by EH = D† EL,n . We want to thank the reviewer of a previous version of our article for calling Hile’s work to our attention. We would also like to point out that in contrast to Hile’s paper, our approach does not rely on the use of the fundamental solution EL,n , it enables us to generalize both the Cauchy-Pompeiu and the Bochner-Martinelli-Koppelman formulas, and, as yet another highlight, it works in two directions, in the sense that the existence of such integral representation formulas completely characterizes the diﬀerential operators under investigation. To be speciﬁc, in our article we will consider triples (A, σ, σ† ) consisting of a real unital Banach algebra A with identity e, that may or may not have an

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involution, and two real linear embeddings σ, σ † : Rn → A, such that either σ(ξ) · σ † (ξ) = μL |ξ|2 e,

σ† (ξ) · σ(ξ) = μR |ξ|2 e,

ξ ∈ Rn ,

(1.6)

or

σ(ξ) · σ † (ξ) + σ† (ξ) · σ(ξ) = μ|ξ|2 e, ξ ∈ Rn , (1.7) where μL , μR , or μ are some elements of A, and | · | is the Euclidian norm on Rn . By regarding σ and σ† as symbol mappings, we introduce a pair (D, D† ) of ﬁrst-order homogeneous diﬀerential operators on Rn with coeﬃcients in A, which according to (1.6) or (1.7) has the property DD† = μL Δeuc,n ,

or

D† D = μR Δeuc,n ,

DD† + D† D = μΔeuc,n ,

(1.8) (1.9)

†

respectively. Every pair (D, D ) as in (1.8) is called a Dirac pair of diﬀerential operators, and every pair (D, D† ) with property (1.9) is called a semi-Dirac pair. Two important examples of a Dirac, or semi-Dirac pair of operators on Cn ≡ R2n are given by D = D† = ∂¯ + ∂¯∗ , or D = ∂¯ and D† = ∂¯∗ , where ∂¯ is the (0, 1)-component of the operator of exterior diﬀerentiation acting on diﬀerential forms on Cn , and ∂¯∗ is its formal adjoint. These operators have well-known integral representation formulas. Our main goal is to prove that for any Dirac, or semi-Dirac pair, (D, D† ), there are two interrelated Cauchy-Pompeiu type integral representation formulas, and, respectively, two interrelated Bochner-MartinelliKoppelman type formulas, one for D and another for D† . In addition, we will show that the existence of such integral representation formulas characterizes the two classes of pairs of diﬀerential operators. The remainder of the article is organized as follows. In Section 2 we will brieﬂy discuss several prerequisites and state the two main results, Theorem A and Theorem B. Section 3 is concerned with some auxiliary results that eventually are used to prove Theorems A and B. In Section 4 we present some consequences and reﬁnements of the main results in both an Euclidean and a Hermitian setting.

2. Pairs of ﬁrst-order diﬀerential operators This section introduces the main objects studied in our article, pairs of homogeneous ﬁrst-order diﬀerential operators with coeﬃcients in a Banach algebra, notation, and some related integral operators. 2.1. Prerequisites To begin with, we assume that A is a real unital Banach algebra with identity e, equipped with two real linear embeddings σ, σ† : Rn → A, n ≥ 2. The coeﬃcients of σ and σ † form two n-tuples A = (a1 , a2 , . . . , an ) and A† = (a†1 , a†2 , . . . , a†n ) of elements of A, namely, σ(ξ) = sA (ξ) = ξ1 a1 + ξ2 a2 + · · · + ξn an ,

ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn ,

(2.1)

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351

and σ † (ξ) = sA† (ξ) = ξ1 a†1 + ξ2 a†2 + · · · + ξn a†n ,

ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn . (2.2)

Suppose next that M is a real Banach left or right A-module. In other words, we assume that M is a real Banach space and A is realized as a subalgebra of L(M), the algebra of all bounded linear operators on M. We let C ∞ (Rn , M) be the space of all smooth M-valued functions on Rn , that becomes an A-module by extending the action of A to M-valued functions pointwise. In particular, given the n-tuples A = (a1 , a2 , . . . , an ) and A† = (a†1 , a†2 , . . . , a†n ) of elements of A, we introduce the diﬀerential operators D = DA and D† = DA† on C ∞ (Rn , M) by setting D = a1 D1 + a2 D2 + · · · + an Dn ,

(2.3)

and

D† = a†1 D1 + a†2 D2 + · · · + a†n Dn , (2.4) where, depending on context, Di = ∂/∂xi , or Di = ∂/∂ξi , with 1 ≤ i ≤ n. Clearly, the embeddings σ and σ † are the symbol mappings of D and D† , respectively.

2.2. Spherical means Associated with σ and σ † , or D and D† , we deﬁne two elements μR , μL ∈ A as 1 σ † (ξ) · σ(ξ) darea(ξ), (2.5) μR = n−1 |S | Sn−1 and 1 μL = n−1 σ(ξ) · σ † (ξ) darea(ξ), (2.6) |S | Sn−1 where |Sn−1 | stands for the total surface area of the unit sphere in Rn , and darea is the surface area measure on Sn−1 . We will refer to μR and μL as the right and left spherical means of (σ, σ† ). We also introduce μ ∈ A, given by μ = μ R + μL . A simple calculation shows that 1 μR = (a†1 a1 + a†2 a2 + · + a†n an ), n and

(2.7)

(2.8)

1 (2.9) (a1 a†1 + a2 a†2 + · + an a†n ). n For convenience, we are going to assume that M is a left A-module, and denote the action of D on a function u ∈ C ∞ (Rn , M) by ∂u ∂u ∂u Du = a1 + a2 + · · · + an . ∂x1 ∂x2 ∂xn However, if M is a right A-module, then the action of D on u ∈ C ∞ (Rn , M) will be denoted by ∂u ∂u ∂u uD = a1 + a2 + · · · + an . ∂x1 ∂x2 ∂xn μL =

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Similar conventions apply to D† . We should notice that even when M is an A-bimodule, as for instance in the special case when M = A, we do not expect Du and uD to be equal. For example, from (2.1), (2.2), (2.3), and (2.4), we obviously get D† σ(ξ) = σ † D(ξ) = nμR , ξ ∈ Rn , (2.10) and Dσ† (ξ) = σD† (ξ) = nμL , ξ ∈ Rn . (2.11) † Though we are interested in properties of the pair (D, D ), we will usually regard D as the primary component of that pair, and employ D† as an auxiliary object that merely helps in studying D. The next deﬁnitions take this distinction into account. We continue by introducing the kernel Φ : Rn0 → A given by Φ(ξ) =

σ † (ξ) , |ξ|n

ξ ∈ Rn0 = Rn \ {0}.

(2.12)

Obviously, Φ is a smooth function homogeneous of degree 1 − n, that is, Φ(tξ) = t1−n Φ(ξ),

t ∈ (0, ∞), ξ ∈ Rn .

In addition, by a direct calculation and using (2.10) and (2.11), from (2.12) we get n ξ ∈ Rn0 , (2.13) ΦD(ξ) = n+2 [ |ξ|2 μR − σ† (ξ) · σ(ξ) ], |ξ| as well as n ξ ∈ Rn0 . (2.14) DΦ(ξ) = n+2 [ |ξ|2 μL − σ(ξ) · σ † (ξ) ], |ξ| 2.3. Related integral operators Further, let us suppose that X ⊂ Rn is a bounded open set with a smooth and oriented boundary ∂X. To D, Φ, X, and ∂X, we now associate four integral operators, IX , RR,X , RL,X , I∂X : C ∞ (Rn , M) → C ∞ (Rn \ ∂X, M), deﬁned by 1

Φ(ξ − x) · u(ξ) dvol(ξ), |Sn−1 | X 1 RR,X u(x) = n−1 p.v. ΦD(ξ − x) · u(ξ) dvol(ξ), |S | X 1 RL,X u(x) = n−1 p.v. DΦ(ξ − x) · u(ξ) dvol(ξ), |S | X IX u(x) =

and I∂X u(x) =

1 |Sn−1 |

(2.15) (2.16) (2.17)

Φ(ξ − x) · σ(ν(ξ)) · u(ξ) darea(ξ), ∂X n

(2.18)

for any u ∈ C ∞ (Rn , M) and x ∈ R \ ∂X, where dvol is the Lebesgue measure on X, p.v. stands for the principal value, darea is the surface area measure on ∂X,

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and, for each point ξ ∈ ∂X, ν(ξ) = (ν1 (ξ), ν2 (ξ), . . . , νn (ξ)) ∈ Rn denotes the unit outer normal vector to ∂X at ξ. The fact that the integral operators IX , RR,X , RL,X transform smooth functions into smooth functions is a consequence of the general Calder´on-Zygmund theory, as presented for instance in the two treatises by Stein [S1, 2]. Finally, we deﬁne a truncation operator associated with X, TX : C ∞ (Rn , M) → C ∞ (Rn \ ∂X, M), by setting TX u(x) = u(x) if x ∈ X, and TX u(x) = 0 if x ∈ Rn \ (X ∪ ∂X). 2.4. Integral representation formulas We are now in a position to state the two main results of our article. Complete proofs of both theorems will be given in Section 3. The ﬁrst result deals with a generalized Cauchy-Pompeiu type representation formula for arbitrary pairs (D, D† ) of ﬁrst-order homogeneous diﬀerential operators on Rn with coeﬃcients in a Banach algebra. Theorem A. Suppose that (D, D† ) is a pair of ﬁrst-order homogeneous diﬀerential operators on Rn , n ≥ 2, with coeﬃcients in a Banach algebra, and let μR be their associated right spherical mean. The following two statements are equivalent: (i) If Δ = Δeuc,n is the standard Laplace operator on Rn, then D† D = μR Δ.

(2.19)

(ii) If X ⊂ Rn is a bounded open set with a smooth oriented boundary ∂X, then μR TX u(x) = I∂X u(x) − IX Du(x), ∞

(2.20)

for any u ∈ C (R , M) and x ∈ R \ ∂X. n

n

The second result provides a generalized form of the Bochner-MartinelliKoppelman formula in a several real variables setting. Theorem B. Suppose that (D, D† ) is a pair of ﬁrst-order homogeneous diﬀerential operators on Rn , n ≥ 2, with coeﬃcients in a Banach algebra, and let μL and μR be their associated left and right spherical means. The following two statements are equivalent: (i) If Δ = Δeuc,n is the standard Laplace operator on Rn, then DD† + D† D = (μL + μR )Δ.

(2.21)

(ii) If X ⊂ Rn is a bounded open set with a smooth oriented boundary ∂X, then (μL + μR )TX u(x) = I∂X u(x) − IX Du(x) − DIX u(x), ∞

(2.22)

for any u ∈ C (R , M) and x ∈ R \ ∂X. n

n

Before concluding this section, we want to make a short comment regarding the equations in parts (i) and (ii) of Theorem A. We claim that the assumption that μR needs to be the right spherical mean of (D, D† ) is redundant. Actually, what really matters is the existence of an element μR that makes the two equations

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true, because from these equations we can prove that such an element must be the right spherical mean. To make a point, we notice that equation (2.19) in part (i) implies ξ ∈ Rn , σ† (ξ) · σ(ξ) = μR |ξ|2 , whence, by integrating over the unit sphere Sn−1 , and then comparing with (2.5), we get that μR needs to be the right spherical mean. With regard to equation (2.20) in part (ii), if we assume that X ⊂ Rn is the open unit ball, let u ∈ C ∞ (Rn , A) be the constant function u(ξ) = e, ξ ∈ Rn , and select x = 0, then once more a comparison with (2.5) shows that μR must be the right spherical mean. A similar observation can be made for the equations in Theorem B.

3. Auxiliary results and proofs This section provides some technical results and proofs of Theorems A and B. The setting and the notation are the same as in Section 2. 3.1. An integral formula Suppose A = (a1 , a2 , · · · , an ) is the n-tuple that deﬁnes D = DA as in equation (2.3). For each 1 ≤ i ≤ n, we denote by dξic the (n − 1)-form on Rn deﬁned by dξic = dξ1 ∧ · · · ∧ dξi−1 ∧ dξi+1 ∧ · · · ∧ dξn , where ξi , 1 ≤ i ≤ n, are the standard coordinate functions on Rn , and let ω = ωA be the A-valued form on Rn given by ω=

n (−1)i−1 ai dξic .

(3.1)

i=1

Assume now that Ω ⊂ Rn is a compact smooth submanifold of Rn of dimension n, with smooth oriented boundary Σ. Given two smooth functions ϕ ∈ C ∞ (Ω, A) and u ∈ C ∞ (Ω, M), we introduce the M-valued (n − 1)-form ϕ · ω · u on Ω, and observe that its exterior derivative equals d(ϕ · ω · u) = (ϕ · Du + ϕD · u)dξ,

(3.2)

where dξ = dξ1 ∧ · · · ∧ dξi ∧ · · · ∧ dξn is the volume form on Ω. We next apply Stokes’ Theorem, by using the compact manifold Ω with boundary Σ, and get ϕ · ω · u = (ϕ · Du + ϕD · u)dξ. (3.3) Σ

Ω

Both sides of (3.2), which are integrals of M-valued diﬀerential forms, can be expressed as integrals of M-valued functions, by taking the surface area measure darea on Σ, and the volume measure dvol on Ω. For each ξ ∈ Σ, we let ν(ξ) = (ν1 (ξ), ν2 (ξ), · · · , νn (ξ)) ∈ Rn be the unit outer normal vector to Σ at ξ. Then, (−1)i−1 dξic |ξ = ν(ξ) · darea(ξ),

1 ≤ i ≤ n,

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whence, by (3.1) and (2.1), we get that equation (3.3) amounts to ϕ(ξ) · σ(ν(ξ)) · u(ξ)darea(ξ) = [ ϕ(ξ) · Du(ξ) + ϕD(ξ) · u(ξ) ]dvol(ξ), (3.4) Σ

Ω

an equation that could be regarded as an integral deﬁnition of D. 3.2. Two lemmas The next technical results point out relationships between the integral operators, the truncation operator, and the spherical means introduced in Section 2. They will prove quite useful in completing the proofs of Theorems A and B. The notation and the assumptions are the same as in Section 2. Lemma 1. Suppose that (D, D† ) is a pair of ﬁrst-order homogeneous diﬀerential operators on Rn , n ≥ 2, with coeﬃcients in a Banach algebra, and let μR be their associated right spherical mean. If X ⊂ Rn is a bounded open set with a smooth oriented boundary ∂X, then μR TX = I∂X − IX D − RR,X ,

(3.5)

as operators from C ∞ (Rn , M) to C ∞ (Rn \ ∂X, M). Lemma 2. Suppose that (D, D† ) is a pair of ﬁrst-order homogeneous diﬀerential operators on Rn , n ≥ 2, with coeﬃcients in a Banach algebra, and let μL be their associated left spherical mean. If X ⊂ Rn is a bounded open set with boundary ∂X, then μL TX = −DIX −RL,X , (3.6) as operators from C ∞ (Rn , M) to C ∞ (Rn \ ∂X, M). Proof of Lemma 1. Let u ∈ C ∞ (Rn , M) be a given function. We need to show that (3.7) μR TX u(x) = I∂X u(x) − IX Du(x) − RR,X u(x), for each x ∈ Rn \ ∂X. We assume ﬁrst that x ∈ Rn \(X∪∂X), set Ω = X∪∂X, and let ϕ ∈ C ∞ (Ω, A) be the function given by ϕ(ξ) = Φ(ξ − x),

ξ ∈ Ω.

(3.8)

Since the boundary Σ of Ω equals ∂X, equation (3.4) reduces to Φ(ξ−x)·σ(ν(ξ))·u(ξ)darea(ξ) = [ Φ(ξ−x)·Du(ξ)+ΦD(ξ−x)·u(ξ) ]dvol(ξ). ∂X

X

Using equations (2.15), (2.16), (2.18), and the deﬁnition of the truncation operator, we notice that the last equation leads to (3.7). Let us next suppose that x ∈ X. We choose ε > 0 such that Bn (x, ε) ⊂ X, where Bn (x, ε) ⊂ Rn is the closed ball of center x and radius ε, and deﬁne the compact manifold Ω as the closure of the open set X \ Bn (x, ε). Its boundary Σ consists of ∂X with the standard orientation, and the sphere Sn−1 (x, ε) of center

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x and radius ε, with the opposite orientation. We deﬁne ϕ(ξ) for ξ ∈ Ω as in (3.8) and then, by applying (3.4) we get Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) − Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) ∂X Sn−1 (x,ε) Φ(ξ − x) · Du(ξ)dvol(ξ) + ΦD(ξ − x) · u(ξ)dvol(ξ). (3.9) = X\Bn (x,ε)

X\Bn (x,ε)

Further, we observe that the second integral in the left-hand side of (3.9) can be changed using the transformation ξ = x + ε ν,

ν ∈ Sn−1 .

Based on some simple calculations we have Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) = Sn−1 (x,ε)

Sn−1

Therefore,

σ † (ν) · σ(ν) · u(x + ε ξ)darea(ν).

lim ε↓0

Sn−1 (x,ε)

Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) = |Sn−1 | μR u(x).

Equation (3.7) now follows, since obviously lim Φ(ξ − x) · Du(ξ)dvol(ξ) = |Sn−1 | IX Du(x), ε↓0

X\Bn (x,ε)

and lim ε↓0

X\Bn (x,ε)

ΦD(ξ − x) · u(ξ)dvol(ξ) = |Sn−1 | RR,X u(x).

The proof of Lemma 1 is complete. The proof of Lemma 2 is left to our reader. It amounts to showing that μL TX u(x) = −DIX u(x) − RL,X u(x), ∞

(3.10)

for any u ∈ C (R , M) and x ∈ R \ ∂X, an equation that can be easily deduced, for instance, from a proof outlined in Tarkhanov [T, Section 2.1.7], or based on a reasoning similar to the proof of Lemma 1 above. n

n

3.3. Proofs of Theorems A and B Proof of Theorem A. We start the proof by observing that equation (2.19) is equivalent to ξ ∈ Rn , (3.11) σ † (ξ) · σ(ξ) = μR |ξ|2 , which, due to (2.13), is equivalent to ΦD(ξ) = 0,

ξ ∈ Rn0 .

(3.12)

On the other hand, from (3.5) in Lemma 1 it follows that equation (2.20) is equivalent to RR,X ≡ 0,

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for any open and bounded set X ⊂ Rn with a smooth oriented boundary, a property that according to (2.16), the deﬁnition of RR,X , is also equivalent to (3.12). The proof of Theorem A is complete. Proof of Theorem B. We ﬁrst observe that equation (2.21) is equivalent to σ(ξ) · σ † (ξ) + σ† (ξ) · σ(ξ) = (μL + μR )|ξ|2 ,

ξ ∈ Rn ,

(3.13)

which, due to (2.13) and (2.14), is equivalent to DΦ(ξ) + ΦD(ξ) = 0,

ξ ∈ Rn0 .

(3.14)

Next, by combining (3.5) and (3.6) from Lemma 1 and Lemma 2 we get (μL + μR )TX = I∂X − IX D − (RL,X + RR,X ),

(3.15)

whence we conclude that equation (2.21) in Theorem B is equivalent to RL,X + RR,X ≡ 0, for any open and bounded set X ⊂ Rn with a smooth oriented boundary, a property that according to (2.16) and (2.17)), the deﬁnitions of RR,X and RL,X , is also equivalent to (3.15). The proof of Theorem B is complete.

4. Concluding remarks We end our investigations with some direct consequences and reﬁnements of the theorems stated and proved above. Theorems A and B take simpler and more familiar forms under additional assumptions. 4.1. Reﬁning Theorem A For instance, using the same notation as in the previous sections, if D = DA is elliptic and Φ is its fundamental solution, satisfying ΦD(ξ) = DΦ(ξ) = 0,

ξ ∈ Rn0 ,

then Theorem A is true and μR = e, so we get a genuine Cauchy-Pompeiu representation formula. Moreover, in this case the ﬁrst term I∂X u in formula (2.20) has the property DI∂X u(x) = 0, x ∈ Rn \ ∂X. Returning to the general setting, let us take both operators D = DA and D† = DA† , with A = (a1 , . . . , an ) and A† = (a†1 , . . . , a†n ). Direct calculations show that condition (2.19) in Theorem A is equivalent to CP(A, A† ) : a†i · aj + a†j · ai = 2δij μR ,

1 ≤ i, j ≤ n.

To correct the lack of symmetry in the last equation, we make another assumption, CP(A† , A) : ai · a†j + aj · a†i = 2δij μL , By retracing our previous reasoning we now get DD† = μL Δ,

1 ≤ i, j ≤ n.

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and Theorem A shows that D† also has a Cauchy-Pompeiu representation formula, where the kernel Φ† is associated with σ = sA by an equation similar to (2.12) The simplest example of a pair of operators (D, D† ) with all properties indicated above is provided by the Cauchy-Riemann operator and its formal adjoint, when n = 2 and A = Aeuc,1 = C. As indicated in Section 1, in higher dimension we can take the Euclidean Dirac operators D = D† = Deuc,n . As speciﬁc realizations of Dirac operators we should mention that Deuc,n ∼ = ∗ d + d , where d is the operator of exterior diﬀerentiation acting on smooth diﬀerential forms on Rn and d∗ is its formal adjoint, or, √ if we use the ∂¯ operator on Cn ∗ ∼ ¯ and its formal adjoint ∂ , then we have Deuc,2n = 2(∂¯ + ∂¯∗ ). These examples and the previous observations motivate the following Deﬁnition. A pair (D, D† ) of ﬁrst-order diﬀerential operators with coeﬃcients in a Banach algebra A is called a Dirac pair, provided there exist μR , μL ∈ A such that D† D = μR Δeuc,n ,

DD† = μL Δeuc,n .

Based on what we already mentioned, the characteristic property of a Dirac pair (D, D† ) of diﬀerential operators with symbols σ and σ† is the existence of two Cauchy-Pompeiu formulas, one for D and another for D† , whose kernels are associated with σ † and σ, respectively. 4.2. Reﬁning Theorem B Theorem B can be analyzed in a similar way. In contrast to the case of Dirac pairs, the symmetry of equation (2.21) in Theorem B shows that, without any other assumptions, operator D† also has a Bochner-Martinelli-Koppelman formula, where the kernel Φ† is associated with the symbol mapping σ of D. This nice feature is summed up in the next Deﬁnition. A pair (D, D† ) of ﬁrst-order diﬀerential operators with coeﬃcients from a Banach algebra A is called a semi-Dirac pair, provided there exists μ ∈ A such that D† D + DD† = μΔ. Referring to our previous examples of self-adjoint Dirac pairs, as typical examples ¯ ∂¯∗ ). of semi-Dirac pairs we may take (d, d∗ ), or (∂, We also want to emphasize that the characteristic property of a semi-Dirac pair (D, D† ) of diﬀerential operators with symbols σ and σ† is the existence of two Bochner-Martinelli-Koppelman formulas, one for D and another for D† , whose kernels are associated with σ† and σ, respectively. To single out the most striking consequences of Theorem B, and to show its natural relationship with the classical Bochner-Martinelli-Koppelman formula, we should switch from real to complex variables.

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We will assume that A is a complex algebra, and let A = (a1 , . . . , an ) and A = (a†1 , . . . , a†n ) be two n-tuples of elements of A, n ≥ 1. We next take the real linear mappings σ, σ † : Cn → A associated with A and A† and given by †

σ(ζ) = ζ1 a1 + ζ2 a2 + · · · + ζn an , and σ† (ζ) = ζ¯1 a†1 + ζ¯2 a†2 + · · · + ζ¯n a†n , for each ζ = (ζ1 , ζ2 , . . . , ζn ) ∈ Cn . There are of course two diﬀerential operators D and D† acting on C ∞ (Cn , M) with symbols σ and σ † , respectively, deﬁned as D=2

n

ai ∂/∂ ζ¯i ,

D† = 2

i=1

n

a†i ∂/∂ζi .

i=1

The spherical means of (σ, σ† ) are given by 1 † a · ai , n i=1 i n

μR =

1 ai · a†i . n i=1 n

μL =

We conclude that (D, D† ) is a semi-Dirac pair, if and only if BMK(A, A† ) : a†i · aj + aj · a†i = (μR + μL )δij ,

1 ≤ i, j ≤ n,

a symmetric set of commutation relations. ¯ ∂¯∗ ), Theorem B makes it possible to recover In the case when (D, D† ) = (∂, the classical Bochner-Martinelli-Koppelman formula. Relevant results regarding this formula are presented in Aizenberg and Dautov [AD], Henkin and Leiterer [HL], Krantz [K], and Range [Ra]. In our more general setting, though it is quite possible to have μR + μL = e, we want to mention that μR , μL , or μR + μL are not expected to be invertible in A. As a ﬁnal remark, we would like to point out that other reﬁnements of Theorems A and B come from assumptions regarding the spectrum of μR , or the spectrum of μL + μR , respectively, and by using appropriate closed subspaces of M. To make a point, let us assume that the operator D, the kernel Φ : Rn0 → A given by (2.12), and the A-module M are such that either (i) D has a Cauchy-Pompeiu representation formula with kernel Φ as in Theorem A, and μR as an operator on M has an invariant closed subspace X such that μR determines an invertible operator on X; or, (ii) D has a Bochner-Martinelli-Koppelman representation formula with kernel Φ as in Theorem B, such that the A-module M has a closed subspace X that consists of eigenvectors of μL + μR associated with a non-zero eigenvalue. Referring now to Theorems A or B, we should notice that under such assumptions we can modify the kernel Φ and get, using the modiﬁed kernel, genuine representation formulas for functions u ∈ C ∞ (Rn , X), in which on the left-hand side we only have TX u without any coeﬃcient from A.

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References [AD]

Aizenberg, L.A. and Dautov, Sh.A., Diﬀerential Forms Orthogonal to Holomorphic Functions or Forms, and Their Properties, Transl. Math. Monographs, 56, Amer. Math. Soc., Providence, RI, 1983. [B] Bernstein, S., A Borel-Pompeiu formula in Cn and its applications to inverse scattering theory, in Progress in Mathematical Physics Series, Volume 19: Clifford Algebras and Their Applications in Mathematical Physics, Birkh¨ auser Verlag, 2000, pp. 117–185. [BDS] Brackx, F., Delanghe, R., and Sommen, F., Cliﬀord Analysis, Pitman Research Notes in Mathematics Series, 76, 1982. [DSS] Delanghe, R., Sommen, F., and Souˇcek, V., Cliﬀord Algebra and Spinor-Valued Functions, Kluwer Academic Publishers, 1992. [GM] Gilbert, J.E. and Murray, M.A.M., Cliﬀord Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 26, Cambridge University Press, 1991. [GS] G¨ urlebeck, K. and Spr¨ ossig, W., Quaternionic and Cliﬀord Calculus for Physicists and Engineers, John Wiley & Sons, New York, 1997. [HL] Henkin, G.M. and Leiterer, J., Theory of Functions on Complex Manifolds, Birkh¨ auser Verlag, 1984. [Hi] Hile, G. N., Representations of solutions of a special class of ﬁrst order systems, Journal of Diﬀerential Equations, 25 (1977), 410–424. [H¨ o] H¨ ormander, L., The Analysis of Linear Partial Diﬀerential Operators, Vol. II: Diﬀerential Operators with Constant Coeﬃcients, Springer-Verlag, Berlin, 1983. [K] Krantz, S. G., Function Theory of Several Complex Variables, John Wiley & Sons, 1982. [M1] Martin, M., Higher-dimensional Ahlfors-Beurling inequalities in Cliﬀord analysis, Proc. Amer. Math. Soc., 126 (1998), 2863–2871. [M2] Martin, M., Convolution and maximal operator inequalities, in Progress in Mathematical Physics Series, Volume 19: Cliﬀord Algebras and Their Applications in Mathematical Physics, Birkh¨ auser Verlag, 2000, pp. 83–100. [M3] Martin, M., Self-commutator inequalities in higher dimension, Proc. Amer. Math. Soc., 130 (2002), 2971–2983. [M4] Martin, M., Spin geometry, Cliﬀord analysis, and joint seminormality, in Trends in Mathematics Series, Volume 1: Advances in Analysis and Geometry, Birkh¨ auser Verlag, 2004, pp. 227–255. [M5] Martin, M., Uniform approximation by solutions of elliptic equations and seminormality in higher dimensions, Operator Theory: Advances and Applications, 149, Birkh¨ auser Verlag, 2004, 387–406. [M6] Martin, M., Uniform approximation by closed forms in several complex variables, in Proceedings of the 7th International Conference on Cliﬀord Algebras and Their Applications, Toulouse, France, 2005, to appear. [M7] Martin, M., Deconstructing Dirac operators. I: Quantitative Hartogs-Rosenthal theorems, Proceedings of the 5th International Society for Analysis, Its Applications and Computation Congress, ISAAC 2005, Catania, Italy, 2005, to appear.

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[M8]

Martin, M., Deconstructing Dirac operators. II: Integral representation formulas, Preprint 2008.

[Mi]

Mitrea, M., Singular Integrals, Hardy Spaces, and Cliﬀord Wavelets, Lecture Notes in Mathematics, 1575, Springer-Verlag, Heidelberg, 1994.

[Ra]

Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, 1986.

[RSS1] Rocha-Chavez, R., Shapiro M., and Sommen, F., On the singular BochnerMartinelli integral, Integral Equations Operator Theory, 32 (1998), 354–365. [RSS2] Rocha-Chavez, R., Shapiro M., and Sommen, F., Analysis of functions and diﬀerential forms in Cm , in Proceedings of the Second ISAAC Congress, Kluwer, 2000, pp. 1457–1506. [RSS3] Rocha-Chavez, R., Shapiro M., and Sommen, F., Integral theorems for solutions of the complex Hodge-Dolbeaut system, in Proceedings of the Second ISAAC Congress, Kluwer, 2000, pp. 1507–1514. [RSS4] Rocha-Chavez, R., Shapiro M., and Sommen, F., Integral Theorems for Functions and Diﬀerential Forms in Cm , Research Notes in Mathematics 428, Chapman & Hall, 2002. [R1]

Ryan, J., Applications of complex Cliﬀord analysis to the study of solutions to generalized Dirac and Klein-Gordon equations, with holomorphic potential, J. Diﬀ. Eq. 67 (1987), 295–329.

[R2]

Ryan, J., Cells of harmonicity and generalized Cauchy integral formulae, Proc. London Math. Soc., 60 (1990), 295–318.

[R3]

Ryan, J., Plemelj formulae and transformations associated to plane wave decompositions in complex Cliﬀord analysis, Proc. London Math. Soc., 64 (1991), 70–94.

[R4]

Ryan, J. (Ed.), Cliﬀord Algebras in Analysis and Related Topics, CRC Press, Boca Raton, FL, 1995.

[R5]

Ryan, J., Intrinsic Dirac operators in Cn , Advances in Mathematics, 118 (1996), 99–133.

[RS]

Ryan, J. and Spr¨ oßig, W. (Eds.), Cliﬀord Algebras and Their Applications in Mathematical Physics, Volume 2: Cliﬀord Analysis, Progress in Physics 19, Birkh¨ auser, Basel, 2000.

[Sh]

Shapiro, M., Some remarks on generalizations of the one-dimensional complex analysis: hypercomplex approach, in Functional Analytic Methods in Complex Analysis and Applications to Partial Diﬀerential Equations, World Sci., 1995, pp. 379–401.

[S1]

Stein, E.M., Singular Integrals and Diﬀerentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.

[S2]

Stein, E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

[So1]

Sommen, F., Martinelli-Bochner formulae in complex Cliﬀord analysis, Zeitschrift f¨ ur Analysis und ihre Anwendungen, 6 (1987), 75–82.

[So2]

Sommen, F., Deﬁning a q-deformed version of Cliﬀord analysis, Complex Variables: Theory and Applications, 34 (1997), 247–265.

362 [T] [VS]

M. Martin Tarkhanov, N.N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995. Vasilevski, N. and Shapiro, M., Some questions of hypercomplex analysis, in Complex Analysis and Applications, Soﬁa, Bulgaria, 1987, 1989, pp. 523–531.

Mircea Martin Department of Mathematics Baker University Baldwin City, 66006 Kansas, USA e-mail: [email protected] Received: March 8, 2009 Accepted: May 26, 2009

Operator Theory: Advances and Applications, Vol. 203, 363–407 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Mapping Properties of Layer Potentials Associated with Higher-order Elliptic Operators in Lipschitz Domains Irina Mitrea Abstract. The method of layer potentials has been applied with tremendous success in the treatment of boundary value problems for second-order diﬀerential operators for a very long time; the literature on this topic is enormous. By way of contrast this method is disproportionally underdeveloped in the case of higher-order operators; the diﬀerence between the higher-order and the second-order settings is striking in term of the scientiﬁc output. This paper presents new results which establish mapping properties of multiple layer potentials associated with higher-order elliptic operators in Lipschitz domains in Rn . Mathematics Subject Classiﬁcation (2000). Primary: 35C15, 78A30, 78A45; Secondary 31B10, 35J05, 35J25. Keywords. Multiple layers, higher-order operators, Lipschitz domains, Calder´ on-Zygmund Theory.

1. Introduction As is well known, many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic diﬀerential operator L in a domain Ω. When L is a diﬀerential operator of second order a variety of tools are available for dealing with such problems including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. The situation when the diﬀerential operator has higher order (as is the case for instance with anisotropic plate bending when one deals with fourth order) stands in sharp contrast with this as only fewer options This work was supported in part by the NSF Grant DMS 0547944 and by the Ruth Michler Prize from the Association of Women in Mathematics. Communicated by J.A. Ball.

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could be successfully implemented. While the layer potential method has proved to be tremendously successful in the treatment of second-order problems (see the comprehensive theory of integral equations on the boundaries of Lipschitz graph domains developed by R. Brown, A. Calder´on, R. Coifman, B. Dahlberg, E. Fabes, D. Jerison, C. Kenig, A. McIntosh, Y. Meyer, J. Pipher, G. Verchota, Z. Shen, M. Taylor and their collaborators), until now this approach has been insuﬃciently developed to deal with the intricacies of the theory of higher-order operators. The main goal of this paper is to show that a remarkable number of signiﬁcant results from the layer potential theory for the second-order case continue to hold when suitably formulated – and this is where the main diﬃculty lies – for higherorder diﬀerential operators as well. This is a survey of new results which can be proved by systematically employing recent advances that have been registered in Harmonic Analysis, particularly for the theory of Calder´on-Zygmund operators. For maximal applicability it is important to allow non-smooth (Lipschitz) domains and general higher-order operators. While providing complete proofs would make this paper unreasonably long (full proofs will appear elsewhere) for the beneﬁt of the reader we outline the main steps and highlight the novel technical diﬃculties encountered as well as the ideas and tools employed to overcome them. Thus, we set to develop a multiple layer potential theory for the treatment of boundary value problems associated with a higher-order, constant coeﬃcient (possibly matrix-valued), homogeneous elliptic diﬀerential operator Lu = ∂ α Aαβ ∂ β u (1.1) |α|=|β|=m

in a Lipschitz domain Ω ⊂ R (see Section 4 for details). This falls within the scope of the program outlined by A.P. Calder´on in his 1978 ICM plenary address in which he advocates the use of layer potentials “for much more general elliptic systems [than the Laplacian]” – see p. 90 in [3]. In contrast with the situation for second-order operators, such as say, the Laplacian, for which it has long been understood how to recover a harmonic function from its boundary trace or its normal derivative using singular integral operators, the case of higher-order operators presents signiﬁcant diﬃculties of both algebraic and analytic nature. To illustrate them, let us consider the classical Dirichlet problem for a diﬀerential elliptic operator of order 2m:

Lu = 0 in Ω, (1.2) ∂νj u = fj on ∂Ω, 0 ≤ j ≤ m − 1, n

where ∂νj denotes iterated normal derivatives of order j. When Ω is a Lipschitz domain, ν ∈ L∞ (∂Ω) exhibits no smoothness and one needs to be careful about deﬁning ∂νj . One way around this diﬃculty is to consider α! ∂νj u := ν α ∂ α u, (1.3) j! |α|=j

Mapping Properties of Multiple Layers

365

where α = (α1 , . . . , αn ) ∈ Nn0 and ν α := ν1α1 ν2α2 . . . νnαn . Here, for each 1 ≤ k ≤ n, νk denotes the kth component of the normal vector ν. Thus, in the light of (1.3) it is more appropriate to work with the boundary value problem Lu = 0

in Ω,

∂ α u = fα

on ∂Ω,

α ∈ Nn0 , |α| ≤ m − 1,

(1.4) ˙ instead of (1.2). Hereafter the array f := {fα }|α|≤m−1 is referred to as the Dirichlet data. Since the elements of the array f˙ (called Whitney arrays) are derivatives of various orders of the same function u it is necessary that the array f˙ satisﬁes a certain set of compatibility conditions (denoted by f˙ ∈ CC): f˙ ∈ CC ⇔ ∂τij fγ = νi fγ+ej − νj fγ+ei ∀ |γ| ≤ m − 2, ∀ i, j ∈ {1, . . . , n}, (1.5) where ∂τij := νi ∂j − νj ∂i with ek denoting the kth canonical multi-index in Nn0 . The set of compatibility conditions is a concept introduced by H. Whitney in his 1934 paper [40]. Since the crux of the matter in the Calder´ on-Zygmund theory of singular integral operators on Lipschitz domains is the study of boundedness on Lp (∂Ω), it is natural to introduce Lp -based arrays (and later on, Sobolev, Besov, TriebelLizorkin, Sobolev-Hardy, H¨ older, BMO, VMO – also play a crucial role) on ∂Ω. One of our ﬁrst aims is to identify the correct analogues of such classical spaces for the higher-order Dirichlet problem in a domain Ω. Inspired by the work of [40], [2] and, more recently, [38], [26], we shall work with Whitney arrays adapted to various types of scales on which the smoothness of scalar-valued functions is traditionally measured. To deﬁne these, we shall use the same basic recipe. Namely, given m ∈ N and a space of scalar functions X(∂Ω) → L1loc (∂Ω), we let X˙ m−1 (∂Ω) be the space of all families f˙ = {fα }|α|≤m−1 , indexed by multi-indices of length ≤ m − 1, with the properties that f˙ ∈ CC and fα ∈ X(∂Ω) ∀ α ∈ Nn with |α| ≤ m − 1. (1.6) 0

The case X = Lp was introduced by Cohen and Gosselin in [5] when L = Δ2 (in particular, m = 2); G. Verchota considered the case X = Lp1 in [38]; V. Maz’ya et al. set forth the case X = Bsp,q when p = q > 1 in [22]; the case X = C α appeared 1,p ﬁrst in the pioneering work of S. Agmon in [2] (where n = 2); the cases X = Hat and X = BMO were ﬁrst dealt with in the work of J. Pipher and G. Verchota, see [26]. Having identiﬁed the natural function spaces of higher-order smoothness on Lipschitz surfaces, the next order of business is to ﬁnd the correct concept of multiple layer associated with an elliptic, constant coeﬃcient diﬀerential operator L as in (1.1). To this end, recall ﬁrst that the classical (harmonic) double layer potential operator in Ω, along with its principal value version are deﬁned as 1 ν(Y ), Y − X (DΔ f )(X) := f (Y ) dσ(Y ), X ∈ Rn \ ∂Ω, (1.7) ωn−1 ∂Ω |X − Y |n ν(Y ), Y − X 1 (KΔ f )(X) := p.v. f (Y ) dσ(Y ), X ∈ ∂Ω, (1.8) ωn−1 ∂Ω |X − Y |n

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where ν = (νj )1≤j≤n is the outward unit normal deﬁned a.e. with respect to the surface measure dσ, ·, · denotes the scalar product in Rn , and ωn−1 denotes the surface area of the unit sphere in Rn . The modern study of these operators and the key role they play in the treatment of boundary value problems for the Laplacian in C 1 domains originates with the celebrated 1979 Acta Mathematica paper by E. Fabes, M. Jodeit Jr. and N. Rivi`ere ([11]). There, these authors have established many basic properties such as nontangential maximal function estimates, jump relations, and the fact that the operator KΔ : Lp(∂Ω) −→ Lp (∂Ω) is compact, for each p ∈ (1, ∞). Although striking advances have been made since its publication, this paper has served as a blue-print for the study of a great many other types of elliptic systems of second order. In this paper we take the next natural step and explore the extent to which a parallel theory can be developed for multiple layer potentials associated with a diﬀerential operator L of order 2m (m ≥ 1), deﬁned as in (1.1), in Lipschitz domains in Rn . Along the way, a substantial portion of the classical Calder´ onZygmund theory has to be reworked in this higher-order setting. The multiple layer potential theory developed here has the same trademark features as the one corresponding to second-order operators (nontangential maximal estimates, jump relations, Carleson estimates, estimates on Besov and Triebel-Lizorkin spaces), indeed the latter becomes a particular case of the former. See Section 4 for the discussion about the double layer potential operator D˙ and Section 5 where we discuss the single layer potential operator. If E denotes a fundamental solution for the operator L, we deﬁne the action of the double layer potential operator on a Whitney array f˙ = {fδ }|δ|≤m−1 by setting ˙ D˙ f(X) := −

m

|α|=m

|β|=m

α!(m−k)!(k−1)! m!γ!δ!

k=1 |δ|=m−k

×

|γ|=k−1

γ+δ+ej =α

νj (Y )Aαβ (∂ β+γ E)(X − Y ), fδ (Y ) dσ(Y )

(1.9)

∂Ω

for X ∈ R \ ∂Ω. In the same context, we deﬁne the principal value multiple layer K˙ by 5. / J (1.10) K˙ f˙ := K˙ f˙ γ n

|γ|≤m−1

where, for each γ ∈ Nn0 of length ≤ m − 1, we have set .

|γ| / K˙ f˙ γ (X) := |α|=m

|β|=m

=1

δ+η+ek =α

θ+ω+ej =γ

|δ|=−1,|η|=m− |θ|=−1,|ω|=|γ|−

α!|δ|!|η|!γ!|θ|!|ω|! m! δ! η! |γ|! θ! ω!

Mapping Properties of Multiple Layers

K

× lim+ ε→0

367

L Aαβ ∂τkj (Y ) (∂ δ+ω+β E)(X − Y ) , fθ+η (Y ) dσ(Y )

Y ∈∂Ω

|X−Y |>ε

−

|α|=m

|β|=m

m

=|γ|+1

δ+η+ek =α

α!|δ|!|η|! m! δ! η!

(1.11)

|δ|=−1,|η|=m−

× lim+ ε→0

K

L νk (Y )Aαβ (∂ δ+β E)(X − Y ), fγ+η (Y ) dσ(Y ),

Y ∈∂Ω

|X−Y |>ε

for X ∈ ∂Ω. In spite of their intricate nature, the deﬁnitions (1.9)–(1.11) are natural. For example, they reduce precisely to (1.7)–(1.8) when L = Δ. They also contain as particular cases the multiple layer potentials introduced by S. Agmon in [2]. Most importantly, these operators satisfy properties similar to those proved by E. Fabes, M. Jodeit Jr. and N.M. Rivi`ere in [11] for the classical harmonic layer potentials (1.7)–(1.8). Also, the so-called single layer operator associated with L and Ω is introduced as G F J 5 ˙ , X ∈ Ω, (1.12) (SΛ)(X) := Λ(·), ∂.α [E(X − ·)] ∂Ω

|α|≤m−1

%∗ $ p,q (∂Ω) and where ·, · denotes the pairing between elements of B˙ m−1,−s+1/p p,q ˙ Bm−1,−s+1/p (∂Ω). We also develop a trace theory for spaces of higher-order smoothness (see Section 3.3 for statements of the main results). For the purpose of this introduction, recall that given a bounded Lipschitz domain Ω in Rn , the map Tr u := u|∂Ω , ¯ extends to a bounded linear operator u ∈ C 0 (Ω), p,q Tr : Bs+1/p (Ω) −→ Bsp,q (∂Ω),

(1.13)

if n−1 n < p ≤ ∞, 0 < q < ∞, and max {0, (n − 1)(1/p − 1)} < s < 1. In addition, the operator in (1.13) is onto – indeed, it has a linear, bounded right inverse – and J 5 p,q (Ω). (1.14) u ∈ Bsp,q (∂Ω) : Tr u = 0 = the closure of Cc∞ (Ω) in Bs+1/p When 1 ≤ p, q ≤ ∞, these claims have been proved in [19], [18]. The above, more general, version of these results has been obtained in [21]. For the problems we have in mind, we are naturally led to considering traces from spaces exhibiting a larger amount of smoothness than the above results could handle. Thus, the very nature of such trace results changes in the higher smoothness setting. Given m ∈ N, we deﬁne the higher-order trace operator by setting 5 J trm−1 u := Tr [∂ α u] , (1.15) |α|≤m−1

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I. Mitrea

whenever meaningful. We are able to generalize the above trace result and establish the following. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and 0 < p, q ≤ ∞, max {0, (n − 1)(1/p − 1)} < s < 1, and m ∈ N. Then the higher trace operator (1.15) induces a well-deﬁned, linear and bounded mapping p,q p,q trm−1 : Bm−1+s+1/p (Ω) −→ B˙ m−1,s (∂Ω).

(1.16)

This is onto – in fact, has a bounded, linear right-inverse – and its null-space is p,q (Ω). the closure of Cc∞ (Ω) in Bm−1+s+1/p

(1.17)

The organization of the paper is as follows. Section 2 contains basic deﬁnitions and notation used throughout the paper as well a brief review of smoothness spaces in Rn , in Lipschitz domains, and on their boundaries. In Section 3 we discuss Whitney arrays and multi-trace theory results. Section 4 deals with mapping properties of the double layer potential operator while Section 5 is concerned with the single layer and the conormal derivative. Full proofs of the results presented in this paper will be published elsewhere.

2. Preliminaries 2.1. Lipschitz domains and nontangential maximal function Throughout this paper, by an unbounded Lipschitz domain Ω in Rn we understand the upper-graph of a Lipschitz function ϕ : Rn−1 → R. Also, we shall call Ω a bounded Lipschitz domain in Rn if there exists a ﬁnite open covering {Oj }1≤j≤N of ∂Ω with the property that, for every j ∈ {1, . . . , N }, Oj ∩ Ω coincides with the portion of Oj lying in the upper-graph of a Lipschitz function ϕj : Rn−1 → R (where Rn−1 × R is a new system of coordinates obtained from the original one via a rigid motion). As is well known, for a Lipschitz domain Ω (bounded or unbounded), the surface measure dσ is well deﬁned on ∂Ω and there exists an outward pointing normal vector ν = (ν1 , . . . , νn ) at almost every point on ∂Ω. Given a Lipschitz domain Ω, we shall set ¯ Ω+ := Ω and Ω− := Rn \ Ω.

(2.1)

Then, for a ﬁxed parameter κ > 0 deﬁne the nontangential approach regions with vertex at X ∈ ∂Ω (corresponding to Ω± ) as Rκ± (X) := {Y ∈ Ω± : |X − Y | ≤ (1 + κ)dist (Y, ∂Ω)},

(2.2)

and, further, the nontangential maximal operator of a given function u in Ω± by Nκ± (u)(X) := sup{|u(Y )| : Y ∈ Rκ± (X)}.

(2.3)

When unambiguous, we agree to drop the superscripts ±. In fact, it can be shown that the dependence of Rκ± and Nκ± on κ plays only an auxiliary role and will be eventually dropped.

Mapping Properties of Multiple Layers

369

Moving on, given a Lipschitz domain Ω ⊂ Rn , a Borelian measure μ on Ω is called a Carleson measure provided 5 J μCar := sup R1−n μ(B(X, R) ∩ Ω) : X ∈ ∂Ω, 0 < R < diam (∂Ω) (2.4) is ﬁnite. In the sequel, we shall refer to μCar as the Carleson constant of μ. Next, we introduce a related concept. Given a Lipschitz domain Ω ⊂ Rn , a Borelian measure μ on Ω is called a vanishing Carleson measure provided it is a Carleson measure and $ 5 J% lim+ sup r1−n μ(B(X, r) ∩ Ω) : X ∈ ∂Ω, 0 < r < R = 0. (2.5) R→0

Going further, let us deﬁne the nontangential boundary trace of a function u deﬁned in Ω± as u (X) := lim u(Y ), X ∈ ∂Ω, (2.6) Y →X

∂Ω

Y ∈R± κ (X)

whenever meaningful. 2.2. Smoothness spaces in Rn For each 1 < p < ∞ and s ∈ R, we denote by Lps (Rn ) the classical Bessel potential space with integrability index p and smoothness s. As is well known, when the smoothness index is a natural number, say s = k ∈ N, this can be identiﬁed with the classical Sobolev space 5 J W k,p (Rn ) := f ∈ Lp (Rn ) : f W k,p (Rn ) := ∂ γ f Lp(Rn ) < ∞ , (2.7) |γ|≤k

Lpk (Rn )

= W (R ) for k ∈ No and 1 0, s ∈ / N, consisting of functions f for which |∂ α f (x) − ∂ α f (y)| ∂ α f L∞ (Rn ) + sup < ∞. (2.8) f C s(Rn ) := |x − y|s−[s] x=y k,p

n

|α|≤[s]

|α|=[s]

Next we turn our attention to Hardy-type spaces in *Rn . Fix a function ψ in Cc∞ (Rn ) with supp (ψ) ⊂ {x ∈ Rn : |x| < 1} and Rn ψ(x) dx = 1, and set ψt (x) := t−n ψ(x/t) for each t > 0. Given a tempered distribution u ∈ S (Rn ) we deﬁne its radial maximal function and its truncated version, respectively, by setting u++ := sup0
h (R ) := {u ∈ S (R ) : uhp (Rn ) := u Lp (Rn ) < ∞}. p

n

n

+

(2.9) (2.10)

Diﬀerent choices of the function ψ yield equivalent quasi-norms so (2.9), (2.10) viewed as topological spaces, are intrinsically deﬁned. Also, H p (Rn ) = hp (Rn ) = Lp (Rn ), for all 1 < p < ∞.

370

I. Mitrea

Next, we make the convention that a barred integral indicates averaging. Then, a function f ∈ L2loc (Rn ) is said to belong to the space BMO(Rn ) if 1/2 2 f BMO(Rn ) := sup − |f (x) − fQ | dx < ∞, (2.11) Q

Q

* where the supremum is taken over all cubes in Rn and fQ := −Q f (x) dx. As is well known, (H 1 (Rn ))∗ = BMO(Rn ) (see [14]). The local version of BMO(Rn ) is deﬁned as follows. A function f ∈ L2loc (Rn ) belongs to bmo(Rn ) if the quantities 1/2 1/2 − |f (x) − fQ |2 dx − |f (x)|2 dx sup , sup (2.12) Q: l(Q)≤1

Q: l(Q)>1

Q

Q

are ﬁnite (we denote by f bmo(Rn ) the maximum of these expressions). Here l(Q) stands for the length of the sides of the cube Q. Then (h1 (Rn ))∗ = bmo(Rn ) (see [17]). Following [28], [34] we let Fsp,q (Rn ), s ∈ R, 0 < p < ∞ and 0 < q ≤ ∞, and Bsp,q (Rn ), s ∈ R, 0 < p ≤ ∞ and 0 < q ≤ ∞, stand for the classical TriebelLizorkin and Besov scales, respectively. It has long been known that many classical smoothness spaces are encompassed by these. For example, C s (Rn ) = Bs∞,∞ (Rn ) for 0 < s ∈ / Z; Lp (Rn ) = F0p,2 (Rn ) for 1 < p < ∞; Lps (Rn ) = Fsp,2 (Rn ) for 1 < p < ∞ and s ∈ R; W k,p (Rn ) = Fkp,2 (Rn ) for 1 < p < ∞ and k ∈ N; hp (Rn ) = F0p,2 (Rn ) for 0 < p ≤ 1; bmo(Rn ) = F0∞,2 (Rn ); and BMO(Rn ) = F˙0∞,2 (Rn ). For a more detailed discussion of the Besov and Triebel-Lizorkin scales of spaces in Rn , including the deﬁnition and properties of their homogeneous counterparts, B˙ sp,q (Rn ) and F˙sp,q (Rn ), the interested reader is referred to [34], [15], [16], [28], and the references therein. 2.3. Smoothness spaces in Lipschitz domains Given an arbitrary open subset Ω of Rn , we denote by f |Ω the restriction of a distribution f in Rn to Ω. For 0 < p, q ≤ ∞ and s ∈ R we then set Bsp,q (Ω) := {f distribution in Ω : ∃ g ∈ Bsp,q (Rn ) such that g|Ω = f }, f Bsp,q (Ω) := inf {gBsp,q (Rn ) : g ∈ Bsp,q (Rn ), g|Ω = f },

f ∈ Bsp,q (Ω). (2.13) A similar deﬁnition is given for Fsp,q (Ω) in the case when p < ∞. From the corresponding density result in Rn , it follows that for any bounded Lipschitz domain Ω and any 0 < p, q < ∞, s ∈ R, C ∞ (Ω) → Bsp,q (Ω) and C ∞ (Ω) → Fsp,q (Ω) densely. The existence of a universal extension operator for Besov and TriebelLizorkin spaces in an arbitrary Lipschitz domain Ω ⊂ Rn has been established by V. Rychkov in [29]. To state this result, let RΩ denote the operator of restriction to Ω, which maps distributions from Rn into distributions in Ω, RΩ (u) := u , u distribution in Rn . (2.14) Ω

Mapping Properties of Multiple Layers

371

Theorem 2.1 ([29]). Let Ω ⊂ Rn be either a bounded Lipschitz domain, the exterior of a bounded Lipschitz domain, or an unbounded Lipschitz domain. Then there exists a linear, continuous operator EΩ , mapping distributions in Ω into tempered distributions in Rn , and such that whenever 0 < p, q ≤ +∞, s ∈ Rn , p,q n p,q EΩ : Ap,q s (Ω) −→ As (R ) boundedly, satisfying RΩ (EΩ f ) = f, ∀ f ∈ As (Ω), (2.15) for A = B or A = F , in the latter case assuming p < ∞.

Recall now the standard Lp -based Sobolev spaces in a Lipschitz domain Ω: 5 J W k,p (Ω) := f ∈ Lp (Ω); ∂ γ f ∈ Lp (Ω), ∀ γ : |γ| ≤ k , 1 ≤ p ≤ ∞, k ∈ N0 , (2.16) " equipped with the norm f W k,p (Ω) := |γ|≤k ∂ γ f Lp (Ω) . In view of Theorem 2.1, for any Lipschitz domain Ω we have W k,p (Ω) = Fkp,2 (Ω), for all 1 < p < ∞ and k ∈ N0 . Going further, for 0 < p, q ≤ ∞, s ∈ R, we set p,q n Ap,q s,0 (Ω) := {f ∈ As (R ) : supp f ⊆ Ω},

f Ap,q := f Ap,q n , s (R ) s,0 (Ω)

(2.17)

p,q p,q (Ω), Fs,0 (Ω) are where, as usual, either A = F and p < ∞ or A = B. Thus, Bs,0 p,q p,q n n closed subspaces of Bs,0 (R ) and Fs,0 (R ), respectively. It follows that if Ω is a bounded Lipschitz domain in Rn and 0 < p, q < ∞, p,q ∞ ∞ s ∈ R, then the following dense embeddings hold C c (Ω) → As,0 (Ω), C (Ω) → p,q ∞ p,q As (Ω), and Cc (Ω) → As,z (Ω), where, as before, tilde denotes the extension by zero outside Ω and A stands for either B or F . Next, for 0 < p, q ≤ ∞ and s ∈ R, we introduce p,q Ap,q s,z (Ω) := {f distribution in Ω : ∃ g ∈ As,0 (Ω) with g|Ω = f }, p,q := inf {gAp,q f Ap,q n : g ∈ As,0 (Ω), g|Ω = f }, s,z (Ω) s (R )

(2.18)

(where, as before, A = F and p < ∞ or A = B) and, in keeping with earlier conventions, p,2 Lps,z (Ω) := Fs,z (Ω) = {f distribution in Ω : ∃ g ∈ Lps,0 (Ω) with g|Ω = f }, (2.19) if 1 < p < ∞, s ∈ R. For further use, let us also make the simple yet important observation that the operator of restriction to Ω induced linear, bounded and onto mappings in the following settings p,q n p,q n p,q RΩ : Ap,q s (R ) −→ As (Ω) and RΩ : As,0 (R ) −→ As,z (Ω)

for 0 < p, q ≤ ∞, s ∈ R.

(2.20)

$ %∗ = Ap−s,q (Ω) If 1 < p, q < ∞ and 1/p+ 1/p = 1/q + 1/q = 1, then Ap,q s,z (Ω) $ %∗ ,q if s > −1 + p1 and Ap,q (Ω) = Ap−s,z (Ω) if s < 1p . Furthermore, for each s ∈ R s

p,q and 1 < p, q < ∞, the spaces Ap,q s (Ω) and As,0 (Ω) are reﬂexive.

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I. Mitrea

2.4. Smoothness spaces on Lipschitz boundaries We start the discussion in this subsection by considering the case of Sobolev spaces on the boundary of a Lipschitz Ω ⊂ Rn . To this end, consider the ﬁrst-order tangential derivative operators ∂τjk acting on a compactly supported function ψ of class C 1 in a neighborhood of ∂Ω by ∂τjk ψ := νk (∂j ψ) −νj (∂k ψ) , j, k = 1, . . . , n. (2.21) ∂Ω ∂Ω * * Then it can be shown that, ∂Ω ϕ (∂τjk ψ) dσ = ∂Ω (∂τkj ϕ) ψ dσ for every ϕ, ψ ∈ Cc1 (Rn ) and j, k ∈ {1, . . . , n}. In turn, one can make use of this identity to prove that, if ψ ∈ Cc1 (Rn ), then ∂τjk ψ actually depends only on ψ|∂Ω . That is, if ψ ∈ Cc1 (Rn ) satisﬁes ψ ≡ 0 on ∂Ω, then ∂τjk ψ ≡ 0 on ∂Ω for every j, k ∈ {1, . . . , n}. Inspired by the discussion above, for every f ∈ L1loc (∂Ω) we next deﬁne the functional ∂τkj f by setting 1 n f (∂τjk ψ) dσ. (2.22) ∂τkj f : Cc (R ) & ψ → ∂Ω

has ∂τkj f ∈* L1loc (∂Ω), the following integration by Indeed, when f ∈ parts formula holds ∂Ω f (∂τjk ψ) dσ = ∂Ω (∂τkj f ) ψ dσ, for every ψ ∈ Cc1 (Rn ). Let us also point out that the “distributional” version of ∂τkj from (2.22) agrees with the “point-wise” deﬁnition considered in (2.21). Moving on, for each p ∈ (1, ∞) we can then deﬁne the Sobolev type space 5 J Lp1 (∂Ω) := f ∈ Lp (∂Ω) : ∂τjk f ∈ Lp (∂Ω), j, k = 1, . . . , n , (2.23) L*1loc (∂Ω)

which becomes a Banach space when equipped with the natural norm f Lp1 (∂Ω) := "n f Lp(∂Ω) + j,k=1 ∂τjk f Lp (∂Ω) . In addition, it can be shown that Lp1 (∂Ω) is densely embedded into Lp (∂Ω). We now turn our attention to other smoothness spaces on Lipschitz boundaries. We start by setting (a)+ := max{a, 0} for any a ∈ R. Consider next three parameters p, q, s subject to % $ <s<1 (2.24) 0 < p, q ≤ ∞, (n − 1) 1p − 1 +

and assume that Ω ⊂ R is the upper-graph of a Lipschitz function ϕ : Rn−1 → R. We then deﬁne Bsp,q (∂Ω) as the space of locally integrable functions f on ∂Ω for which the assignment Rn−1 & x → f (x, ϕ(x)) belongs to Bsp,q (Rn−1 ) (cf. Section 2.2). We then set n

f Bsp,q (∂Ω) := f (·, ϕ(·))Bsp,q (Rn−1 )

(2.25)

As far as Besov spaces with a negative amount of smoothness are concerned, in the same context as above we set ) p,q p,q f ∈ Bs−1 (∂Ω) ⇐⇒ f (·, ϕ(·)) 1 + |∇ϕ(·)|2 ∈ Bs−1 (Rn−1 ), (2.26) ) 2 p,q p,q (2.27) f Bs−1 (∂Ω) := f (·, ϕ(·)) 1 + |∇ϕ(·)| Bs−1 (Rn−1 ) .

Mapping Properties of Multiple Layers

373

As is well known, the case when p = q = ∞ corresponds to the usual (nonhomogeneous) H¨older spaces C s (∂Ω), deﬁned by the requirement that f C s (∂Ω) := f L∞(∂Ω) +

sup X=Y

X,Y ∈∂Ω

|f (X) − f (Y )| < +∞. |X − Y |s

(2.28)

All the above deﬁnitions then readily extend to the case of (bounded) Lipschitz domains in Rn via a standard partition of unity argument. The following trace result holds. Theorem 2.2 (See [21]). Let Ω be a Lipschitz domain in Rn and assume that the indices p, s satisfy n−1 < p ≤ ∞ and (n − 1)( p1 − 1)+ < s < 1. Then the following n hold: (i) The restriction to the boundary extends to a linear, bounded operator p,q p,q (∂Ω) Tr : Bs+ 1 (Ω) −→ Bs p

for 0 < q ≤ ∞.

(2.29)

Moreover, for this range of indices, Tr is onto and has a bounded right inverse p,q Ex : Bsp,q (∂Ω) −→ Bs+ 1 (Ω).

(2.30)

p

(ii) Similar considerations hold for p,q p,p (∂Ω) Tr : Fs+ 1 (Ω) −→ Bs

(2.31)

p

with the convention that q = ∞ if p = ∞. More speciﬁcally, Tr in (2.31) is linear, bounded, operator which has a linear, bounded right inverse p,q Ex : Bsp,p (∂Ω) −→ Fs+ 1 (Ω).

(2.32)

p

We now proceed to discuss Triebel-Lizorkin spaces deﬁned on the boundary of a bounded Lipschitz domain Ω ⊂ Rn , denoted in the sequel by Fsp,q (∂Ω). Compared with the Besov scale, the most important novel aspect here is the possibility of allowing the endpoint case s = 1 as part of the general discussion if q = 2. To discuss this in more detail, retain the same notation as in the ﬁrst part of Section 3.4 and assume that either $ % 1 0 < p < ∞, 0 < q ≤ ∞, (n − 1) −1 < s < 1, (2.33) min {p, q} + or n−1 < p < ∞, q = 2, s = 1. (2.34) n In this scenario, the Triebel-Lizorkin scale in Rn−1 is invariant under pointwise multiplication by Lipschitz maps as well as composition by Lipschitz diﬀeomorphisms. When Ω is the upper-graph of a Lipschitz function ϕ : Rn−1 → R, we may therefore deﬁne the space Fsp,q (∂Ω) as the collection of all locally integrable functions f on ∂Ω such that f (·, ϕ(·)) ∈ Fsp,q (Rn−1 ), endowed with the norm

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I. Mitrea

p,q (∂Ω) is deﬁned as the collection of f Fsp,q (∂Ω) := f (·, ϕ(·))Fsp,q (Rn−1 ) . Also, Fs−1

all functionals f ∈ (Lipc (∂Ω)) such that ) p,q f (·, ϕ(·)) 1 + |∇ϕ(·)|2 ∈ Fs−1 (Rn−1 ), (2.35) ) 2 p,q p,q with the quasi-norm f Fs−1 (∂Ω) := f (·, ϕ(·)) 1 + |∇ϕ(·)| Fs−1 (Rn−1 ) . Finally, n when Ω ⊂ R is a bounded Lipschitz domain and (s, p, q) are as in (2.33)–(2.34), p,q (∂Ω) via localization (using a smooth, ﬁnite partiwe deﬁne Fsp,q (∂Ω) and Fs−1 tion of unity) and pull-back to Rn−1 (in the manner described above, for graphLipschitz domains). When equipped with the natural quasi-norms, the TriebelLizorkin spaces just introduced are quasi-Banach, and diﬀerent partitions of unity yield equivalent quasi-norms. Introduce next

p hat (∂Ω) if n−1 n < p ≤ 1, p h (∂Ω) := (2.36) Lp (∂Ω) if 1 < p < ∞,

and

hp1 (∂Ω)

:=

h1,p at (∂Ω) if Lp1 (∂Ω)

n−1 n

< p ≤ 1,

(2.37)

if 1 < p < ∞.

Then, two basic identities, relating Triebel-Lizorkin spaces to Sobolev spaces on ∂Ω read as follows: F0p,2 (∂Ω) = hp (∂Ω),

F1p,2 (∂Ω) = hp1 (∂Ω),

n−1 n

< p < ∞.

(2.38)

p Also, it. can/ be shown that, for each p ∈ ( n−1 n , 1], the dual space of hat (∂Ω) is 1 (n−1) p −1 (∂Ω), and C ) (2.39) f ∈ hpat (∂Ω) ⇐⇒ f (·, ϕ(·)) 1 + |∇ϕ(·)|2 ∈ F0p,2 (Rn−1 ).

Next, let us recall the local BMO space. For some ﬁxed 0 < η < diam (∂Ω), def

this is introduced by requiring that f ∈ bmo (∂Ω) ⇐⇒ f ∈ L2 (∂Ω) and − |f − fΔr | dσ < ∞ sup Δr surface ball with r ≤ η

(2.40)

Δr

* where fΔr := −Δr f dσ, and is equipped with the natural norm ⎛ ⎞ ⎜ ⎟ sup − |f − fΔr | dσ ⎟ f bmo (∂Ω) := f L2 (∂Ω) + ⎜ ⎝ ⎠. Δr surface ball with r ≤ η

Δr

(2.41)

Mapping Properties of Multiple Layers

375

%∗ $ %∗ $ Then (cf. [7]), h1at (∂Ω) = bmo (∂Ω) and h1at (∂Ω) = vmo (∂Ω) , where f ∈ def

vmo (∂Ω) ⇐⇒ f ∈ bmo (∂Ω) and ⎛ ⎜ lim ⎝

R→0

sup

Δr surface ball with r ≤ R

⎞

− Δr

⎟ |f − fΔr | dσ ⎠ = 0.

(2.42)

The space vmo (∂Ω) is Sarason’s space of functions of vanishing mean oscillation. Let us point out that, for each s ∈ (0, 1), an alternative characterization of the latter space is (2.43) vmo (∂Ω) = the closure of C s (∂Ω) in bmo (∂Ω). . n−1 Moving on, for each p ∈ n , 1 , ﬁx 1 < po ≤ ∞ and an arbitrary small η > 0

and then deﬁne"the space h1,p at (∂Ω) as the collection of distributions f ∈ Lipc (∂Ω) p such that f = j λj aj with (λj )j ∈ , and for each j aj is a regular (p, po )-atom supported in a surface ball of radius ≤ η,

(2.44)

where the series converges in Lipc (∂Ω) , and equip it with the natural inﬁmum quasi-norm. Above, a function a ∈ Lp1o (∂Ω) is called a regular (p, po )-atom if there exists a surface ball Δr so that $ % supp a ⊆ Δr ,

∇tan aLpo (∂Ω) ≤ r

Let us also record here the fact that, if

n−1 n

(n−1)

1 po

1 −p

.

(2.45)

< p ≤ 1, we have

p,2 n−1 ). f ∈ h1,p at (∂Ω) ⇐⇒ f (·, ϕ(·)) ∈ F1 (R

(2.46)

In particular, various choices of the parameter po in (2.44) yield the same vector space and topology on h1,p at (∂Ω). The equivalence (2.46) also shows that the space 1,p n−1 h1,p (∂Ω), p ∈ ( , 1], is local, in the sense that f ∈ h1,p at at (∂Ω) =⇒ ψf ∈ hat (∂Ω), n plus a natural estimate, for every function ψ ∈ Lipc (∂Ω). < p ≤ 1, and p∗ ∈ (1, ∞) When Ω ⊂ Rn is a bounded Lipschitz domain, n−1 n 1 1 1 such that p∗ = p − n−1 , it can be shown that 5 J p p∗ h1,p at (∂Ω) = f ∈ L (∂Ω) : ∂τjk f ∈ hat (∂Ω), 1 ≤ j, k ≤ n . In addition, f h1,p (∂Ω) ≈ f Lp∗ (∂Ω) + at

n " j,k=1

(2.47)

∂τjk f hp (∂Ω) .

As already mentioned, the Besov and Triebel-Lizorkin spaces have been deﬁned in such a way that a number of basic properties from the Euclidean setting carry over to spaces deﬁned on ∂Ω in a rather direct fashion.

376

I. Mitrea

3. Function spaces of Whitney arrays Here we discuss how to adapt the traditional ways of measuring smoothness for scalar functions (deﬁned on the boundary of a Lipschitz domain) to the case of Whitney arrays. Throughout the section, we ﬁx a Lipschitz domain Ω ⊂ Rn (bounded or unbounded). 3.1. Whitney-Lebesgue, Whitney-Sobolev and Whitney-Besov spaces Given m ∈ N, we say that a family f˙ := {fα : α ∈ Nn0 , |α| ≤ m − 1} of reasonably well-behaved functions on ∂Ω is a Whitney array if the components satisfy certain compatibility conditions, henceforth abbreviated as CC. More speciﬁcally, f˙ ∈ CC ⇐⇒ ∂τjk fγ = νj fγ+ek − νk fγ+ej whenever |γ| ≤ m − 2 and 1 ≤ j, k ≤ n,

(3.1)

where ej := (0, . . . , 1, . . . , 0) ∈ Nn0 has the only nonzero component on the jth position. For each 1 ≤ p ≤ ∞, we then deﬁne the Whitney-Lebesgue space 5 L˙ pm−1,0 (∂Ω) := f˙ = {fα }|α|≤m−1 ∈ CC : fα ∈ Lp (∂Ω) if |α| ≤ m − 1}, (3.2) ˙ ˙p which we equip with the norm f L

"

|α|≤m−1 fα Lp (∂Ω) . Note that, p ˙ in the light of (3.1), the membership f ∈ L˙ m−1,0 (∂Ω) implies fα ∈ Lp1 (∂Ω), whenever |α| ≤ m − 2 and, hence, ˙ ˙p f fα Lp1 (∂Ω) + fα Lp (∂Ω) . (3.3) L (∂Ω) ≈ m−1,0 (∂Ω)

m−1,0

|α|≤m−2

:=

|α|=m−1

We shall also work with a more regular version of (3.2), namely the WhitneySobolev space 5 J (∂Ω) := f˙ = {fα }|α|≤m−1 ∈ CC : fα ∈ Lp (∂Ω) if |α| ≤ m − 1 , (3.4) L˙ p 1

m−1,1

which, for each 1 < p < ∞, we endow with the norm given by f˙L˙ p := m−1,1 (∂Ω) " p f . α L1 (∂Ω) |α|≤m−1 We next elaborate on the operation of multiplication between scalar functions and Whitney arrays. In this regard, we make the following. Deﬁnition 3.1. Given a bounded Lipschitz domain Ω ⊂ Rn and a scalar-valued function η ∈ Cc∞ (Rn ), for each family f˙ = {fα }|α|≤m−1 with locally integrable components on ∂Ω, set 5 α! J fγ ∂ β η . (3.5) η f˙ := β!γ! ∂Ω |α|≤m−1 β+γ=α

Mapping Properties of Multiple Layers

377

It is then easy to check that for each η, ξ ∈ Cc∞ (Rn ) there holds η f˙ +ξ f˙ = (η +ξ)f˙. Also, it can be shown that for each m ∈ N and 1 < p < ∞, the spaces L˙ pm−1,0 (∂Ω), L˙ pm−1,1 (∂Ω) are modules over Cc∞ (Rn ), in the sense described in Deﬁnition 3.1. We now present a basic density result involving Whitney array spaces. Proposition 3.2. Let Ω be a bounded Lipschitz in Rn , and ﬁx 1 < p < ∞, 5 domain J p α m ∈ N. Then L˙ m−1,1 (∂Ω) is the closure of (∂ F ∂Ω )|α|≤m−1 : F ∈ Cc∞ (Rn ) 5 in Lp1 (∂Ω) ⊕ · · · ⊕ Lp1 (∂Ω), and L˙ pm−1,0 (∂Ω) is the closure of (∂ α F ∂Ω )|α|≤m−1 : J F ∈ Cc∞ (Rn ) in Lp1 (∂Ω) ⊕ · · · ⊕ Lp1 (∂Ω) ⊕ Lp (∂Ω). Due to the stability of the hypotheses and the conclusions in Proposition 3.2 to multiplication by smooth, compactly supported functions, a useful observation is that matters reduce to considering the case when the domain Ω is the uppergraph of a Lipschitz function. In this setting, when m = 1, a simple molliﬁcation argument would do. However, in the general higher-order case, the challenging technical diﬃculty is raised by the compatibility conditions that the elements of the Whitney-Lebesgue and Whitney-Sobolev spaces L˙ pm−1,j (∂Ω), j ∈ {0, 1}, satisfy. We shall now consider Besov spaces exhibiting higher-order smoothness by adopting a point of view similar to (3.2)–(3.4). Concretely, for m ∈ N and p, q, s p,q as in (2.24), we deﬁne the Whitney-Besov space B˙ m−1,s (∂Ω) via the requirement p,q (∂Ω) ⇐⇒ fα ∈ Bsp,q (∂Ω) if |α| ≤ m − 1, and f˙ ∈ CC. (3.6) f˙ ∈ B˙ m−1,s " p,q For each f˙ ∈ B˙ m−1,s (∂Ω) we then set f˙B˙ p,q (∂Ω) := |α|≤m−1 fα Bsp,q (∂Ω) . m−1,s It can then be shown that the Besov space B˙ p,q (∂Ω) is a module over Cc∞ (Rn ) m−1,s

(where the multiplication is in the sense of (3.5)). The case 1 ≤ p = q ≤ ∞ has been studied more in the literature ([1], [19], [22]). When 1 < p < ∞, we clearly have f˙B˙ p,p (∂Ω) ≈ fα Lp1 (∂Ω) + fα Bsp,p (∂Ω) . (3.7) m−1,s

|α|≤m−2

|α|=m−1

3.2. Whitney-Hardy and Whitney-Triebel-Lizorkin spaces Let Ω be a bounded Lipschitz domain in Rn and, for 1 < p < ∞ and 0 ≤ s ≤ 1, abbreviate Lps (∂Ω) := Fsp,2 (∂Ω),

(3.8)

which, by (2.38), is in agreement with the notation already employed for s ∈ {0, 1}. Going further, we also deﬁne 5 J L˙ pm−1,s (∂Ω) := f˙ = {fα }|α|≤m−1 ∈ CC : fα ∈ Lps (∂Ω) if |α| ≤ m − 1 , (3.9) " which we equip with the norm f˙L˙ p := |α|≤m−1 fα Lps (∂Ω) . Once again, m−1,s (∂Ω) the notation for L˙ p (∂Ω) is consistent with that used for the spaces (3.2) and m−1,s

378

I. Mitrea

(3.4), which occur for s = 0 and s = 1, respectively. A natural continuation of the , 1] can scale L˙ pm−1,1 (∂Ω), originally deﬁned for 1 < p < ∞, to the range p ∈ ( n−1 n be constructed using Hardy-Sobolev spaces. More speciﬁcally, if n−1 < p < ∞, set n 5 J h˙ pm−1,1 (∂Ω) := f˙ = {fα }|α|≤m−1 ∈ CC : fα ∈ hp1 (∂Ω) if |α| ≤ m − 1 , (3.10) " ˙ ˙p p endowed with the quasi-norm f |α|≤m−1 fα h1 (∂Ω) , where the hm−1,1 (∂Ω) := p p p space h1 (∂Ω) is as in (2.36). Clearly, h˙ m−1,1 (∂Ω) = L˙ m−1,1 (∂Ω) if 1 < p < ∞. Finally, it is convenient to introduce an even larger scale of Whitney arrays, based on Triebel-Lizorkin spaces. Concretely, for p, q, s as in (2.33)–(2.34), we deﬁne the space 5 J p,q F˙m−1,s (∂Ω) := f˙ = {fα }|α|≤m−1 ∈ CC : fα ∈ Fsp,q (∂Ω) if |α| ≤ m − 1 , (3.11) " equipped with the quasi-norm f˙F˙ p,q (∂Ω) := |α|≤m−1 fα Fsp,q (∂Ω) . m−1,s

For further reference, we conclude with a brief discussion of the operation of multiplication between scalar functions and Whitney arrays belonging to various smoothness spaces, in the sense of Deﬁnition 3.1. We have Proposition 3.3. Assume that Ω ⊂ Rn is a bounded Lipschitz domain. Then, for each m ∈ N the spaces h˙ pm−1,1 (∂Ω), n−1 n < p < ∞, .1 / p,q B˙ m−1,s (∂Ω), n−1 n < p ≤ ∞, 0 < q ≤ ∞, (n − 1) p − 1 + < s < 1, . / p,q 1 F˙ m−1,s (∂Ω), n−1 n < p ≤ ∞, 0 < q ≤ ∞, (n − 1) min {p,q} − 1 + < s < 1, are all modules over Cc∞ (Rn ), in the sense described in Deﬁnition 3.1. Furthermore, if f˙ := trm−1 u where u ∈ Ap,q s (Ω), with A ∈ {B, F } and the indices p, q, s exists, as in Theorem 2.2, then η f˙ = trm−1 (ηu). Also, if u is such that g˙ := u|m−1 ∂Ω . then η g˙ := (ηu)|m−1 ∂Ω 3.3. Multi-trace theory in %Rn , for indices m, n, p, s such that m ∈ N, If Ω ⊂ Rn is a Lipschitz domain $ n−1 < p ≤ ∞, and (n − 1) p1 − 1 < s < 1, deﬁne the higher-order trace n + ∞ ¯ operator trm−1 mapping C (Ω) into Whitney arrays of length m by setting J 5 trm−1 F := [∂ α F ] . (3.12) ∂Ω

|α|≤m−1

The following trace result is of independent interest and is the complete/sharp result of this type in the context of Lipschitz domains. For more general sets but for a more restrictive set of indices related results have been proved by A. Jonsson and H. Wallin in [19]. n Theorem 3.4. Assume $ that % Ω ⊂ R is a bounded Lipschitz domain and that 0 < 1 p, q ≤ ∞, (n − 1) p − 1 < s < 1. Also, ﬁx m ∈ N. Then the higher-order +

Mapping Properties of Multiple Layers

379

trace operator introduced above extends by continuity to a well-deﬁned, linear and bounded operator p,q p,q trm−1 : Bm−1+s+1/p (Ω) −→ B˙ m−1,s (∂Ω).

(3.13)

In addition, (3.12) is onto and, in fact, has a bounded, linear right-inverse. That is, there exists a linear, continuous operator p,q p,q (∂Ω) −→ Bm−1+s+1/p (Ω) E : B˙ m−1,s

(3.14)

such that p,q (∂Ω) =⇒ Tr [∂ α (E f˙)] = fα , f˙ = {fα }|α|≤m−1 ∈ B˙ m−1,s

∀ α : |α| ≤ m − 1. (3.15) Similar results are valid on the Triebel-Lizorkin scale. Concretely, with the convention that q = ∞ if p = ∞, the operator p,q p,p trm−1 : Fm−1+s+1/p (Ω) −→ B˙ m−1,s (∂Ω)

(3.16)

is also well deﬁned, linear and bounded. Moreover, this operator has a linear, bounded right inverse. The most diﬃcult part in the proof of Theorem 3.4 is the construction of the right-inverse of the multi-trace operator and there are three major ingredients that enter this construction. The ﬁrst one has to do with the fact that D˙ ± , the versions of the multiple layer potential operator D˙ (associated with an elliptic constant coeﬃcient diﬀerential operator of order 2m, say L = Δm ) corresponding to the domains Ω± maps the space B p,q (∂Ω) in a suitable Besov space in Ω± (recall that Ω± have been introduced in (2.1)). This is a mapping property established in Theorem 4.11. The second important fact is the existence of a universal extension operator from Ω± in Rn with preservation of smoothness. There are many forerunners to such a result (A. Calder´on and E. Stein among others) but the most general result in the class of Lipschitz domains is due to V. Rychkov in [29] (see Theorem 2.1). Finally, the last ingredient in the construction of the aforementioned right-inverse is the fact that f˙ = trm−1 D˙ + f˙ − trm−1 D˙ − f˙,

(3.17)

which follows from the jump relations established in Theorem 4.9. n−1 Corollary 3.5. Fix an $ integer % m ∈ N along with n < p < ∞, 0 < q < ∞ and assume that (n − 1) 1p − 1 < s < 1. Also, suppose that Ω ⊂ Rn is a bounded +

Lipschitz domain. Then J 5 p,q (∂Ω). (∂ α F |∂Ω )|α|≤m−1 : F ∈ Cc∞ (Rn ) is dense in B˙ m−1,s

(3.18)

Proof. This is a direct consequence of the fact that C ∞ (Ω) → Bsp,q (Ω) and the fact that the trace map (3.13) is bounded and onto for the range of indices speciﬁed in the statement of the corollary.

380

I. Mitrea

It is important to remark here that the extension operator given by (3.14) is universal, i.e., it does not depend on m, p, q, s. Also we would like to mention here a related result in [22] in which, for each 1 < p < ∞ and 0 < s < 1, the authors have constructed extension and trace operators p,p m,p E : B˙ m−1,s (∂Ω) −→ W1−s−1/p (Ω), (3.19) m,p p,p trm−1 : W1−s−1/p (Ω) −→ B˙ m−1,s (∂Ω). The following result gives an alternative, useful for practical purposes, characterization of the higher-order smoothness spaces Ap,q m−1+s+1/p,z for certain ranges of indices, where as is customary, A ∈ {B, F }. Theorem 3.6. Let Ω be a bounded Lipschitz domain in Rn , and assume that m ∈ N, n−1 n < p < ∞, (n − 1)(1/p − 1)+ < s < 1, and min {1, p} ≤ q < ∞. Then 5 J p,q p,q (Ω) = u ∈ Fm−1+s+1/p (Ω) : trm−1 u = 0 and (3.20) Fm−1+s+1/p,z 5 J p,q Cc∞ (Ω) → u ∈ Fm−1+s+1/p (Ω) : trm−1 u = 0 densely. (3.21) Furthermore, a similar result is valid for the scale of Besov spaces. Specif< p < ∞, (n − 1)(1/p − 1)+ < s < 1 and 0 < q < ∞, ically, if m ∈ N, n−1 n then 5 J p,q p,q Bm−1+s+1/p,z (Ω) = u ∈ Bm−1+s+1/p (Ω) : trm−1 u = 0 and (3.22) 5 J p,q (Ω) : trm−1 u = 0 densely. (3.23) Cc∞ (Ω) → u ∈ Bm−1+s+1/p While the left-to-right inclusions in both identities (3.20) and (3.22) are straightforward, the reverse inclusions rely on a delicate argument which involves the Calder´ on extension operator of order m. In the second part of this subsection, we shall discuss certain trace results involving some of the spaces introduced earlier. To set the stage, for a suﬃciently nice function u deﬁned in Ω set m−1 5 J u (X) := (∂ α u) : |α| ≤ m − 1 , (3.24) ∂Ω

∂Ω

where the boundary traces in the right-hand side are taken in the sense of (2.6). Proposition 3.7. Let Ω be a bounded Lipschitz domain in Rn and ﬁx m ∈ N. Assume that u ∈ C ∞ (Ω) is such that N (∇m−1 u) ∈ Lp (∂Ω) for some p ∈ (0, ∞), exists. Then and u|m−1 ∂Ω u|m−1 ∂Ω Lp (∂Ω) ≤ C

m−1

N (∇j u)Lp (∂Ω) .

(3.25)

j=0

In addition,

m−1 u ∈ L˙ pm−1,0 (∂Ω) whenever 1 < p < ∞. ∂Ω

(3.26)

Mapping Properties of Multiple Layers

381

exists is that u ∈ C ∞ (Ω) is A suﬃcient condition that guarantees that u|m−1 ∂Ω m−1 p such that Lu = 0 in Ω and N (∇ u) ∈ L (∂Ω) for some p ∈ (0, ∞), where L is a homogeneous, constant (real) coeﬃcient, symmetric, strongly elliptic diﬀerential operator of order 2m. Proposition 3.8. Let Ω be a bounded Lipschitz domain in Rn . Assume that u ∈ C ∞ (Ω) satisﬁes N (∇m u) ∈ Lp (∂Ω) for some p ∈ ( n−1 n , ∞) and m ∈ N. Then m−1 ∈ h˙ pm−1,1 (∂Ω) and u|m−1 u ˙p ∂Ω h

m−1,1 (∂Ω)

∂Ω

≤C

m

N (∇j u)Lp (∂Ω) . (3.27)

j=0

The limiting cases s ∈ {0, 1} in Theorem 3.4 fail in the general setting of all bounded Lipschitz domains. Nonetheless, if addition information is available, then certain versions of (3.16) do hold. Next, we wish to elaborate on this issue. As a preamble, we shall state a result linking the membership of a function to the Triebel-Lizorkin scale to the integrability properties of its nontangential maximal function. Proposition 3.9. Let Ω be a bounded Lipschitz domain in Rn and assume that L is a homogeneous, constant (real) coeﬃcient, symmetric, strongly elliptic diﬀerential operator of order 2m. Also, ﬁx k ∈ N0 , n−1 0 with the property that m−1+k

p,q N (∇j u)Lp (∂Ω) ≤ CuFm−1+k+1/p (Ω)

(3.28)

j=0 p,q (Ω) ∩ Ker L. for each u ∈ Fm−1+k+1/p

Our next result addresses the converse direction discussed in Proposition 3.9, i.e., passing from nontangential maximal function estimates to membership to Besov spaces. Proposition 3.10. Let Ω be a bounded Lipschitz domain in Rn and assume that 1 < p ≤ 2. Also, let L be a homogeneous, constant (real) coeﬃcient, symmetric, strongly elliptic system of diﬀerential operators of order 2m. If the function u is p,2 (Ω), plus a such that Lu = 0 in Ω and N (∇m−1 u) ∈ Lp(∂Ω), then u ∈ Bm−1+1/p natural estimate. The result below can be viewed as the limiting case s ∈ {0, 1} of (3.16) in Theorem 3.4. Theorem 3.11. Let Ω be a bounded Lipschitz domain in Rn and assume that L is a homogeneous, constant (real) coeﬃcient, symmetric, strongly elliptic diﬀerential operator of order 2m. Then the higher-order trace operator trm−1 introduced in (3.12) extends to a bounded mapping in each of the following cases: p,q trm−1 : Fm−1+1/p (Ω) ∩ Ker L −→ L˙ pm−1,0 (∂Ω),

(3.29)

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I. Mitrea

if 1 < p ≤ 2, 0 < q < ∞, and p,q (Ω) ∩ Ker L −→ h˙ pm−1,1 (∂Ω), trm−1 : Fm+1/p

if

n−1 n

(3.30)

< p ≤ 2, 0 < q < ∞.

In particular, using (3.30), if 1 < p ≤ 2 and 0 < q < ∞, p,q (Ω) ∩ Ker L −→ L˙ pm−1,1 (∂Ω) boundedly, trm−1 : Fm+1/p

(3.31)

and a related result is discussed below. Theorem 3.12. Let Ω ⊂ Rn be a bounded Lipschitz domain satisfying a uniform exterior ball condition. Then for each m ∈ N, the higher-order trace operator trm−1 : F p,2 (Ω) −→ L˙ p (∂Ω) (3.32) m−1,1

m+1/p

is well deﬁned and bounded whenever 1 < p ≤ 2. Theorem 3.12 should be compared with Proposition 3.2 on p. 176 in [18], according to which the following is true. If 1 < p ≤ 2 then there exists a bounded p,2 C 1 domain Ω in the plane and u ∈ F1+1/p (Ω) such that Tr u ∈ / Lp1 (∂Ω). On the 1,1 other hand, any bounded C domain is Lipschitz and satisﬁes a uniform exterior ball condition. Hence, the conclusion in Theorem 3.12 holds for such domains. Other classes of domains for which the conclusion in Theorem 3.12 is valid is the family of bounded, convex domains, and the family of polygons in the plane. p,2 The issue of characterizing Tr F1+1/p (Ω) , when 1 < p < ∞ and Ω is an arbitrary p,2 bounded Lipschitz domain and, more generally, the trace of Fm+1/p (Ω) for m ∈ N, is listed as Problem 3.2.20 on p. 121 of [20]. Related results are in [19]. Here we are able to prove the following.

Proposition 3.13. Let Ω be a bounded Lipschitz domain in Rn . Then the extension operator induces a bounded mapping p,q (Ω) if 2 ≤ p < ∞, 0 < q ≤ ∞, s ∈ {0, 1}. E : L˙ pm−1,s (∂Ω) −→ Fm−1+s+1/p (3.33) As a corollary, if 2 ≤ p < ∞ and 0 < q ≤ ∞, then the following inclusion p,q p,2 (Ω) . In particular, Lp1 (∂Ω) ⊆ Tr F1+1/p (Ω) holds: L˙ pm−1,1 (∂Ω) ⊆ trm−1 Fm+1/p whenever 2 ≤ p < ∞. 3.4. Whitney-BMO and Whitney-VMO spaces Throughout this section let Ω be a bounded Lipschitz domain in Rn . For each m ∈ N, we denote by trm−1 the multi-trace operator onto the boundary of Ω and introduce Pm−1 := the set of polynomials in Rn of degree ≤ m − 1, J 5 P˙ m−1 (E) := trm−1 P : P ∈ Pm−1 , E ⊆ ∂Ω. E

To proceed, recall that barred integrals denote integral averages.

(3.34) (3.35)

Mapping Properties of Multiple Layers Deﬁnition 3.14. For each f˙ ∈ L˙ 2m−1,0 (∂Ω) and X ∈ ∂Ω set ⎧ , 12 ⎫ + ⎨ ⎬ f˙# (X) := sup |f˙(Q) − P˙ (Q)|2 dσ(Q) inf , − ⎭ r>0 ⎩P˙ ∈P˙ m−1 (∂Ω) Sr (X)

383

(3.36)

where Sr (X) := ∂Ω ∩ B(X, r) denotes the surface ball of radius r > 0 centered at X. The higher-order version of the space of bounded mean oscillations is then deﬁned as ˙# ∈ L∞ (∂Ω)}, ˙ m−1 (∂Ω) := {f˙ ∈ L2 BMO m−1,0 (∂Ω) : f

(3.37)

˙ ˙# equipped with the norm f BMO ˙ m−1 (∂Ω) := fL2m−1,0 (∂Ω) + f L∞ (∂Ω) . We next turn our attention to deﬁning Whitney-VMO spaces. As a prelimi∞,∞ (∂Ω) → L˙ ∞ nary observation, we note that for each s ∈ (0, 1), B˙ m−1,s m−1,0 (∂Ω) → ˙ BMOm−1 (∂Ω), linearly and continuously. Inspired by (2.43) we make the following deﬁnition. Deﬁnition 3.15. Let Ω be a bounded Lipschitz domain in Rn , and assume that ˙ m−1 (∂Ω) is the closed subspace of BMO ˙ m−1 (∂Ω) given, for m ∈ N. Then VMO some s ∈ (0, 1), by ˙ m−1 (∂Ω). ˙ m−1 (∂Ω) := the closure of B˙ ∞,∞ (∂Ω) in BMO VMO m−1,s

(3.38)

˙ m−1 (∂Ω) in (3.38) does not actually The fact that the deﬁnition of VMO depend on s ∈ (0, 1) is a direct consequence of the following result. Proposition 3.16. Assume that Ω is a bounded Lipschitz domain in Rn , and let m ∈ N. Then ˙ m−1 (∂Ω) = the closure of trm−1 C ∞ (Rn ) in BMO ˙ m−1 (∂Ω). VMO c

(3.39)

4. The double layer potential 4.1. The fundamental solution For some ﬁxed M ∈ N and m ∈ N, consider a M × M system of homogeneous diﬀerential operators of order 2m with complex constant coeﬃcients: $ % L= ∂ α Aαβ ∂ β = Ljk (4.1) |α|=|β|=m

1≤j,k≤M

$ % where, for each α, β multi-indices of length m, Aαβ = aαβ ∈ CM×M jk 1≤j,k≤M " β and Ljk := |α|=|β|=m ∂ α aαβ jk ∂ . Throughout, we shall use the superscript t to indicate transposition. We now proceed to deﬁne several concepts of ellipticity.

384

I. Mitrea

Deﬁnition 4.1. The operator L in (4.1) is said to be W -elliptic provided that ⎤ ⎡ % $ α β ⎦= aαβ (4.2) 0, ∀ ξ ∈ Rn \ {0}. det ⎣ jk ξ ξ |α|=|β|=m

1≤j,k≤M

Also, the operator L in (4.1) is said to be elliptic whenever it satisﬁes the LegendreHadamard condition. That is, there exists C > 0 such that ⎛ ⎞ M α β ⎠ ≥ C|ξ|2m |η|2 , ∀ ξ ∈ Rn , ∀ η ∈ CM . (4.3) Re ⎝ aαβ jk ξ ξ ηj ηk |α|=|β|=m j,k=1

Finally, L is called S-elliptic provided there exists C > 0 such that ⎛ ⎞ M M α! α 2 α β⎠ aαβ ζ ζ Re ⎝ ≥ C |ζ | , j jk k m! j j=1 |α|=|β|=m j,k=1

|α|=m

(4.4)

for all families of numbers ζjα ∈ C, |α| = m, 1 ≤ j ≤ M . Notice that the Legendre-Hadamard property (4.3) implies W -ellipticity, as introduced in (4.2). Also, (4.3) is equivalent to the strict positivity condition ⎞ ⎛ Aαβ ξ α ξ β ⎠ ≥ C|ξ|2m IM×M , ∀ ξ ∈ Rn , (4.5) Re ⎝ |α|=|β|=m

in the sense of M × M matrices with complex entries (in which case Re A := ¯ , generally speaking). Let us also note that being S-elliptic (A + A∗ )/2, A∗ := (A) is a stronger concept than mere ellipticity. Theorem 4.2. Let L be a homogeneous diﬀerential operator L in Rn of order 2m, m ∈ N, with complex constant coeﬃcients as in (4.1)$ which % satisﬁes (4.2). Then, there exists a matrix of tempered distributions, E = Ejk , such that the 1≤j,k≤M

following hold: (1) For each 1 ≤ j, k ≤ M , Ejk ∈ C ∞ (Rn \ {0}) and Ejk (−X) = Ejk (X) for each X ∈ Rn \ {0}. (2) For each 1 ≤ j, k ≤ M ,

M 0 if j = k, X (4.6) Ljr Erk (X − Y ) = δY (X) if j = k, r=1 where δ stands for the Dirac delta distribution at 0 and the superscript X denotes the fact that the operator Ljs in (4.6) is applied in the variable X. $ % (3) If 1 ≤ j, k ≤ M , then Ejk (X) = Φjk (X) + log |X| Pjk (X) for each X ∈ Rn \ {0}, where Φjk ∈ C ∞ (Rn \ {0}) is a homogeneous function of degree 2m − n, and Pjk is identically zero when either n is odd, or n > 2m, and

Mapping Properties of Multiple Layers

385

is a homogeneous polynomial of degree 2m − n when n ≤ 2m. In fact, with , for each X ∈ Rn there holds C(m, n) := (2π i)n−1 (2m−n)! $ % $ %−1 Pjk (X) = C(m, n) (X · ξ)2m−n ξ α+β Aαβ dσξ . (4.7) 1≤j,k≤M

|α|=|β|=m

S n−1

(4) For each α ∈ Nn0 there exist Cα > 0 such that if either n is odd, or n > 2m, or if |α| > 2m − n then |∂ α E(X)| ≤

Cα n−2m+|α| |X|

(4.8)

or |∂ α E(X)| ≤

Cα (1 + | log |X||) |X|n−2m+|α|

if 0 ≤ |α| ≤ 2m − n,

(4.9)

uniformly for X ∈ Rn \ {0}. (5) One can assign to each diﬀerential operator L as in (4.1)–(4.2) a fundamental $ %t solution EL which satisﬁes (1)–(4) above and, in addition, EL = ELt . (6) The matrix-valued function E has entries which are tempered distributions in is a C ∞ Rn . When restricted to Rn \ {0}, the (matrix-valued) distribution E function and, moreover, $ %−1 E(ξ) = (−1)m ξ α+β Aαβ for each ξ ∈ Rn \ {0}. (4.10) |α|=|β|=m

We conclude with a few conventions which are going to play a role later on. $ % Given a coeﬃcient tensor A = Aαβ , deﬁne At by |α|=|β|=m

t

t

(A )αβ := (Aβα ) ,

∀ α, β ∈ Nn0 : |α| = |β| = m.

(4.11)

In particular, A = At ⇐⇒ Aαβ = (Aβα )t A = At =⇒ L is self-adjoint.

∀ α, β ∈ Nn0 : |α| = |β| = m,

(4.12) (4.13)

4.2. Deﬁnition and nontangential maximal estimates Let L be a homogeneous diﬀerential operator of order 2m, with constant (possibly matrix-valued) coeﬃcients: ∂ α Aαβ ∂ β (4.14) L := |α|=|β|=m

and consider the quadratic form associated with (the representation of) L in (4.14) B(u, v) := Aαβ ∂ β u, ∂ α v. (4.15) |α|=|β|=m

386

I. Mitrea

Also, ﬁx a bounded Lipschitz domain Ω ⊂ Rn with outward unit normal ν and surface measure dσ, as well as two arbitrary functions u, v, which are suﬃciently * well behaved on Ω. Starting with Ω B(u, v)dX and successively integrating by parts (as to passing on to u all partial derivatives acting on v), we eventually arrive at Aαβ ∂ β u, ∂ α v dX = (−1)m Lu, v dX + boundary terms. (4.16) Ω

|α|=|β|=m

Ω

It is precisely the boundary terms which will determine the actual form of the double potential layer operator, which we shall deﬁne momentarily. For now, we record the following. Proposition 4.3. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and that L is as in (4.14). Then Aαβ ∂ β u, ∂ α v dX = (−1)m Lu, v dX (4.17) |α|=m

|β|=m

|α|=|β|=m Ω

Ω

−

m

|α|=|β|=m k=1

(−1)k α!(m−k)!(k−1)! m!γ!δ!

γ+δ+ej =α

νj Aαβ ∂ β+γ u, ∂ δ v dσ, ∂Ω

|γ|=k−1, |δ|=m−k

for any CM -valued functions u, v which are reasonably behaved in Ω. The following deﬁnition is inspired by the format of the boundary integral in (4.17), in which formally, the function u is replaced by E(X − ·), the family of derivatives (∂ δ v)|δ|≤m−1 is replaced by the Whitney array f˙ = (fδ )|δ|≤m−1 . Deﬁnition 4.4. Let Ω ⊂ Rn be a Lipschitz domain with outward unit normal ν = (νj )1≤j≤n and surface measure dσ. Also, let L be a constant coeﬃcient, homogeneous, elliptic diﬀerential operator of order 2m and denote by E a fundamental solution for L in Rn . Then the double layer potential operator acting on f˙ = {fδ }|δ|≤m−1 is given by D˙ f˙(X):= −

m

|α|=m

|β|=m

k=1

α!(m−k)!(k−1)! m!γ!δ!

(4.18)

|δ|=m−k |γ|=k−1

γ+δ+ej =α

×

νj (Y )(∂ β+γ E)(X − Y )Aβα fδ (Y ) dσ(Y )

∂Ω

for X ∈ Rn \ ∂Ω. As a practical matter, Df˙(X) is obtained by formally replacing, in the boundary integral in (4.17), the function u by E(X − ·), the family of derivatives (∂ δ v)|δ|≤m−1 by the Whitney array f˙ = (fδ )|δ|≤m−1 , and then multiplying everything by (−1)m−1 . The reason for which this deﬁnition is natural is as follows.

Mapping Properties of Multiple Layers

387

Proposition 4.5. There holds L(D˙ f˙)(X) = 0 for each X ∈ Rn \ ∂Ω. Also, for each F ∈ Cc∞ (Rn ), there holds ˙ m−1 (F ))(X) (4.19) (∂ β E)(X − Y )Aβα (∂ α F )(Y ) dY = F (X) − D(tr |α|=|β|=m Ω

if X ∈ Ω, and

˙ m−1 (F ))(X) (∂ β E)(X − Y )Aβα (∂ α F )(Y ) dY = −D(tr

(4.20)

|α|=|β|=m Ω

¯ if X ∈ Rn \ Ω. When attempting to solve the Dirichlet problem for a higher-order diﬀerential operator L as in (1.1) in the sense of B. Dahlberg (i.e., with nontangential maximal function estimates, see [8]) using the method of layer potentials (as it was done by G. Verchota in [39] in the case when L = Δ), the following result is of most signiﬁcant importance. Up to now such results have been known only for layer potential operators associated with second-order operators. Theorem 4.6. For Ω and L as above, consider the double layer potential operator deﬁned in (4.18). Then for each 1 0 such that m−1 j=0 m

N (∇j D˙ f˙)Lp (∂Ω) ≤ Cf˙L˙ p

m−1,0 (∂Ω)

, (4.21)

N (∇ D˙ f˙)Lp (∂Ω) ≤ Cf˙L˙ p j

m−1,1 (∂Ω)

j=0

.

. Also, for any p ∈ n−1 n , 1 , there exists a ﬁnite constant C = C(Ω, p, L) > 0 such that m ˙ ˙p N (∇j D˙ f˙)Lp (∂Ω) ≤ Cf (∂Ω). (4.22) h m−1,1

j=0

Proof. We outline here the general course of the proof and we start with the ﬁrst estimate in (4.21). The starting point is the identity ˙ f˙)(X) = D(

n ¯ |α|=|β|=m R \Ω

(∂ β E)(X − Y )Aβα (∂ α F )(Y ) dY,

∀ X ∈ Ω, (4.23)

valid whenever F ∈ Cc∞ (Rn ) and f˙ := trm−1 (F ). This is readily seen from (4.20) ¯ reversed (here it is useful to keep in mind that with the roles of Ω and Rn \ Ω the outward unit normal of the latter domain is −ν). While our goal is to put derivatives up to order m − 1 on D˙ in (4.23) and establish nontangential maximal function estimates, it should be noted that, based on (4.18), D˙ appears to be

388

I. Mitrea

already saturated (as the worst type of singularity for which one can prove nontangential maximal function estimates is |X − Y |−(n−1) which occurs when k = m in (4.18)). The ﬁrst order of business is to arrange matters in the expression of ∂ γ D˙ f˙(X), for multi-indices γ of length ≤ m−1, in such a way that the extra partial derivatives γ with respect to X, ∂X , acting on the fundamental solution E are replaced by partial derivatives with respect to the variable of integration Y (we need a partial derivative with respect to Y if there is any hope to integrate by parts to transfer the extra derivatives ∂Yγ acting on E onto the derivatives of F ). This is followed by an iterative trade of derivatives (done via integration by parts): one derivative from ∂Yα is moved from F onto E while one derivative from ∂Yγ is moved from E onto F . When all ∂ γ is exhausted, the remaining derivatives from ∂ α (which have not been already moved onto E) are ﬁnally transfered onto E via integration by parts. This ultimately leads to γ ˙ ˙ ∂ D f (X) = (LE)(X − Y )(∂ γ F )(Y ) dY + boundary terms ¯ Rn \Ω

=

boundary terms

(4.24)

¯ as (LE)(X − Y ) = 0 for X = Y , which is the case as X ∈ Ω and Y ∈ Rn \ Ω. α γ The delicate exchange of derivatives (one at a time from ∂ and ∂ ) is needed to insure that the resulting integral kernels in the boundary terms in (4.24) are not hypersingular. The idea is to employ the following result to establish nontangential maximal function estimates for the most singular boundary terms. This is essentially due to R. Coifman, A. McIntosh and Y. Meyer, see [6], and states that if Ω is a bounded Lipschitz domain in Rn and T is a singular integral operator given by k(X − Y )f (Y ) dσ(Y ), X ∈ Ω, (4.25) T f (X) := ∂Ω

where k ∈ C N (Rn \ {0}) such that k(−X) = −k(X)

(4.26) and k(λX) = λ−(n−1) k(X) ∀ λ > 0, ∀ X ∈ Rn \ {0}. . / then for each p ∈ n−1 n , ∞ there exists a ﬁnite constant C = C(∂Ω, p, n) > 0 such that N (T g)Lp (∂Ω) ≤ Ck|S n−1 C N ghpat(∂Ω) , ∀ g ∈ hpat (∂Ω).

(4.27)

Another important feature of the swapping of derivatives process described earlier is that at the end of each swapping cycle the boundary terms combine in such a fashion as to produce a tangential derivative acting on derivatives of order up to m + |γ| − 1 of the fundamental solution E. All in all the treatment of ∂ γ D˙ f˙(X)

Mapping Properties of Multiple Layers

389

as described above leads to the ability to express it as C(j, k, ω, δ) ∂τjk (∂ ω E)(X − Y )fδ (Y ) dσ(Y ) (4.28) ∂ γ D˙ f˙(X) = ∂Ω

1≤j,k≤n,|δ|≤m−1

|ω|≤m+|γ|−1

+

˜ η, δ) C(j,

νj (Y )(∂ η E)(X − Y )fδ (Y ) dσ(Y ), ∂Ω

1≤j≤n,|δ|≤m−1

m+|γ|≤|η|≤2m−1

˜ η, δ) are constants depending explicitly on j, k, ω, δ where C(j, k, ω, δ) and C(j, and j, η, δ, respectively. The top singularities (|ω| = 2m − 2 when |γ| = m − 1 and |η| = 2m − 1) in (4.28) are then handled using (4.27). This ﬁnishes the proof of the ﬁrst estimate in (4.21). Turning our attention to proving the second estimate in (4.21) and (4.22) we remark that when f˙ ∈ L˙ pm−1,1 or f˙ ∈ h˙ pm−1,1 (∂Ω) all the components of the array f˙ can absorb an extra tangential derivative via integration by parts. This is the reason for which the terms in (4.28) containing ∂τjk are well suited for this process. Indeed, if Ω is a bounded Lipschitz domain in Rn , and if 1 < p, p < ∞ satisfy 1/p + 1/p = 1, and 1 ≤ j, k ≤ n, then (∂τjk f ) g dσ = f (∂τkj g) dσ (4.29) ∂Ω

∂Ω

for every f ∈ Lp1 (∂Ω) and g ∈ Lp1 (∂Ω). This fact allows us to consider |γ| ≤ m in this case (i.e., include one extra derivative when compared to the previous L˙ pm−1,0 (∂Ω) case). Once again we appeal to (4.27) to ﬁnish the proof of this case. The Dirichlet problem with BMO data in C 1 domains has been solved in the case of the Laplacian using the layer potential method by E. Fabes and C. Kenig in [12]. A fundamental role in their proof is played by the fact that the classical harmonic double layer potential operator D maps BMO(∂Ω) into densities of Carleson measures. A version of this result when m = 2 and L = Δ2 has been considered by J. Cohen in [4]. Our goal is to deal with the more general setting, i.e., that of operators of any order. Before stating this result, recall (2.4). We have Theorem 4.7. Consider an elliptic homogeneous diﬀerential operator L of order 2m with real constant coeﬃcients which is symmetric. Let Ω be a bounded Lipschitz domain in Rn and for each X ∈ Ω denote by ρ(X) its distance to ∂Ω. Let D˙ be a double layer potential associated with L. Then there exists a ﬁnite constant C = C(Ω, L) > 0 with the property that |∇m D˙ f˙(X)|2 ρ(X) dX

is a Carleson measure on Ω, with ˙ m−1 (∂Ω). Carleson constant ≤ Cf˙2BMO for each f˙ ∈ BMO ˙ (∂Ω) m−1

(4.30)

390

I. Mitrea

Also, |∇m D˙ f˙(X)|2 ρ(X) dX is a vanishing Carleson measure on Ω for each f˙ ∈ ˙ m−1 (∂Ω). VMO Proof. Fix Xo ∈ ∂Ω and r > 0 and let Δr (Xo ) := Ω∩B(Xo , r) be the Carleson box associated with the surface ball Sr (Xo ), and denote by |Δr (Xo )| its n-dimensional Lebesgue measure. As in the past, |Sr (Xo )| denotes the surface measure of Sr (Xo ). ˙ m−1 (∂Ω) We need to show that there exists C > 0 such that for each f˙ ∈ BMO 1 ρ(X)|∇m D˙ f˙(X)|2 dX ≤ C(f˙# (Xo ))2 . (4.31) |Sr (Xo )| Δr (Xo ) The idea is to split f into several pieces f˙ = η(f˙ − ω˙ 2r )+(1−η)(f˙ − ω˙ 2r )+ ω˙ 2r where the function η deﬁned in Rn is supported in B4r (Xo ), is identically one in B2r (Xo ), and ∂ α η ≈ r−|α| for each multi-index α ∈ Nn0 . Also, ω˙ 2r ∈ P˙ m−1 := trm−1 Pm−1 is a so-called ‘best ﬁt’ polynomial array to f˙ on S2r (Xo ). Since D˙ reproduces the highest degree polynomial when acting on polynomial arrays from P˙ m−1 , we obtain ˙ f˙ − ω˙ 2r )] + ∇m D[(1 ˙ ∇m D˙ f˙ = ∇m D[η( − η)(f˙ − ω˙ 2r )], as ∇m D˙ ω˙ 2r = ∇m ω2r = 0. 2 Consequently, using that (a + b) ≤ 2a2 + 2b2 , matters reduce to proving that ˙ m−1 (∂Ω), I ≤ C(f˙# (Xo ))2 and II ≤ C(f˙# (Xo ))2 , ∀ f˙ ∈ BMO (4.32) where I and II are obtained by replacing in the left-hand side of the inequality (4.31) the array f˙ by η(f˙ − ω˙ 2r ) and by (1 − η)(f˙ − ω˙ 2r ), respectively. Since the contribution to the Carleson expression we need to estimate looks ˙ f˙ − ω˙ 2r )], I is handled via area function estimates like the area function of u := D[η( due to B. Dahlberg, C. Kenig, J. Pipher and G. Verchota (see p. 1432 in [9]) which guarantees that given an elliptic system L of homogeneous, constant realcoeﬃcient, diﬀerential operators of order 2m, there exists C = C(Ω, L) > 0 such that ρ(X)|∇m u(X)|2 dX ≤ C |N (∇m−1 u)(Q)|2 dσ(Q), (4.33) Ω

∂Ω 2m

for any vector-valued function u ∈ C (Ω) such that Lu = 0 in Ω. We apply this to u as introduced above and then we use the nontagential maximal function estimates (with p = 2) established in (4.21), the fact that η is supported in B4r (Xo ) and the deﬁnition of the sharp maximal function from (3.36) to obtain the ﬁrst estimate in (4.32). As for handling the term II, the approach using the area function estimate is no longer helpful due to the large support of the components of the array (1 − η)(f˙ − ω˙ 2r ). However, it is straightforward to see that in this case it suﬃces to establish that there exists a ﬁnite constant C = C(Ω, L) > 0 such that for each ˙ m−1 (∂Ω) f˙ ∈ BMO C ˙ − η)(f˙ − ω˙ 2r )](X)| ≤ f˙BMO |∇m D[(1 if X ∈ Δr (Xo ). (4.34) ˙ m−1 (∂Ω) r ˙ − η)(f˙ − ω˙ 2r )](X) into a telescopic sum This is done by decomposing ∇m D[(1 of integrals over boundary dyadic cubes and using decay estimates, the key fact

Mapping Properties of Multiple Layers

391

being the separation between X and the variable of integration Y (which, due to the presence of the factor 1 − η is away from S2r (Xo )). Here the properties of ω˙ 2j r of being ‘best ﬁt’ to f on S2j r (Xo ), j = 1, 2, . . . , are used in order to enhance the decay of the integral kernels for the telescopic sums. In the case of the Laplacian this idea was ﬁrst carried out by E. Fabes and C. Kenig (see [12]) but in the present case we have to overcome several signiﬁcant diﬃculties (for example we need to establish a Poincar´e type inequality but not for functions but rather for Whitney arrays). 4.3. Jump relations Our goal is to study the boundary behavior of the multiple layer potential operator D˙ which is given by the so-called jump relations. The jump relations for the double layer potential operator associated with second-order elliptic systems/equations are a trademark feature of the layer potential theory in that setting and play a key role in its successful implementation for the treatment of the Dirichlet problem for second-order diﬀerential operators in smooth and Lipschitz domains. In this subsection we establish jump relations in the higher degree setting, a process in which we need to overcome a number of diﬃculties of both algebraic and analytic nature. Throughout this subsection, we assume that Ω ⊂ Rn is a bounded, Lipschitz domain, and that L is a diﬀerential operator of order 2m as discussed in detail in Section 4.1. We start with the following deﬁnition. Deﬁnition 4.8. For each f˙ ∈ L˙ pm−1,0 (∂Ω), 1 < p < ∞, deﬁne 5. / J (4.35) K˙ f˙ := K˙ f˙ γ |γ|≤m−1

where, for each γ ∈ Nn0 of length ≤ m − 1, we have set |γ| . / K˙ f˙ γ (X) := |α|=m

|β|=m

=1

δ+η+ek =α

θ+ω+ej =γ

C1 (m, , α, δ, η, γ, θ, ω)

|δ|=−1,|η|=m− |θ|=−1,|ω|=|γ|−

$ % ∂τkj (Y ) (∂ δ+ω+β E)(X − Y ) Aβα fθ+η (Y ) dσ(Y )

× lim

ε→0+ Y ∈∂Ω

|X−Y |>ε

−

m

|α|=m

=|γ|+1

δ+η+ek =α

|β|=m

C2 (m, , α, δ, η)

(4.36)

|δ|=−1,|η|=m−

νk (Y )(∂ δ+β E)(X − Y )Aβα fγ+η (Y ) dσ(Y ),

× lim

ε→0+ Y ∈∂Ω

|X−Y |>ε

for X ∈ ∂Ω. Above C1 (m, , α, δ, η, γ, θ, ω) := C2 (m, , α, δ, η) :=

α!(m−)!(−1)! . m! δ! η!

α!(m−)!(−1)!γ!(−1)!(|γ|−)! , m! δ! η! |γ|! θ! ω!

and

392

I. Mitrea

The essence of the method of layer potentials resides in reducing the Dirichlet problem (1.4) for the operator L as in (1.1) to a boundary integral equation using a multiple layer representation of the solution u := D˙ g˙ where g˙ belongs to a suitable Whitney arrays space (this choice ensures Lu = 0 in Ω regardless of g). ˙ The m−1 ? Here ·| denotes the nontangential question then becomes: what is D˙ g| ˙ m−1 ∂Ω ∂Ω boundary multi-trace from (3.24). It should be noted that trace results of this nature have only been known for second-order operators (i.e., when m = 1) and when m = 2 and L = Δ2 (see J. Cohen and J. Gosselin’s paper [5] for this latter case). Our next theorem provides an answer to this basic question. To state it, recall the double layer potential operator deﬁned in (4.18), the convention (2.1), and set $ % D˙ ± f˙ := D˙ f˙ . (4.37) Ω±

˙ In the sequel we will sometimes abbreviate D˙ + by D. Theorem 4.9. For each f˙ ∈ L˙ pm−1,0 (∂Ω), 1 < p < ∞, the expression K˙ f˙(X) is meaningful at almost every X ∈ ∂Ω. The following operators are well deﬁned, linear and bounded K˙ : L˙ pm−1,0 (∂Ω) −→ L˙ pm−1,0 (∂Ω), K˙ : h˙ pm−1,1 (∂Ω) −→ h˙ pm−1,1 (∂Ω), Moreover the following jump-relation holds m−1 ˙ f˙, ∀ f˙ ∈ L˙ p D˙ ± f˙ = (± 12 I + K) m−1,0 (∂Ω), ∂Ω

p ∈ (1, ∞), p∈

(4.38)

( n−1 , ∞). n

(4.39)

1 < p < ∞.

(4.40)

In particular, for each p ∈ (1, ∞), the operator K˙ : L˙ pm−1,1 (∂Ω) −→ L˙ pm−1,1 (∂Ω)

(4.41)

is also well deﬁned, linear and bounded. As a corollary, for each j ∈ {0, 1} the operator $ %∗ $ %∗ K˙ ∗ : L˙ pm−1,j (∂Ω) −→ L˙ pm−1,j (∂Ω) (4.42) is also well deﬁned, linear, bounded operators for each p ∈ (1, ∞). Proof. At the core of the proof of the jump formula (4.40) is the following result, see, e.g., [10], [23]. If T is a singular integral operator as in (4.25)–(4.26) and for each ε > 0 we set k(X − Y )f (Y ) dσ(Y ), X ∈ ∂Ω, (4.43) Tε f (X) := Y ∈∂Ω

|X−Y |>ε

then, for each p ∈ (1, ∞), f ∈ Lp (∂Ω), the limit T f (X) := lim Tε f (X) ε→0+

(4.44)

Mapping Properties of Multiple Layers exists for a.e. X ∈ ∂Ω, and the jump-formula ˆ (X) + T f (X) (X) = ± 2√1−1 k(ν(X))f T f

393

(4.45)

∂Ω±

is valid at a.e. X ∈ ∂Ω, where the boundary trace in the left-hand side is taken in the nontangential sense from (2.6). Here the ‘hat’ denotes Fourier transform. When we take the multi-trace of D˙ f˙ we have to consider ordinary nontangential traces to the boundary of ∂ γ D˙ f˙ for |γ| ≤ m − 1. The idea is to appeal to the representation formula (4.28) and to notice that the integral operators in (4.28) are either weakly singular (thus they do not jump) or amenable to (4.45). A careful analysis based on (4.45) shows that, due to weak singularity or to the presence of ∂τjk in the kernel, the integral operators in the ﬁrst sum in (4.28) do not jump. As for the Calder´ on-Zygmund integral operators in the second sum in (4.28), which are the most singular in this sum and which correspond to |η| = 2m − 1, their jump is calculated using (4.45) and the fact that, in general, −1 √ √ |η| η ˆ |η|+2m η α+β η 7 ∂ E(ξ) = ( −1) ξ E(ξ) = ( −1) ξ Aαβ ξ . (4.46) |α|=|β|=m

Thus, matters reduce to showing that for each multi-index γ of length ≤ m − 1, there holds √ fδ (X)ν(X)η+ej 1 ˜ η, δ)( −1)|η|+2m f (X) = . (4.47) C(j, γ 2 Aαβ ν(X)α+β 1≤j≤n,|δ|≤m−1 |η|=2m−1

|α|=|β|=m

What is challenging is the fact that is of paramount importance to have ˜ η, δ) (as opposed to the proof of Theoexplicit expressions for the constants C(j, rem 4.6 where just the existence of such constants will do as far as the nontangential estimates (4.21) proved there are concerned) and the fact that they turn out to have a complicated combinatorial ﬂavor. Turning attention to (4.38), this is a consequence of (4.40) and the nontangential maximal function estimates proved in Theorem 4.6, since generally speaking, on the boundary, the Lp norm of the trace of a function is ≤ the Lp norm of the nontangential maximal operator of that function. The following result regards the mapping properties of the boundary operator K˙ on the space of BMO and VMO Whitney arrays. Theorem 4.10. Let Ω be a bounded Lipschitz domain in Rn and let K˙ be the boundary double layer potential operator given in (4.36). Then, ˙ m−1 (∂Ω), ˙ m−1 (∂Ω) −→ BMO K˙ : BMO ˙ m−1 (∂Ω) −→ VMO ˙ m−1 (∂Ω), K˙ : VMO are well deﬁned, linear and bounded.

(4.48) (4.49)

394

I. Mitrea

Proof. Since due to (4.38) we have that K˙ f˙ ∈ L˙ 2m−1,0 (∂Ω) for each array f˙ ∈ ˙ m−1 (∂Ω), matters reduce to showing that the sharp maximal function of BMO ˙ f˙, is in L∞ (∂Ω) whenever f˙ ∈ BMO ˙ m−1 (∂Ω). K˙ f˙, or equivalently that of ( 12 I + K) When m = 1 this argument is due to E. Fabes and C. Kenig (see [12]). Here we sketch the proof of the boundedness of the operator (4.48) in the ˙ m−1 (∂Ω). Much as in the proof higher-order setting. Fix Xo ∈ ∂Ω and f˙ ∈ BMO of Theorem 4.7, and with the same notation, we split f˙ into several pieces and use the fact that 12 I + K˙ reproduces polynomial arrays from P˙ m−1 . Thus, up to a ˙ f˙ can be expressed as I + II where polynomial array, ( 12 I + K) . / ˙ η(f˙ − ω˙ 2r ) (·) I(·) := ( 12 I + K) (4.50) . / ˙ (1 − η)(f˙ − ω˙ 2r ) (·). II(·) := ( 1 I + K) 2

The conclusion that the operator in (4.48) is bounded follows then from the claims that there exists C > 0, independent of Xo , f˙ and r > 0, such that ,1/2 + 2 |I(Y )| dσ(Y ) ≤ C f˙# (Xo ), (4.51) − Sr (Xo )

and such that there exists a polynomial array P˙ ∈ P˙ m−1 satisfying +

,1/2 |II(Y ) − P˙ (Y )| dσ(Y )

−

2

≤ C f˙# (Xo ),

(4.52)

Sr (Xo )

The treatment of (4.51) relies on the boundedness of the operator K˙ on L˙ 2m−1,0 (∂Ω) established in Theorem 4.9 and the special properties of the ‘best ﬁt’ polynomial ω˙ 2r . The remaining challenge now is the construction of the polynomial array P˙ for which (4.52) holds and this is done using (up to degrees . Taylor like expansions / ˙ (1 − η)(f˙ − ω˙ 2r ) . ≤ m − 1) about Xo of the array ( 12 I + K) 4.4. Estimates on Besov and Triebel-Lizorkin spaces Throughout this subsection, we shall continue to assume that Ω is a bounded Lipschitz domain in Rn , and that the diﬀerential operator L is as in Section 4.1. Also, recall that D˙ and K˙ have been deﬁned in (4.4) and (4.8), respectively. The following result is important for solving the Dirichlet problem with data in Besov spaces (when m = 1 and L = Δ this has been established by D. Jerison and C. Kenig in [18] using harmonic measure techniques) and by E. Fabes, O. Mendez and M. Mitrea in [13] using layer potential techniques. Also, these mapping properties of the multi-layer potential operator D˙ on Besov spaces are used in the proof of the multi-trace result from Theorem 3.4.

Mapping Properties of Multiple Layers Theorem 4.11. For operators

n−1 n

395

< p ≤ ∞, 0 < q ≤ ∞, (n − 1)( p1 − 1)+ < s < 1, the

p,q p,q D˙ : B˙ m−1,s (∂Ω) −→ Bm−1+s+1/p (Ω) p,p p,q D˙ : B˙ (∂Ω) −→ F (Ω) m−1,s

(4.53)

m−1+s+1/p

are well deﬁned, linear and bounded (in the second case, with the additional convention that q = ∞ if p = ∞). Finally, similar properties hold for ψ D˙ − (cf. the convention (4.37)), for any cutoﬀ function ψ ∈ Cc∞ (Rn ). Our next result deals with the mapping properties of the boundary multiple ˙ when considered on Whitney-Besov spaces. Recall layer K˙ and its relation with D, (4.37). ˙ originally introTheorem 4.12. The boundary multiple layer potential operator K, duced in (4.8), extends to a bounded mapping p,q p,q (∂Ω) −→ B˙ m−1,s (∂Ω) (4.54) K˙ : B˙ m−1,s . / whenever n−1 n < p ≤ ∞, (n − 1) 1/p − 1 + < s < 1, and 0 < q ≤ ∞. In addition, with the same assumptions on p, q and s, p,q trm−1 ◦ D˙ ± = ± 21 I + K˙ in B˙ m−1,s (∂Ω).

(4.55)

Proof. The boundedness of the operator K˙ from (4.54) follows in the case when 1 < p = q < ∞ and s ∈ (0, 1) via interpolation by the real method between (4.38) and (4.41) in Theorem 4.9. In order for this to work, as a preamble and of independent interest, we need to establish an interpolation result according to which real interpolation between Whitney-Lebesgue and Whitney-Sobolev spaces gives rise to diagonal Whitney-Besov spaces. Furthermore, the jump formula (4.55) p,p (∂Ω) and coincide holds since the two operators involved are bounded on B˙ m−1,s p on L˙ m−1,1 (∂Ω) which, under the current assumptions on the indices, is densely p,p (∂Ω), thanks to (3.18). The challenge here is to extend embedded into B˙ m−1,s these results to the non-diagonal case and to indices p < 1 which is done relying on a more general version of the aforementioned interpolation result. For further reference, let us also record here the following useful result. ˙ originally Corollary 4.13. Recall the boundary multiple layer potential operator K, introduced in (4.8). Then its adjoint extends to a bounded mapping $ %∗ $ %∗ p,q p,q K˙ ∗ : B˙ m−1,s (∂Ω) −→ B˙ m−1,s (∂Ω) , (4.56) whenever 1 ≤ p, q ≤ ∞ and 0 < s < 1. Our last result in this subsection deals with the limiting cases s = 0 and s = 1 in (4.53). To state it, recall that p ∨ 2 = max {p, 2}.

396

I. Mitrea

Theorem 4.14. The operators p,p∨2 D˙ : L˙ pm−1,s (∂Ω) −→ Bm−1+s+1/p (Ω),

1 < p < ∞,

p,q (Ω), D˙ : L˙ pm−1,s (∂Ω) −→ Fm−1+s+1/p

(4.57)

2 ≤ p < ∞, 0 < q ≤ ∞, (4.58)

are well deﬁned, linear and bounded if either s = 0, or s = 1.

5. The single layer operator 5.1. Deﬁnition and nontangential maximal estimates The aim of this section is to deﬁne and study the main properties of the single layer potential operator. As before, we continue to assume that the diﬀerential operator L of order 2m is as in the ﬁrst part of Section 4.1. Deﬁnition 5.1. Let Ω be a bounded Lipschitz in Rn , and ﬁx p ∈ (1, ∞). The action of the%single layer potential operator S on an arbitrary functional Λ ∈ $ ∗ (∂Ω) is given by L˙ p m−1,1

˙ (SΛ)(X) := =

K

L trm−1 [E(X − ·)] , Λ K5 (−1)|α| (∂ α Ej• )(X − ·)

(5.1) L

J ∂Ω

|α|≤m−1

,Λ

, 1≤j≤M

where X ∈ Rn \ ∂Ω, and Ej• is the jth row in E. In (5.1), trm−1 E is deﬁned as a M × M matrix with entries arrays, whose rows are obtained by applying trm−1 to each row of E. Upon recalling the convention (2.1), we set $ % ˙ . S˙ ± Λ := SΛ (5.2) Ω±

˙ We shall occasionally abbreviate S˙ + by S. Next, it can then be easily checked that, if 1/p + 1/p = 1, ˙ (−1)|α| (∂ α E)(X − Y )fα (Y ) dσ(Y ), ∀ X ∈ Rn \ ∂Ω, SΛ(X) = |α|≤m−1

∂Ω

whenever Λ = (fα )|α|≤m−1

$ %∗ (5.3) p ˙ ∈ L (∂Ω) ⊕ · · · ⊕ L (∂Ω) → Lm−1,1 (∂Ω) . p

p

For future reference, let us point out here that, as a simple consequence of the Hahn-Banach theorem, we have that the space 5 J . /∗ Λ ∈ L˙ pm−1,1 (∂Ω) : Λ = (fα )|α|≤m−1 ∈ Lp (∂Ω) ⊕ · · · ⊕ Lp (∂Ω) (5.4) %∗ $ ˙ = 0 in Rn \ ∂Ω, for every funcembeds densely in L˙ pm−1,1 (∂Ω) . Clearly, L(SΛ) $ %∗ $ %∗ $ %∗ tional Λ ∈ L˙ pm−1,1 (∂Ω) . Since L˙ pm−1,0 (∂Ω) → L˙ pm−1,1 (∂Ω) , it follows

Mapping Properties of Multiple Layers

397

%∗ $ ˙ =0 that S˙ has also a well-deﬁned pointwise action on L˙ pm−1,0 (∂Ω) , and L(SΛ) holds in that context as well. G. Verchota has used the single layer potential operator S to solve the regularity problem for the Laplacian with Lp1 (∂Ω) data. When L = Δ, fundamental properties of the single layer potential operator such as nontangential maximal function estimates and jumps can be deduced without much eﬀort from the results of R. Coifman, A. McIntosh and Y. Meyer in [6]. By way of contrast, when dealing with a general higher-order diﬀerential operator L, the nature of S˙ is considerably more subtle. For example, the natural setting in which the boundary version of the single layer potential operator% acts naturally is that of a dual of a $ ∗ p ˙ Whitney array space, namely Lm−1,0 (∂Ω) . Note that when m = 1 (i.e., for . /∗ second-order diﬀerential operators) this dual space is Lp (∂Ω) = Lp (∂Ω) with 1/p + 1/p = 1, i.e., a simple Lebesgue space. In the higher degree case this simple identiﬁcation is lost. The next two results regard nontangential maximal estimates, mapping properties, and jump formulas for the operator S˙ in the Lp context. Theorem 5.2. Let Ω be a bounded Lipschitz in Rn and assume that 1 < p, p < ∞ satisfy 1p + p1 = 1. Then there exists a ﬁnite constant C = C(Ω, p) > 0 such that m

˙ N (∇j SΛ) Lp (∂Ω) ≤ CΛ(L ˙ p

m−1,0 (∂Ω))

j=0 m−1

˙ N (∇j SΛ) Lp (∂Ω) ≤ CΛ(L ˙ p

m−1,1 (∂Ω))

j=0

(5.5)

∗

∗

.

(5.6)

Proof. As a preliminary matter we show that, based on the Riesz Representation /∗ . Theorem, for each Λ ∈ L˙ pm−1,0 (∂Ω) there exists a unique N -tuple (where N is p the cardinality of the set {α ∈ Nn0 : |α| ≤ m − 1}) g := (gγ )|γ|≤m−1 ∈ ⊕N 1 L (∂Ω), such that g⊕N ≤ Λ(L˙ p p 1 L (∂Ω)

m−1,0 (∂Ω))

∗

.

(5.7)

and, for each X ∈ Rn \ ∂Ω, ˙ SΛ(X) = trm−1 [E(X − ·)] , Λ (trm−1 [E(X − ·)])α gα dσ. = |α|≤m−1

(5.8)

∂Ω

Then (5.5) is a direct consequence of the classical nontangential maximal function estimates (4.27) and (5.7). Consider next Λ ∈ (L˙ pm−1,1 (∂Ω))∗ . Much as before, relying on Hahn-Banach and Riesz Theorems, we may conclude the existence of g = {gα }|α|≤m−1 , g ∈

398

I. Mitrea

p g is controlled by that of Λ and ⊕N 1 L−1 (∂Ω) with the property that the norm of that, for each, X ∈ Rn \ ∂Ω, there holds n L K α ˙ (−1)|α| (∂ α E)(X − ·) , (∂τij gij ) SΛ(X) = |α|≤m−1 i,j=1

+

(−1)

|α|

∂Ω

|α|≤m−1

(∂ α E)(X − Y )g∗α (Y ) dσ(Y ).

(5.9)

α ∈ Lp (∂Ω) such that Here, for each α ∈ Nn0 with |α| ≤ m − 1, there exist g∗α , gij "n α α α α gα = i,j=1 ∂τij gij + g∗ , with the norms of gij and g∗ controlled by that of gα . Integration by parts in the ﬁrst term of the right-hand side of (5.9) and classical Calder´ on-Zygmund theory (see again (4.27)) ﬁnish then the proof.

Theorem 5.3. Lipschitz domain in Rn . Then, for each 1 < p < %∗ $ Let Ω be a bounded p ∞ and Λ ∈ L˙ m−1,1 (∂Ω) , m−1 m−1 ˙ ˙ := SΛ ˙ = SΛ , (5.10) SΛ ∂Ω+

∂Ω−

is well deﬁned. Moreover, if 1/p + 1/p = 1, then the following operators are bounded %∗ $ S˙ : L˙ pm−1,1 (∂Ω) −→ L˙ pm−1,0 (∂Ω) $ %∗ (5.11) S˙ : L˙ p (∂Ω) −→ L˙ p (∂Ω). m−1,0

m−1,1

$ %t Also, if S˙ L indicates that the single layer is associated with L, etc., then S˙ L = S˙ Lt where S˙ L , S˙ Lt are taken in the context of (5.11). In particular, when A = At (cf. (4.12)), then S˙ is invariant under transposition (in the sense that the transposed of the ﬁrst operator in (5.11) is the second one). Proof. To prove this result we make use of the representation formulas (5.8) and (5.9) established in Theorem 5.2. Then we apply the jump formula (4.45) and this has to be done in a controlled fashion so that we can identify the collective contributions of all the jump terms. We conclude this section by discussing Carleson measure estimates for the single layer potential. Theorem 5.4. Let L be an elliptic homogeneous diﬀerential operator of order 2m with real constant coeﬃcients which is symmetric. Assume that Ω is a bounded Lipschitz domain in Rn and for each X ∈ Ω denote by ρ(X) its distance to ∂Ω. Then there exists a ﬁnite constant C = C(Ω, L) > 0 with the property that 2 ˙ |∇m−1 SΛ(X)| ρ(X) dX is a Carleson measure on Ω, with . / /∗ for each Λ ∈ h˙ 1m−1,1 (∂Ω) ∗ . (5.12) Carleson constant ≤ CΛ.2 ˙ 1 hm−1,1 (∂Ω)

Mapping Properties of Multiple Layers

399

5.2. Estimates on Besov and Triebel-Lizorkin spaces In this subsection we study the mapping properties of the single layer-type operators introduced in (5.1), (5.10) on Besov and Triebel-Lizorkin scales on Lipschitz domains. Proposition 5.5. Let Ω be a bounded Lipschitz domain in Rn and assume that 1 < p, q < ∞, s ∈ (0, 1). Then %∗ $ p,q p ,q (∂Ω) −→ B˙ m−1,1−s (∂Ω), (5.13) S˙ : B˙ m−1,s in% this where 1/p + 1/p = 1/q + 1/q = 1. Furthermore, S˙ is formally $ self-adjoint ∗ p ,q ˙ ˙ context, in the sense that the dual of the operator (5.13) is S : B (∂Ω) → m−1,1−s

p,q (∂Ω). B˙ m−1,s

Theorem 5.6. Let Ω be a bounded Lipschitz domain in Rn and for each 1 ≤ p, q ≤ ∞ let p , q be such that 1/p + 1/p = 1/q + 1/q = 1. Then the following operators are bounded %∗ $ 1 ≤ p, q < ∞, p,q p ,q ˙ ˙ (5.14) S : Bm−1,1−s (∂Ω) −→ Bm−1+s+1/p (Ω) for s ∈ (0, 1), ⎧ ⎨ 1 ≤ p < ∞, $ %∗ p,p p ,q 0 < q < ∞, S˙ : B˙ m−1,1−s (5.15) (∂Ω) −→ Fm−1+s+1/p (Ω) for ⎩ s ∈ (0, 1), $ %∗ ∞,∞ 1,1 S˙ : B˙ m−1,1−s (∂Ω) −→ Bm+s (Ω). (5.16) In addition S˙ = trm−1 ◦ S˙ on

$ %∗ p,q B˙ m−1,s (∂Ω) ,

(5.17)

Finally, similar properties hold for ψ S˙ − (cf. the convention (5.2)), for any cutoﬀ function ψ ∈ Cc∞ (Rn ), and $ %∗ p,q trm−1 ◦ S˙ + = trm−1 ◦ S˙ − on B˙ m−1,s (∂Ω) . (5.18) We can now further augment the results in Proposition 5.5 and Theorem 5.6 with the following. Corollary 5.7. The operator (5.15) is in fact well deﬁned and bounded for 1 ≤ p ≤ ∞, 0 < q < ∞, 0 < s < 1. Furthermore, the operator (5.13) is in fact well deﬁned and bounded for 1 ≤ p, q ≤ ∞, 0 < s < 1, and (5.17) holds in this range. 5.3. The conormal derivative Let Ω be a bounded Lipschitz domain in Rn . Then, for any bilinear form B(u, v) = (−1)m Aαβ ∂ β u(X) , ∂ α v(X) dX |α|=|β|=m

Ω

(5.19)

400

I. Mitrea

there exists a unique diﬀerential operator L of order 2m – in fact, given by (1.1) – such that ∀ u, v ∈ Cc∞ (Ω). (5.20) B(u, v) = (Lu)(X) , v(X) dX, Ω

Here we further develop this point of view, by considering the conormal derivative associated with such a bilinear form. Speciﬁcally, given a family of constant coefﬁcients A = (Aαβ )α,β , we make the following deﬁnition (the reader is reminded that the coeﬃcients Aαβ are matrix-valued and that the functions involved are vector-valued). Deﬁnition 5.8. For a suﬃciently nice function u in Ω, deﬁne J 5 ∂νA u = (∂ν u)δ with the δ-component given by (∂νA u)δ :=

|δ|≤m−1 n

(−1)|δ|

|α|=|β|=m j=1

$ % (5.21) α!|δ|!(m − |δ| − 1)! νj Aαβ ∂ α+β−δ−ej u , m!δ!(α − δ − ej )! ∂Ω

where we make the convention that α − δ − ej = ∅ if any of its components are negative. Equivalently, if u and v are suﬃciently well behaved functions in Ω, then K m L A ∂ν u, trm−1 v dσ = C(k, m, α, δ, γ) (5.22) ∂Ω

|α|=|β|=m k=1 |γ|=k−1,|δ|=m−k γ+δ+ej =α

× ∂Ω

where C(k, m, α, δ, γ) :=

K

$ % νj Aαβ ∂ β+γ u

∂Ω

$ % , ∂δv

L dσ, ∂Ω

(−1)m−k α!(m−k)!(k−1)! . m!δ!γ!

Proposition 5.9. If u and v are two reasonably behaved functions in Ω, the following Green formula holds: K L L K β α m Aαβ ∂ u(X), ∂ v(X) dX = (−1) Lu(X), v(X) dX |α|=|β|=m

Ω

Ω

K

+(−1)m+1

L ∂νA u(Y ), trm−1 v(Y ) dσ(Y ).

(5.23)

∂Ω

Recall the multiple layer potential operator from (4.18) and the fact that the number N stands for the cardinality of the set of multi-indices with n components and of length ≤ m − 1. To state our next result, we shall need the following convention. Given a M × M -matrix-valued function E, the conormal ∂νA E is the (N M ) × M -matrix whose ith column is ∂νA acting (according to (5.21)) on the ith column in E. It is then elementary to check that for any M × M -matrix-valued function E and any η ∈ CM there holds (∂νA E)η = ∂νA (Eη).

Mapping Properties of Multiple Layers

401

Proposition 5.10. The integral kernel of the multiple layer is the conormal derivative of the fundamental solution. More precisely, t At t (X − Y )) (E (5.24) ∂ν(Y f˙(Y ) dσ(Y ), X ∈ Rn \ ∂Ω. D˙ f˙(X) = L ) ∂Ω

Recall the conventions and results from (4.11)–(4.13). In the sequel, the notat tion ∂νA is chosen to indicate that the conormal is taken with respect to the bilinear form associated with At . Finally, recall that Lt is the transposed of L. Proposition 5.11. For any reasonably well-behaved functions u, v in Ω, there holds K K L L t A ∂ν u(Y ), trm−1 v(Y ) dσ(Y ) − trm−1 u(Y ), ∂νA v(Y ) dσ(Y ) ∂Ω

K =

∂Ω

K L L Lu(X), v(X) dX − u(X), Ltv(X) dX.

Ω

(5.25)

Ω

for any two reasonably behaved functions u and v in Ω. Let us also associate with L a Newtonian-like potential, by setting ΠΩ u(X) := E(X − Y )u(Y ) dY, X ∈ Rn ,

(5.26)

Ω

for any reasonable function u in Ω. Corollary 5.12. For any suﬃciently nice function u in Ω, the following integral representation formula holds: ˙ A u) + ΠΩ (Lu) ˙ m−1 u) − S(∂ u = D(tr ν

in Ω.

(5.27)

In particular, if u is also a null-solution of L in Ω, then ˙ m−1 u) − S(∂ ˙ A u) u = D(tr ν

in Ω.

(5.28)

Proposition 5.13. Let Ω be a Lipschitz domain in Rn and assume that u, v ∈ Cc∞ (Rn ). Also, let Λ be a reasonable vector-valued function deﬁned on ∂Ω. Consider L a constant coeﬃcient elliptic diﬀerential operator of order 2m, m ∈ N, which is self-adjoint. If D˙ is as in (4.18) and S˙ is as in (5.1), then the following hold: K K L L . / ˙ SΛ (X), v(X) dX = Λ(Y ), trm−1 (ΠΩ v)(Y ) dσ(Y ), (5.29) Ω

K

Ω

∂Ω

K L L ΠΩ u(X), v(X) dX = u(X), ΠΩ v(X) dX. Ω

(5.30)

402

I. Mitrea

In addition, K K L L ˙ m−1 v)(X) dX, (5.31) ∂νA ΠΩ u(Y ), trm−1 v(Y ) dσ(Y ) = u(X), D(tr ∂Ω

K

Ω

L ˙ m−1 u))(Y ), trm−1 v(Y ) dσ(Y ) ∂νA (D(tr

∂Ω

K =

L ˙ m−1 v))(Y ) dσ(Y ). trm−1 u(Y ), ∂νA (D(tr

(5.32)

∂Ω

Deﬁnition 5.14. Let Ω be a bounded Lipschitz domain and consider L a constant coeﬃcient elliptic diﬀerential operator of order 2m. For each 1 < p, q < ∞, 0 < s < 1 we introduce the conormal derivative operator ∂νA acting on the set $ %∗ 5 p,q p ,q (Ω) ⊕ Bm+s−1+1/p (Ω) : (5.33) (u, w) ∈ Bm−s+1/p J Lu = w as distributions in Ω , (i.e., the distributions above on C%c∞ (Ω)), 1/p + 1/p = 1/q + 1/q = 1, with $ act ∗ p ,q p ,q (∂Ω), values in the dual space B˙ m−1,s (∂Ω) , by setting for each f˙ ∈ B˙ m−1,s K L L K L K Aαβ ∂ β u, ∂ α F ∂νA (u, w), f˙ := (−1)m+1 + w, F , (5.34) |α|=|β|=m

Ω

p ,q ˙ where F ∈ Bm−1+s+1/p (Ω) is such that trm−1 F = f . In (5.34), by p,q (Ω) B−s+1/p

K L ·, ·

we Ω

and elements denote the duality pairing between elements of the space K L p ,q in its dual, Bs+1/p (Ω), and, respectively, by ·, · we denote the duality pairing

p ,q between elements of the space Bm−1+s+1/p (Ω) and its dual. Similarly, one can introduce a conormal derivative for the exterior domain ¯ in place of Ω (in which case (5.34) is altered by changing the sign Ω− := Rn \ Ω of the left-hand side). When necessary to distinguish this from (5.35), we shall denote this by ∂νA− , and denote the former by ∂νA+ .

It is important to point out that the deﬁnition (5.34) is independent on the p ,q ˙ choice of F ∈ Bm−1+s+1/p (Ω) such that trm−1 F = f . Indeed, in order to see this it suﬃces to show that the right-hand side in (5.34) equals zero whenever p ,q F ∈ Bm−1+s+1/p (Ω) is such that trm−1 F = 0. This latter fact easily follows from integration by parts whenever F ∈ Cc∞ (Ω). Since, due to (3.23), Cc∞ (Ω) is dense p ,q in the subspace of Bm−1+s+1/p (Ω) consisting of functions with vanishing trace, it

p ,q follows that the right-hand side in (5.34) vanishes whenever F ∈ Bm−1+s+1/p (Ω), as desired.

Mapping Properties of Multiple Layers

403

Proposition 5.15. Let Ω be a bounded Lipschitz domain in Rn and consider L a constant coeﬃcient elliptic diﬀerential operator of order 2m. For each 1 < p, q < ∞, 0 < s < 1, the conormal derivative operator ∂νA from Deﬁnition 5.14 induces a linear, bounded operator 5 J $ %∗ p,q p ,q (Ω) : Lu = 0 in Ω −→ B˙ m−1,s (∂Ω) , (5.35) ∂νA : u ∈ Bm−s+1/p where 1/p + 1/p = 1/q + 1/q = 1, according to K L L K Aαβ ∂ β u, ∂ α F , ∂νA u, f˙ = (−1)m+1 |α|=|β|=m

(5.36)

Ω

p ,q p ,q ˙ (∂Ω) and F ∈ Bm−1+s+1/p for each f˙ ∈ B˙ m−1,s (Ω) such that trm−1 F = f (as K L p,q denote the duality pairing between elements of the space B−s+1/p (Ω) before, ·, · Ω

p ,q and elements in its dual, Bs+1/p (Ω)). Similarly, one can consider the conormal derivative acting on null-solutions ¯ in place of Ω. When necessary to distinguish this from of L in Ω− := Rn \ Ω, (5.35), we shall denote this by ∂νA− , and denote the former by ∂νA+ .

5.4. Jump relations for the conormal derivative The ﬁrst order of business is to actually deﬁne the conormal derivative of the single layer potential operator. Proposition 5.16. Assume that Ω is a bounded Lipschitz domain in Rn , and ﬁx 1 < p, q < ∞ and 0 < s < 1. Then one can deﬁne the conormal derivative of the single layer potential (associated with Ω) in such a way that %∗ $ %∗ $ p,q p,q (∂Ω) −→ B˙ m−1,s (∂Ω) (5.37) ∂νA S˙ : B˙ m−1,s becomes a linear, bounded operator. Similarly, one can deﬁne the conormal derivative of the single layer associated ¯ When necessary to distinguish this from (5.37), we shall denote with Ω− := Rn \ Ω. ˙ and denote the former by ∂ A S. ˙ this by ∂ A S, ν−

ν+

The following result is signiﬁcant for the treatment of Neumann type boundary value problems for the higher-order diﬀerential operator L as in (4.14). n Theorem 5.17. Let Ω be a Lipschitz %∗ in R and ﬁx 1 < p, q < ∞ and $ domain p ,q 0 < s < 1. Then for each Λ ∈ B˙ m−1,s (∂Ω) one has K % L L K $ p ,q ˙ g˙ = Λ, ∓ 1 I + K˙ g˙ , ∂νA± SΛ, ∀ g˙ ∈ B˙ m−1,s (∂Ω). (5.38) 2

Here 1/p + 1/p = 1/q + 1/q = 1. In particular,

$ %∗ p ,q (∂Ω) , ∂νA± S˙ = ∓ 12 I + K˙ ∗ as operators on B˙ m−1,s %∗ $ p ,p (∂Ω) . ∂νA+ S˙ − ∂νA− S˙ = I on B˙ m−1,s

(5.39) (5.40)

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Corollary 5.18. Let Ω be a Lipschitz domain in Rn . Then for each 1 < p < ∞ the conormal derivative of the single layer,$initially considered in the sense of %∗ p (5.37), extends to a bounded operator from L˙ m−1,0 (∂Ω) into itself, and from %∗ $ L˙ pm−1,1 (∂Ω) into itself. Hence, ∂νA± S˙ = ∓ 21 I + K˙ ∗ (5.41) $ %∗ $ %∗ considered as either operators on L˙ pm−1,0 (∂Ω) , or on L˙ pm−1,1 (∂Ω) . Proposition 5.19. Let Ω be a Lipschitz domain in Rn and assume that 1 < p, p <

(0, 1), S˙ K˙ ∗ = K˙ S˙ as (linear, bounded) ∞ satisfy 1/p+1/p %∗ for each s ∈ $ = 1. Then, p,q p ,q (∂Ω) into B˙ (∂Ω). As a consequence, operators from B˙ m−1,s

m−1,1−s

$ %∗ p,q p ,q the dual of S˙ K˙ ∗ : B˙ m−1,s (∂Ω) −→ B˙ m−1,1−s (∂Ω) $ %∗ p ,q p,q is the operator S˙ K˙ ∗ : B˙ m−1,1−s (∂Ω) −→ B˙ m−1,s (∂Ω)

(5.42)

and, hence, $ %∗ 2,2 2,2 (∂Ω) −→ B˙ m−1,1/2 (∂Ω) is self-adjoint. S˙ K˙ ∗ : B˙ m−1,1/2

(5.43)

Finally, the intertwining $formula S˙ K˙ ∗%= K˙ S˙ is also valid when both sides ∗ into L˙ pm−1,1 (∂Ω), or as operators are viewed as operators from L˙ pm−1,0 (∂Ω) $ %∗ from L˙ pm−1,1 (∂Ω) into L˙ pm−1,0 (∂Ω). We now proceed to deﬁne the conormal derivative of the double layer potential operator. Proposition 5.20. Suppose that Ω is a bounded Lipschitz domain in Rn , and ﬁx 1 < p, q < ∞ and 0 < s < 1. Assume 1/p + 1/p = 1/q + 1/q = 1. Then it is possible to deﬁne the conormal derivative of the double layer potential (associated with Ω) in such a way that $ %∗ p,q p ,q (∂Ω) −→ B˙ m−1,1−s (∂Ω) (5.44) ∂νA D˙ : B˙ m−1,s becomes a linear, bounded operator. Analogously, one can deﬁne the conormal de¯ When necessary to rivative of the double layer associated with Ω− := Rn \ Ω. A ˙ distinguish this from (5.44), we shall denote this by ∂ν− D, and denote the former ˙ by ∂νA+ D. We now describe some of the basic properties of the conormal derivative of the double layer potential introduced above.

Mapping Properties of Multiple Layers

405

Theorem 5.21. Let Ω be a Lipschitz domain in Rn , ﬁx s ∈ (0, 1), and assume that 1 < p, p , q, q < ∞ satisfy 1/p + 1/p = 1. Then $ %∗ f˙ ∈ B˙ p,q (∂Ω) =⇒ ∂ A D˙ f˙ = ∂ A D˙ f˙ in B˙ p ,q (∂Ω) . (5.45) ν+

m−1,s

ν−

m−1,1−s

Also, the conormal derivative of the double layer potential is a formally selfp ,q (∂Ω) −→ adjoint operator in the sense that the dual of (5.44) is ∂νA D˙ : B˙ m−1,1−s $ %∗ p,q ˙ B (∂Ω) . m−1,s

Finally,

%∗ % $ % $ p,q + K˙ ∗ ◦ − 12 I + K˙ ∗ on B˙ m−1,s (∂Ω) , % $ % $ p,q S˙ ◦ ∂νA D˙ = 12 I + K˙ ◦ − 12 I + K˙ on B˙ m−1,s (∂Ω). ∂νA D˙ ◦ S˙ =

$

1 I 2

(5.46) (5.47)

Acknowledgment The author would like to take this opportunity to thank the Department of Mathematics at Cornell University, where she has written this paper, for its hospitality during her stay as a Ruth Michler Fellow in the Fall of 2008.

References [1] V. Adolfsson and J. Pipher, The inhomogeneous Dirichlet problem for Δ2 in Lipschitz domains, J. Funct. Anal., 159 (1998), no. 1, 137–190. [2] S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane. I., Comm. Pure Appl. Math., 10 (1957), 179–239. [3] A.P. Calder´ on, Commutators, singular integrals on Lipschitz curves and applications, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 85–96, Acad. Sci. Fennica, Helsinki, 1980. [4] J. Cohen, BMO estimates for biharmonic multiple layer potentials, Studia Math., 91 (1988), no. 2, 109–123. [5] J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a C 1 domain in the plane, Indiana Univ. Math. J., 32 (1983), no. 5, 635–685. [6] R.R. Coifman, A. McIntosh and Y. Meyer, L’int´egrale de Cauchy d´eﬁnit un op´erateur born´e sur L2 pour les courbes lipschitziennes, Annals of Math., 116 (1982), 361–387. [7] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), no. 4, 569–645. [8] B.E. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal., 65 (1977), no. 3, 275–288. [9] B.E. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier, (Grenoble) 47 (1997), no. 5, 1425–1461. [10] B.E. Dahlberg and G.C. Verchota, Galerkin methods for the boundary integral equations of elliptic equations in nonsmooth domains, Harmonic analysis and partial diﬀerential equations (Boca Raton, FL, 1988), 39–60, Contemp. Math., 107, Amer. Math. Soc., Providence, RI, 1990.

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[11] E.B. Fabes, M. Jodeit Jr. and N.M. Rivi`ere, Potential techniques for boundary value problems on C 1 -domains, Acta Math., 141 (1978), no. 3-4, 165–186. [12] E.B. Fabes and C.E. Kenig, On the Hardy space H 1 of a C 1 domain, Ark. Mat. 19 (1981), no. 1, 1–22. [13] E.B. Fabes, O. Mendez, and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), no. 2, 323–368. [14] C. Feﬀerman and E.M. Stein, H p spaces of several variables, Acta Math., 129 (1972), no. 3-4, 137–193. [15] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., Vol. 93 No. 1 (1990), 34–170. [16] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, Vol. 79, AMS, Providence, RI, 1991. [17] D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27–42. [18] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), no. 1, 161–219. [19] A. Jonsson and H. Wallin, Function spaces on subsets of Rn , Math. Rep., Vol. 2, 1984. [20] C.E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, Vol. 83, AMS, Providence, RI, 1994. [21] S. Mayboroda and M. Mitrea, Green potential estimates and the Poisson problem on Lipschitz domains, preprint, 2005. [22] V. Maz’ya, M. Mitrea and T. Shaposhnikova, The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coeﬃcients, preprint, 2005. [23] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163 (1999), no. 2, 181–251. [24] J. Pipher and G. Verchota, The Dirichlet problem in Lp for the biharmonic equation on Lipschitz domains, Amer. J. Math., 114 (1992), no. 5, 923–972. [25] J. Pipher and G.C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, Potential Anal., 4 (1995), no. 6, 615–636. [26] J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and C 1 domains, Comment. Math. Helv., 68 (1993), no. 3, 385–414. [27] J. Pipher and G.C. Verchota, Dilation invariant estimates and the boundary G˚ arding inequality for higher order elliptic operators, Ann. of Math., (2) 142 (1995), no. 1, 1–38. [28] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Diﬀerential Operators, de Gruyter, Berlin, New York, 1996. [29] V. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc., (2) 60 (1999), no. 1, 237– 257.

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[30] Z. Shen, The Lp Dirichlet Problem for Elliptic Systems on Lipschitz Domains, Math. Research Letters, 13 (2006), 143–159. [31] Z. Shen, Necessary and Suﬃcient Conditions for the Solvability of the Lp Dirichlet Problem on Lipschitz Domains, to appear in Math. Ann., (2006). [32] Z. Shen, The Lp Boundary Value Problems on Lipschitz Domains, preprint (2006). [33] E.M. Stein, Singular integrals and diﬀerentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970. [34] H. Triebel, Theory of function spaces. II, Monographs in Mathematics, 84. Birkh¨ auser Verlag, Basel, 1992. viii+370 pp. [35] G. Verchota, The biharmonic Neumann problem in Lipschitz domains, Acta Math., 194 (2005), 217–279. [36] G. Verchota, The Dirichlet problem for the biharmonic equation in C 1 domains, Indiana Univ. Math. J., 36 (1987), no. 4, 867–895. [37] G.C. Verchota, Potentials for the Dirichlet problem in Lipschitz domains, pp. 167– 187 in Potential Theory - ICPT 94, de Gruyter, Berlin, 1996. [38] G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J., 39 (1990), no. 3, 671–702. [39] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), no. 3, 572–611. [40] H. Whitney, Analytic extensions of diﬀerentiable functions deﬁned in closed sets, Trans. Amer. Math. Soc., 36 (1934), no. 1, 63–89. Irina Mitrea Department of Mathematical Sciences Worcester Polytechnic Institute Worcester, MA 01609-2280, USA e-mail: [email protected] Received: February 28, 2009. Accepted: June 9, 2009.

Operator Theory: Advances and Applications, Vol. 203, 409–426 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Applications of a Numerical Spectral Expansion Method to Problems in Physics; a Retrospective George H. Rawitscher Abstract. A long collaboration between Israel Koltracht and the present author resulted in a new formulation of a spectral expansion method in terms of Chebyshev polynomials appropriate for solving a Fredholm integral equation of the second kind, in one dimension. An accuracy of eight signiﬁcant ﬁgures is generally obtained. The method will be reviewed, and applications to physics problems will be described. Mathematics Subject Classiﬁcation (2000). 41A10, 45B05, 65Rxx, 65Z05. Keywords. Spectral expansions, integral equations, numerical methods.

1. Introduction More than 12 years ago professor Israel Koltracht informed me that the solution of an integral equation is numerically more stable than the solution of a diﬀerential equation. Our collaboration had its aim to verify this statement for the case of the Schr¨odinger equation, that is the basic pillar of quantum mechanics. It is a second-order diﬀerential equation, that for a partial wave of angular momentum number = 0 and a spherically symmetric potential V (r) has the form 2 d 2 (1) + k ψ(r) = V (r)ψ(r), dr2 Here ψ(r) is the wave function to be obtained, r is the radial distance deﬁned in the range from 0 to ∞, k 2 is the energy in units of inverse length squared (assumed given), k is the wave number (in units of inverse length), and V (r) (assumed given) is also in units of inverse length squared, while ψ is dimensionless. The connection between the energy units and length units involves Planck’s constant and the reduced mass of the two interacting objects, as is described in Appendix A, Communicated by L. Rodman.

410

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Eq. (5A). The connection of ψ(r) with the overall quantum mechanical wave function Ψ(x, y, z, t) is also described in Appendix A, where it is shown that ψ(r) is the radial part of a partial wave of angular momentum number = 0 in the center of mass frame. For positive energies ψ(r) is not determined by a two-point boundary condition, but by the condition that it vanish at r = 0, with a normalization that is initially arbitrary. The latter is subsequently determined by the physical application envisaged. There exists the equivalent integral equation, denoted as Lippmann-Schwinger (L − S), of the form ∞ G0 (k; r, r ) V (r ) ψ(r ) dr . (2) ψ(r) = sin(kr) + 0

The Green’s function G0 (k; r, r ), the boundary conditions for ψ, and the origin of Eq. (2) are described in Appendix A. Physicists prefer to solve the diﬀerential equation because of the simplicity of the numerical recurrence relation algorithm, and shy away from solving integral equations because the matrices are non-sparse and cumbersome to handle as well as memory intensive. The object of our investigations was to show that according to our scheme the accuracy of the numerical solution of Eq. (2) is substantially larger than the solution of Eq. (1) for the same number of mesh points, that the non-sparseness objection of the matrices involved in Eq. (2) can be overcome by implementing a division of the radial interval into suitable partitions, and that the choice of partitions can be obtained adaptively for a given accuracy requirement if the solution in each partition is spectrally expanded in a ﬁxed number of Chebyshev polynomials. These investigations beneﬁtted greatly from the interaction between the very gifted mathematician, Professor Koltracht, and this physicist, and led to a number of publications. It is the purpose of the present contribution, written in memory of Professor Koltracht, to give an account of the basic results of our many investigations in a readable logical sequence, the main purpose having been to replace the numerical solution of Eq. (1) by the numerical solution of Eq. (2). Since in the literature the L − S equation (2) is usually formulated and solved in momentum space, our results obtained in conﬁguration space contain novel features, not previously known. The main diﬀerence between the momentum space and the conﬁguration space solutions of the L − S equation is that the Green’s function in the former space has a pole singularity, while in the latter space it is continuous but has a derivative singularity.

2. The spectral expansion method (IEM) The present method of solving an integral equation via a spectral expansion in terms of Chebyshev polynomials is based on a procedure developed by Greengard and Rokhlin [GR 91]. The basic mathematical properties of our method, investigated in Refs. [Gonz97], [Gonz99], consist in dividing the radial interval [0, Rmax ]

Spectral Expansions for Physics Applications

411

into a number m of partitions i, i = 1, 2, . . . , m with lower and upper radial limits (i) (i) (i+1) (i) b1 and b2 , such that b1 = b2 . The Green’s function G(r, r ) is given by 1

G0 (r, r ) = − F (r< )G(r> ) (3) k r< and r> being the lesser and larger values of r and r , respectively, where F (r) = sin(kr), G(r) = cos(kr) (4) and where k is the wave number deﬁned in Eq. (1), such that k 2 is equal to the energy of the incident wave. In view of the semi-separable nature of G(r, r ) one can show [Gonz97] that the solution of Eq. (2), in the partition i, ψ (i) (r) is a linear combination of two independent functions Y (i) (r) and Z (i) (r) ψ (i) (r) = A(i) Y (i) (r) + B (i) Z (i) (r)

(5)

each of which is the solution of a L − S equation restricted to the partition i, with diﬀerent driving terms F (r) and G(r) b(i) 2 (i) Y (r) = F (r) + G0 (k; r, r ) V (r ) Y (i) (r ) dr . (6) (i)

b1

Z (i) (r) = G(r) +

(i)

b2

(i)

G0 (k; r, r ) V (r ) Z (i) (r ) dr .

(7)

b1

The decomposition of ψ (i) (r) given by Eq. (5) is reminiscent of a decomposition into splines, for example, but has the advantage that there are only two functions Y (i) and Z (i) , and that these functions and their derivatives can be calculated with spectral accuracy from Eqs. (6) and (7). Equations (2), 6 and 7 are Fredholm integral equation of the second kind, and the respective solutions are unique under certain conditions [Fredh1903]. The numerical solution of Eqs. (6) and (7) is performed by expanding an approximation to the unknown solutions Y (r) or Z(r) into a ﬁnite set of N + 1 Chebyshev polynomials of order 0, 1, . . . , N. Such expansions and their errors can be understood as follows: For a given function f (x), if the expansion is truncated at an upper value N of n, one obtains a truncation error εN (x) f (x) = fN (x) + εN (x) where fN (x) =

N

ai Ti (x) − 1 ≤ x ≤ 1.

(8)

(9)

i = 0

The values of r in Eqs. (6) and (7) contained in the partition i are changed into the variable x in Eq. (9) by means of an appropriate linear transformation such that (i) (i) b1 and b2 are transformed into −1 and 1, respectively. The Chebyshev-Fourier coeﬃcients ai are obtained by making use of the relation between the coeﬃcients

412

G.H. Rawitscher

a0 , a1 , a2 . . . , aN and the value of the function f at the zeros τ1 , τ2 , τ3 . . . , τN +1 of the Chebyshev polynomial of order N + 1 [a0 , a1 , a2 , . . . , aN ]T = C −1 [f (τ1 ), f (τ2 ), f (τ3 ), . . . , f (τN +1 )]T ,

(10)

where the elements (i, j) of the discrete cosine transform matrix C are given by Ci,j = Ti (τj ) and the elements of C −1 can be obtained [GvL83] in terms of the transposed matrix C T . If the function f (x) has only a ﬁnite number p of continuous derivatives, then "N the truncation error εN (x) = |f (x) − i = 0 ai Ti (x)| decreases with N according to N (−p+1) , as described on page 181 in Ref.[Gonz99], or in [GoOr77]. If f (x) has an inﬁnite number of continuous derivatives, then the convergence is superalgebraic, i.e., faster than any power of N. An example is given in [RaKo05a] for the function f (x) = exp(x), for which |εN (x)| ≤ (e/2)N × N −(N +1/2) . By contrast, since exp(x) is not a periodic function, its Fourier expansion converges more slowly. For this example it is found that for N = 8, | εN | ∼ = 2 × 10−7 , and further that |εN (x)| ) |aN +1 |. (11) This last result is a general property of Chebyshev expansions, and permits one to construct an adaptive choice of the size of each partition i, such that the error of the functions Y (i) and Z (i) has a numerical value less than a pre-established accuracy parameter “tol”. It is suﬃcient to solve Eqs. (6) and (7) for a guessed size of a partition, for a ﬁxed number of Chebyshev support points N + 1 (for example N = 16). If the sum of the absolute values of the three last coeﬃcient is larger than tol, then the size of the partition is cut in half. As a result, many partitions accumulate in the regions where the solutions Y or Z change with position very rapidly, but in the regions where the potential V is small, i.e., where the values of Y or Z are close to their respective driving terms F or G, the size of the partitions can be of the order of hundred units of length. In addition to the expansion properties described above, the solution of Eqs. (6) and (7) makes use of the property * y that the Chebyshev expansion coeﬃcients ci of an indeﬁnite integral FL (y) = −1 f (x)dx FLN (y) =

N +1

ci Ti (y)

(12)

i = 0

can be expressed in terms of the expansion coeﬃcients ai of f (x) [CC60] by means of the matrix relation (13) [c0 , c1 , c2 , . . . , cN +1 ]T = SL [a0 , a1 , a2 , . . . , aN ]T . *1 and likewise, the expansion of the indeﬁnite integral FR (y) = y f (x)dx can be given in terms of a matrix SR . Expressions for SL and SR can be found in various references, including Ref. [Gonz97] . The error of these expansions (12) is of the same general magnitude as the error εN of the expansion of f (x), as is veriﬁed in

Spectral Expansions for Physics Applications

413

an explicit example in Table II of Ref. [RaKo05a]. By repeated use of Eqs. (10), 12 and 13 one ﬁnally arrives at a matrix equation for the functions Y (i) calculated at the Chebyshev support points τ1 , τ2 , . . . , τN +1 in the partition i of the form T

(1 + K(i) ) [Y (τ1 ), Y (τ2 ), Y (τ3 ), . . . , Y (τN +1 )] T

(14)

= [F (τ1 ), F (τ2 ), F (τ3 ), . . . , F (τN +1 )] . The equation for Z (i) is of similar form, obtained by replacing F (τj ) by G(τj ), j = 1, 2, . . . , N + 1 on the right-hand side of Eq. (14). The (N + 1, N + 1) matrix K(i) is obtained in terms of products of the matrices C, C −1 , SL , SR and diagonal matrices of the potential V and of the functions F and G, calculated at the support points τj in partition i [Gonz97]. In addition the matrix K(i) also contains the (i) (i) factor (b2 − b1 )/2, that takes into account the linear transformation from the coordinates r in partition i to the coordinates x. In view of this last factor, the smaller the size of the partition i, the smaller is the norm of K (i) , and hence Eq. (14) can be made to have a unique solution. The accuracy of the solution is given by the tolerance parameter tol , which, in the applications described in here, is of order 10−8 . A description of the expansion and accuracy properties described above is also given in an extensive review by Deloﬀ [Del77]. To calculate the factors A(i) and B (i) , which are needed in order to determine the global solution ψ in all partitions i = 1, 2, . . . , m according to Eq. (5), one inserts into the global L − S Eq. (2) the expressions for ψ (i) of Eq. (5), and after carrying out the Green’s function integrals over all partitions, one obtains linear relations between all the coeﬃcients. There are 2m such relations, whose factors involve integrals of the type 1 (ξη)i = k

(i)

b2

(i)

ξ(r )V (r )η(r ) dr ,

(15)

b1

where ξ(r ) represents either the functions F (r ) or G(r ), and η(r ) represents either the functions Y (i) (r ) or Z (i) (r ). Since all these functions are known to the accuracy speciﬁed by the parameter tol, and the integrals themselves can be carried out without loss of accuracy by means of the matrix SL of the Curtis-Clenshaw method [CC60], one obtains these factors (ξη)i with the accuracy speciﬁed by tol. The 2m linear relations between the coeﬃcients A and B can be put into the matrix form [Gonz97] ⎞ ⎞⎛ ⎞ ⎛ ⎛ α1 0 ζ I M12 ⎟ ⎜ α2 ⎟ ⎜ ζ ⎟ ⎜ M21 I M23 ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎜ α3 ⎟ ⎜ ζ ⎟ ⎜ M I M . . . 32 34 ⎟ ⎟⎜ ⎟=⎜ ⎜ ⎟⎜ ... ⎟ ⎜ ... ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎝ I Mm−1,m ⎠ ⎝ αm−1 ⎠ ⎝ ζ ⎠ Mm−1,m−2 ω 0 Mm,m−1 I αm (16)

414

G.H. Rawitscher

where I and 0 are two by two unit and zero matrices, respectively, the αi , ω and ζ are column vectors (i) A 0 1 αi = (17) ; ζ = ; ω = 0 0 B (i) and where

Mi−1,i =

and

Mi,i−1 =

(GY )i − 1 0

0 (F Y )i−1

(GZ)i 0

, i = 2, 3, . . . , m

(18)

0 (GZ)i−1 − 1

, i = 2, 3, . . . , m.

(19)

Note that Eq. (16) generally connects the A and B’s of three contiguous partitions. One can also rearrange the system of linear equations for the A’s and B’s by ﬁrst writing them into a (2 × 1) column form involving the vectors αi , and subsequently subtracting equations with contiguous i-values from each other, however leaving the last equation in its original form. The result is [Sim07] ⎞ ⎞ ⎛ ⎞⎛ ⎛ ζ α1 Γ1 −Ω2 ⎟ ⎜ α2 ⎟ ⎜ ζ ⎟ ⎜ Γ2 −Ω3 ⎟ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎜ α3 ⎟ ⎜ ζ ⎟ ⎜ Γ −Ω . . . 3 4 ⎟ ⎟=⎜ ⎟⎜ ⎜ ⎟ ⎜ . . . ⎟ ⎜ . . . ⎟ , (20) ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎜ ⎝ Γm−1 −Ωm ⎠ ⎝ αm−1 ⎠ ⎝ ζ ⎠ ω γ1 γ2 γ3 ... γm−1 I αm and is denoted as method B in Ref. [RaKo05a]. In the above 1 0 Γi = , −(F Y )i 1 − (F Z)i 1 − (GY )i −(GZ)i , Ωi = 0 1 and

γi =

0 (F Y )i

0 (F Z)i

(21) (22)

.

(23)

It is noteworthy that the ﬁrst m − 1 equations in (20), Γi αi = Ωi+1 αi+1 , i = 1, 2, . . . , m − 1

(24)

are equivalent to matching the wave function ψ at the end of partition i to ψ at the (i) start of partition i + 1. This can be seen by imposing the two conditions ψi (b2 ) = (i+1) (i) (i+1)

) and ψi (b2 ) = ψi+1 (b1 ) where ψi is the wave function in partition ψi+1 (b1

i given by Eq. (5) and where ψi is the corresponding derivative with respect to r. The resulting equations Ai = Ai+1 [1 − (GY )i+1 ] − Bi+1 (GZ)i+1 Bi+1 = −Ai (F Y )i + Bi [1 − (F Z)i ].

(25)

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415

are equivalent to Eq. (24). The important reason why integrals of the type of Eq. (25) are related to the derivative of the solution of an integral equation (6) and (7) is due to the separable form of the Green’s function. In this case a derivative of a function that obeys a L − S integral equation can itself be expressed in terms of integrals together with derivatives of known analytical functions, and no loss in accuracy results. For example, by taking the derivative relative to r on both (i) sides of Eq. (6), and letting r assume the value of b2 , one obtains dY (i) /dr = dF/dr − (dG/dr)(F Y )i . By successive applications of Eq. (24) −1

αi+1 = (Ωi+1 )

Γi αi

one can relate the values of αi , i = 2, 3, . . . , m, to α1 and then use the last of the (20) equations m−1 1 (26) γi αi +αm = 0 i=1

in order to ﬁnd the value of A1 . It can be shown that Eq. (26) is compatible with the requirement that B1 = 0.

3. Numerical properties of the method Some of the most important features of the present spectral integral method (S − IEM ) are as follows: a) The matrices that determine the coeﬃcients A and B in Eqs. (20) or (16) are sparse, and can be solved by Gaussian elimination. This sparseness property results from the semi-separable nature of the integration kernel G0 (k; r, r ) V (r ), as is shown in Refs. [Gonz97], [Gonz99], which however applies only in the conﬁguration representation of the Green’s function. This part of our procedure also diﬀers substantially from that of Ref. [GR 91]. The matrices that determine the solution of the functions Y and Z in each partition are not sparse, however, the size (N + 1) × (N + 1) of the matrices is small since a preferred choice for N is 16 for which the computational complexity, of order (N + 1)3 , is not large. Therefore, the computational complexity of the S −IEM is comparable to that of the solution of the diﬀerential equation by ﬁnite diﬀerence methods. b) The scattering boundary conditions can be implemented reliably. This is because the Green’s function incorporates the asymptotic boundary conditions automatically. This is particularly important for the solution of coupled integral equations [Gonz99]. By contrast, the solution of the corresponding coupled diﬀerential equations are prone to developing numerical instabilities [Gonz99]. c) The equations (25) that link the coeﬃcients A and B from one partition to the immediate neighboring one [RaKo05a] is more transparent than the equations that connect three contiguous partitions, and is more versatile as well [Sim07]. For example, it enables one to use diﬀerent and more suitable Green’s functions in each

416

G.H. Rawitscher

Figure 1. Comparison of the accuracy of two methods for solving the diﬀerential equation for the Riccati-Bessel function. The “Fin. Diﬀ” method uses a 6th order Numerov method, while “Int. Eq.” displays the result for the spectral integral equation method (S-IEM), which solves the corresponding Lippmann-Schwinger integral equation. Both numerical results are normalized to a bench-mark value at one particular radial point. One sees that the accumulation of round-oﬀ errors is slower for the S-IEM method than for the Numerov method (described as Milne’s method in Ref. [AbSt72], also known as Cowell or Fox-Goodwin) , and requires fewer radial meshpoints to achieve the same accuracy. partition, whose choice depends on the nature of the potential in each particular partition. d) The known error of the truncation of the Chebyshev expansion at N + 1 terms permits the numerical algorithm to choose the size of each partition adaptively. This feature, together with the inherently rapid convergence of spectral expansions, leads to a large economy of meshpoints for a given required precision. e) The solution of the integral equation can be obtained at any radial point, rather than at ﬁxed support points, because the solution provides the coeﬃcients for the Chebyshev polynomial expansions at each partition. Since the Chebyshev polynomials can be evaluated precisely at any point, the overall solution can also be found at any point and not just at the support points, as is common with other methods. The main numerical features of the S − IEM can be seen from a sample calculation of a Riccati-Bessel function [AbSt72], displayed in Fig. (1). In this ﬁgure the numerical error of the result is plotted as a function of the total number of mesh points in a ﬁxed radial interval of 50 units of length. The angular momen-

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417

tum number L = 6, i.e., the potential V in Eqs. (1) or 2 is given by L(L + 1)/r2 , and the wave number k is 1 in units of inverse length. The error, plotted on the vertical axis, is the maximum discrepancy between the numerical and the analytic result across the radial range. Both calculations are done in FORTRAN in double precision. The curve marked as “Fin Diﬀ” is obtained via the ﬁnite diﬀerence Numerov method, with an error of order h6 in each three-point recurrence relation, where h is the distance between radial mesh points. In Ref. [AbSt72] this method is described as Milne’s and is given as formula 25.5.21 C. It is also known as Cowell or Fox-Goodwin method. The curve labeled “Int Eq” was calculated with the S − IEM, without the imposition of an accuracy parameter tol. The number of Chebyshev support points in each partition was N + 1 = 17, and the number of partitions m was progressively increased, and its size accordingly reduced, so as to produce the total number of support points contained in the radial interval [0, 50] displayed on the x-axis. The S − IEM error decreases very rapidly with the number of support points, which demonstrates the super-algebraic reduction of the truncation error with the size of the partition, described above. Another property displayed in Fig. (1) is the slow accumulation of the numerical truncation error, which occurs in the region where the error increases with increasing number of meshpoints nm . This error is due to the ﬁnite number of decimal places provided by the computer, 14 to 15, in this case. For the S−IEM the truncation errors overwhelm the algorithm errors for nm = 3 × 103 , but when the number of points in the interval [0, 50] is increased further, the corresponding truncation error increases much more slowly with nm than is the case for the Fin. Diﬀ. method. The capability to adaptively determine of the size of each partition is illustrated in Fig. (8) of Ref. [Gonz99] and also in Figs. (3) and (4) in Ref. [RaKo06]. The latter refers to a calculation of the bound state energy eigenvalue of a Helium di-atom, and Fig. (4) of Ref. [RaKo06] is reproduced here as Fig. (2). In that case, because the binding energy is so small and the corresponding wave function extends to large distances, the bound state wave function has to be calculated out to 3, 000 units of length (Bohr radii a0 ) for the required accuracy of 1 : 108 , hence an adaptive choice of the partition sizes is very useful. The distribution of partitions depends of the value of the accuracy parameter tol, as can be seen in Fig. (2), and is particularly dense in the region where the potential (or its derivative) changes rapidly, which occurs near 2.5 a0 . The numbers in the right-hand side of the ﬁgure represent the total number of partitions needed to cover the whole radial region [0, 3000 a0 ], which in turn shows that approximately half of the total number of partitions are located in the small radial region [0, 4]. The maximum accuracy of the S − IEM method is larger than that of the F in.Dif f. method, and is reached for a substantially smaller number of mesh points nm . This feature, already evident from Fig. (1), is also demonstrated in several other investigations, such as in Fig. (5) of Ref. [RaEsTi99], or in Fig. (2) of Ref. [RaKaKo03]. A particularly cogent example for demonstrating the high accuracy of the S − IEM as compared with other methods, can be seen from Fig. (3), taken from Fig. (2) of Ref. [RaKo05a]. The numerical calculation [RaKo05a] is compared to

418

G.H. Rawitscher

Figure 2. The distribution of partition end points is shown only in the radial region from 0 to 4 atomic units a0 . The larger the imposed value of the tolerance parameter, the larger is the size of the partitions. The numbers written in the right end for each curve represent the total number of partitions required to cover the total radial range form 0 to 3000 atomic units (a0 ). An accumulation of partitions occurs near 2.5 a0 where the potential has a “kink”. This ﬁgure is taken from Fig. 4 of Ref. [RaKo06], for the calculation of the bound state energy of the He-He diatom molecule. an analytical calculation [RaMe02], from which the numerical errors are be obtained. This test is especially rigorous because it involves a resonance situation, where the wave function decreases in the region of the repulsive barrier, while the corresponding numerical errors increase in the same barrier region. The comparison between various computational methods is illustrated in Fig. (3). The x-axis displays the incident wave number k in the vicinity of the center of the resonance, and the y-axis displays the absolute value of the error for the corresponding = 0 scattering phase shift, deﬁned near Eq. (7A) in the Appendix A. The S − IEM results give the smallest error, approximately six orders of magnitude smaller than a Numerov ﬁnite diﬀerence calculation [EZ04]. S − IEM results denoted as A use the three-partition recurrence relation (16), and are obtained with FORTRAN in double precision, while results B are based on the two-partition recurrence relation (20), and are obtained with MATLAB. The latter has 16 signiﬁcant ﬁgure of precision, and hence is more accurate that the FORTRAN result. In summary, the essential numerical features of the S − IEM are illustrated above for the case of solving the Schr¨odinger equation.

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419

Figure 3. Comparison of the accuracy of three numerical methods for the calculation of the S-wave phase shift for a Morse potential in a resonant incident energy region. Methods LD and NUM are ﬁnite diﬀerence methods, and IEM is the integral method S − IEM , as obtained by two diﬀerent realizations, A and B, as explained in the text.

4. Retrospective of the work with Israel Koltracht The initial aim of our collaboration was to conﬁrm the assertion that the solution of the integral L − S equation (2) is numerically more stable than the solution of the diﬀerential equation (1). The initial investigation [Gonz97] conﬁrmed the assertion, and introduced our version of the S − IEM . The extension to the coupled Schr¨ odinger equation also showed [Gonz99] that the implementation of the boundary conditions can be achieved more stably with the solution of the coupled integral L − S equations than with the diﬀerential equation. Comparison with other methods of calculation were made in the study of the collision between two atoms [RaEsTi99]. That calculation involved the coupling between two channels: one with positive energy, another with negative energy. An extension was made [RaKaKo03] to the case that the potential V in Eqs. (1) and (2) is non-local, i.e., * the product V (r )ψ(r ) is replaced by V (r , r

)ψ(r

) dr

. In the case considered, namely the scattering of an electron by a Hydrogen atom [RaKaKo03], the potential is semi-separable, as given by Eqs. (11) and (12)*of Ref. [RaKaKo03] with the consequence that the integration kernel F(r, r

) = G(r, r )V (r , r

)dr also becomes semi-separable, Eqs. (19) and (20) of Ref. [RaKaKo03]. The semi-separable property of F allows one to solve the integral equations with only minimal modiﬁcations of the original S − IEM method for local potentials. The case of a more general kernel F(r, r

) that is either discontinuous or not smooth along the main

420

G.H. Rawitscher

diagonal has been treated in Ref. [KaKoRa03]. A stringent accuracy test of the S − IEM was provided [RaKo05a] in a study of the resonant scattering from a potential with a repulsive barrier for which analytical results are available [RaMe02] for comparison. The comparison of the accuracy with two other methods is shown in Fig. (3). The cases discussed so far refer to positive incident energies. For negative energies, i.e., when the quantity k2 in Eq. (1) is replaced by −κ2 , the functions F and G that deﬁne the Green’s function in Eq. (4) are replaced by sinh(κr) and exp(−κr), respectively. For large values of κr these functions generate integrals deﬁned in Eq. (15) which become unacceptably large or small, leading to numerical errors in the solution of the matrix equations (16) or (20). In this case the function Y and Z, which solve the negative energy equivalent of equations (6) and (7), have to be rescaled, as is described in Ref. [Gonz99], [Eqs. (35) through (37)], and in Ref [RaKo06], [Eq. (13) and Appendix A]. In this negative energy case Eq. (1) becomes an eigenvalue equation for κ2 , and Eq. (2) with the driving term F (r) removed, is satisﬁed only for a discrete set of κ values. The latter are found one by one iteratively [RaKo06], and not as the eigenvalues of a matrix. As a result, the discrete values of κ that correspond to weakly bound states, can be found with the same accuracy of 7 or 8 signiﬁcant ﬁgures, as the ones that correspond to strongly bound states, contrary to what is the case for the small eigenvalues of a truncated matrix. Various other applications were developed and presented as contributions to specialized nuclear physics meetings [RaKo04], [RaKo05b], but they are likely to be supplanted by subsequent work done in collaboration with W. Gl¨ ockle [GlRa07].

5. Subsequent developments An important and long standing problem is the calculation of the quantum mechanical wave function for a system of three particles, for example the bound states for a molecule containing three atoms, or the scattering of a proton from a deuteron target. One popular approach is to solve the three-body diﬀerential Schr¨ odinger equation in conﬁguration space, another is to solve the integral Faddeev equations in momentum space [GlWi96], [GoSk05], [WiGo06]. A formulation of the solution of the integral Faddeev equations in conﬁguration space, using the highly accurate spectral expansion methods described here, is in progress [GlRa07]. Unfortunately Professor Koltracht could no longer participate in these developments. As preliminary steps required to implement the thee-body calculation, a method was developed to adapt the S − IEM method to the calculation in conﬁguration space of the scattering K-matrix [Ra09], as well as integrals over the scattering matrix [GlRa07]. Such calculations have up to now been done only in momentum space. In conﬁguration space the K-matrix, touched upon at the end of Appendix A, is a function of two radial variables, and it obeys the L − S integral equation ∞

K(E; r, r ) = V (r) δ(r − r ) + V (r) K(E; r, r¯)G0 (E; r¯, r ) d¯ r, (27) 0

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421

where δ(r−r ) is the Dirac delta function. As a result the driving term V (r) δ(r−r ) is highly discontinuous. This problem can be circumvented by deﬁning a new function R(r, r ) K(E; r, r ) = V (r) δ(r − r ) + R(E; r, r ). which obeys a two-variable L − S integral equation ∞ G0 (E; r, r¯) R(E; r¯, r ) d¯ r, R(E; r, r ) = V (r)G0 (E; r, r )V (r ) + V (r)

(28)

(29)

0

whose driving term V (r)G0 (E; r, r )V (r ) has a discontinuity in its derivative when r = r . This diﬃculty can be overcome by the S − IEM method by means of a judicious choice of the boundaries of the partitions [Ra09]. By contrast, the integral of the K-matrix over some given function Φ(r) ∞ K(E; r, r )Φ(r ) dr

(30) ϕ(E, r) = 0

obeys a L − S equation

∞

ϕ(E; r) = V (r) Φ(r) + V (r)

G0 (E; r, r¯) ϕ(E; r¯) d¯ r,

(31)

0

whose driving term V (r) Φ(r) is continuous, provided that both V (r) and Φ(r) are continuous. In applications to the three-body problem in physics these conditions are usually met. The ability to calculate such integrals is important because they provide the basic ingredients for the solution of the three-body Faddeev integral equations in conﬁguration space. Numerical examples are presented in Ref. [RaGl08]

6. Summary and conclusions A review of the work by Israel Koltracht and the present author for developing and applying a spectral expansion method (S − IEM ) to the solution of Fredholm integral equations of the second kind is presented. For applications in physics these equations are denoted as Lippmann-Schwinger (L − S), but they are seldom solved numerically in conﬁguration space since physicists are more familiar with solving diﬀerential equations. The various numerical-mathematical properties of the S − IEM were reviewed, and a retrospective of the work with I. Koltracht was presented. Further developments of the method (in progress), aimed at solving the three-body problem in physics with an accuracy not previously achieved, are brieﬂy reviewed. It is believed that the association of a mathematician with a physicists was very fruitful since it had the eﬀect of introducing new methods into the domain of physics.

422

G.H. Rawitscher

Appendix A A short description of the general quantum mechanical Schr¨odinger equation will be given here, in order to elucidate the meaning of the function ψ(r) described in the text. A book on quantum mechanics that is very elegant and speaks in the language of operator theory is by Dirac [Di47]. Of the many other books which exist in the literature, the one by Rubin Landau [La90] will be referred to here because it addresses in simple terms topics on scattering theory, relevant for the present review. The Schr¨odinger equation is a partial diﬀerential equation for a wave function Ψ(x, y, z, t) that describes the behavior of a particle subject to a ﬁeld of forces. For example, the behavior of an electron incident on a given target, such as a hydrogen atom in the ground state. The motion of the electron is not deterministic, but can be predicted to occur only with a certain probability in terms of |Ψ|2 . Contrary to what is the case for a “classical” particle well localized on a trajectory of motion, a quantum mechanical particle proceeds without being narrowly localized in space. This is because its behavior is described by a function that satisﬁes a wave equation, which is a disturbance that extends over a ﬁnite region in space. The equation for Ψ is 2 ∂ 2 ∂2 ∂2 ∂Ψ ¯ . (1A) − + 2 + 2 + V (r) Ψ(x, y, z, t) = i 2 2μ ∂x ∂y ∂z ∂t The Cartesian coordinates of the displacement vector r of the projectile relative to the target are given by x, y, z, and t is the time. The coordinates of the center of mass of the system have already been discarded, since it moves with uniform velocity. The spherical polar coordinates of the displacement vector r are r, θ, and ϕ. The reduced mass of the projectile-target system is denoted as μ, and is Planck’s constant divided by 2π ( has units of energy times time). If m and M are the masses of the projectile and target, respectively, then μ−1 = m−1 + M −1 . The potential that describes the interaction of the projectile with the target is given by V¯ (r). This function is assumed to vanish at large values of r faster than 1/r (hence the Coulomb case is excluded here), but it may have points of discontinuity. If the energy of the projectile-target system is well deﬁned, and has the value E, then the system is in a “stationary” state Ψ = u(r) exp(−iE t/) (stationary because then |Ψ|2 = |u|2 becomes independent of time), and the equation for u(r) becomes 2 2 ∂ ∂2 ∂2 ¯ (r) uE (r) = E uE (r). + V − + + (2A) 2μ ∂x2 ∂y 2 ∂z 2 The energy E is an eigenvalue of the operator in square brackets. The positive values form a continuum, while the negative values are discrete. For positive energies u has to be ﬁnite for all values of r, but it generally is not square integrable, while for the negative discrete energies u has to be ﬁnite at the origin, and has to decrease exponentially at large distances (hence, belong to L2 ). Thus, the negative

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423

part of the energy spectrum describes the bound states, while the positive energy part describes the scattering states. If one expresses the operator in round brackets in terms of spherical polar coordinates, then the angular and radial parts separate, and the eigenfunctions of the angular part, Y,m (θ, ϕ), (called spherical harmonics) form a complete set, in terms of which the expansion of u can be written as uE (r) =

∞ 1 ψ,m (E, r)Y,m (θ, ϕ). r

(3A)

=0 m=−

In the above = 0, 1, 2, . . . , ∞, and for each there are 2 + 1 values of m = −, − + 1, . . . , − 1, . The above is called a partial wave expansion of uE (r) which generally converges well, and each partial wave ψ,m obeys a second-order diﬀerential equation in r. If V¯ (r) is not spherically symmetric, then this equation contains coupling terms to other partial waves. If however V¯ (r) = V¯ (r) is spherically symmetric, then ψ,m becomes independent of m and obeys 2 2 d2 ¯ (r) + ( + 1) ψ (E, r) = Eψ (E, r). − + V (4A) 2μ dr2 2μ r2 By rearranging this equation, setting = 0, dropping the labels and E in ψ (E, r), multiplying both sides of the equation (4A) by 2μ/2 , and deﬁning V (r) = (2μ/2 ) V¯ (r); k 2 = (2μ/2 ) E (5A) one obtains Eq. (1) in the text. The above describes the transformation of energy units to inverse length units, and assumes that the energy is positive, and k is the wave number at inﬁnity. The boundary conditions will be described next. For r ) 0, ψ has to vanish at least as fast as r because of the factor 1/r in Eq. (3A), and for distances r > R beyond which V¯ (r) becomes negligible (it is now assumed that V¯ (r) decreases faster than 1/r2 ), ψ (r) = α F (r) + βG (r), r > R, is given as a linear combination of Riccati-Bessel functions F (r) = (kr)j (kr); G (r) = −(kr)y (kr),

(6A)

deﬁned in Eq. (10.3.1) in [AbSt72]. The functions F and G obey Eq. (4A) with V¯ (r) set equal to 0 and j and y are spherical Bessel functions [AbSt72]. In numerical calculations based on Eq. (4A) the coeﬃcients α and β are usually obtained by matching the numerical solution for ψ (r) to F (r) and G (r) at a point r > R. In view of the asymptotic behavior of the Riccati-Bessel functions F (r) ) sin(kr − π/2) and G (r) ) cos(kr − π/2) the asymptotic behavior of ψ is proportional to ψ (r) ) eiδl sin(kr − π/2 + δ ), (7A) with the phase shift δ = tan−1 (β/α). This normalization is suitable for scattering situations which involves an incident plane wave and an outgoing scattering wave, as given in Ref.[La90], Eqs. (1.11) and (3.21). However, since the normalization of the numerical solution of Eq. (4A) can be ﬁxed arbitrarily, the phase shift δ is the main result of the solution of Eq. (4A) for positive energies. The scattering cross

424

G.H. Rawitscher

section (the distribution of scattered projectiles as a function of the scattering angle can be determined in terms of the δ , = 0, 1, 2, . . . , as is given in Eq. (3.36) of Ref. [La90]. In our numerical calculations with = 0 [Gonz97] we use G0 and replace the potential by V + ( + 1)/r2 and ﬁnd that the singularity near r = 0 does not cause loss of accuracy. Coupled equation versions of Eq. (4A) occur in the case that the potential is not spherically symmetric, and also if the target can be excited into diﬀerent states other than the ground state during the collision process. One example with two coupled equations is given in Ref. [RaEsTi99]. In addition to the bound states also continuum (break-up of the target) states can be excited. If the latter excitations are included, the calculation becomes much more complicated, and falls into the domain of three-body physics. Professor Koltracht made several contributions in this area [RaKo04], [RaKo05b]. The L − S equation ∞ ¯ ψ (r) = F (r) + G (k; r, r ) V (r ) ψ¯ (r ) dr

0 r 1 = F (r) − G (r) F (r ) V (r )ψ¯ (r ) dr

k 0∞ 1 − F (r) G (r ) V (r )ψ¯ (r ) dr

(8A) k r has a real solution ψ¯ that is proportional to the real solution of Eq. (4A) provided that the Green’s function is given by 1 G (r, r ) = − F (r< ) G (r> ). (9A) k (For a complex solution G is replaced by G ± i F .) This can be seen by verifying that ψ¯ (r) satisﬁes Eq. (4A), as follows by taking ﬁrst and second derivatives with respect to r of Eq. 8A), and making use of the Wronskian between the functions F and G . Further, the boundary conditions of the function ψ¯ (r) are proportional to the ones described for Eq. (4A). This can be seen by choosing for r in Eq. (8A) a value r > R, where R is the point beyond which the potential is negligible. As a result the second integral in Eq. (8A) is negligible, and ∞ 1 F (r ) V (r )ψ¯ (r ) dr , r > R. (10A) ψ¯ (r) = F (r) − G (r) k 0 ¯ (r) one has α Hence in the expression ψ¯ (r) = α ¯ F (r) + βG ¯ = 1 and ∞ 1 F (r ) V (r )ψ¯ (r ) dr

(11A) β¯ = tan(δ ) = − k 0 This equation avoids the need to match ψ¯ to the Ricatti Bessel functions in order to ﬁnd a phase shift, thus avoiding a loss of accuracy. Near the origin (r → 0) the ﬁrst integral in Eq. (8A) vanishes faster than G increases, and ψ¯ becomes proportional to F , which vanishes as r → 0. In the text the solution of Eq. (2) for = 0 is denoted as ψ(r).

Spectral Expansions for Physics Applications If a K(r, r )-matrix is introduced such that ∞ V (r )ψ¯ (r ) = K (r , r

)F (r

)dr

425

(12A)

0

then tan(δ ) can be expressed by a double integral involving ∞ ∞ dr F (r )K (r , r

)F (r

)dr

. 0

0

If the wave function is complex, and if all partial waves are included, then the K-matrix is replaced by a T -matrix, as is described by Eq. (6.1) in Ref. [La90], and the integral above is replaced by a matrix element of T taken between the initial and ﬁnal plane waves.

References [AbSt72]

Abramowitz, M. and Stegun, I., “Handbook of Mathematical Functions”, Dover, 1972, p. 445. [CC60] Clenshaw, C.C and Curtis, A.R., Numer. Math., 1960, 2, 197. [Del77] Deloﬀ A., “Semi-spectral Chebyshev method in quantum mechanics”, 2007, Annals of Phys. 322, 1373–1419. [Di47] Dirac, P.A.M., The principles of Quantum Mechanics, Oxford, Clarendon Press 3rd edition, 1947. [EZ04] The author thanks Dr. Essaid Zerrad for performing the sixth order Numerov calculation. [FaMe93] Faddeev, L.D. and Merkuriev, S.P. , 1993 “Quantum Scattering Theory for Several Particle Systems”, Kluwer Academic Publishers, Dordrecht 1993. [Fredh1903] Fredholm, I., “Sur une classe d’´equations fonctionelles”, Acta math., 1903, 27, 365–390. [GlWi96] Gl¨ ockle, W., Witala, W.H., H¨ uber, D., .Kamada, H., Golak, “The threenucleon continuum: achievements, challenges and applications”, 1996, J. Phys. Rep. 274, 107–285. [GlRa07] Gloeckle, W. and Rawitscher, G., “Scheme for an accurate solution of Faddeevv integral equations in conﬁguration space”, Proceedings of the 18th International Conference on Few-Body Problems in Physics, Santos,Brazil, Nucl. Phys. A, 790, 282–285 (2007). [Gonz97] Gonzales, R.A., Eisert, J., Koltracht, I., M. Neumann, M. and Rawitscher, G., “Integral Equation Method for the Continuous Spectrum Radial Schr¨ odinger Equation”, J. of Comput. Phys., 1997, 134, 134–149. [Gonz99] R.A. Gonzales, R.A., Kang, S.-Y., Koltracht, I. and Rawitscher G., “Integral Equation Method for Coupled Schr¨ odinger Equations”, J. of Comput. Phys., 1999, 153, 160–202. [GoOr77] Gottlieb, D. and Orszag, S., “Numerical Analysis of Spectral Methods”, SIAM, Philadelphia, 1977. [GR 91] Greengard, L. and Rokhlin, V. Commun. Pure Appli. Math 1991, 44, 419. [GoSk05] Golak, J., Skibinski, R., Witala, H., Gl¨ockle, W., Nogga, A., Kamada, H., “Electron and photon scattering on three-nucleon bound A states”, 2005, Phys. Rep. 415, 89–205.

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G.H. Rawitscher

[GvL83]

Golub, G.H., and Van Loan, C.H., “Matrix Computations”, page 10, Johns Hopkins Press, Baltimore, 1983. [KaKoRa03] Kang, S.-Y., I. Koltracht, I. and Rawitscher, G., “Nystr¨om-ClenshawCurtis Quadrature for Integral Equations with Discontinuous Kernels”, 2002, Math. Comput. 72, 729–756. [La90] Landau, R.H., Quantum Mechanics II, John Wiley & Sons, 1990. [RaEsTi99] Rawitscher G.H. et al., “Comparison of Numerical Methods for the Calculation of Cold Atom Collisions,” J Chem. Phys., 1999, 111, 10418–10426. [RaGl08] Rawitscher, G. and Gloeckle, W., “Integrals of the two-body T matrix in conﬁguration space”, 2008, Phys. Rev A 77, 012707 (1–7). [Ra09] Rawitscher, G. “Calculation of the two-body scattering K-matrix in conﬁguration space by an adaptive spectral method”, 2009, J. Phys. A: Math. Theor. 42, 015201. [RaKaKo03] Rawitscher, G., Kang, S.-Y. and I. Koltracht, I. , “A novel method for the solution of the Schr¨ odinger equation in the presence of exchange terms”, 2003, J. Chem. Phys., 118, 9149–9156. [RaKo04] Rawitscher, G. and Koltracht, I., “A spectral integral method for the solution of the Faddeev equations in conﬁguration space”, Proceedings of the 17th International IUPAP conference on Few Body problems in physics, 2004, Nucl. Phys. A, 737 CF, pp. S314–S316. [RaKo05a] Rawitscher, G. and I. Koltracht, “Description of an eﬃcient Numerical Spectral Method for Solving the Schr¨ odinger Equation”, Computing in. Sc. and Eng., 2005, 7, 58. [RaKo05b] Rawitscher, G. and I. Koltracht, “Can the CDCC be improved? A proposal”, Proceedings of the NUSTAR05 conference, J. of Phys. G: Nuclear and particle physics, 31, p. S1589–S1592. [RaKo06] Rawitscher, G. and I. Koltracht I., “An economial method to calculate eigenvalues of the Schr¨ odinger equation”, Eur. J. Phys. 2006, 27, 1179–1192. [RaMe02] Rawitscher, G., Merow C., Nguyen M., Simbotin, I., “Resonances and quantum scattering for the Morse potential as a barrier”, Am. J. Phys. 2002, 70, 935–944. [Sim07] The author acknowledges useful discussion with Dr. Ionel Simbotin at the University of Connecticut concerning the feasibility and the advantages of connecting two (rather than three) neighboring partitions to each other. [WiGo06] Witala, H. Golak, J.; Skibinski, R.; Glockle, W.; Nogga, A.; Epelbaum, E.; Kamada, H.; Kievsky, A.; Viviani, M., “Testing nuclear forces by polar reactions at E (lab) = ization transfer coeﬃcients in d( p, p)d and d( p, d)p P 22.7 M eV ”, 2006, Phys. Rev. C (Nuclear Physics), 73, 44004-1-7. George H. Rawitscher Department of Physics, University of Connecticut Storrs, CT 06268, USA e-mail: [email protected] Received: February 22, 2009 Accepted: June 25, 2009

Operator Theory: Advances and Applications, Vol. 203, 427–444 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Regularized Perturbation Determinants and KdV Conservation Laws for Irregular Initial Proﬁles Alexei Rybkin Dedicated to Israel Gohberg on the occasion of his 80th birthday. We ﬁrst learned about regularized determinants from one of his books with Mark Krein.

Abstract. In the context of the Korteweg-de Vries equation we put forward some new conservation laws which hold for real initial proﬁles with low regularity. Some applications to spectral theory of the one-dimensional Schr¨odinger operator with singular potentials are also considered. Mathematics Subject Classiﬁcation (2000). Primary 35Q53, 37K15; Secondary 34L40, 34B20. Keywords. Korteweg-de Vries equation, modiﬁed perturbation determinants, conservation laws.

1. Introduction The present paper is an extended exposition of the talk given by the author at the 2008 International Workshop on Operator Theory and Applications (IWOTA), Williamsburg, Virginia. We retain the structure of the talk expanding only on some crucial ingredients. It is a fundamental fact of soliton theory that the Cauchy problem for the Korteweg-de Vries (KdV) equation on the full line Vt − 6V Vx + Vxxx = 0 V (x, 0) = V0 (x)

(1.1)

Based on research supported in part by the US National Science Foundation under Grant DMS 070747. Communicated by I.M. Spitkovsky.

428

A. Rybkin

has inﬁnitely many conservation laws: d σn (x, t)dx = 0, dt R

n ∈ N.

(1.2)

The functions σn (x, t), called conserved densities, represent a sequence of diﬀerential polynomials in V obtained by the recursion formula d σn−k−1 σk , σn−1 − dx n−2

σ1 = V, σn = −

n ≥ 2.

k=1

Explicitly: dV d2 V d3 V dV , σ3 = −V 2 + , etc. , σ4 = − 3 + 4V 2 dx dx dx *dx In fact, all {σ2l } are complete derivatives and therefore R σ2l (x, t)dx = 0 do not contribute to (1.2). For n = 2l − 1, l ∈ N, the integrals (1.2) admit nice representations σ2l−1 (x, t)dx (1.3) σ2 = −

R

∞ N 4l 2l−1 l κn − (−4) k 2l−2 f (k)dk, l ∈ N, 2l − 1 n=1 0 2 in terms of bound states −κn and certain scattering quantity f (k) associated with the Schr¨odinger operator −∂x2 +V0 (x) on the full line which potential V0 is the initial proﬁle in (1.1). Formulas (1.3) are usually referred to as Faddeev-Zakharov trace formulas. Diﬀerential polynomials σ2l−1 (x, t) can be rearranged in diﬀerent ways by adding full x-derivatives to σ2l−1 (x, t). With this in mind for the ﬁrst three trace formulas in the chain (1.1) one has: ∞ V (x, t) dx = −4 κn + 4 f (k) dk (1.4) =−

R

0

n

(the 1st conservation law),

16 3 V (x, t) dx = κ + 16 3 n n R

2

∞

k 2 f (k) dk

(1.5)

0

(the 2nd conservation law), $ % 3 2 2V (x, t) + Vx (x, t) dx = R

∞ N 64 5 κ + 64 k 4 f (k) dk (1.6) − 5 n=1 n 0 (the 3rd conservation law).

We emphasize that existence of inﬁnitely many conservation laws is directly linked to the complete integrability of (1.1) by the so-called inverse scattering

KdV Conservation Laws

429

transform (IST). The latter was originally developed for inﬁnitely diﬀerentiable smooth initial data V0 rapidly decaying at inﬁnity (the Schwartz class.) Kappeler [4] proved that IST also works for V0 ’s that are measures (including the δ function) satisfying a certain rapid decay assumption at inﬁnity. However only the ﬁrst conservation law (1.4) actually holds for such solutions V (x, t). On the other hand, as it was established by Collianger et al [1], the Cauchy problem (1.1) is globally well posed for V0 ’s from the L2 base Sobolev space H −3/4+ . Thus the Cauchy problem for the KdV equation with a distributional initial data V0 ∈ H −3/4+ has a unique global solution but conservation laws (1.3) need not hold for any natural l. This fact may look disturbing as conservation laws are a principle ingredient of the IST method. It appears unknown if the IST can be extended to solve (1.1) with H −3/4+ initial proﬁles. We do not tackle this problem here but study some other conserved quantities that are well deﬁned for a variety of initial data V0 from certain Sobolev spaces with negative indices.

2. Notation and preliminaries We will follow standard notation: R± = (±∞, 0) , C± = {z ∈ C : ± Im ≥ 0} . ·X stands for the norm in a Banach (Hilbert) space X. We use standard Lebesgue spaces (1 ≤ p < ∞)

& 1/p

Lp (Δ) =

p

f : f Lp (Δ) ≡

|f (x)| dx

< ∞ , Lp (R) ≡ Lp ,

Δ

5 J L∞ (Δ) = f : f L∞ (Δ) ≡ ess sup |f (x)| < ∞ , x∈Δ

Lploc = {∩Lp (Δ) : Δ is compact} . We agree to write

≡

Given distribution f , let 1 f(λ) = √ 2π

R

, Lp ≡ Lp (R) .

1 e−iλx f (x) dx, f ∨ (λ) = √ 2π

eiλx f (x) dx

be the standard Fourier transform and its inverse. Hps , s ∈ R, p ≥ 1, denote the Sobolev spaces of distributions $ ( %∨ . / s 2 s/2 p Hp = f : 1 + λ f (λ) ∈ L , H s ≡ H2s We will be particularly concerned with s < 0. For s = −n, n ∈ N , one has f ∈ Hp−n ⇐⇒ f (x) =

n m=0

∂xm fm (x) with some fm ∈ Lp .

(2.1)

430

A. Rybkin

Note that functions fm in the decomposition (2.1) are not unique. Moreover, there is no standard choice of the norm in Hps . For instance, in H −1 any # # # f(λ) # # # #√ 2 # # a + λ2 # 2 L

deﬁnes a norm of f which are all equivalent, due to # # # # # # # f(λ) # # # f(λ) # b# # # f(λ) # # # # 0 < a < b =⇒ # √ < #√ < #√ # # # . # b2 + λ2 # 2 # a2 + λ2 # 2 a # b2 + λ2 # 2 L L L Spec (A) denotes the spectrum of an operator A, Rz (A) = (A − zI)−1 is its resolvent and A is its uniform norm. Next, Spec d (A) and Specac (A) are, respectively, the discrete and absolutely continuous spectrum of a (self-adjoint) operator A. Sp , p > 0, denote Schatten-von Neumann classes of linear operators A: A ∈ Sp ⇐⇒ ASp := tr (A∗ A) p

p/2

< ∞.

In particular, S1 is the trace class and S2 is the Hilbert-Schmidt class. The following assertion will be frequently used [8]: for p ≥ 2 1/p 1 f (x) g (−i∂x )Sp ≤ f Lp gLp . (2.2) 2π For an operator A from Sp (p ∈ N) we deﬁne the regularized p-determinant ,&

+ p−1 (−1)n n (2.3) A detp (I + A) := det (I + A) exp n n=1 with the convention det1 (I + A) = det (I + A) . From (2.3)

+ detp (I + A) = detp−1 (I + A) exp

p−1

(−1) Ap−1 p−1

, .

(2.4)

3. Regularized perturbation determinants Through this paper we deal with pairs (H, H0 ) of operators H0 = −∂x2 , H = H0 + V (x) , on the full line. Introduce the notion of the regularized perturbation p-determinant Δp of such a pair (H, H0 ) as follows: Δp (z) := detp (I + Q (z))

(3.1)

Q (z) := Rz1/2 (H0 ) V Rz1/2 (H0 ) ,

(3.2)

where

KdV Conservation Laws 1/2

431

1/2

is ﬁxed so that Im Rz ≥ 0. We use the short-hand notation . / . /−1/2 . /−1/2 V H0 + a2 . Qa (V ) := Q −a2 = H0 + a2

and the branch of Rz

Since H0 = −∂x2 will always be the same, for a ﬁxed Qa depends only on V . When needed, we indicate this by writing Qa (V ). Perturbation determinants play an important role in perturbation and scattering theory (see, e.g., [8], [9] ). Our choice (3.2) of Q in (3.1) is not the only possible and the following expressions V Rz (H0 ) , Rz (H0 ) V, V 1/2 Rz (H0 ) V 1/2 are also widely used in the literature. These expressions are not equivalent and a particular choice is determined by the speciﬁc setting. For instance, in the context of the Birman-Schwinger principle V 1/2 Rz (H0 ) V 1/2 is typically used. The motivation for our choice will become transparent from the following convenient criterion for existence of the regularized perturbation p-determinant in terms of the potential V. Theorem 1. If V ∈ Hp−1 , p = 2, 3, . . . then Δp (z) is well deﬁned. The proof follows from the assertion: Proposition 1. If V ∈ Hp−1 with some real p ≥ 2 and V = u + v is the decomposition (2.1) then Qa (V ) ∈ Sp and 1/p # /−1/2 # 1 1 2C #. 2 # ||Qa (V )||Sp ≤ 1−1/p ||u||Lp + ||v||Lp , C := # x +1 # p. 2a 2π L a (3.3) Moreover tr Qpa (V ) can be evaluated by one of the formulas: p/2 = p 1 Vˆ (λn − λn+1 ) p tr Qa (V ) = dλ1 · · · dλp (3.4) 2π λ2n + a2 Rp n=1 p p = "p 1 e−a n=1 |xn −xn+1 | V (xn ) dx1 . . . dxp , (3.5) = 2a Rp n=1 where xp+1 = x1 (λp+1 = λ1 ) and the integrals are understood in the distributional sense. Proof. By (2.1) V = u + v where u, v ∈ Lp and hence Qa (V ) = Qa (u) + Qa (v ) . For Qa (u) we have Qa (u)Sp

(3.6)

# # # 1/2 # 1/2 = #R−a2 (H0 ) uR−a2 (H0 )# Sp # ## # # 1/2 ## # 1/2 ≤ #R−a2 (H0 )# #u (x) R−a2 (H0 )# Sp ## #. . / /−1/2 # −1/2 # # # # = # −∂x2 + a2 # #u (x) −∂x2 + a2 #

Sp

.

(3.7)

432 But and by (2.2)

A. Rybkin # #. /−1/2 # /−1/2 # # #. 2 # # # = # x + a2 # # −∂x2 + a2 # . /−1/2 # # # #u (x) −∂x2 + a2 # ≤ =

1 2π 1 2π

1/p

L∞

= 1/a

(3.8)

(3.9)

Sp

#. /−1/2 # # # uLp # x2 + a2 #

Lp

1/p uLp

1 a1−1/p

#. /−1/2 # # 2 # # x +1 #

Lp

.

Combining (3.7)–(3.9) we have ||Qa (u)||Sp ≤ where

C a2−1/p

uLp

(3.10)

1/p # /−1/2 # 1 #. 2 # # x +1 # p. 2π L

We now turn now to Qa (v ). By the product rule for diﬀerentiation 5 . /−1/2 J ∂x v (x) −∂x2 + a2 (3.11) / . / . −1/2 −1/2 = v (x) −∂x2 + a2 + v (x) ∂x −∂x2 + a2 , . 2 / −1/2 is understood as the product of two commuting operators where ∂x −∂x + a2 . 2 . / /−1/2 −1/2 . Multiplying (3.11) on the left by −∂x2 + a2 and then ∂x and −∂x + a2 . 2 . 2 / /

2 −1/2

2 −1/2 v (x) −∂x + a , we get solving for Qa (v ) = −∂x + a . . /−1/2 /−1/2 Qa (v ) = ∂x −∂x2 + a2 v (x) −∂x2 + a2 /−1/2 /−1/2 . . − −∂x2 + a2 v (x) −∂x2 + a2 ∂x (3.12) J 5 . 2 . / / −1/2 −1/2 v (x) −∂x2 + a2 . = 2 Im (−i∂x ) −∂x + a2

C=

Formula (3.12) readily implies # . . /−1/2 /−1/2 # # # v (x) −∂x2 + a2 ||Qa (v )||Sp ≤ 2 #(−i∂x ) −∂x2 + a2 # Sp # # # # . 2 . 2 / / # 2 −1/2 # # 2 −1/2 # ≤ #(−i∂x ) −∂x + a # #v (x) −∂x + a # .(3.13) Sp

But

# # . /−1/2 # /−1/2 # # # . 2 # # # = #x x + a2 # #(−i∂x ) −∂x2 + a2

and by (2.2), similarly to (3.9), one has # . /−1/2 # # # #v (x) −∂x2 + a2 #

Sp

≤

C a1−1/p

L∞

vLp .

=1

(3.14)

(3.15)

KdV Conservation Laws

433

Combining (3.13)–(3.15) we have ||Qa (v )||Sp ≤

2C uLp a1−1/p

(3.16)

with C being the same as in (3.10). From (3.6), (3.10) and (3.16) ||Qa (V )||Sp

≤ ≤

||Qa (u)||Sp + ||Qa (v )||Sp 2C 1 + ||v|| ||u|| Lp Lp a1−1/p 2a

and (3.3) is proven. It remains to prove (3.4)–(3.5). Since Qa (V ) ∈ Sp , we have Qpa (V ) ∈ S1 and p tr Qa (V ) can be easily evaluated in terms of the kernel of Qa (V ). To show (3.4) compute tr Qpa (V ) in terms of the potential V by using the Fourier representation. := F QF ∗ , where F is the Fourier transform, one has Denoting Q $ . $ . % % / / a = F −∂x2 + a2 −1/2 F ∗ (F V F ∗ ) F −∂x2 + a2 −1/2 F ∗ . Q . /−1/2 ∗ /−1/2 . F is the multiplication operator by λ2 + a2 and Since F −∂x2 + a2 1 ∗ F V F is an integral operator with the kernel √2π V (λ − ω), one concludes that Oa is an integral operator with the kernel the operator Q 1 V (λ − ω) √ √ √ 2π λ2 + a2 ω 2 + a2

$ %p Oa is also an integral operator, the kernel of which is and hence Q

1 2π ×

p/2

1 (3.17) λ2 + a2 1 Vˆ (λ − λ1 )Vˆ (λ1 − λ2 ) · · · Vˆ (λp−1 − ω) / . 2 . dλ1 · · · dλp−1 √ 2 2 2 2 (λ1 + a ) · · · λp−1 + a ω + a2

√ Rp−1

Setting in (3.17) λ = ω and then integrating with respect to λ yields (3.4). Equation (3.5) follows from (3.4) if one goes back from Vˆ to V : p/2 = p 1 Vˆ (λn − λn+1 ) dλ1 · · · dλp 2π λ2n + a2 Rp n=1 * i(λ −λ )x p/2 = p e n+1 n n V (xn ) dxn 1 1 √ = dλ1 · · · dλp 2π λ2n + a2 2π Rp n=1 p = p ei(xn −xn+1 )λn+1 1 = V (xn ) dx1 . . . dxp . (3.18) 2π λ2n+1 + a2 Rp n=1

434

A. Rybkin

where at the last step we have used the convention λp+1 = λ1 and the Abel transformation p p (λn+1 − λn )xn = λn+1 (xn − xn+1 ). n=1

n=1 1

But by the calculus of residues

eiαλ dλ π = e−a|α| 2 2 λ +a a

and (3.18) immediately yields the required formula (3.5). Remark 1. For p = 2 inequality (3.3) turns into the equality. Namely, 2 V (k) 1 2 dk ||Qa (V )||S2 = tr Q2a (V ) = a k 2 + 4a2

(3.19)

and hence Qa (V ) ∈ S2 ⇐⇒ V ∈ H −1 . . /−1 is Remark 2. Note that an analog of Proposition 1 for Qa (V ) = V −∂x2 + a2 much weaker, as we only have # . /−1 # C # # #V −∂x2 + a2 # ≤ 2−1/p ||V ||Lp Sp a which of course assumes local integrability of V. On the other hand, the integral in (3.5) becomes absolutely convergent.

4. The regularized perturbation 2-determinant is a KdV invariant In this section we show that the regularized perturbation 2-determinant is invariant under the KdV ﬂow for very irregular initial data. Consider the pair (H, H0 ) of operators H0 = −∂x2 , H = H0 + V (x) , on the full line under a very generous assumption for the time being that V = V ∈ C0∞ , where C0∞ is the class of smooth compactly supported function on R. Under such conditions on the potential one has a typical scattering theoretical situation which means that there exist all four wave operators and the scattering operator for the pair (H, H0 ). In particular, the absolutely continuous part of H is unitarily equivalent to H0 . The spectrum Spec (H) of H consists of two components: twofold absolutely continuous (a.c.) Specac (H) ﬁlling R+ and simple discrete Specd (H) N containing a ﬁnite number of negative eigenvalues −κn2 n=1 . 1 Recall

our convention

*

:=

*∞

−∞

KdV Conservation Laws

435

The a.c. spectrum of H is of uniform multiplicity two and hence the scattering matrix S is a two by two unitary matrix t (k) r+ (k) , k 2 ∈ Specac (H) = R+ , S (k) = r− (k) t (k) where t and r± denote the transmission and reﬂection coeﬃcients from the left (right) incident. Due to unitarily of S one has 2

2

|t| + |r± | = 1,

(4.1)

(4.2) t (−k) = t (k), r± (−k) = r± (k). The quantities t and r± are related to the existence of special solutions ψ± to the stationary Schr¨ odinger equation −u

+ v (x) u = k 2 u, k ∈ R, asymptotically behaving as ψ+ (x, k) ∼

x → ∞, t (k) eikx , e + r− (k) e−ikx , x → −∞, ikx

e−ikx + r+ (k) eikx , x → ∞, x → −∞. t (k) eikx , The scattering matrix S is pertinent to the a.c. spectrum of H. A fundamental fact of the short-range scattering theory is that $ % 2 t (k) = Δ−1 (k + i0) , k ∈ R, (4.3) ψ− (x, k) ∼

where (4.4) Δ (z) = det{I + V Rz (H0 )}. The following function will play an important role: 1 f (k) := log |t (k)|−1 . (4.5) π Due to (4.1) and (4.2), f (−k) = f (k) ≥ 0. (4.6) The function f is also integrable. Note that the determinant in (4.4) exists due to V ∈ C0∞ =⇒ V Rz (H0 ) ∈ S1 and is an analytic function on C R+ with a ﬁnite number N of simple zeros −κn2 1/2 1/2 on R− . Observe that since Rz (H0 ) V Rz (H0 ) is also in S1 , we have % $ Δ (z) = det I + Rz1/2 (H0 ) V Rz1/2 (H0 ) . The transmission coeﬃcient t (k) can then be analytically extended from R to C+ by (4.3) and for suﬃciently large |z| −1

|t (z)| ≤ C |z|

.

436

A. Rybkin

What has been said actually implies the well-known representation (see, e.g., [3]) ( N √ = 1 z − iκn f (k) √ dk . √ exp Δ (z) = (4.7) z + iκn i k− z n=1 Since due to (4.6)

√ f (k) dk √ =2 z k− z

∞ 0

f (k) dk, k2 − z

(4.7) takes the form ( ∞ N √ = √ z − iκn f (k) √ dk . exp −i z Δ (z) = k2 − z z + iκn 0 n=1 Setting in (4.8) z = −a2 , a > κ1 , one arrives at the dispersion relation ∞ N . / a − κn f (k) log + 2a dk = log Δ −a2 . 2 2 a + κn k +a 0 n=1

(4.8)

(4.9)

Dispersion relations like (4.9) turn out to be very useful. A version of (4.9) was ﬁrst used by Faddeev-Zakharov [3] to derive their famous trace formulas (sum rules) (1.3). It was also used in [6] in the setting of the Lieb-Thirring inequality and spectral analysis of the Schr¨ odinger operator with H −1 potentials. We employ it here in the context of the KdV equation with highly irregular initial data. From the deﬁnition (3.1) of Δ2 . / . / Δ2 −a2 = Δ −a2 e− tr Qa , and hence

. / log Δ −a2 =

. / log Δ2 −a2 + tr Qa . / 1 = log Δ2 −a2 + V (x) dx 2a N . 2/ 2 2 ∞ κn + f (k) dk, = log Δ2 −a − a n=1 a 0

(4.10)

where we have used the well-known fact that 1 V (x) dx (4.11) tr Q = 2a and (1.4). Substituting (4.10) into (4.9) and rearranging the terms we get that for any a > κn N . / 1 + κn /a −2κn /a 2 ∞ k 2 f (k) log + dk = − log Δ2 −a2 . (4.12) e 2 + a2 1 − κ /a a k n 0 n=1 Since κn and f (k) are time conserved under the KdV ﬂow, (4.12) immediately implies that det2 {I + Qa (V (·, t))} = det2 {I + Qa (V (·, 0))} = det2 {I + Qa (V0 )}

(4.13)

KdV Conservation Laws

437

where V (x, t) is the solution to the KdV equation with the initial proﬁle V0 ∈ C0∞ . The Cauchy problem for the KdV equation (1.1) is globally well posed for any real V0 ∈ H −3/4+ ⊂ H −1 . Approximate now V0 ∈ H −3/4+ by C0∞ functions. Note that det2 (I + A) is continuous with respect to A in the Hilbert-Schmidt norm, and hence by (3.19) det2 {I + Qa (V )} is also continuous with respect to V ∈ H −1 . Therefore det2 {I + Qa (V )} is also continuous with respect to V ∈ H −3/4+ since H −3/4+ ⊂ H −1 . Equation (4.13) can then be established for any V0 ∈ H −3/4+ and we arrive at the following result: Theorem 2. Let V (x, t) be the solution to the Cauchy problem for the KdV equation with some initial data V0 from H −3/4+ . Then 5 . . /−1/2 /−1/2 J d V (x, t) −∂x2 + a2 = 0. det2 I + −∂x2 + a2 dt It is reasonable to ask if equation (4.12) could be extended to any V from H −1 . Since the terms on the left-hand side of (4.12) are positive they both have ﬁnite limits as we approximate a real V ∈ H −1 by C0∞ functions. The main question is if f (k) dk will tend to an absolutely continuous measure. We do not know if the answer is aﬃrmative. In this connection we oﬀer one curious assertion which we do not actually use in our exposition. Proposition 2. Let H0 = −∂x2 on L2 and H = H0 + V (x) where V is real and lies in2 H −1 . Then for any complex z away from Spec (H) the regularized perturbation 2-determinant Δ2 of the pair (H, H0 ) admits the following representation ( ∞ = √z − iκn √ 1 dμ −2iκn / z √ , (4.14) e exp √ Δ2 (z) = z + iκn i z 0 k−z n where −κn2 is the negative discrete spectrum of H and dμ is a ﬁnite non-negative measure. This proposition easily follows from (4.12), and we omit its proof. We only mention that Spec (H) ∩ (−∞, 0) is purely discrete, subject to κn3 < ∞ (4.15) n

and Specac (H) = R+ (see, [6]). The absolutely continuous component of μ is supported on R+ . Besides these two components the positive spectrum of H may have a singular component which is likely to present the main diﬃculties in extending the IST method to initial proﬁles from L2 . Note that Δ2 (z) does not have a bounded characteristic (e.g., the Blaschke type product in (4.14) " converges under condition (4.15) that is weaker than the Blaschke condition n κn2 < ∞). Nevertheless (4.14) immediately implies that Δ2 (z) has boundary values almost everywhere on the real line. It is an important feature of regularized perturbation 2-determinants associated with Schr¨ odinger operators. In general such determinants need not have boundary values. 2 The

sense in what H0 + V (x) is deﬁned will be presented in the appendix.

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In Theorem 2 we assumed that V0 ∈ H −3/4+ to make sure that V (x, t) exists. If (1.1) was well posed for H −1 initial data then the extra assumption V0 ∈ H −3/4+ could be removed. It does not appear to be known how far beyond H −3/4+ the problem (1.1) remains well posed (even locally).

5. Almost conserved quantities By looking at Theorem 2 one could ask if higher-order regularized perturbation determinants are also KdV invariants. Although it is not the case but in a way they are almost conserved. Lemma 1. Let A be a selfadjoint Hilbert-Schmidt operator such that AS2 < 1. Then 2 1 1 − A2S2 ≤ det2 (I + A) ≤ e− 6 AS2 . (5.1) Proof. Since log det = tr log, we have log det (I + A)−1 eA

= tr log (I + A)−1 eA = tr (A − log (I + A)) = tr

(−1)n An n n≥2 A2n+1

1 1 A2n − 2n 2n + 1 n≥1 2n = tr I− A A2n . 2n + 1

= tr

(5.2)

n≥1

Since AS2 < 1, (5.2) implies −1 A

log det (I + A)

e ≥ tr

n≥1

1 1 1 2 A2n ≥ tr A2 = AS2 . 2n (2n + 1) 6 6

Hence

1

2

det2 (I + A) = det (I + A) e−A ≤ e− 6 AS2 , and the estimate from above in (5.1) is proven. For the estimate from below: # # # tr A2n # 2n 2n 2n # I− A A A# tr I− ≤ # 2n + 1 2n + 1 # 2n n≥1 n≥1 $ %n 2 2n A AS2 S2 2n ≤ I+ ≤ 2n + 1 2n n n≥1

=

log

n≥1

1 2

1 − AS2

.

(5.3)

Combining (5.2) and (5.3) yields det2 (I + A) ≥ 1 − A2S2 .

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Applying Lemma 1 with A = Qa (V ) and equation (3.19) yields Theorem 3. Under the conditions of Theorem 2, if a > 0 is chosen so that 2 V O0 (k) 1 dk < 1 a k 2 + 4a2 then for any t > 0 2 V (k, t) 1 dk ≤ 6 log Δ−1 (5.4) 1 − Δ2 ≤ 2 a k 2 + 4a2 where 5 . /−1/2 . 2 /−1/2 J V0 −∂x + a2 . Δ2 = det2 I + −∂x2 + a2 * |V (k,t)|2 Note that by Theorem 2 the double-sided estimate (5.4) means that dk k2 +4a2 is bounded between two conserved quantities that justiﬁes its name – almost con* |VO0 (k)|2 * |V (k,t)|2 dk served quantity. In particular if a1 k2 +4a2 dk is small enough then a1 k2 +4a2 will also be small. For instance, 2 2 V V (k, t) O0 (k) 1 1 dk < 1 − e−1/6 =⇒ dk < 1 for any t > 0. a k 2 + 4a2 a k 2 + 4a2 * |V (k,t)|2 Note that k2 +4a2 dk is actually conserved for solutions of the form V (x, t) = f (x − ct) with some f , e.g., a one soliton solution. Due to (2.4) 2

Δ3 = Δ2 eQa S2 /2 and by Lemma 1, under the conditions of Theorem 3 we have 0 < Δ3 ≤ 1/Δ22 . The latter means that the perturbation 3-determinant Δ3 is also an almost conserved quantity. Same conclusion can be made regarding any Δp . Due to (3.5) they are all conserved for solutions of the form V (x, t) = f (x − ct).

6. Applications to spectral theory of the Schr¨ odinger operator Formulas (1.3) among other types of trace formulas appeared predominantly in the context of completely integrable systems and direct/inverse scattering theory. In the context of spectral theory their use was somewhat limited until Deift-Killip [2] employed (1.5) as the main ingredient in proving the stability of the absolutely continuous spectrum of H0 = −∂x2 under L2 perturbations. Paper [2] was followed by a number of works among which we mention only [5] by Molchanov-NovitskiiVainberg where the whole hierarchy (1.3) was used to derive some optimal statements regarding absolutely continuous spectrum preservation under certain long

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range perturbations. In this section we show that the presence of a free parameter a in the dispersion relation (4.12) makes it particularly useful in applications to spectral analysis of Schr¨ odinger operators with singular potentials. Here we restrict ourselves to a few examples; the full exposition will appear elsewhere. Assuming for the time being that V ∈ C0∞ , it follows from (4.12) that N = . 2/ 1 + κn /a −2κn /a −a . e ≤ Δ−1 2 1 − κ /a n n=1

The latter inequality can be extended to any V ∈ H −1 = 1 + κn /a . 2/ e−2κn /a ≤ Δ−1 −a . 2 1 − κn /a n

(6.1)

It immediately from (6.1) that if V ∈ H −1 the product in (6.1) converges " follows 3 and hence n κn < ∞. By this reason (6.1)5can be called J a generalized Lieb2 corresponding to the Thirring inequality for rescaled bound states − (κn /a) Schr¨ odinger operator with a singular potential from H −1 . The energy −a2 in 2 * ∞ dk (6.1) should be chosen so that a2 0 V (k) k2 +4a 2 < 1 which automatically forces supn κn < a. This choice of a also warrants by (4.12) that . 2/ 2 ∞ k 2 f (k) dk ≤ log Δ−1 −a . (6.2) 2 2 2 a 0 k +a Inequality (6.2) implies (see, [6]) that Specac (H) = R+ for any real V ∈ H −1 . While regular (L1loc ) potentials cover most of realistic situations there are some physically interesting H −1 potentials. Examples include Coulomb, delta, oscillatory Vigner von-Neumann (irregular behavior at inﬁnity) to name just a few. Assuming for the time being that V ∈ C0∞ , it follows from (4.12), (2.3), (3.19) and the Taylor expansion log

α2m+1 1+α − 2α = 2 ≥ 0, 0 ≤ α < 1, 1−α 2m + 1 m≥1

that N $ 2 κn %2m+1 2 ∞ k 2 f (k) + dk 2m + 1 n=1 a a 0 k 2 + a2 m≥1 2 ∞ V (k) . / 1 dk − log Δ3 −a2 . = 2 2 a 0 k + 4a

(6.3)

Observe that if we multiply (6.3) by a3 and then let a → ∞, (6.3) transforms into the second conservation law (1.5). Although the latter looks nicer than (6.3), it holds under much stronger conditions on V . The presence of an extra parameter a

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makes (6.3) more ﬂexible for applications to spectral theory. For example, repeatedly multiplying (6.3) by a and then diﬀerentiating with respect to a yields that for any a > κ1 and for all natural p: ∞ N $ (m + p − 1)! k 2 f (k)dk κn %2m+1 + a2p−1 p!(m − 1)!(2m + 1) n=1 a (k 2 + a2 )p+1 0 m≥1 ∞ 1 |Vˆ (k)|2 = (2a)2p−1 dk + p+1 fp (a), (6.4) 2 2 p+1 (k + 4a ) 2 p! 0 where fp (a) are recursively obtained from f0 (a) =

− log Δ3 (−a2 ),

fp (a) =

d 2(p − 1)fp−1 (a) − (afp−1 (a)) . da

(6.5)

While derived from (6.3), equations (6.4) are not equivalent to (6.3) as they can be extended to hold for broader classes of potentials. Indeed, it can be easily shown that all fp (a) are ﬁnite if V ∈ H3−1 . The integral on the right-hand side of (6.4) is ﬁnite if and only if V ∈ H −p . Since both terms on the left-hand side of (6.4) are non-negative, equation (6.4) can be suitably extended to any V ∈ H3−1 ∩H −p . Since H3−1 ∩ H −p1 ⊂ H3−1 ∩ H −p2 if p1 < p2 , this actually implies that Specac (H) = R+ if V ∈ H3−1 ∩ H −p for some natural p. Yet another rearrangement of (6.3) yields ∞ 4 $ κn %2m+1 2 k f (k) − a3 +2 dk (6.6) 2m + 1 a k 2 + a2 0 m≥2 n≥1 2 ∞ V O (k) . / 1 dk + a3 log Δ3 −a2 = 4 0 k 2 + 4a2 " which means that if V ∈ H3−1 , V ∈ H −1 and n≥1 κn5 < ∞, then Specac (H) = R+ . Observe that if we multiply (6.6) by a3 and then let a → ∞, (6.6) transforms into the third conservation law (1.6). Note that (6.6) does not hold for a delta potential as the integral ∞ 2 dk O

V (k) 2 k + 4a2 0 clearly diverges if V (x) = δ (x). However the diﬀerentiated equation (6.6) m − 1 $ κn %2m+1 ∞ k 4 f (k) + dk −a 2 2m + 1 a (k 2 + a2 ) 0 m≥2 n≥1 2 ∞ V O (k) . / 1 1 d 3 = a log Δ3 −a2 dk + 2 2 0 (k 2 + 4a2 ) a da

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A. Rybkin

already admits delta potentials. This demonstrates once again that the presence of an extra parameter a in the trace formulas makes them easily adjustable to low regularity potentials. In general, (6.3) and (6.6) can be included in an inﬁnite chain of relations ∞ 2p N $ m! k f (k)dk κn %2m+1 + (−1)p+1 p!a (m − p)!(2m + 1) n=1 a (k2 + a2 )p+1 0 m≥p p+1 −1 = fp (a) 2

(6.7)

where fp (a) are recursively obtained from f0 (a) =

(−1)n+1 tr Qna n

n≥1

fp (a) = (2p − 1)fp−1 (a) + a

d fp−1 (a). da

Each of equalities (6.7) reproduces in the limit the corresponding relation (1.3). But as opposed to (1.3) they hold for singular potentials. This could be used to push the results of Molchanov-Novitskii-Vainberg [5] to broader classes of singular potentials. For instance it can be shown that for p = 3 equation (6.7) implies the assertion V ∈ H4−1 , V

∈ H −1 =⇒ Specac (H) = R+ .

(6.8)

This improves [5] where the stronger condition V ∈ L , V ∈ L is imposed. Note that, since for p = 3 the both terms on the right-hand side of equation (6.7) are non-negative, one also has κn7 < ∞ (6.9) 4

2

n≥1

under the assumptions of (6.8) which could produce a Lieb-Thirring 7/2 inequality for singular potentials. It is reasonable to assume that V ∈ H4−1 would be suﬃcient for (6.9) to hold but we don’t have a proof. We hope to return to all these issue elsewhere.

7. Appendix: Impedance form of Schr¨ odinger operators with singular potentials Particular cases of singular (i.e., not locally integrable) potentials like delta and Coulomb potentials were considered by many authors. However a systematical treatment of H −1 potentials appears to have been originated by Savchuk-Shkalikov around 1998 (see, e.g., [7] and the literature therein). We emphasize that singular perturbations of self-adjoint operators have been studied even earlier but the

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443

author was unable to ﬁnd out if a general theory of singular perturbations was linked to singular potentials. The Savchuk-Shkalikov’s idea was to rewrite H = −∂x2 + V (x) on L2 with V ∈ H −1 in the impedance form H = −∂x (∂x − v (x)) − v (x) ∂x + u (x) with some u, v ∈ L

2

(7.1) 3

from decomposition (2.1). On the domain Dom H = y ∈ L2 : y, y − v (x) y ∈ ACloc , Hy ∈ L2 ,

(7.2)

the operator H is self-adjoint in L25and J Dom H does not depend on a speciﬁc choice of u, v ∈ L2 in (2.1). Moreover, if V is a sequence of real-valued functions from 5 J ,H = −∂x2 + V (x) , converges C0∞ converging in H −1 to V then the sequence H in the uniform resolvent sense to H deﬁned by (7.1). That is, # # # $ %# # # # # (7.3) → 0 =⇒ #Rz (H) − Rz H # → 0, Im z = 0. #V − V # H −1

The corresponding details can be found in [7]. We only note that the key ingredient here is the following representation y v (x) 1 Y, Y := Y = (7.4) u (x) − v 2 (x) − z −v (x) y [1] of the Schr¨ odinger equation −y

+ V (x) y = zy, V (x) = u (x) + v (x) , where y [1] := y − v (x) y is the so-called quasi-derivative of y. Since u, v ∈ L2 , equation (7.4) is solvable and Y = (y, y − v (x) y) ∈ ACloc (although y need not be continuous). The deﬁnition of the Wronskian W [y1 , y2 ] of two functions y1 , y2 should be modiﬁed to read [1]

[1]

W [y1 , y2 ] = y1 y2 − y1 y2 , which of course agrees with the usual Wronskian if y1 , y2 ∈ ACloc .

References [1] Colliander, J.; Keel, M.; Staﬃlani, G.; Takaoka, H.; Tao, T. Sharp global wellposedness for KdV and modiﬁed KdV on R and T . J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. [2] Deift, P.; and Killip, R. On the absolutely continuous spectrum of one-dimensional Schr¨ odinger operators with square summable potentials, Commun. Math. Phys. 203 (1999), 341–347. [3] Faddeev, L.D.; and Zakharov, V.E. Kortevew-de Vries equation: A completely integrable Hamiltonian system, Funt. Anal. Appl., 59, 280 (1971), 280–287. 3 AC loc

denotes the set of locally a.c. functions on R.

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[4] Kappeler, Thomas Solutions to the Korteweg-de Vries equation with irregular initial proﬁle. Comm. Partial Diﬀerential Equations 11 (1986), no. 9, 927–945. [5] Molchanov, S.; Novitskii, M.; and Vainberg, B. First KdV integrals and absolutely continuous spectrum for 1-D Schr¨ odinger operator, Comm. Math. Phys. 216 (2001), no. 1, 195–213. [6] Rybkin, Alexei On the spectral L2 conjecture, 3/2−Lieb-Thirring inequality and distributional potentials. J. Math. Phys. 46 (2005), no. 12, 123505, 8 pp. [7] Savchuk, A. M.; Shkalikov, A.A. Sturm-Liouville operators with distribution potentials. (Russian) Tr. Mosk. Mat. Obs. 64 (2003), 159–212; translation in Trans. Moscow Math. Soc. 2003, 143–192. [8] Simon, Barry, Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005. viii+150 pp. [9] Yafaev, D.R. Mathematical scattering theory. General theory. Translated from the Russian by J.R. Schulenberger. Translations of Mathematical Monographs, 105. American Mathematical Society, Providence, RI, 1992. x+341 pp. Alexei Rybkin Department of Mathematical and Statistics University of Alaska Fairbanks PO Box 756660 Fairbanks, AK 99775, USA e-mail: [email protected] Received: October 21, 2008 Accepted: January 17, 2009