Mathematical Physics, Analysis and Geometry 5: 1–63, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis N. N. KHURI Department of Physics, The Rockefeller University, New York, NY 10021, U.S.A. (Received: 28 December 2001) Abstract. It is well known that the s-wave Jost function for a potential, λV , is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V , and at zero energy, these zeros of the ‘coupling constant’, λ, will all be real and negative, λn (0) < 0. By rescaling λ, such that λn < −1/4, and changing variables to s, with λ = s(s − 1), it follows that as a function of s the Jost function has only zeros on the line sn = 1/2 + iγn . Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search. In this paper we make a significant enlargement of the class of potentials needed for a generalization of the above idea. We also make this new class amenable to construction via inverse scattering methods. We show that all one needs is a one parameter class of potentials, U (s; x), which are analytic in the strip, 0 Re s 1, Im s > T0 , and in addition have an asymptotic expansion in powers of [s(s − 1)]−1 , i.e. U (s; x) = V0 (x) + gV1 (x) + g 2 V2 (x) + · · · + O(g N ), with g = [s(s − 1)]−1 . The potentials Vn (x) are real and summable. Under suitable conditions on the Vn s and the O(g N ) term we show that the condition, 0∞ |f0 (x)|2 V1 (x) dx = 0, where f0 is the zero energy and g = 0 Jost function for U , is sufficient to guarantee that the zeros gn are real and, hence, sn = 1/2 + iγn , for γn T0 . Starting with a judiciously chosen Jost function, M(s, k), which is constructed such that M(s, 0) is Riemann’s ξ(s) function, we have used inverse scattering methods to actually construct a U (s; x) with the above properties. By necessity, we had to generalize inverse methods to deal with complex potentials and a nonunitary S-matrix. This we have done at least for the special cases under consideration. For our specific example, 0∞ |f0 (x)|2 V1 (x) dx = 0 and, hence, we get no restriction on Im gn or Re sn . The reasons for the vanishing of the above integral are given, and they give us hints on what one needs to proceed further. The problem of dealing with small but nonzero energies is also discussed. Mathematics Subject Classifications (2000): 81U40, 11M26, 11M06, 81U05. Key words: Riemann hypothesis, inverse scattering.
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1. Introduction Many physicists have been intrigued by the Riemann conjecture on the zeros of the zeta function. The main reason for this is the realization that the validity of the hypothesis could be established if one finds a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros. The hope is that this operator could be the Hamiltonian for some quantum mechanical system. Results by Dyson [1], and Montgomery [2] first made the situation more promising. The pair distribution between neighboring zeros seemed to agree with that obtained for the eigenvalues of a large random Hermitian matrix. But later numerical work showed correlations between distant spacings do not agree with those of a random Hermitian matrix. The search for such a Hamiltonian in physical problems has eluded all efforts. Berry [3] has suggested the desired Hamiltonian could result from quantizing some chaotic system without time reversal symmetry. This seems to be in better agreement with numerical work on the correlations of the Riemann zeros, but one is still far from even a model or example. It is useful to explore new ideas. Our choice for this paper is an idea originating from Chadan [4]. In this approach, one tries to relate the zeros of the Riemann zeta function to the ‘coupling constant spectrum’ of the zero energy, S-wave, scattering problem for repulsive potentials. We sketch this idea briefly. The Schrödinger equation on x ∈ [0, ∞) is −
d2 f (λ; k; x) + λV (x)f (λ; k; x) = k 2 f (λ; k; x), 2 dx
(1.1)
where k is the wave number, λ a parameter physicists call the coupling constant, V (x) is a real potential satisfying an integrability condition as in Equation (2.2) below, and f is the Jost solution determined by a boundary condition at infinity, (e−ikx f ) → 1 as x → +∞. The Jost function, M(λ; k), is defined by limx→0 f (λ; k; x) = M(λ; k). It is well known that M is also the Fredholm determinant of the Lippmann–Schwinger scattering integral equation for S-waves. Both f (λ; k; x) and M(λ; k) are, for any fixed x 0, analytic in the product of the half plane, Im k > 0, and any large bounded region in the λ plane. In fact, it is known that for any fixed k, Im k 0, M(λ; k) is entire in λ and of finite order. Thus M(λ; k) has an infinite number of zeros, λn (k), with λn (k) → ∞ as n → ∞. Starting with Equation (1.1), and its complex conjugate with k = iτ , τ > 0, and setting λ = λn (iτ ), we obtain ∞ |f (λn (iτ ); iτ ; x)|2 V (x) dx = 0. (1.2) [Im λn (iτ )] 0
For the class of potentials, we deal with V = O(e−mx ) as x → ∞. Thus, we can take the limit τ → 0, and get ∞ |f (λn (0); 0; x)|2 V (x) dx = 0. (1.3) [Im λn (0)] 0
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INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Hence, for repulsive potentials, V (x) 0, all the zeros λn (0) are real. For any τ, τ > 0, the same is true for all λn (iτ ). But λn (iτ ) must be negative, since the potential [λn (iτ )V ] will have a bound state at E = −τ 2 , and that could not happen if V 0 and λn (iτ ) > 0. Hence, by continuity, λn (0), for all n, is real and negative [5]. The zero energy coupling constant spectrum, λn (0), lies on the negative real line for V 0. Chadan’s idea is very simple. He introduces a new variable, s, and defines λ ≡ s(s − 1).
(1.4)
Thus, one can write M(λ, 0) = M(s(s − 1); 0) ≡ χ(s).
(1.5)
It is easy to see now that, for | Im s| > 1, the zeroes, sn , of χ(s) are all such that sn =
1 2
+ iγn ;
λn (0) ≡ sn (sn − 1).
(1.6)
The problem is actually somewhat simplified by noting that first we do not need the condition λn < 0 as long as we restrict ourselves to the strip 0 Re s 1, and Im s > 1. Second, it is sufficient to prove that the integral in Equation (1.3) does not vanish. Thus, one does not need a fully repulsive potential for the Riemann problem. One might comment that it is very difficult to find a potential with λn (0) = sn (sn − 1)
and
sn =
1 2
± iγn ,
sn being the Riemann zeros. But it is probably as difficult as finding an Hermitian operator whose eigenvalues are γn . Indeed, the latter may be impossible without introducing chaotic systems. The results mentioned above also apply when V = V0 + λV1 , with only V1 0, and V0 , V1 both real and satisfying Equation (2.2) and with certain restrictions on V0 . This remark leads directly to the basic idea of this paper the of objective of which is to show that the coupling constant approach can be significantly simplified and made amenable to inverse scattering methods. Our first remark is that one does not need a potential, V = V0 + λV1 , depending linearly on the coupling parameter λ. Given a one-parameter family of complex potentials, U (s; x), x ∈ [0, ∞), which for fixed x are analytic in s in the strip, 0 Re s 1, Im s > T0 > 2, we can, following similar arguments as above, obtain, for s = sn , sn being a zero of the zero energy Jost function, (1.7) |f (sn ; 0; x)|2 [Im U (sn ; x)] dx ≡ 0, where f is the zero energy Jost solution evaluated at s = sn . Next, suppose in addition to the above properties, U has an asymptotic expansion in inverse powers of s, actually better, s(s − 1), i.e. U (s, x) = V0 (x) + gV1 (x) + g 2 V2 (x) + · · · + g N VR(N) (g; x),
(1.8)
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N. N. KHURI
where g≡
1 . s(s − 1)
(1.9)
Under suitable conditions on the Vn (x) and estimates of the O(g N ) term, and its phase, one again gets ∞ [Im gn ] |f (0; 0; x)|2 V1 (x) dx = 0, (1.10) 0
with gn = [sn (sn − 1)]−1 , the sn ’s are the zeros of M(s, 0) the zero energy Jost function, and f (g; k; x) is the Jost solution with the full U . The result (1.10) is only established for zeros with Im sn > T0 , where T0 is large enough for the V1 contribution to Equation (1.7) to dominate the integral in Equation (1.7). However, this is sufficient, since the Riemann hypothesis has already been proved for zeros with | Im sn | < T , where T could be as large as 105 . Again, all we need for sn = 1/2 + iγn is to have the integral in (1.10) not vanishing. In the end, only the properties of V1 matter. In this paper we will use inverse scattering methods, albeit for complex potentials, to actually prove the existence of such a U (s; x). By construction, this potential has the additional property that the zero energy Jost function is Riemann’s ξ function, lim M(s; k) ≡ 2ξ(s).
k→0
(1.11)
We will also give explicit expressions for V0 , V1 , V2 , and bounds on VR(N) . The difficult point turns out to be that, in our specific example, ∞ |f (0; 0; x)|2 V1 (x) dx ≡ 0. (1.12) 0
Thus, we get no information on [Im gn ], or Re(sn − 1/2). We shall discuss what one needs to proceed further. This will require working with small, but nonzero energy values. We start by introducing a special class of Jost functions, M ± , which depend on an extra parameter ν = s − 1/2, with the property that the zero energy limit, lim M ± (ν, k) = 2ξ(ν + 12 ).
k→0
For fixed ν, the Jost functions are taken to be of the Martin [8] type, i.e. having cut plane analyticity in the momentum variable k. This is the class of Jost functions that results when the potential is a superposition of Yukawa potentials. We then use inverse scattering methods to prove the existence of a complex potential U (ν, τ ) which is determined uniquely by the initial S-matrix. We do carry out the analysis for ν in the truncated critical strip, i.e. −1/2 < Re ν < 1/2, and Im ν > T0 , with
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
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T0 > 16π 2 . This is, of course, the domain most relevant to the Riemann problem. Standard techniques of inverse scattering are not immediately applicable, because S(ν, k) does not satisfy the reality condition, and is not unitary for complex ν. However, we shall see that, in our specific case, we can bypass these difficulties and carry out an inverse scattering procedure anyway. We have attempted to make the paper self-contained and do not rely on results that need the unitarity of S in the proof. In Section 2, we give a brief review of relevant scattering theory results intended for mathematicians not familiar with them. This review also helps define our physics terminology. Section 3 is devoted to the introduction of our special class of Jost functions, M ± (ν; k). Following that, in Section 4, we briefly discuss the real ν case, which is a standard inverse scattering case covered by well-known results. This section is instructive, even though real ν is uninteresting for the Riemann problem. The next step, Section 5, is to study in more detail the properties of M ± . The main result is an asymptotic expansion in powers of a variable, g ≡ (ν 2 − 1/4)−1 , which gives M (±) = M0(±) (k) + gM1(±) (k) + g 2 M2(±) + · · · + g N RN(±) (g; k).
(1.13)
Mn(±)
can be computed exactly via recursion formulae, and in addition, Here all the for real k, they satisfy [Mn(+) (k)]∗ = Mn(−) (k) and
Mn(+) (−k) = Mn(−) (k).
The remainder functions, RN(±) , are given explicitly and are O(g) as g → 0. In Section 6, with fixed ν in the strip, we determine the number and positions of zeros in the upper half k-plane. It turns out that there is at most one such zero and it lies close to the origin. In fact we can give a good estimate of its position. Section 7 is devoted to the study of the case |ν| → ∞, i.e. |g| → 0. Here (±) M (ν, k) → M0(±) (k) which is a known rational function in k. This leads to an exactly soluble Marchenko equation and an exact result for the corresponding V0 (x). Section 8 is devoted to proving the existence of solutions of the Marchenko equation for our specific class of S-matrices. With the resulting Marchenko operator, A(ν; x, y), which is now complex, we proceed to define in the standard way a potential U (ν, x) and corresponding Jost solutions, f (±) (ν; k, x), of the Schrödinger equation. Finally, we check directly that indeed f (±) are solutions of the Schrödinger equation with the desired asymptotic properties. The main difference from the standard case is that U (ν; x) is now complex unless ν is purely imaginary. In Section 9 we discuss the case ν = it, t real. This is a standard inverse problem with S(it, k) unitary for k real, and the resulting U (it; x) is real. More detailed properties of V (ν, x) are given in Section 10. There we give an asymptotic expansion, V (ν, x) = V0 (x) + gV1 (x) + g 2 V2 (x) + · · · + g N VR(N) (g, x)
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with all Vn ’s real and all representable by superpositions of Yukawa potentials. Also Vn (x) is continuous and differentiable for x ∈ [0, ∞), and Vn (0) is finite. For completeness we calculate V1 (x) explicitly, and indicate how Vn (x), n > 1, can easily be computed. We also give some needed properties of VR(N) and of [Im VR(N) ] for small Re ν. In Section 11, we study the zeros, νn (k), of M − (ν, k) for small fixed k with Im k 0. We prove that νn (0) are the standard Riemann zeros, and also that |νn (k) − νn (0)| = O(k 1/p ) for small k. Here p is the multiplicity of the Riemann zero νn = νn (0). We also prove that any Riemann zero, νj , is the limit of a zero of M (−) (ν; k), νj (k), as k → 0. Finally, in Section 12 we discuss the relation of our potential, V (g; x), and its Jost solutions to the Riemann hypothesis. We prove that in this case ∞ |f (0; 0; x)|2 V1 (x) dx = 0 0
and, hence, no information on the Riemann hypothesis can result directly from this example at zero energy. But the reasons for the failure are clear, and they indicate the properties of a desired Jost function that will be sufficient to make the important step. The fact that one can set k = iτ , τ > 0 but small, and try to prove the hypothesis for νn (τ ), τ arbitrarily small, but τ = 0, provides a significant simplification of the problem. 2. A Sketch of Scattering Theory This section is intended to facilitate the reading of this paper by those mathematicians (or physicists) who are not familiar with elementary scattering theory in quantum mechanics. At the end of this section we will give a list of books and review papers where more information can be obtained. The Schrödinger equation for s-waves is given by d2 f + gV (x)f = k 2 f, k = κ + iτ. (2.1) dx 2 Here x ∈ [0, ∞), V (x) is real, g is a parameter that physicists call a coupling constant. The reason for introducing it will become apparent below. One studies the class of real potentials, V (x), which are locally summable functions and satisfy the condition, ∞ x|V (x)|eαx dx = C < ∞, 0 α m. (2.2) −
0
For scattering theory, the important solutions of Equation (2.1) are the so-called Jost solutions [9]. These are the two linearly independent solutions, f (±) (g, k, x) with boundary values at infinity given by lim e±ikx f (±) = 1.
x→∞
(2.3)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
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Using the method of variation of parameters, we can replace Equation (2.1) and the condition (2.3) by an integral equation: ∞ sin k(x − x) (±) ∓ikx V (x )f (±) (g; k; x ) dx . (2.4) +g f (g; k; x) = e k x Starting with the papers of Jost [9] and Levinson [10], the existence of solutions to Equation (2.4) and their properties have been well established for V (x) satisfying the condition (2.2). The basic input needed is the upper bound on the kernel, sin k(x − x) e|τ ||x −x| C1 , Im k ≡ τ, (2.5) k(x − x) 1 + |k||x − x| where C1 is O(1). With this bound and the bound (2.2) one proves the absolute convergence of the iterative series of the Volterra equation (2.4) for any x 0, and k with Im k > −(m/2) for f (−) , and Im k < m/2 for f (+) . Also it is easy to prove that, for any finite g and x 0, f (+) (g; k; x) is an analytic function of k for Im k < m/2. Similarly, f (−) (g; k; x) is analytic in Im k > −(m/2). In addition, for k in the analyticity domain, the power series in g obtained by iterating Equation (2.4) is absolutely and uniformly convergent for g inside any finite region in the g-plane. Thus both f ± (g; k; x) are entire functions of g. The scattering information is all contained in the Jost functions, denoted by M (±) (g; k) and defined by M (±) (g; k) ≡ lim f (±) (g; k; x). x→0
(2.6)
Both limits in Equation (2.6) exist for finite |g|, and k in the respective domain of analyticity, for all potentials satisfying the condition (2.2). The S-matrix is given by S(g; k) ≡
M (+) (g; k) . M (−) (g; k)
(2.7)
For real g and Im k > 0, M (−) (g, k) has no zeros except for at most a finite number on the imaginary k-axis. These zeros, kn = iτn , give the point spectrum of the 2 Hamiltonian ∞ of (2.1) with En = −τn . Their number cannot exceed the value of the integral 0 x|V | dx, a result due to Bargmann [11]. Another important property of M (−) (g; k) was first obtained by Jost and Pais [12]. The regular solution of Equation (2.1), φ(g; k; x), with φ(g; k; 0) = 0, is φ(g; k; x) ≡
1 [M (+) (g; k)f (−) (g; k; x) − M (−) (g; k)f (+) (g; k; x)]. (2.8) 2ik
The solution φ satisfies a Fredholm type integral equation which, for potentials satisfying (2.2), was studied in [12]. Jost and Pais demonstrate explicitly that
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M (−) (g, k) is identical to the Fredholm determinant of the scattering integral equation for φ. Hence, for any fixed k, with Im k > 0, the zeros of M (−) (g; k) in the g-plane, gn (k), give the ‘coupling constant eigenvalues’ at which the homogeneous Fredholm equation has solutions, φ = gn (k)Kφ. Since, M (−) (g, k) is an entire function of finite order in g, the sequence g1 (k), g2 (k), . . . , gn (k) tends to infinity as n → ∞. For the purposes of this paper a result of Meetz [5] is instructive. Let us consider a potential which is repulsive, i.e. V > 0 for all x ∈ [0, ∞). Then for k = iτ, τ > 0, the coupling constant spectrum, gn (iτ ), is real and negative. This result is implicitly contained in [12]. In this brief review we need to make an important remark about complex potentials, V = V ∗ . Mathematicians and mathematical physicists often ignore these potentials. The Hamiltonian is no longer self-adjoint if V = V ∗ , with g = 1. But physicists, especially those who work on nuclear physics, do not have such a luxury. There are many interesting and useful models, especially in nuclear physics, where V is complex. Of course, the general and beautiful results which hold for real V do not all apply for complex V . But many survive, and one has just to be careful which to use and to establish alternative ones when needed. There are many books that cover inverse scattering. But for the purposes of this paper, we recommend the book of Chadan and Sabatier [13], since it also discusses the superposition of the Yukawa case and the Martin results. For the standard results on inverse scattering, the review paper by Faddeev [14] is highly recommended.
3. A Special Class of Jost Functions In this section we will combine two results whose progeny could not be more different to obtain a representation for a class of Jost functions that we shall study in detail. The first is Martin’s representation for the Jost functions of the class of potentials that can be represented as a Laplace transform. The second is Riemann’s formula for the function ξ(s) defined below. Starting 40 years ago, physicists [15, 16], for reasons not relevant to this paper, studied the class of potentials that, in addition to satisfying Equation (2.2), have a Laplace transform representation, i.e. for all x > 0, V (x) =
∞
C(α)e−αx dα,
m > 0,
(3.1)
m
∞ where C(α) is summable and restricted to satisfy m |C(α)|α −2 dα < ∞. This last condition guarantees that x|V (x)| is integrable at x = 0.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
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For these potentials Martin) proved that the Jost functions M (±) (k) have the representation ∞ w(α) (±) M (k) = 1 + dα. (3.2) m α ± ik 2 Here w is real and summable and is such that M (±) → 1 as |k| → ∞. We have set g = 1 here. Note that not any arbitrarily chosen summable w(α) is acceptable. M (−) (k) must have no zeros for Im k > 0 except for a finite number on the imaginary k-axis corresponding to the point spectrum. For our purposes, here we choose a specific family of functions M (±) (ν; k) defined such that M (±) (ν; 0) ≡ 2ξ(ν + 12 ), where ξ(s) =
1 s(s 2
− 1)π
−s/2
(3.3)
s ζ(s) + 2
(3.4)
and ζ(s) =
∞
n−s ,
Re s > 1.
(3.5)
n=1
Riemann’s formula for ξ(s) defines an entire function of order one in s, and is given by [18] ∞ ψ(α)[α s/2−1 + α −1/2−s/2] dα, (3.6) 2ξ(s) = 1 + s(s − 1) 1
where ψ(α) =
∞
e−πn α , 2
α 1.
(3.7)
n=1
We also have the symmetry relation ξ(s) = ξ(1 − s). For convenience we define the variable, ν, as s≡
1 2
+ ν.
(3.8)
With this variable ξ(1/2 + ν) is symmetric in ν, and we have ∞ 1 2 1 ψ(α)α −3/4[α ν/2 + α −ν/2 ] dα. 2ξ( 2 + ν) = 1 + (ν − 4 ) 1
) See [8] and [17].
(3.9)
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Our starting point is to define two functions, M ± (ν; k), as ∞ ψ(α)α 1/4[α ν/2 + α −ν/2 ] dα. M (±) (ν; k) ≡ 1 + (ν 2 − 14 ) α ± ik 1
(3.10)
This definition holds for any finite, real or complex, ν, and for any k excluding the cuts on the imaginary k-axis, k = iτ , 1 τ < ∞, for M (+) , and −1 τ > −∞, for M (−) . Obviously, we have M (±) (ν; 0) ≡ 2ξ( 12 + ν).
(3.11)
In addition, the fact that ψ(α) = O(e−πα ) as α → +∞, guarantees that for any finite |ν|, lim M ± (ν; k) = 1.
(3.12)
|k|→∞
This is true along any direction in the complex k-plane excluding the pure imaginary lines. But even for arg k = ±π/2, the limit holds using standard results. The immediate question that faces us at this stage is: for which regions in the νplane, if any, can one use the functions M (±) (ν; k) defined in Equation (3.10) as Jost functions and proceed to use the resulting S-matrix, S(ν; k), as the input in an inverse scattering program. There are two issues involved. The first, and most important, is to make sure that M (−) (ν; k), has no complex zeros in k for Im k > 0, except for a finite number on the imaginary axis. This is not true for any ν. But fortunately for the set of ν’s most important to the Riemann hypothesis, M (−) (ν; k) has at most one zero close to the origin with Im k > 0. This will be shown in Section 6. The second issue relates to the question of reality. For real potentials V and real k, we have the relations [M (+) (k)]∗ = M (−) (k) and
|S(k)| = 1.
Clearly, for complex ν, this does not hold for M (±) (ν; k). However, we will prove that for those values of ν in the truncated critical strip, one can still carry out the inverse scattering program and obtain a unique and well-defined V (ν; x) which, of course, could now be complex. Since the old results of inverse scattering theory all use the fact that, |S(k)| = 1, we have to go back to square one and prove every step anew for the present case. Our task is tremendously simplified by the fact that, even though S(ν; k) is not unitary, we still have |S(ν; k)| = 1 + O(1/|ν|2 ), and we are only interested in |ν| > 103 .
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INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
4. The Real ν Case For potentials satisfying the representation (3.1), Martin) , in addition to the results summarized in Equation (3.2), developed an iterative scheme which enables one to reconstruct the measure, C(α) in Equation (3.1) from the knowledge of the discontinuity of S(k) along the branch cut on the imaginary axis, k = iτ,
m τ < ∞. 2
This gives an inverse scattering method that, at first sight, looks quite different from the standard ones of Gelfand and Levitan [6] and Marchenko [7]. The relation between these two methods was first clarified by Gross and Kayser [19] and independently by Cornille [20]. They showed that for potentials of the form (3.1) the Marchenko kernel is a Laplace transform of the discontinuity of S(k), and they carried out an extensive analysis of the relation between Martin’s and Marchenko’s methods. These results were reviewed and enlarged in a more recent paper by the author [21]. For ν real and |ν| > 1/2, the functions M (±) (ν; k) defined in Equation (3.10) are indeed bona fide Jost functions with (M (+) (ν; k))∗ = M (−) (ν; k) for real k. The positivity of ψ(α) guarantees the absence of a point spectrum. The S-matrix is S(ν; k) ≡
M (+) (ν; k) . M (−) (ν; k)
(4.1)
We define the discontinuity D(ν; τ ) as D(ν, τ ) = lim [S(ν; iτ + ε) − S(ν; iτ − ε)], ε→0
τ > 1.
(4.2)
From Equation (3.10), one obtains D(ν, τ ) =
1+
1 π
ω(ν; τ ) ∞ ω(ν,β) 1
β+τ
dβ
,
(4.3)
with ω(ν, τ ) = π(ν 2 − 1/4)ψ(τ )τ 1/4 [τ ν/2 + τ −ν/2 ].
(4.4)
For real ν > 1/2, D(ν, τ ) and ω(ν, τ ) are positive for all τ 1. The case with ω(ν, τ ) 0 is the easiest to handle by the Martin inverse method, and it can be done explicitly. Although having ν real and ν > 1/2 is of little direct interest to the Riemann problem, we give the results here as they might be helpful to the reader. For details, one should consult [21]. ) See [8] and [13].
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The S-matrix, S(ν, k), uniquely determines a potential, V (ν; x), and its Jost solutions f (±) (ν; k; x). V is given by ∞ −1 n+1 ∞ dα0 . . . × V (x) = 4 π 1 n=0 n n ∞ −2αj x j =0 D(ν; αj )e dαn n−1 αj . (4.5) × (α + α ) 1 j j +1 j =0 j =0 This series for V is absolutely and uniformly convergent for all x 0, and ν > 1/2. This follows from the positivity in (4.3), 1 ∞ ω(ν, α)/α dα 1 ∞ |D(ν, α)| π 1 < 1. (4.6) dα ∞ 1 π 1 α+τ 1 + π 1 ω(ν,β) dβ β The Jost solutions, f (±) (ν; k; x) are given by ∞ ∞ −1 n+1 ∞ (±) ∓ikx ∓ikx = e +e dα0 . . . dαn × f π 1 1 n=0 n −2αj x j =0 D(ν; αj )e . × n−1 [ j =0 (αj + αj +1 )][α0 ± ik]
(4.7)
Again this last series is absolutely and uniformly convergent for all x 0, and k in a compact domain inside the respective regions of analyticity. One can check directly that f ± given by Equation (4.6) are solutions of the Schrödinger equation with V (ν; x) of Equation (4.4) as potential, see [21] for more details. 5. Some Properties of M (±) (ν, k) for |ν| > 1 To proceed further and study M (±) (ν, k) for complex ν, and more specifically ν in the critical strip, −1/2 < Re ν < 1/2; Im ν > 1, the defining representation (3.10) is not fully instructive. This is because the behavior of M ± for large Im ν is not adequately shown by Equation (3.10). Our final result in this section is to obtain an asymptotic expansion of M ± (ν, k) for fixed k in inverse powers of (ν 2 − 1/4). We need to carry out integrations by parts on the integrand in Equation (3.10) analogous to those performed in Titchmarsh’s book [18], for Equation (3.9). The following lemma will prove extremely useful: LEMMA 5.1. Let W (α), α ∈ [1, ∞), be a C ∞ function, and W (α) = O(e−α ) as α → ∞, then given the integral ∞ W (α)[α ν/2 + α −ν/2 ] dα, (5.1) I (ν) = 1
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
one has, after two integrations by parts,
1 I (ν) = 2 1 8[W (1) + (W (α))α=1 ] + ν −4 ∞ ν/2 −ν/2 W1 (α)[α + α ] dα , +
13
(5.2)
1
where W1 (α) =
15 W (α) 4
+ 12αW (α) + 4α 2 W (α).
Proof. We rewrite Equation (5.1) as ∞ dα(W (α)α 3/4 )[α ν/2−3/4 + α −ν/2−3/4]. I (ν) =
(5.3)
(5.4)
1
Integrating by parts, we get ν/2+1/4
∞ d α α −ν/2+1/4 2W (1) 3/4 [W (α)α ] dα − ν 1 . (5.5) I (ν) = 2 1 − ν dα ν −4 + 14 −4 1 2 2 This again can be rewritten as ν/2−5/4
∞ α α −ν/2−5/4 d 2W (1) 3/2 3/4 [W (α)α ] dα α − ν 1 . (5.6) I (ν) = 2 1 − ν dα ν −4 + 14 −4 1 2 2 Carrying out a second integration by parts, we obtain d 1 (W (α)α 3/4) + I (ν) = 2 1 2W (1) + 8 dα ν −4 α=1
∞ d 4 −1/4 3/2 d 3/4 α (W (α)α dαα ) × + dα dα (ν 2 − 14 ) 1 × [α ν/2 + α −ν/2 ].
(5.7)
Performing the differentiations in (5.7) easily leads to Equation (5.2).
✷
We can apply this lemma to the integral in Equation (3.10) which defines M (±) (ν, k). Setting W (±) (α; k) ≡
ψ(α)α 1/4 , α ± ik
(5.8)
and restricting k to the corresponding domain of analyticity in k P (+) = {k | Im k < 1};
P (−) = {k | Im k > −1},
(5.9)
14
N. N. KHURI
we get M
(±)
(ν; k) =
M0(±) (k)
∞
+ 1
dαW1(±) (α; k)[α ν/2 + α −ν/2 ],
(5.10)
with W1(±) =
3 6(1) 5 (α) , (α ± ik)5 5=1
(5.11)
and 1/4 + 14ψ (α)α 5/4 + 4ψ (α)α 9/4, 6(1) 1 (α) = 6ψ(α)α 5/4 6(1) − 8ψ (α)α 9/4, 2 (α) = −14ψ(α)α 9/4 . 6(1) 3 (α) = 8ψ(α)α
(5.12)
The first term in (5.10) is independent of ν, and given by M0(±) (k) = 1 +
a1 a2 , + 1 ± ik (1 ± ik)2
(5.13)
with a1 = −1 + 8ψ(1);
a2 = −8ψ(1).
(5.14)
In obtaining (5.14), we have used the identity 4ψ (1) + ψ(1) = − 12 .
(5.15)
It is important to note that both a1 and a2 are negative and that a1 + a2 = −1. This leads to lim M0(±) (k) = 0.
(5.16)
k→0
As a check on Equation (5.10), we take the k → 0 limit ∞ 3 (1) dα (65 /α 5 ) [α ν/2 + α −ν/2 ]. M (±) (ν; 0) = 1
(5.17)
5=1
Substituting the expressions for 6(1) 5 given in Equation (5.12), we get ∞ dα(ψ (α)α 5/4 + 32 ψ (α)α 1/4)[α ν/2 + α −ν/2 ] M (±) (ν, 0) = 4 1
= 2ξ( 12 + ν).
(5.18)
Lemma 5.1 can be used repeatedly to give an asymptotic expansion of M ± (ν; k) in inverse powers of (ν 2 − 1/4). Recursion formulae can be given to give each term from the preceding one.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Indeed given 1 = 2 1 n−1 (ν − 4 )
In(±) (ν; k)
∞ 1
dαWn(±) (α; k)[α ν/2 + α −ν/2 ],
15
(5.19)
with Wn(±) =
2n+1
6(n) 5 (α) , [α ± ik]5
5=1
(5.20)
±
and k ∈ P , one obtains (±) In+1
1 dWn(±) (±) = 8 Wn (1; k) + + dα (ν 2 − 14 )n α=1 ∞ (±) dαWn+1 (α; k)[α ν/2 + α −ν/2 ] , +
(5.21)
1
and (±) = Wn+1
15 Wn(±) 4
+ 12α(Wn(±) ) + 4α 2 (Wn(±) ) ,
(5.22)
(±) will be where the primes indicate differentiation with respect to α. Again W(n+1) as in Equation (5.20) ± W(n+1)
=
2n+3 5=1
6(n+1) (α) 5 . [α ± ik]5
(5.23)
For each n we have 1 5 2n + 1, and the functions 6(n) 5 satisfy a recursion formula, which follows from (5.22). (α) = 6(n+1) 5
15 (n) 65 4 2
(n) + 12α(6(n) 5 ) − 12α(5 − 1)65−1 +
(n) + 4α [(6(n) 5 ) − 2(5 − 1)(65−1 ) +
+ (5 − l)(5 − 2)6(n) 5−2 ].
(5.24)
All the 6(n) 5 can thus be determined by iteration starting from 1/4 . 6(0) 1 (α) ≡ ψ(α)α
(5.25)
The general form of 6(n) 5 (α) is easily determined to be 6(n) 5 (α)
=
2n+1−5
C (n) (5; j ) α 1/4+5+j −1ψ (j ) (α).
(5.26)
j =0
The coefficients C (n) (5; j ) are real, and C (0) (1; 0) = 1 determines all the others. Also j d ψ(α). (5.27) ψ (j ) (α) ≡ dα
16
N. N. KHURI
At this point we can substitute Equation (5.26) in (5.24) and obtain a recursion formula for C (n) (5; j ), C (n+1) (5, j ) + 12( 14 + 5 + j − 1) + = C (n) (5; j )[ 15 4 + 4( 14 + 5 + j − 1)( 14 + 5 + j − 2)] − − C (n) (5 − 1, j )[12(5 − 1) + 8(5 − 1)( 14 + 5 + j − 2)] + + C (n) (5, j − 1)[12 + 8( 14 + 5 + j − 2)] + 4C (n) (5, j − 2) − − 8(5 − l)C (n) (5 − 1; j − 1) + 4(5 − 1)(5 − 2)C (n) (5 − 2; j ).
(5.28)
Here we have 1 5 2n + 1,
0 j 2n + 1 − 5.
(5.29)
For all other values of 5 and j , C (n) (5, j ) ≡ 0. Starting with C (0) (1, 0) ≡ 1,
(5.30)
we can compute all other C (n) (5, j ). For example, C (1) (1, 0) = 6, C (1) (1, 1) = 14, and C (1) (1, 2) = 4. This agrees with the direct calculation given in Equation (5.12). In Table I, we give all the coefficients C (n) (5, j ) up to n = 4. All the coefficients are integers. Finally, we give the general form of the surface term in Equation (5.21). We define Mn(±) (k), dWn(±) (±) (±) . (5.31) Mn (k) = 8 Wn (1, k) + dα α=1 From Equations (5.20) and (5.26) we obtain, after some algebra, Mn(±) (k) =
2n+2 5=1
χ5(n) , [1 ± ik]5
(5.32)
with χ5(n)
= 8
2n+2−5
C (n) (5, j ){( 14 + 5 + j )ψ (j ) (1) + ψ (j +1) (1)} −
j =0
−8
2n+2−5
(5 − 1)C (n) (5 − 1, j )ψ (j ) (1).
(5.33)
j =0
For the purposes of this paper, it is sufficient to apply our lemma up to the n = 3 level. We introduce g, as a new variable: g≡
ν2
1 . − 14
(5.34)
17
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Table I. Values of C (n) (l; j ) for n 3 C (0) l=1
j =0 1
C (1) l=1 2 3
j =0 6 −14 8
C (2) l=1 2 3 4 5
j =0 36 −364 1000 −1056 384
C (3) l=1 2 3 4 5 6 7
j =0 216 −7784 58304 −174768 250752 −172800 46080
1 14 −8
2 4
1 364 −1000 1056 −384
2 500 −528 192
1 7784 −58304 174768 −250752 172800 −46080
3 176 −64
2 29152 −87384 125376 −86400 23040
4 16
3 29128 −41792 28800 −7680
4 10448 −7200 1920
5 1440 −384
6 64
Our final result for M (±) (ν, k) with k ∈ P (±) and |ν| 1, M (±) (ν; k) = M0(±) (k) + gM1(±) (k) + g 2 M2(±) (k) + g 2 R2(±) (ν, k).
(5.35)
Here we have M1(±) (k)
=
4 5=1
b5 , [1 ± ik]5
(5.36)
c5 , [1 ± ik]5
(5.37)
with b5 ≡ χ5(1) , and M2(±) (k) =
6 5=1
c5 ≡
χ5(2) .
The remainder function R2(±) is given by ∞ 7 6(3) (±) 5 (α) [α ν/2 + α −ν/2 ] dα. R2 (ν; k) = 5 [α ± ik] 1 5=1
(5.38)
18
N. N. KHURI (n)
Table II. Values of χl
for n = 1, 2, 3
(n)
n=1
n=2
l=1 2 3 4 5 6 7 8
10.9973 4.7050 −7.4045 −8.2977
−460.8231 −309.0434 451.5522 910.6755 71.4580 −663.8194
χl
n=3 28967.9828 36560.3049 −18626.4002 −114291.8885 −76110.8714 131495.9200 123526.6033 −111521.6508
For |Im ν| > 103 , the first two terms of Equation (5.35) give a very good estimate for M (±) . We shall explore this in much more detail later. One can go to higher orders in g, but the resulting series is only asymptotic. For our purposes, here Equation (5.35) is enough. It is important to stress another property of M1(±) and M2(±) , namely as k → 0, M1(±) (0) = 0,
M2(±) (0) = 0.
(5.39)
We have already shown that M0(±) (0) = 0. To check this, we give the explicit form of the coefficients b5 in (5.36). Using χ5(1) = b5 , Equation (5.33), and Table I, we get b1 b2 b3 b4
= = = =
60ψ(1) + 300ψ (1) + 216ψ (1) + 32ψ (1), −300ψ(1) − 432ψ (1) − 96ψ (1), 432ψ(1) + 192ψ (1), −192ψ(1).
(5.40)
Numerically, the b’s are given in Table II. We now have M1(±) (0) =
4
b5
5=1
= 32[ψ (1) +
15 ψ (1) 4
+
15 ψ (1)]. 8
(5.41)
But ψ (1) +
15 ψ (1) 4
+
15 ψ (1) 8
= 0.
This identity follows from the relation [18] √ α(2ψ(α) + 1) = 2ψ(1/α) + 1.
(5.42)
(5.43)
19
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Differentiating (5.40) once and setting α = 1 immediately gives Equation (5.15). Differentiating three times leads to Equation (5.42). Indeed, there is an infinite sequence of identities like Equation (5.42), always starting with ψ 2n+1 (1), odd derivatives, which result from differentiating Equation (5.40) (2n + 1) times. Thus, again M2(±) (0) =
6
c5 = 0,
(5.44)
5=1
depends on the next identity: ψ (5) (1) +
45 (4) ψ (1) 4
+
235 (3) ψ (1) 4
+
975 (2) ψ (1) 8
+
1635 (1) ψ (1) 32
= 0.
(5.45)
Of course, only the first two coefficients in (5.45) are unique, since we can always add a multiple of the left-hand side of (5.42) to (5.45). The vanishing of Mj± (0), j = 0, 1, 2, is indeed necessary since ∞ 7 6(3) 5 (α) 2 (±) 2 dα [α ν/2 + α −ν/2 ] g R2 (ν; 0) = g 5 α 1 5=1 = 2ξ( 12 + ν),
(5.46)
which is the result of carrying out four more differentiations by parts on the formula for ξ(1/2 + ν) given on page 254 of [18]. In Table II, we give the numerical values of χ5(n) , for n = 1, 2, and 3, and b5 ≡ χ5(1) while c5 ≡ χ5(2) . For the convenience we summarize the results of this section: M (±) (ν, k) =
N
g n Mn(±) (k) + g N RN(±) (ν, k),
(5.47)
n=0
where g = (ν 2 − 14 )−1 , Mn(±) (k) =
2n+2 5=1
χ5(n) , [1 ± ik]5
(5.48) (5.49)
and χ5(n) are real numbers given in Equation (5.33). In addition, we have 2n+2
χ5(n) ≡ 0,
(5.50)
5=1
which guarantees that Mn(±) (k) → 0 as k → 0. For real k, we have [Mn(+) (k)]∗ = Mn(−) (k).
20
N. N. KHURI
Finally the remainder term RN(±) is given explicitly by ∞ 2N+3 6(N+1) (α) (±) 5 dα [α ν/2 + α −ν/2 ], RN (ν, k) = 5 (α ± ik) 1 5=1
(5.51)
with 6(n) 5 (α)
=
2n+1−5
C (n) (5; j )α 1/4+5+j −1ψ (j ) (α).
(5.52)
j =0
The C (n) (5, j ) are integers determined by a recursion formula given in Equation (5.28), with C (0) (1, 0) ≡ 1, and ψ (j ) (α) are the j th derivatives of ψ(α), Equation (5.27). For k = 0, we have M (±) (ν, 0) = 2ξ(ν + 12 ).
(5.53)
From Equations (5.47) and (5.51) we then have, for any integer n 0, ∞ 2n+3 6(n+1) (α) 5 dα [α ν/2 + α −ν/2 ]. 2ξ(ν + 12 ) = g n 5 α 1 5=1
(5.54)
For n = 0, this formula is given in [18], page 225. The results for larger n can be obtained by successive integrations by parts. 6. The Zeroes of M (−) (ν; k) for Im k > 0, and Fixed ν To study the Riemann hypothesis we need only to focus on the truncated critical strip, S(T0 ), S(T0 ) = {ν | Im ν > T0 , − 12 < Re ν < 12 }.
(6.1)
Since the Riemann hypothesis has already been rigorously established up to Im ν = O(106 ), we can simplify the calculations of this paper tremendously by taking T0 to be large. Initially, we take T0 ∼ = 103 . The following lemma will be quite useful: LEMMA 6.1. For any ν ∈ S(T0 ), and k such that Im k > −1/4, we have |M (−) (ν, k) − M0(−) (k)|
C2 T02
(6.2)
and C2 103 .
(6.3)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
21
Proof. Taking the expansion of M (−) (ν; k) in powers of g = (ν 2 − 1/4)−1 to first order, we have M (−) (ν, k) − M0(−) (k) = gM1(−) (k) + gR1(−) (ν, k), where M1(−) (k) is given by Equation (5.36), and ∞ 5 6(2) (−) 5 (α) [α ν/2 + α −ν/2 ]. dα R1 (ν; k) = 5 [α − ik] 1 5=1 Here 6(2) 5 is given by Equation (5.26) and Table I. First, we have for Im k > −1/4, 4 4 |b | 5 |M1(−) (k)| |b5 | · ( 43 )5 . 5 |1 − ik| 5=1 5=1
(6.4)
(6.5)
(6.6)
Using Table II, we get |M1(−) (k)| < 68,
Im k > − 14 .
(6.7)
The upper bound on R1(−) for Im k > −1/4 is 5 ∞ (2) (−) 1/4 −5 4 5 α |65 (α)|α ( 3 ) , |R1 (ν, k)| 2 1
5=1
where we have used α 4 α − ik < 3 for α 1 and
Im k > − 14 .
Using Equation (5.26), we have ∞ 5 5−5 (−) (2) −1/2+j (j ) 4 5 dα (3) | |C (5, j )|α ψ (α)| , |R1 (ν, k)| 2 1
(6.8)
5=1
(6.9)
j =0
where we note that C (n) (5, j ) = (−1)5+1 · |C (n) (5, j )| as can be seen from Table I. The series in Equation (3.7) that defines ψ(α) is highly convergent for α 1. Indeed the first term gives a good approximation to it and to its first six derivatives. One can easily derive the bounds, 0 j 7, π j e−πα |ψ (j ) (α)| π j e−πα (1 + ;(j )),
(6.10)
where ;(j ) is ;(j ) = e−3π [1 + (2)2j ].
(6.11)
22
N. N. KHURI
For j 4, ;(j ) < 0.021. Thus it is sufficient for the purposes of this estimate to use ψ (j ) (α) ∼ = (−1)j π j e−πα . Substituting this in (6.9) and carrying out the α integration, we get |R1(−) (ν, k)|
5 5−5 2(1.1) 4 5 (2) √ (3) | |C (5, j )|(−1)j +(j + 12 ; π )| ≡ C2 , π 5=1 j =0
(6.12)
where +(j, β) is the incomplete gamma function. From Table I, it is now easy to check our bound of C2 ≡ C2 +68 < 200. This completes the proof of Lemma 6.1. ✷ It should be apparent to the reader that one could use more refined methods to obtain a much better bound on R1(−) . We do not do this at this stage. Our most important task is to study the Riemann conjecture for Im ν > T0 with T0 taken below the maximum for which the hypothesis has been rigorously established. In a future paper, we will try to find the lowest value of T0 for which our method works. The function M0(−) (k) given by Equation (5.13) is a rational function of k −1 − a1 a1 + 1 − ik (1 − ik)2 −k[k + i(2 + a1 )] = . (1 − ik)2
M0(−) (k) = 1 +
(6.13)
Here a1 = −1 + 8ψ(1) = −0.6543.
(6.14)
Obviously, M0(−) has two zeros, k1 = 0
and
k2 = −i(2 + a1 ) = −i(1 + 8ψ(1)).
Thus, Im k2 < −1. Hence, M0(−) (k) has only one zero in the half-plane, Im k 0. Focusing on the domain Im k > −1/4, and |k| > 1/4 we get, with k = κ + iτ , √ κ 2 + τ 2 κ 2 + (τ + η)2 (−) , (6.15) |M0 (k)| = κ 2 + (τ + 1)2 where η = 2 + a1 > 1. It is easy to find a lower bound for |M0(−) | in the above domain. Setting q = (0, −η), and p = (0, −1) we get |M0(−) (k)| =
|k| · |k − q| |k|2 1 . 2 2 |k − p| |k − p| 25
We can now prove the following lemma:
(6.16)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
23
LEMMA 6.2. For any ν ∈ S(T0 ), and k such that Im k 0, with |k| 1/4, M (−) (ν, k) has no zeros and has a lower bound 1 (−) . (6.17) |M (ν, k)| > 0.04 − T0 Proof. From Lemma 6.1 we have |M (−) (ν, k)| > |M0(−) (k)| −
C2 , T02
(6.18)
with C2 /T0 < 2. Using (6.16) we get 1 1 (−) > . |M (ν, k)| > 0.04 − T0 26
(6.19)
Thus any zeros, k0 of M (−) (ν, k) in the upper half k-plane must have |k0 | < 1/4. ✷ Proceeding further, we have the following lemma: LEMMA 6.3. For any fixed ν ∈ S(T0 ), the maximum number of zeros of M (−) (ν; k) with Im k 0 is one. Proof. From Lemma 6.1 we get, for ν ∈ S(T0 ), |M (−) (ν, k) − M0(−) (k)| <
1 , T0
|k| = 14 .
By very similar arguments, we can also show that dM (−) (ν, k) dM0(−) < λ , |k| = 1 , − 4 dk dk T0
(6.20)
(6.21)
where λ = O(1). denote the number of zeros of M0(−) (k) in the disc |k| 1/4, and N 1 Let N (0) 1
4
4
be the corresponding number for M (−) (ν, k), ν ∈ S(T0 ), then (−) 1 [M (ν, k)] [M0(−) (k)] (0) , − dk N1 − N1 = 4 4 2π i C 1 M (−) (ν, k) M0− (k)
(6.22)
4
where C 1 is the circle |k| = 1/4, and the prime indicates differentiation with 4 respect to k. Hence, |N 1 − N (0) 1 | 4
4
(26)2 {Max|k|= 1 |M0(−) (k)[M (−) (ν, k)] − [M0(−) (k)] M (−) (ν, k)|} 4 4 169λ < 1, (6.23) T0
24
N. N. KHURI
where we have used Equations (6.16), (6.19), and (6.21). Since M0(−) (k) has only one zero in the disc, so does M (−) (ν, k) and Lemma 6.2, which proves the absence of zeros with Im k 0, and |k| 1/4, completes our proof. ✷ The next question is where is this one zero of M (−) (ν; k)? To answer this question we first-order expansion for M (−) (ν, k), M (−) (ν, k) = M0(−) (k) + gM1(−) (k) + ∞ 5 6(2) 5 (α) [α ν/2 + α −ν/2 ]. dα +g 5 (α − ik) 1 5=1 With k = 0, we have
∞
2ξ(ν + 1/2) = g
dα
5 6(2)(α)
1
5
5=1
α5
· [α ν/2 + α −ν/2 ].
(6.24)
(6.25)
Subtracting these two equations, we get M (−) (ν, k) = 2ξ(ν + 1/2) + M0(−) (k) + gM1(−) (k) +
∞ 5 1 1 (2) ν/2 −ν/2 dα − ] . 65 (α)[α + α +g (α − ik)5 α 5 1 5=1
(6.26)
Note that now the integral on the right-hand side is also O(k) as k → 0. The ξ function for large values of | Im ν| is exponentially small, mainly due to the +(s/2) factor in Equation (3.4). In fact given standard results on the order of ζ(s) in the critical strip we have ξ( 12 + ν) = O(|ν|p e
−| Im ν|π 4
),
where p = 2 + δ and 0 < δ < 1. Thus, from Equation (6.26) and the exact expression for M0(−) (k) in Equation (6.13), we get the position of the zero, k0 , near the origin, k0 ∼ =
−2iξ( 12 + ν) [(2 + a1 ) + O( T12 )]
+ O(k02 ),
(6.27)
0
where (2 + a1 ) > 1. Hence, −2 Re ξ( 12 + ν) . Im k0 ∼ = 2 + a1
(6.28)
If Re ξ( 12 + ν) 0, M (−) (ν, k) has no zeros for Im k > 0. On the other hand, if ν ∈ S(T0 ) is such that Re ξ(1/2 + ν) < 0 there will be one zero close to the origin, but in the upper half k-plane.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
25
Finally, we note one important fact, namely, if (ν0 + 1/2) is a zero of the ξ function, then M (−) (ν0 , k) has no zeroes for Im k > 0, and its only zero with Im k > −; occurs exactly at k = 0. In summary M (−) (ν; k) , with ν ∈ S(T0 ), has most of the properties of a Martin type Jost function with the exception of one, i.e. reality. We list these properties: (i) (ii) (iii) (iv)
M (−) (ν, k) analytic in the cut k-plane with a cut for k = iτ , −∞ < τ < −1. lim|k|→∞ M (−) (ν, k) = 1. M (−) (ν, k) has no zeros for Im k > 1/4, and ν ∈ S(T0 ). If we write ∞ 2 1 dαψ(α)α 1/4[α ν/2 + α −ν/2 ]e−αu , (6.29) >(ν, u) ≡ (ν − 4 ) 1
then
M
(±)
(ν, k) = 1 +
∞
>(ν, u)e∓iku du.
(6.30)
0
In fact we could have used Equations (6.29) and (6.30) as the starting definitions of M (±) (ν, k). (v) We define the S-matrix S(ν, k) =
M (+) (ν, k) , M (−) (ν, k)
and it follows that +∞ |S(ν, k) − 1|2 dk < ∞.
(6.31)
(6.32)
−∞
(vi) The reality condition does not hold for all ν. If ν is purely imaginary, i.e. ν = it, then for real k, (M (+) (ν, k))∗ = M (−) (ν, k) and, hence, |S(ν, k)| = 1. However, if ν is complex, i.e. Re ν = 0, then the above relation does not hold. However, we still have for real k 1 . (6.33) |S(ν, k)| = 1 + O |ν|2 For an arbitrary ν, ν ∈ S(T0 ), we can still carry out the inverse scattering program by properly handling the one zero in the upper half k plane. We will do that in Section 8, where the resulting potential is complex. 7. The Limit Case, |ν| → ∞ Before we proceed to the main proof, we shall solve exactly the limiting case |ν| → ∞. This result will be extremely useful in the rest of this paper.
26
N. N. KHURI
We start with M (±) (ν, k) → M0(±) (k),
|ν| → ∞,
(7.1)
where, from Equation (6.13), we have M0(±) (k) =
−k[k ∓ i(2 + a1 )] . (1 ± ik)2
(7.2)
We thus have Jost functions which are rational in k. This is the case first studied by Bargmann [22] in the paper which gave the famous phase equivalent potentials. The S-matrix is also rational,
M0(+) k − i(2 + a1 ) 1 − ik 2 , (7.3) S0 (k) ≡ (−) = k + i(2 + a1 ) 1 + ik M0 where (2+a1 ) > 1, a1 = −1+8ψ(1). One can use Bargmann’s method to uniquely determine a potential V0 (x) which has the S-matrix given here. But we prefer to determine V0 by using Marchenko’s method. The Marchenko kernel F0 is ∞ 1 (S0 (k) − 1)eikx dk. (7.4) F0 (x) = 2π −∞ This Fourier transform converges in the mean, (S0 − 1) → O(1/k) as k → ±∞. By contour integration F0 (x) = λ0 e−x + λ1 xe−x ,
(7.5)
where λ0 =
8a1 + 4a12 − 4 (3 + a1 )2
(7.6)
λ1 =
−4(1 + a1 ) . (3 + a1 )
(7.7)
and
Both λ0 and λ1 are negative. The Marchenko equation is
∞
A0 (x, y) = F0 (x + y) +
A0 (x, u)F0 (u + y) du
(7.8)
x
and A0 (x, y) ≡ 0 for y < x. With F0 as defined by Equation (7.5), one can easily obtain the exact solution of the integral equation (7.8). From Bargmann’s paper [22], it is clear that we have the ansatz A0 (x, y) ≡ [B(x) + (y − x)C(x)]e−(y−x) .
(7.9)
27
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Substituting this trial solution in Equation (7.8) and carrying out the integration over u, we get B(x) C(x) −(y−x) −2x + (λ0 + λ1 x) + e A0 (x, y) = e 2 4
B(x) + C(x) λ1 + λ0 + xλ1 + + 4
B(x)λ1 C(x)λ1 + + λ1 . (7.10) + ye−2x 2 4 But from Equation (7.9) we also have A0 (x, y) = [(B − xC) + yC]e−(y−x) .
(7.11)
Comparing the coefficients of y in Equations (7.10) and (7.11) we get B(x)λ1 C(x)λ1 + + λ1 , 2 4 and the terms to zero-order in y give C Bλ1 Cλ1 B + (λ0 + λ1 x) + + + λ0 + xλ1 = e2x [B − xC]. 2 4 4 4 e2x C(x) =
(7.12)
(7.13)
These last two equations determine B(x) and C(x) giving C(x) =
λ1 − (λ21 /4)e−2x
(7.14)
λ21
[e2x − ( 16 )e−2x − 12 (λ0 + λ1 ) − λ1 x]
and λ2
B(x) =
λ0 + 2λ1 x + ( 41 )e−2x λ2
[e2x − ( 161 )e−2x − 12 (λ0 + λ1 ) − λ1 x]
.
(7.15)
Since λ1 < 0, λ0 < 0, and (λ21 /16) 1, the denominators in Equations (7.14) and (7.15) do not vanish for any x 0. One can simplify Equations (7.14) and (7.15) by defining 1 + a1 λ1 = 0.1474. (7.16) ρ≡− = 4 3 + a1 Then its is easy to show that − 12 (λ0 + λ1 ) = 1 − ρ 2 .
(7.17)
We obtain C(x) =
[e2x
−(4ρ + 4ρ 2 e−2x ) − ρ 2 e−2x + (1 − ρ 2 ) + 4ρx]
(7.18)
28
N. N. KHURI
and B(x) =
(2ρ 2 + 4ρ − 2) − 8ρx + 4ρ 2 e−2x . [e2x − ρ 2 e−2x + (1 − ρ 2 ) + 4ρx]
(7.19)
From Equation (7.9) we see that the potential, V0 (x), is V0 (x) = −2
dB(x) dA0 (x, x) = −2 . dx dx
(7.20)
This leads to (4 − 2ρ 2 − 16ρ)e2x − 4(1 − ρ 2 )ρ 2 e−2x − − 16ρx(e2x + ρ 2 e−2x ) − 64ρ 2 x − 16ρ 2 . V0 (x) = [e2x − ρ 2 e−2x + (1 − ρ 2 ) + 4ρx]2 Note that V0 (x) = O(e−2x ) as x → +∞. The two Jost solutions f0(±) are given by ∞ (±) ∓ikx + dyA0 (x, y)e∓iky . f0 (k, x) = e
(7.21)
(7.22)
x
Substituting Equation (7.9) for A0 , we get C(x) B(x) + . f0± = e∓ikx 1 + (1 ± ik) (1 ± ik)2
(7.23)
One can now check directly that −d2 f0± + V0 (x)f0± = k 2 f0± , dx 2
(7.24)
also f0(±) → M0± (k) as x → 0. The results of this section can also be obtained by using the technique developed by Bargmann [22] which preceded the results of [6] and [7]. The Jost functions defined in Equation (7.2) do indeed uniquely determine the potential V0 (x) and its solutions f0(±) (k, x). The fact that M0(±) (k) = O(k), as k → 0, leads to the degenerate case using Bargmann’s method, but the final results agree with those by Marchenko’s method.) Finally, it should be remarked that the full scattering amplitude for V0 (x) has a pole at k = 0. However, this is not part of the point spectrum. There is no L2 (0, ∞) solution of the Schrödinger equation with V0 (x) for k = 0.
) We thank H. C. Ren for clarifying and checking this point.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
29
8. The Marchenko Equation We define an S-matrix as in Equation (6.34) S(ν, k) =
M (+) (ν, k) , M (−) (ν, k)
(8.1)
where ν ∈ S(T0 ). S(ν, k) is, for any ν, analytic in k in the strip −1 < Im k < +1. Next we define a Marchenko kernel for S(ν, k), 1 dk[S(ν, k) − 1]eikx , x > 0, (8.2) F (ν; x) = 2π L where L is a line Im k = δ > 0, 1 > δ > 0, and without loss of generality we fix δ, δ = 1/4. This Fourier transform is convergent in the mean (S − 1) = O(1/k), as Re k → ±∞. Actually, we can also perform an integration by parts on (8.2) for any x > ε > 0. Since dS/dk is bounded and (dS/dk) = O(1/k 2 ) as k → ±∞, we have absolute convergence for any x > 0. It is important to note here that as shown in Equation (6.32), [S(ν, k) − 1] ∈ L2 (−∞, +∞), along the line Im k = 1/4. Of course, Equation (8.2) is not the standard definition of the Marchenko kernel. In the standard case, one integrates along the real k-axis. If we move the contour in (8.2) to the real axis, then there could be an extra contribution from the pole produced by the zero of M (−) (ν, k) when Re ξ(ν + 1/2) < 0. But for a Martin type S-matrix, all the scattering data, including that coming from the point spectrum, is contained in the discontinuity across the branch cut on the imaginary k-axis (see [19–21]). We prove the following lemma: LEMMA 8.1. F (ν, x) is (a) continuous and differentiable in x, x ∈ [0, ∞); (b) F (ν, x) = O(e−x ) as x → +∞; (c) Both F (ν, 0) and F (ν, 0) are finite and ∞ ∞ |F (ν, x)| dx < ∞, |F (ν, x)| dx < ∞; 0
(8.3)
0
(d) F (ν, x) is analytic in ν, for ν ∈ S(T0 ), and fixed x 0. Proof. In the Appendix, we prove that F (ν, x) can be written as 1 ∞ D(ν, α)e−αx dα, F (ν, x) = π 1
(8.4)
where D(ν, α) =
π(ν 2 − 1/4)ψ(α)α 1/4[α ν/2 + α −ν/2 ] . M (−) (ν, iα)
(8.5)
30
N. N. KHURI
This result is obtained by deforming the contour in Equation (8.2) and using the original representation (3.10) for M (±) (ν, k). We note that S(ν, k) is analytic for Im k > 0, except on the cut k = iτ ; 1 τ < ∞. Next we note that α(α + (2 + a1 )) (−) − |g||M (−) (iα) + R (−) (ν; iα)|. (8.6) |M (ν; iα)| 1 1 (1 + α)2 This follows from Equations (6.4) and (6.13). Since (2 + a1 ) > 1, we get
C2 (−) 1 1 2 + a1 |M (ν, iα)| 2 + 2 − 2 , α 1, (1 + α) T0
(8.7)
where the last term comes from Lemma 6.1. Hence, we have |M (−) (ν, iα)| 12 .
(8.8)
Finally, we obtain from (8.5), with |Re ν| < 1/2, |D(ν, α)| 2π(ν 2 − 14 )e−πα α 1/2,
α 1.
(8.9)
This bound guarantees the absolute and uniform convergence of the Laplace transform in Equation (8.4) for all x ∈ [0, ∞) and, hence, all the assertions (a), (b), and (c) of our lemma are true. Finally, (d) is also true, given the lower bound in ✷ Equation (8.8) and the uniform bound on (α ν/2 + α −ν/2 ) for ν ∈ S(T0 ). The Marchenko equation can now be defined as ∞ duA(ν; x, u)F (ν; u + y), A(ν; x, y) = F (ν; x + y) +
(8.10)
x
with A(ν; x, y) ≡ 0,
y < x.
(8.11)
The integral equation (8.10) is of Fredholm type and the Hilbert–Schmidt norm of F is finite, ∞ ∞ F 2= du dv|F (ν; u + v)|2 < ∞. (8.12) x
x
This follows from Lemma 8.1. We first prove the following lemma: LEMMA 8.2. For all ν ∈ S(T0 ) and x 0, we have |F (ν; x) − F0 (x)| <
C e−x/4 , |Im ν|2
(8.13)
where the constant C is bounded C < 2 × 103 .
(8.14)
31
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Proof. From Equations (6.4) and (6.5), we have M (±) (ν, k) − M0(±) (k) = gM1(±) (k) + gR1(±) (ν; k),
(8.15)
where g = (ν 2 − 1/4)−1 . From the definitions of F (ν, x) and F0 (x), we get
(+) 1 M (ν, k) M0(+) (k) ikx F (ν, x) − F0 (x) = e . − dk (8.16) 2π L M (−) (ν, k) M0(−) (k) This gives F − F0 = (EF )1 + (EF )2 ,
(8.17)
with (EF )1 ≡
g 2π
eikx
(M0(−) M1(+) − M0(+) M1(−) )
eikx
(M0(−) R1(+) − M0(+) R1(−) )
M0(−) M (−)
L
(8.18)
and g (EF )2 ≡ 2π
M0(−) M (−)
L
.
(8.19)
Again, here L is the line k = (λ + (i/4)), and −∞ < λ < +∞. In order to get bounds on (EF )1,2 , we need to separate out the terms which are only conditionally convergent, i.e. O(1/k) as k → ∞, from those which are absolutely convergent and, hence, easier to handle. From Equations (5.36) and (6.5), we write M1(±) (k) =
b1 + Mˆ 1(±) (k) 1 ± ik
(8.20)
and R1(±) (ν, k)
∞
= 1
6(2) (α) ν/2 (α + α −ν/2 ) + Rˆ 1(±) (ν, k), dα 1 α ± ik
(8.21)
where Mˆ 1(±) (k) =
4 5=2
Rˆ 1(±) (ν, k)
b5 , (1 ± ik)5 ∞
=
dα 1
5 6(2) 5 (α) [α ν/2 + α −ν/2 ]. 5 (α ± ik) 5=2
Both Mˆ 1 and Rˆ 1 are O(1/k 2 ) as |k| → ∞.
(8.22)
(8.23)
32
N. N. KHURI
For (EF )1 , we can write (EF )1 ≡ (EF )11 + (EF )12 ,
(8.24)
with (EF )11
gb1 ≡ 2π
ikx
dke L
M0(−) (k) M0(+) (k) − (1 + ik) (1 − ik)
1
(8.25)
M0(−) M (−)
and (EF )12
g = 2π
ikx
dke
M0(−) (k)Mˆ 1(+) (k) − M0(+) (k)Mˆ 1(−) (k) M0(−) (k)M (−) (ν, k)
L
.
(8.26)
The integral in (8.25) is conditionally convergent, |M0± (k)| → 1 as |k| → ∞, and |M (−) (ν, k)| is bounded from below for all k with Im k 1/4. Also |M (−) (ν, k)| → 1 as |k| → ∞. To obtain a bound on (EF )11 we first note the following: (−) M0 (k) 1 ikx = 0. (8.27) dke 1 − ik M0(−) (k)M (−) (ν, k) L This follows from Jordan’s lemma. The integrand in (8.27) is analytic for Im k 0, and the bracketed term is O(1/k) for large |k|. Adding twice the left-hand side of Equation (8.27) to Equation (8.25), one obtains
(−) M0(+) (k) − M0(−) (k) gb1 ikx 2M0 (k) × dke − (EF )11 = 2π L 1 + k2 1 − ik 1 . (8.28) × M0(−) (k)M (−) (ν, k) To obtain an upper bound on |(EF )11 |, we first need lower bounds on M0(−) (k) for k ∈ L. From Equation (5.13), we have |M0(−) (k)| 1 −
|a2 | |a1 | − , |1 − ik| |1 − ik|2
(8.29)
with |a1 | = 0.654,
|a2 | = 0.346
and
i k =λ+ , 4
−∞ < λ < +∞,
we finally have |M0(−) (k)| 0.255 > 14 ,
k ∈ L.
(8.30)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
33
Lemma 6.1 will now give us a lower bound for M (−) (ν, k), ν ∈ S and k = λ + i/4. Since |M (−) − M0(−) | C2 /T02 we get |M (−) (ν, k)| 14 ,
(8.31)
for k = λ + i/4. Finally, we need to bound M0(+) − M0(−) , where from Equation (5.13), M0(+) − M0(−) =
−2ika1 4ika2 − . 2 1+k (1 + k 2 )2
(8.32)
But for k = λ + i/4, |k/(1 + ik)| < 1, we obtain |M0(+) (k) − M0(−) (k)|
δ , |1 − ik|
(8.33)
with δ = 2|a1 | +
64 |a | 15 2
∼ = 2.78.
(8.34)
The above bounds lead us immediately to 4δ 2|g||b1 |e−x/4 +∞ 2 + dλ |EF |11 π |1 + (λ + i/4)2 | |1 − ik|2 −∞
c11 −x/4 e , | Im ν|2
where c11
4|b1 | (2 + 4δ) · = π
(8.36)
∞ 0
(8.35)
16 15
dλ 290. + λ2
(8.37)
The bound for (EF )12 is easier to calculate, from (8.26) (+) M |g|e−x/4 +∞ (+) (−) dλ 4|Mˆ 1 (k)| + 4|Mˆ 1 (k)| · 0(−) . (8.38) |(EF )12 | 2π M0 −∞ For k = λ + i/4, a simple calculation gives (+) M0 (k) (−) 5/3. M0 (k)
(8.39)
Using Equation (8.22), we get |(EF )12 |
c12 e−x/4 , | Im ν|2
(8.40)
34
N. N. KHURI
with c12
4
∞ 4 1 1 5 ∼ = |b5 | dλ 3 2 + (3) 5 2 = 112. π 5=2 [( 4 ) + λ2 ] [( 4 ) + λ2 ] 0
(8.41)
The values of b5 are given in Equation (5.40). To estimate (EF )2 we again split R (±) (ν, k) into two terms as in Equation (8.21). We write (EF )2 ≡ (EF )21 + (EF )22 , where (EF )21
g = 2π
ikx
dke
(EF )22 =
g 2π
dkeikx L
M0(−) (k)r1(+) − M0(+) (k)r1(−)
M0(−) (k)M (−) (ν, k)
L
and
(8.42)
M0(−) (k)Rˆ 1(+) − M0(+) Rˆ 1(−) M0(−) (k)M (−) (ν, k)
(8.43)
,
with Rˆ 1(±) given in Equation (8.23) and ∞ 6(2) (α) ν/2 [α + α −ν/2 ]. dα 1 r1(±) (ν, k) = α ± ik 1
(8.44)
(8.45)
Equation (8.43) can be rewritten as (−) (+) + r1(−) ) (M0(+) − M0(−) )r1(−) g ikx M0 (r1 dke − . (8.46) (EF )21 = 2π L M0(−) M (−) M0(−) M (−) Here we have again used Jordan’s lemma, which implies (−) (−) ikx M0 r1 dke ≡ 0. M0(−) M (−) L From Equations (8.45) and (8.46) we get ∞ |g|e−x/4 +∞ dλ dα(2α 1/4 )|6(2) |(EF )21 | 1 (α)| × 2π −∞ 1 16|M0(+) − M0(−) | 8α + , × |α 2 + k 2 | |α + 1/4 − iλ|
(8.47)
(8.48)
where k = λ + i/4. Using the bound (8.33) we obtain |(EF )21 |
c21 e−x/4 , | Im ν|2
(8.49)
35
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
with
∞ 1 dα(2α 1/4 )|6(2) 1 (α)| × 2π 1
+∞ 8α 16δ dλ + × . 1 (α 2 − 16 ) + λ2 |α + 1/4 − iλ|| 54 − iλ| −∞
c21 =
This leads to c21
∞
dαα
1/4
1
128 |6(2) 1 (α)|[ 15
+
64 δ] 5
∞
(45) 1
dαα 1/4|6(2) 1 (α)|.
(8.50)
(8.51)
Using the definition of 6(2) 1 (α) in Equation (5.26), Table I, and the bounds on ψ (j ) (α) given in Equation (6.10), one can easily get a rough numerical bound on ∞ the above integral, 1 dαα 1/4|6(2) (α)| 10 and, hence, c21 450.
(8.52)
From Equations (8.44), (8.30), (8.31), and (8.23), we get 5 4|g| −x/4 ∞ e dαα 1/4|6(2) |(EF )22 | 5 (α)| × 4 1 5=2 (+) +∞ M0 1 1 dλ + × |α + ik|5 M0(−) |α − ik|5 −∞ with k = λ + i/4. Using the fact that |α ± ik| [(α ∓
1 2 ) 4
+λ ]
2 1/2
and
(+) M0 (−) < M0
5 3
(8.53)
for k ∈ L,
we have |(EF )21 |
c22 −x/4 e |Im ν|2
(8.54)
with c22 where β5 ≡
5−1 5 32 ∞ 1 dαα 1/4 |6(2) (α)| β5 , 5 3π 5=2 1 α − 1/4
+∞
−∞
du . (1 + u2 )5/2
(8.55)
(8.56)
For α 1, |(α/(α − 1/4)| < 4/3 and, hence, c22
5 32 A5 β5 ( 43 )5 , 3π 5=2
(8.57)
36
N. N. KHURI
with
∞
A5 = 1
dαα 5/4−5|6(2) 5 (α)|.
(8.58)
A simple numerical estimate will give c22 103 .
(8.59)
This completes the proof of our lemma with C = c11 + c12 + c21 + c22 < 2 × 103 . ✷ Lemma 8.2 guarantees that for ν ∈ S, F = F0 + O(1/T02 ). Indeed, for the Hilbert–Schmidt norms, we have F F0 + F − F0 . However, from the lemma, ∞ ∞ F − F0 2 = du dv|F (ν, u + v) − F0 (u + v)|2 x x ∞ C2 ∞ du dve−(u+v)/2 |ν|4 x x C 2 ∞ −w/2 we dw. |ν|4 0
(8.60)
(8.61)
Thus F − F0
2C 2 . 2 T0 T0
(8.62)
We have the exact expression for F0 (x) given in Equation (7.5) and we can calculate F0 exactly. The Hilbert–Schmidt norm for F0 is ∞ ∞ 2 F0 x = du dv[F0 (u + v)]2 . (8.63) x
x
Note that here x appears as a parameter (see Equation (8.10)). As x increases the norm of F0 tends to zero. From Equation (7.5), we get ∞ 2 F0 x=0 = dw[λ20 w + 2λ0 λ1 w 2 + λ1 w 3 ]e−2w , (8.64) 0
with λ0 and λ1 given by Equations (7.6) and (7.7). After some algebra, we obtain λ21 , (8.65) 16 with λ1 = −0.590. Thus the Hilbert–Schmidt norm for x = 0 is slightly bigger than one, however, it is easy to show that for x > x0 ∼ = 0.1, F0
2 x=0
=1+
F0 < 1.
(8.66)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
37
Hence, the iterative series for both F and F0 converges for x > 0.1. But to prove the existence of A(ν; x, y) for all x 0, we will use Equation (8.61) and proceed in another way. In Section 7, we gave an explicit solution of the integral equation A0 = F0 + A0 F0 .
(8.67)
We can now write I0 ≡ (1 − F0 )−1 ,
(8.68)
A0 = (1 − F0 )−1 F0 = I0 F0 .
(8.69)
and
This leads to A0 + 1 = I0 .
(8.70)
The kernel A0 (x, y) is given explicitly in Equation (7.9) and (7.18)–(7.19), and thus I0 (x, y) is known. The full Marchenko equation (8.10) can now be written as A = F + AF.
(8.71)
We define I as I = (1 − F )−1
(8.72)
A = (1 − F )−1 F.
(8.73)
and
We prove that both I and A exist and have a finite norm. We have A + 1 = I = (1 − F )−1 .
(8.74)
To show that I (and A) exist, we note first that I = (1 − F )−1 = (1 − F0 − E)−1 ,
(8.75)
where E ≡ F − F0 .
(8.76)
Using the fact that (1 − F0 )I0 = 1, we get I = [(1 − F0 )I0 {(1 − F0 ) − E}]−1 = (1 − I0 E)−1 I0 .
(8.77)
38
N. N. KHURI
Using Equation (8.74), we get A = (1 − I0 E)−1 I0 − 1 = (1 − I0 E)−1 (A0 + I0 E),
(8.78)
where A0 + 1 = I0 . Next we define the operator K as K ≡ I0 E = (A0 + 1)E
(8.79)
and obtain A = A0 + (1 − K)−1 (1 + A0 )K.
(8.80)
The Hilbert–Schmidt norm of E is small. Indeed, using Equation (8.61), we have C2 2 E 2 ≡ F − F0 2 4 2 1. (8.81) |ν| T0 However, K ≡ A0 E + E, and we get K A0 E + E . A0 is known and A0 2 < 5, thus 3C K 2 1, |ν|
(8.82)
(8.83)
for all ν ∈ S(T0 ). Thus the inverse (1 − K)−1 is given by an absolutely convergent series ∞ Kn (8.84) (1 − K)−1 = n=0
and has a bounded norm, (1 − K)−1 2. The final result for A is A = A0 + H + A0 H,
(8.85)
where H ≡
∞
K n,
(8.86)
n=1 2 ˜ . and H 2 K 1. Indeed, we have A − A0 C/|ν| The kernel K(ν; x + y) can be written as
K(ν; x, y) = F (ν; x, y) − F0 (x + y) + ∞ duA0 (x, u)[F (ν; u + y) − F0 (u + y)]. +
(8.87)
x
The properties of the kernel K(ν; x, y) are similar to those of F (ν; x, y). We have the following lemma:
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
39
LEMMA 8.3. K(ν; x, y) is for y x 0, (a) analytic for ν ∈ S(T0 ); (b) differentiable in both x and y; (c) analytic for Re x 0 and Re y 0, when ν ∈ S(T0 ); and x+y (d) |K(ν; x, y)| C/|ν|2 e−( 4 ) . Proof. These results follow from Equation (8.88), Lemma 8.1, and the exact result (7.11) for A0 (x, u). We note that the denominators appearing in the expressions (7.18) and (7.19) for B(x) and C(x) do not vanish for Re x 0. Both B and C are thus analytic in the half plane Re x 0. The full expression for A(ν; x, y) is ∞ A(ν; x, y) = A0 (x, y) + H (ν; x, y) + A0 (x, u)H (ν; u, y), (8.88) x
where H (ν; x, y) =
∞
K (n) (ν; x, y)
(8.89)
n=1
and K (n) (ν; x, y) ∞ du1 . . . = x
∞
dun−1 K(ν; x, u1 )K(ν; u1 , u2 ) . . . K(ν; un−1 , y).
(8.90)
x
The series in (8.89) is absolutely and uniformly convergent for y x 0, and all ✷ ν ∈ S(T0 ). The properties of A(ν; x, y) can be summarized in the following lemma: LEMMA 8.4. (a) For y x 0, A(ν; x, y) is analytic in ν for ν ∈ S(T0 ); (b) A(ν; x, y) is differentiable in both x and y, y x. Also A(ν, 0, 0) and [d/dx(A(ν; x, x)]x=0 are finite; (c) For fixed ν, ν ∈ S(T0 ), A(ν; x, y) is analytic in x and y for Re x 0, Re y Re x 0; and (d) For all ν ∈ S(T0 ), we have the bound |A(ν; x, y) − A0 (x, y)|
C˜ −( x+y ) e 4 . |ν|2
(8.91)
Proof. These results follow immediately from Equation (8.89) and Lemmas 8.1 and 8.3. The bound (8.91) follows from the bound (8.13) of Lemma 8.2. The constant C˜ is certainly such that, C˜ < 104 , which is sufficient for our purposes at this stage, but can be improved with more careful estimates. ✷
40
N. N. KHURI
The next step is to define two functions U (ν; x) and f (±) (ν; k, x) as follows: U (ν; x) = −2
d A(ν : x, x), dx
and f ± (ν; k, x) = e∓ikx +
∞
x0
dyA(ν; x, y)e∓iky .
(8.92)
(8.93)
x
Without recourse to the standard methods of inverse scattering, one can directly prove the next lemma. LEMMA 8.5. For any ν ∈ S(T0 ), f ± satisfy a Schrödinger equation with U (ν, x) as the potential −
d2 f ± + U (ν; x)f ± = k 2 f ± . dx 2
(8.94)
Proof. From (a) and (b) in Lemma 8.4 it follows that U (ν; x) is analytic in ν for ν ∈ S(T0 ) and x 0. Similarly, from Equation (8.94), it follows that f (−) (ν; k, x) (with x 0) and Im k 0 is also analytic in ν in the truncated strip. Similarly, f (+) with Im k 0 is analytic. The same is true for d/dx(f (±) ), and d2 /dx 2 (f (±) ) since absolute and uniform convergence allows us to differentiate under the integral sign in (8.93). ✷ In the next section we will prove the validity of Equation (8.94) on the line ν = it, t T0 . Hence, by analytic continuation, the Schrödinger equation (8.94) holds for all ν ∈ S(T0 ). In the Appendix we will give a more direct proof of Equation (8.94) and also show that one does indeed recover the original Jost function from the potential U (ν; x). 9. The Case ν = it For purely imaginary ν, our S-matrix, S(ν, k), is unitary and satisfies all the properties needed for the standard inverse scattering methods of Gelfand, Levitan, and Marchenko to be applicable. We sketch some relevant results in this section. First, we define S as S(it, k) ≡
M (+) (it, k) , M (−) (it, k)
t > π 2.
(9.1)
For real k, it follows from Equation (3.10) that [M (+) (it, k)]∗ = M (+) (it, −k) = M (−) (it, k).
(9.2)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
41
S(it, k) satisfies all the conditions given in Faddeev’s [14] review paper, which are sufficient to guarantee that the Marchenko equation will lead to a unique real potential U (it, x). We can easily check that for real k, |S(it, k)| = S(it, 0) = S(it, ∞) = 1
(9.3)
[S(it, k)]∗ = S(it, −k).
(9.4)
and
The number of discrete eigenvalues of S for fixed t > π 2 is at most one (see Section 6). In the physicist’s language, we have either one bound state or one antibound state. This is evident from Equation (6.28) and the fact that ξ(it + 1/2) is real. We will discuss this point in more detail at the end of this section. The Marchenko kernel is now given by ∞ 1 F (it, x) = dk[S(it, k) − 1]eikx ; x > 0 (9.5) 2π −∞ for the case ξ(it + 12 ) 0
(9.6)
and 1 F (it, x) = π
+∞ −∞
dk[S(it, k) − 1]eikx + c0 e−τ0 x
(9.7)
for the case where ξ(it + 1/2) < 0, i.e. with a bound state at E = −τ02 . Both Fourier transforms in Equations (9.5) and (9.6) are convergent in the mean, since [S − 1] = O(1/k) for large k. Also, it is clear that F (it, x) is real. πt As noted previously ξ(it + 1/2) = O(t p e− 4 ) and, hence, small for, t > π 2 , this makes τ0 1/4 and the bound state is very shallow for t > π 2 . One can now move the contour of integration up in both Equations (9.5) and (9.6) to obtain 1 dk[S(it, k) − 1]eikx , x > 0 (9.8) F (it, x) = 2π L for both cases. Here L is the line Im k = 1/4. In the case of Equation (9.6), the contribution from the pole at k = iτ0 exactly cancels the second term on the righthand side, see [22]. The solution of the Marchenko equation, A(it; x, y) exists, is real, and differentiable for y x > 0. The resulting potential, U (it, x), is real, continuous for all x 0, and O(e−2x ) for large x. We close this section by calculating the position of the bound state or antibound state for fixed ν = it, t > T0 .
42
N. N. KHURI
Rewriting Equation (6.26) for ν = it, we get M (−) (it, k) = 2ξ(it + 12 ) + M0(−) (k) + gM1(−) (k) +
∞ 5 1 t 1 (2) log α , (9.9) + 2g dα − 5 65 (α) cos 5 (α − ik) α 2 1 5=1 where g=
t2
−1 . + 1/4
(9.10)
Now, ξ(it + 1/2) is real and exponentially small for large t. The three other terms on the right-hand side of Equation (9.8) are all O(k) for small k. From Equation (6.13), we have M0(−) (k) = −ik(2 + a1 ) + O(k 2 ),
(9.11)
a1 = −1 + 8ψ(1) = −0.6543.
(9.12)
with
Thus, for t > T0 , the one zero of M (−) (it, k) will occur only when k = iτ and τ =−
2ξ(it + 1/2) + O(τ 2 ), (2 + a1 ) + O( t12 )
(9.13)
where the O(1/t 2 ) term is real, and the same for the O(τ 2 ) term. Note that M0(−) (iτ ) and M1(−) (iτ ) are both real as is the integral in (9.8) for k = iτ . The resulting potential, or one parameter family of potentials, U (it, x) ≡ V (g, x),
(9.14)
has a remarkable property as t increases, t > T0 . It will have exactly one bound state when ξ(it + 1/2) < 0, with energy E0 = −τ02 , E0 = −
4[ξ(it + 1/2)]2 + O([ξ(it + 1/2)]3 ). [(2 + a1 ) + O( t12 )]2
(9.15)
Then, as we pass a Riemann zero and ξ(it + 1/2) > 0, there will be no bound state until t reaches the next Riemann zero. As t → +∞, the potential, U (it, x), presents us with a seemingly puzzling situation. The bound state, i.e. a point spectrum of one, appears and then as t increases disappears, with this process repeating as t increases until as t → ∞, we reach V0 (x) which has no point spectrum. Schwinger’s theorem relating the number of bound states to the number of nodes of the zero energy regular solution, φ(it; 0; x), defined in Equation (2.8), is instructive for the present case.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
From [13], we have an integral equation for φ, x sin k(x − x ) sin kx + V (x )φ(it; k; x ) dx , φ(it; k; x) = k k 0 where clearly φ(it; k; 0) ≡ 0. The zero energy φ is given by
x V (x )φ(it; 0; x ) dx x − φ(it; 0; x) = 1 + x0 V (x )φ(it; 0, x ) dx . −
43
(9.16)
(9.17)
0
For x not large, x T0 , we can approximate φ by the t → ∞ solution 1 φ(it; k; x) = φ0 (k, x) + O 2 , t
(9.18)
where φ0 is defined as in (2.8) but with f ± replaced by f0± and M ± replaced by M0± . It is easy to check that φ0 (0, x) is positive for x not large, and φ0 (0; 0) = 0. But from Equation (9.17) we see that the large x behavior is φ(it; 0; x) → [C(t) + o(1)] +
∞ V (x )φ(it; 0; x ) dx + o(1) , +x 1 +
(9.19)
0
where
∞
C(t) = −
x V (x )φ(it; 0; x ) dx .
(9.20)
0
But under the integral sign, we can replace V by V0 and φ by φ0 , and obtain ∞ 1 x V0 (x )φ0 (it; 0, x ) dx + O 2 , (9.21) C(t) = − t 0 where V0 and φ0 are known exactly from Section 7. It is a simple matter to check that C(t) > 0,
(9.22)
t > T0 .
Next in [13], we have the result relating the Jost function to φ, and it gives ∞ (−) dx V (it; x )φ(it; k; x ). (9.23) M (it; k) = 1 + 0
Taking the k → 0 limit, we have from Equation (9.19) φ(it; 0; x) → C(t) + 2ξ(it + 12 )x,
x → ∞.
(9.24)
44
N. N. KHURI
We can only have a node in φ if ξ(it + 1/2) < 0, otherwise there is no node and no bound state. For t > T0 and large, the node occurs at large values of x, x0 ∼ =
C(t) ∼ O(e π4t ). 1 = 2ξ(it + 2 )
(9.25)
This discussion shows that, while our asymptotic estimates for f , φ, and V , are good for low values of x, x < T0 , one cannot use them for large values of x except in estimating integrals as in Equation (9.21). The well established results for a unitary S and real potentials now guarantee that Schrödinger’s equation holds for f ± (it; k; x) and U (it; x), i.e. −
d2 ± f (it; k; x) + U (it; x)f ± = k 2 f ± , dx 2
t > T0 .
(9.26)
We are also guaranteed that f ± (it; k; 0) = M (±) (it; k), where M (±) is the original Jost function we started with ∞ ψ(α)α 1/4[α it/2 + α −it/2 ] (±) 2 dα. M (it; k) = 1 − (t + 1/4) α ± ik 1
(9.27)
(9.28)
Thus by analytic continuation the Jost solutions f ± (ν; k; x) given in the previous sections will also give the original Jost function, i.e. (9.28) with it replaced by ν, and ν ∈ S(T0 ). 10. Asymptotic Expansion in Powers of g In this section, we carry out the asymptotic expansion of the kernels F and A and the potential V in powers of g, where g ≡ (ν 2 − 1/4)−1 , and ν ∈ S(T0 ) with T0 > 103 . We start with the definition of the Marchenko kernel F (ν; x) given in Equation (8.2), 1 dk[S(ν, k) − 1]eikx , x > 0, (10.1) F (ν; x) = 2π L where L is the line Im k = 1/4. Using the asymptotic expansion of M (±) (ν, k) given in Equation (5.47), we have N g n Mn(+) (k) + g N RN(+)(ν, k) n=0 − 1, (10.2) S(ν, k) − 1 = N g n Mn(−) (k) + g N RN(−) (ν, k) n=0
where
RN(±) (ν, k)
are both O(g) for small g.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
45
Next we recall the lower bound obtained in Section 8 for M0(−) (k) when k = λ + i(1/4), and −∞ < λ < +∞. This is given in Equation (8.30) |M0(−) (k)| 0.255,
k ∈ L.
(10.3)
From Lemma 6.1, we get also a lower bound on |M (−) (ν, k)| for ν ∈ S(T0 ), and k ∈ L given in Equation (8.31) |M (−) (ν, k)| 1/4,
T0 > 103 .
(10.4)
The last two bounds guarantee that the denominator in Equation (10.2) does not vanish for any k ∈ L and ν ∈ S(T0 ). We can then proceed to expand [S(ν, k) − 1] in powers of g for any k ∈ L and get [S(ν, k) − 1] =
N
g n Hn (k) + g N HR(N) (g, k),
(10.5)
n=0
where from Equation (10.2) we get H0 (k) ≡ S0 (k) − 1 =
H1 (k) ≡
M0(+) (k) M0(−) (k)
− 1,
M0(+) (k)M1(−) (k) (+) M (k) − 1 M0(−) (k) M0(−) (k) 1
(10.6)
(10.7)
and M1(+) (k)M1(−) (k) M2(−) (k)M0(+) (k) (+) M2 (k) − − + H2 (k) ≡ M0(−) (k) M0(−) (k) M0(−) (k) M (+) (k)[M1(−) (k)]2 (10.8) + 0 [M0(−) (k)]2 1
with similar expressions for Hn (k), n > 2, which we will not need in this paper. The remainder term HR(N) is O(g) as g → 0. All Hn (k) are rational functions of k. Equation (10.5) immediately gives us the asymptotic expansion for the kernel F (ν; x), F (ν; x) = F0 (x) + gF1 (x) + g 2 F2 (x) + · · · + g N FR(N) (g; x), where 1 Fn (x) = 2π
(10.9)
dkHn (k)eikx ,
x 0.
(10.10)
L
This last Fourier transform is conditionally convergent since |Hn (k)| = O(1/k) for large k (note that L is the line Im k = 1/4).
46
N. N. KHURI
Next we stress that all the Hn (k) are analytic for 1 > Im k > 0. The denominators in Equations (10.6)–(10.8), do not vanish in Im k 0, except at k = 0. This also holds for HR(N) (g, k). Thus we can shift back the contour in Equation (10.10) to the real k-axis and obtain +∞ 1 dkHn (k)eikx , x 0. (10.11) Fn (x) = 2π −∞ For real k, it follows from Equation (5.49) that [Mn(+) (k)]∗ = M (−) (k), and that Mn(+) (−k) = Mn(−) (k). This leads us to Hn∗ (k) = Hn (−k),
for k real.
(10.12)
Thus it immediately follows from Equation (10.11) that all Fn (x), n = 0, 1, 2, . . . , N, are real functions. However, FR(N) (g, x), is certainly not real for ν ∈ S(T0 ). We have explicitly calculated F0 (x) in Section 7, and obtained F0 (x) = λ0 e−x + λ1 xe−x , with λ0 and λ1 real and given in Equations (7.6) and (7.7). One can also easily calculate explicitly F1 (x) by contour integration.
+∞ (+) 1 M1 (k) M0(+) (k)M1(−) (k) ikx − dk e . F1 (x) = 2π −∞ M0(−) (k) [M0(−) (k)]2 The result is F1 (x) =
3
(10.13)
(10.14)
σn x n e−x ,
(10.15)
n=0
where the constants σn are explicitly given as functions of a1 , and bj , j = 1, . . . , 4. Here we will only give the numerical value of the σn s σ0 = 26.5228, σ2 = −9.3291,
σ1 = 1.7901, σ3 = −2.3582.
The result for F2 (x) will be similar, 5 −x n βn x . F2 (x) ≡ e
(10.16)
(10.17)
n=0
We will not give its explicit value as it is not needed. The Marchenko equation (8.10), for ν ∈ S(T0 ) can be written in operator form A = F + AF.
(10.18)
47
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
Writing A = A0 + gA1 + g 2 A2 + g 2 A(2) R ,
(10.19)
and using the expansion for F given in Equation (10.9) we get by comparing terms A0 = F0 + A0 F0 ,
(10.20)
an equation we solved explicitly in Section 7. In addition, we have A1 = (F1 + A0 F1 ) + A1 F0 ,
(10.21)
as the integral equation for A1 , and A2 = (F2 + A1 F1 + A0 F2 ) + A2 F0 ,
(10.22)
for A2 . It is obvious that the integral equations for An , n = 0, 1, 2, . . . all have the same kernel F0 . Thus, they are all explicitly solvable. Given our solution, A0 , for Equation (10.20), we get A0 = (1 − F0 )−1 F0
(10.23)
and, hence, (1 − F0 )−1 = A0 + 1.
(10.24)
This leads to solutions for A1 , A2 , etc., with A1 = F1 + 2A0 F1 + A0 (A0 F1 )
(10.25)
and, given A1 , we can now get A2 explicitly as A2 = F2 + A1 F1 + 2A0 F2 + A0 (A1 F1 ) + A0 (A0 F2 ).
(10.26)
It is now obvious that all the An ’s are real and continuously differentiable, for y x 0, since from Equation (7.11), A0 (x, y) = [B(x) + (y − x)C(x)]e−(y−x) ,
(10.27)
with B(x) and C(x) given by Equations (7.18) and (7.19) and B, C are O(e−2x ) as x → ∞. The kernels Fn (x) are of the form 2n+1 (n) σj x j . (10.28) Fn (x) = e−x j =0
The potential U (ν; x) is given by U (ν; x) = −2
d A(ν; x, x), dx
x 0.
(10.29)
48
N. N. KHURI
Figure 1. Plot of V1 (x).
Using the variable g ≡ (ν 2 − 1/4)−1 , we write U (ν; x) ≡ V (g; x).
(10.30)
The expansion of A in powers of g, given in Equation (10.19), now gives us, V (g; x) = V0 (x) + gV1 (x) + g 2 V2 (x) + · · · + g N VR(N) (g, x),
(10.31)
where d An (ν; x, x), dx d (ν; x, x) VR(N) (g, x) = −2 A(N) dx R
Vn (x) = −2
(10.32) (10.33)
and VR(N) is O(g). All the Vn ’s are real, continuous for x ∈ [0, ∞), and O(e−2x ) for large x. (N) VR (g, x) is complex for ν ∈ S(T0 ) but also continuous and O(e−2x ) for large x. It is now clear why one can refer to ‘g’ as a coupling constant specially for large values of Im ν, i.e., small g, g = O(|Im ν|−2 ). We have already calculated the first term in the expansion, V0 (x), and it is given explicitly in Equation (7.21). Later, we will need to have V1 (x) and we proceed to calculate it here. From Equation (10.25), we get ∞ dzA0 (x, z)F1 (z + x) + A1 (x, x) = F1 (2x) + 2 x ∞ ∞ dz1 dz2 A0 (x; z1 )A0 (x; z2 )F1 (z1 + z2 ), (10.34) + x
x
where A0 and F1 are given in Equations (10.27) and (10.25). It is now evident that A1 (x, x) is continuously differentiable and V1 (x) = −2
∂A1 (x, x), ∂x
(10.35)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
49
can be explicitly calculated in terms of B(x), C(x), and the real constants σj . The resulting V1 (x) is continuous, finite at x = 0, and V1 = O(x 3 e−2x ) for large x. We do not write down the full result, but exhibit a graph of V1 (x) in Figure 1. In closing we comment on the asymptotic expansion (10.31) for V (g; x). It is not, of course, convergent no matter how small |g| is. This follows from the fact that the expansions for M ± (ν; k) are also divergent. There is an essential singularity at g = 0. The constants χ5(n) , 5 = 1, . . . , 2n + 2, given in Equations (5.32) and (5.33) grow fast. However, Equation (10.31) can still give an extremely good estimate for V (g; x) as long as N is O(1). Indeed it is possible to get a uniform bound on VR(N) (g, x) which is |VR(N) (g, x)|
C(N) −x e , |Im ν|2
x > 0,
(10.36)
where C(N) grows fast with N. For the purposes of this paper we need at most N = 2 or 3. For T0 = 104 , (C(2)/T02 ) 1. This makes (g 2 VR(2) ) smaller than (O(1)/|Im ν|4 )e−x , and thus V0 +gV1 +g 2 V2 , give an excellent estimate of V (g, x) for all ν ∈ S(T0 ), and x not large. However, this estimate is not good for large x where both V and the error are small. Finally, we will need an important result on the phase of VR(N) near the critical line ν = it. As we have shown in Section 9, F (it; x), A(it; x; y), and U (it; x) are all real, for x ∈ [0, ∞) and y x. In addition, in the asymptotic expansion given in Equation (10.31) all the coefficients Vn (x) are real. But the remainder term, VR(N) (g, x), is in general complex for ν = ω + it, and ω = 0. However, for ω = 0, we again have reality UR(N) (it; x) ≡ VR(N) (−(t 2 + 14 )−1 ; x) = (VR(N) )∗ .
(10.37)
In Section 8 we proved that both A(ν; x, y) and U (ν; x) are analytic in ν for ν ∈ S(T0 ). Hence, so is UR(N) (ν; x). This leads us to the following lemma: LEMMA 10.1. For ν ∈ S(T0 ), and ν = δ + it, we have |Im(VR(N) (g; x))| = |Im UR(N) (δ + it; x)| < C(x) · |δ| + O(δ 2 ),
(10.38)
where C(x)
c1 −x e . t2
(10.39)
Proof. [Im UR(N) (ν; x)]
dUR(N) (ν; x) = Im (ν − ν) + O((ν − ν)2 ). dν ν=it
(10.40)
The derivative is finite and setting ν = δ + it, δ 1/2, we get Equation (10.38).
50
N. N. KHURI
The bound on C(x) follows from our previous estimates. We will not give the proof here. ✷
11. The Zeros of M (−) (ν; k) for Fixed k In this section we shall study the properties of the infinite set of zeros, νn (k), of M (−) (ν, k) for fixed k. We prove three lemmas for {νn (k)}. For convenience and without loss of generality, we set k = iτ , and τ 0. We write M(ν; τ ) ≡ M (−) (ν, iτ )
(11.1)
Mn (τ ) ≡ Mn(−) (iτ ).
(11.2)
and
It is clear from the equation defining M(ν, τ ), i.e. Equation (3.10) with k = iτ , that M(ν, τ ), fixed τ > 0, is an entire function of ν with order the same as ξ(ν + 1/2), i.e. order 1. Hence, M(ν, τ ) has an infinite set of zeros, νn (τ ), with |νn (τ )| → ∞ as n → ∞. Next it follows from Section 6 that, νn (τ ), will all be outside the truncated π T0 critical strip, S(T0 ), for τ > T0N e− 4 . As we decrease τ , νn (τ ), will start appearing πT in S(T0 ) for τ < O(T p e− 4 ). Our first lemma is the following: LEMMA 11.1. As τ → 0, lim νn (τ ) = νn ,
(11.3)
τ →0
where (νn + 1/2) is a zero of the zeta function, i.e. ξ(νn + 12 ) = 0.
(11.4)
Proof. From the asymptotic expansion of M(ν; τ ) given in Equation (5.47), we have M(ν; τ ) = 2ξ(ν + 12 ) + M0 (τ ) + gM1 (τ ) + g 2 M2 (τ ) + g 2 R˜ 2 (ν, τ ), (11.5) where R˜ 2 (ν, τ ) =
∞
dα 1
7 5=1
6(3) 5 (α)
1 1 − 5 5 (α + τ ) α
[α ν/2 + α −ν/2 ].
(11.6)
In getting Equations (11.4) and (11.5), we have used Equation (5.46) for the ξ function.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
51
By definition, we have M(νn (τ ), τ ) ≡ 0.
(11.7)
Hence, we get −2ξ(νn (τ ) + 12 ) = M0 (τ ) + gn (τ )M1 (τ ) + gn2 (τ )M2 (τ ) + + gn2 (τ )R˜ 2 (νn (τ ); τ ),
(11.8)
with gn (τ ) =
1 [νn2 (τ )
− 14 ]
.
(11.9)
Now all the terms on the right-hand side of Equation (11.8) are O(τ ) as τ → 0. Hence, we get lim ξ(νn (τ ) + 12 ) = ξ(νn (0) + 12 ) = 0,
τ →0
(11.10)
and, therefore, νn (0) = νn .
✷
The next lemma gives us as estimate of (νn (τ ) − νn (0)) as τ → 0. LEMMA 11.2. If νn is a first-order zero of ξ(ν + 1/2), then as τ → 0 (νn (τ ) − νn ) = O(τ ),
(11.11)
and if νn is of order p, then (νn (τ ) − νn ) = O(τ 1/p ).
(11.12)
Proof. Since ξ(ν + 1/2) is entire we can write for any νn ξ(νn (τ ) + 12 ) = ξ(νn +
1 ) 2
dξ + (νn (τ ) − νn ) + O[(νn (τ ) − νn )2 ]. dν ν=νn
(11.13)
But the first term on the right is zero, and (ξ )ν=νn = 0 for a first-order zero, and from Equation (11.7) we get −2(ξ )ν=νn (νn (τ ) − νn ) = τ [(2 + a1 ) + O(gn (τ ))] + O(τ 2 ). This gives our proof for a first-order zero.
(11.14) ✷
For a zero of multiplicity p, we have by definition (ξ (j ) )ν=νn = 0, for j = 1, 2, . . . , p − 1, and (ξ (p) )ν=νn = 0. Hence, we get −2[ξ (p) ]ν=νn (νn (τ ) − ν)p = τ [(2 + a1 ) + O(gn (τ ))] + O(t 2 ).
(11.15)
52
N. N. KHURI
This leads to (νn (τ ) − νn ) = O(τ 1/p ),
τ → 0.
(11.16)
So far we have shown that every νn (τ ) approaches a Riemann zero as τ → 0, but have not established the converse, i.e. that any νn is the limit of a νn (τ ) as τ → 0. To do this we first define a rectangular region R(T0 , T ) as follows: R(T0 , T ) = {ν | −3/2 Re ν 3/2; T0 Im ν T },
(11.17)
with T T0 . We now prove our third lemma. LEMMA 11.3. Let Nξ (T0 , T ) be the number of zeros of ξ(ν + 1/2) for ν ∈ R(T0 , T ), and NM (T0 , T ; τ ) be the number of zeros, νn (τ ), of M(ν, τ ), with νn (τ ) ∈ R(T0 , T ), then for sufficiently small τ , |NM (T0 , T ; τ ) − Nξ (T0 , T )| < 1.
(11.18)
There exists a small interval in τ , 0 τ τ0 (T ), such that NM (T0 , T ; τ ) = Nξ (T0 , T ).
(11.19)
Proof. We start with the standard expression: 1 M (ν; τ ) ξ (ν + 1/2) − , dν NM − Nξ = 2π i +R M(ν; τ ) ξ(ν + 1/2)
(11.20)
where +R is the boundary of the rectangle R. We also choose T and T0 , such that they both lie between the abscissa of successive zeros νn , i.e. Im νn1 < T < Im νN1 +1 , and Im νn0 < T0 < Im νn0 +1 . Thus, +B never has a zero of ξ on it. The prime in (11.20) denotes (d/dν). We follow the method used to prove theorem 9.3 in [18]. Using the symmetry of ξ in ν, we have 3 +iT iT 3 +iT0 2 2 1 1 1 ξ ξ ξ dν + dν + dν Nξ = π i 32 +iT0 ξ π i 32 +iT ξ π i iT0 ξ 1 (11.21) = { arg ξ(ν + 12 )}, π where E denotes the variation from iT0 to 3/2 + iT0 then from (3/2 + iT0 ) to 3/2 + iT , and thence to iT . But from Equation (11.5), we get {M(ν; τ )/ξ(ν + 12 )} = 1 +
(2 + a1 )τ + cgτ + O(τ 2 ). 2ξ(ν + 12 )
(11.22)
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
53
On the horizontal parts of +R , we can use Theorem 9.7 of [18] to obtain a lower bound on |ξ(ν + 1/2)|. Indeed, there is a constant, A, such that each interval (T , T + 1) contains a value of t for which |ζ(ν + 12 )| > t −A ,
− 32 ω 32 ,
(11.23)
where ν = ω + it. On the vertical parts of +B ,|ζ(ν + 1/2)| is obviously bounded from below. Using the standard asymptotic expression for +(1/4+ν/2) as t → ∞, we finally get τ . (11.24) |[arg M(ν; τ ) − arg ξ(ν + 12 )]| |ξ(ν + 12 )| We can choose τ = T −A−N (e
−π T 4
) with N 3 and obtain
E(arg M(ν, τ ) − arg ξ(ν + 12 ))
1 T N−1
.
(11.25)
Hence, as T → ∞, we get NM − Nξ = 0.
(11.26)
This completes our proof.
✷
We stress one important fact that is a consequence of the results of this section. Namely, we are now not limited to the study of the zero energy zeros, νn (0). One can consider the case for small enough τ , but τ > 0, and obviously if there is an interval 0 < τ < τn for which [Re νn (τ )] = 0, this will be sufficient for the validity of the Riemann hypothesis. In the next section we will see the importance of this remark. 12. The Potential V (g, x) and the Riemann Hypothesis Following the notation of the previous section, we define the Jost solution f (g; τ, x) as f (g; τ ; x) = f (−) (g; iτ, x),
(12.1)
with k = iτ , and ν ∈ S(T0 ). We also recall the result given in Faddeev’s review [14] ∞ −τ x −τ x u|V (g, u)| du, (12.2) |f − e | Ke x
hence, f = O(e−τ x ) as x → ∞, τ 0. We write the Schrödinger equation for f and f ∗ , −d2 f + V (g, x)f = −τ 2 f dx 2
(12.3)
54
N. N. KHURI
and −d2 f ∗ + V ∗ (g, x)f ∗ = −τ 2 f ∗ . dx 2
(12.4)
Multiplying the first equation by f ∗ and the second by f , integrating from x = 0 to x = ∞, and subtracting the two equations, we get ∞ 2i dx|f (g; τ ; x)|2 [Im V (g, x)] 0
= M(ν, τ )K∗ (ν, τ ) − M ∗ (ν, τ )K(ν, τ ), where
K(ν, τ ) =
df (g; τ ; x) dx
(12.5)
,
(12.6)
x=0
and g = (ν 2 − 1/4)−1 . The derivative exists for x → 0 in our present case, since V (g, 0) is finite. One can also check this from the expression for f in terms of A(ν; x, y), ∞ −τ x f (g; τ, x) = e + dyA(ν; x, y)e−τy . (12.7) x
From Section 8, we know that A(ν; 0, 0) is finite, and also (∂A/∂x), for x 0, y x exists and is integrable. Next we set ν = νn (τ ), and g = gn (τ ) in Equation (12.3), and we get ∞ dx|f (gn (τ ); τ ; x)|2 [Im V (gn (τ ); x)] = 0. (12.8) 0
This last integral is absolutely and uniformly convergent, since V = O(e−2x ) as x → ∞. Thus, we can take the limit τ → 0 and obtain ∞ dx|f (gn ; 0; x)|2 [Im V (gn ; x)] = 0. (12.9) 0
Now V (g, x) has an asymptotic expansion in g. V (g; x) = V0 (x) + gV1 (x) + g 2 V2 (x) + g 2 VR(2)(g, x).
(12.10)
Also from the expansion for A(ν; x, y), for the Jost solution we get f (g; τ ; x) = f0 (τ ; x) + gf1 (τ ; x) + g 2 f2 (τ ; x) + g 2 fR (g; τ ; x).
(12.11)
Both VR(2) and fR(2) are O(g) and have bounds in x. It is sufficient, for the validity of the Riemann hypothesis for a constant c > 0, c = O(1), to exist such that ∞ dx|f0 (x)|2 V1 (x) = c > 0, (12.12) 0
55
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
where f0 (x) = f0 (0, x). To prove this last statement, we first note that setting νn = νn (0), we have νn = ωn + itn ,
(12.13)
tn > T0
and Im gn =
[tn2
2ωn tn , − ωn2 + 1/4]2
ωn2 14 .
(12.14)
The vanishing of Im gn with tn > T0 > 0, implies ωn = 0 and, hence, sn = 1/2 + itn . Next, from Equation (12.10), we write Im V (gn ; x) = (Im gn )V1 (x) + (Im gn2 )V2 (x) + + (Im gn2 )(Re VR(2)(gn ; x)) + (Re gn2 )(Im VR(2) (gn ; x)). Also in the integrand in (12.0), we can write 1 2 2 |f (gn ; 0; x)| = |f (0; 0; x)| + O 2 . tn
(12.15)
(12.16)
In Equation (12.15) we note that ωn ωn and Im gn2 = O 5 , (Im gn ) = O 3 tn tn while (Re gN2 ) = O(1/tn4 ). On substituting Equations (12.15) and (12.16) in (12.9), and using the assumption (12.12), we see at first that consistency requires ωn to be small, i.e. ωn = O(1/tn3 ). Here we use the fact that |VR(2) | = O(g) for small g. However, we have more information on VR(2) and specifically its phase for small ωn . This was given in Lemma 10.1, where it was shown that Im VR(2) = O(ω) for small ω and that VR(2) is real for ω = 0. Given the bound on Im VR(2) from this lemma, we see that the leading contribution from (12.15) to Equation (12.9) must come from [(Im gn )V1 ] and cannot be cancelled by the other three terms. We have (Im gn )c + (Im gn2 )X1 + (Im gn2 )X2 + (Re gn2 )X3 = 0, with
∞
|X1 | 2 |X2 | 2
0 ∞ 0 ∞
|f0 (x)|2 |V2 (x)| dx, |f0 (x)|2 | Re VR(2) (gn ; x)|,
|f0 (x)|2 | Im VR(2) (gn , x)| dx 2ωn c1 ∞ |f0 (x)|2 e−x dx. tn2 0
|X3 | 2
(12.17)
0
(12.18)
56
N. N. KHURI
Hence we have constants Bj such that |Xj | < Bj ; j = 1, 2, and |X3 | < (ωn /tn2 )B3 . Note that in the last inequality we used Lemma 10.1. We take T0 large enough such that |Bj |/T02 c, j = 1, 2, 3; and with c = O(1). Next we rewrite Equation (12.17) as (Im gn )[c + 2(Re gn )X1 + 2(Re gn )X2 + (Re gn2 ) · tn Xˆ 3 ] = 0,
(12.19)
where now |Xˆ 3 | < 2B3 . The term in the square bracket cannot vanish and, hence, we obtain ∞ |f0 (x)|2 V1 (x) dx = 0, (12.20) [Im gn ]c = [Im gn ] 0
and thus, if c = 0, then for all tn > T0 , we get Im gn = 0 or
sn =
1 2
+ itn .
(12.21)
We have already calculated f0 (x) exactly in Section 7, and V1 (x) in Section 10, and we can compute the integral in (12.12) directly, the result is ∞ |f0 (x)|2 V1 (x) = 0. (12.22) 0
This can be checked numerically, and indeed can be rigorously proved. So we have no information on (Im gn ) from (12.17). However, the proof of (12.22) suggests to us how we can proceed further. To prove (12.22) we use the Schrödinger equation and the expansions (12.10) and (12.11). We obtain −d2 f0 (τ, x) + V0 (x)f0 (τ, x) = −τ 2 f0 (τ, x) 2 dx
(12.23)
−d2 f1 (τ, x) + V0 (x)f1 (τ, x) + V1 (x)f0 (τ, x) = −τ 2 f1 (τ, x). dx 2
(12.24)
and
Multiplying the first equation above by f1 and the second by f0 , integrating from zero to infinity, and subtracting, we have ∞ [f0 (τ, x)]2 V1 (x) dx = −[f1 (τ, 0)f0 (τ, 0) − f0 (τ, 0)f1 (τ, 0)], (12.25) 0
where the prime denotes (d/dx). By definition, we have f0 (τ, 0) = M0 (τ )
(12.26)
f1 (τ, 0) = M1 (τ ).
(12.27)
and
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
57
Both M0 (τ ) and M1 (τ ) are O(τ ) for small τ and vanish as τ → 0. We then get, after taking the limit, ∞ |f0 (0, x)|2 V1 (x) dx ≡ 0. (12.28) 0
The crucial factor here is the fact that M(ν, 0) = 2ξ(ν + 1/2), and for large |Im ν|, ν ∈ S(T0 ), M(ν, 0) = 2ξ(ν + 12 ) = O(e
−π | Im ν| 4
).
This forces all the coefficients, Mn (τ ), in the asymptotic expansion of M(ν, τ ) in powers of g to vanish as τ → 0. The culprit is the factor +(ν/2 + 1/4) in the Equation (3.4) which relates ξ(s) to ζ(s). We will also see below how this fact hinders us in treating the case τ = 0, but τ small. From the results of Section 11, it is evident that it is sufficient to prove that Re νn (τ ) = 0 in an interval 0 < τ < τ0 (n), where −π | Im νn | . τ0 (n) = O exp 4 From Equation (12.19), one can prove that ∞ [f0 (τ, x)]2 V1 (x) dx = Kτ + O(τ 2 ),
(12.29)
0
where K is a constant, K = O(1). The integral does not vanish if τ > 0. This suggests trying a double expansion in powers of g and τ . However, again this will not lead to any restriction on Im gn . The main problem is the relevant −π t domain in τ is small, i.e. τ = O(e 4 ), and terms of order g 2 are much larger than terms of order τ . To proceed further along the lines suggested by this paper, one has to do two things: (i) First, find an even function h(ν), analytic for ν ∈ S(T0 ), and having no zeros in S(T0 ), such that if we define, χ(ν) as χ(ν) ≡ ξ(ν + 12 )h(ν),
(12.30)
we have χ(ν) = O([t 2 ]−p ),
1 < p < 32 ,
(12.31)
where ν = ω + it, t > T0 . The point here is that χ(ν) is small but not smaller than O(g 2 ). This first step is achievable. For example, we can define χ as χ(ν) ≡
ξ(ν + 12 )[cos π4 ν] (ν 2 − 14 )2+δ
,
1 4
> δ > 0.
(12.32)
58
N. N. KHURI
This will give χ(ν) = O([t 2 ]−(1+δ) ).
(12.33)
The second requirement is much harder to achieve: (ii) One has to construct Jost functions, Mχ(±) (ν, k), preferably of the Martin type, such that lim Mχ(ν) = χ(ν + 12 )
(12.34)
k→0
and for small g, ν ∈ S(T0 ). In addition, Mχ± has to be of the Martin type and it must have an asymptotic expansion in powers of g = (ν 2 − 1/2)−1 . Appendix In this Appendix we first give a proof of the Laplace transform representation for the Marchenko kernel, F (ν; x), x > 0. Starting with the definition (8.2) 1 F (ν; x) = (S(ν; k) − 1)eikx dk, (A.1) 2π L where L is the line Im k = δ, with 1/4 δ < 1, we note that for ν ∈ S(T0 ) we −π T0 have S(ν; k) analytic in k for Im k > δ > O(e 4 ), except for the cut along the positive imaginary k-axis, 1 Im k < ∞. Second, in this region, we have a bound for large |k| |S(ν, k) − 1| <
C , |k|
|k| → ∞.
(A.2)
This bound holds along any radial direction that excludes the cut. We can deform the contour L from along the line Im k = 1/4, to a contour surrounding the cut, i.e. 1 [S(ν; k) − 1]eikx dk, x > 0, (A.3) F (ν; x) = 2π C where C starts at (−ε, +i∞) and descends to (−ε, +i), turns around the point k = i, and then extends from (+ε, +i) to (+ε, +i∞). The contribution from the large semicircle, |k| = K, vanishes as K → ∞. For |K|−1/2 < arg k < π − |K|−1/2 , the contribution is O(exp(−|K|1/2 x)), and vanishes for x > 0. Here we use the bound (A.2). For the regions 0 arg k |K|−1/2 ,
and
π − |K|−1/2 arg k π,
the contribution from the semicircle to (A.1) will be O(|K|−1/2 ) and also vanishes as |K| → ∞.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
From (A.3), we finally obtain 1 ∞ D(ν; α)e−αx da, F (ν; x) = π 1
x > 0,
59
(A.4)
where D is given in Equation (8.5), and from Equation (4.2) is just the discontinuity of S(ν; k) along the cut. Noting that D = O(e−πα ) as α → ∞, we see that (A.4) will also hold for x = 0. The next task for this Appendix is to give a direct proof of the fact that U (ν; x) and f (±) (ν; k; x) as defined in Equations (8.93) and (8.94) do indeed satisfy the Schrödinger equation, i.e. to prove Lemma 8.5 directly. We will also give an explicit expression for U (ν; x). Following [21], we define an operator, Q(ν; x), depending on two parameters, ν and x, with Re x 0, and ν ∈ S(T0 ). Q acts on functions u(β), 1 β < ∞, with u ∈ L2 (1, ∞). We define Q as ∞ Q(ν; x; α, β)u(β) dβ, (A.5) [Q(ν; x)u](α) ≡ 1
where Q(ν; x; α, β) =
1 D(ν; β)e−2βx , π [α + β]
Re x 0,
(A.6)
with D(ν; β) given by Equations (4.3) and (4.4). Q will have a finite Hilbert– Schmidt norm Q(ν; x) Ke−2x ,
(A.7)
where K depends on ν. We introduce a new integral equation, ∞ Q(ν; x; α, β)W (ν; x; β) dβ, W (ν; x; α) = 1 +
(A.8)
1
with W ∈ L2 (1, ∞). This integral equation is equivalent to the Marchenko equation (8.10). To see that, we write Z(ν; x; α) ≡ ˜ x, y) is A(ν; ˜ A(ν; x, y) ≡
1 D(ν; α)e−αx W (ν; x; α). π
∞
Z(ν; x; α)e−αy dα.
(A.9)
(A.10)
1
The Laplace transform exists since D = O(e−πα ) and W ∈ L2 . Substituting (A.9) and (A.10) in Equation (A.8) and using (A.4), we obtain for A˜ ∞ ˜ x, u)F (ν; u + y)a. ˜ A(ν; (A.11) A(ν; x, y) = F (ν; x + y) + x
60
N. N. KHURI
But this is just the Marchenko equation which we have shown in Section 8 does have a unique solution A. Hence, A˜ = A for all ν ∈ S(T0 ). Thus we conclude that Equation (A.8), which is of the Fredholm type, has a unique solution, since the homogeneous equation W = QW cannot have a solution for that will lead to the existence of a solution for A = AF which we have shown in Section 8 is not possible. Given the function W (ν; x; α), we can easily get expressions for U (ν; x) and f ± (ν; k; x). The potential is given by
d 1 ∞ −2αx D(ν, α)e W (ν; x; α) da , Re x 0. (A.12) U (ν; x) = −2 dx π 1 Similarly, from (A.9) and (A.10), we get ∞ D(ν; α)e−2αx W (ν; x; α) (±) ∓ikx ∓ikx 1 dα. (A.13) +e f (ν; k, x) = e π (α ± ik) 1 To check that we recover the same Jost functions we started with, we write ∞ 1 D(ν; α)W (ν; 0; α) (±) (±) dα. (A.14) f (ν; k; 0) ≡ M (ν; k) ≡ 1 + π (α ± ik) 1 However, the integral equation for W for x = 0 is trivially soluble. From Equation (A.8), we have D(ν; β)W (ν; 0; β) 1 dβ. (A.15) W (ν; 0; α) = 1 + π (α + β) Setting W (ν; 0; α) ≡ M (−) (ν; iα) and using Equations (4.3) and (4.4) for D(ν; α), we have ∞ ψ(β)β 1/4 [β ν/2 + β −ν/2 ] . dβ M (−) (ν; iα) = 1 + (ν 2 − 14 ) β +α 1
(A.16)
(A.17)
This is our original expression for M (−) . The operator Q is a Fredholm-type operator, and for Re x 0, and ν ∈ S(T0 ), we proved that there are no nontrivial solutions of the homogeneous equation u = Qu. Hence, the determinant, Det(1 − Q), cannot vanish for any Re x 0. This determinant can be calculated explicitly, as was done in [21], and we obtain n ∞ ∞ ∞ D(ν, αj )e−2αj x 1 dα1 . . . dαn × Det(1 − Q(ν; x)) = 1 + (n!) 2π α j 1 1 n=1 j =1 ×
n (αi − αj )2 . (αi + αj )2 i<j
(A.18)
61
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
This series is absolutely convergent for all Re x 0, ν ∈ S(T0 ), and | Im ν| < ∞. Also as shown in [21], the potential U (ν; x) is now given by Dyson’s [23] formula, i.e. d2 {log[Det(1 − Q(ν; x))]}. (A.19) dx 2 Finally, we give here a direct check on the validity of the Schrödinger equation for f ± and U (ν; x). From Equation (A.7) giving a bound on the norm of Q, we see that for some x0 , Q < 1 for all (Re x) > x0 , x0 log K/2. Thus the iterative series for W (ν; x; α) is absolutely convergent for all Re x > x0 , U (ν; x) = −2
W =1+
∞
Qn .
(A.20)
n=1
Using Equations (A.6), (A.12), and (A.13), we obtain ∞ ∞ n+1 ∞ 1 dα0 . . . dαn × U (ν; x) = 4 π 1 1 n=0 n n −2αj x j =0 D(ν; αj )e αj , Re x > x0 . × n−1 j =0 (αj + αj +1 ) j =0
(A.21)
A similar series holds for f ± : f
(±)
∓ikx
(ν; k; x) = e
∓ikx
+e
n−1
∞ n+1 1 n=0
π
∞
∞
dα0 . . . 1
dαn ×
1
−2αj x j =0 D(ν; αj )e . × n−1 [ j =0 (αj + αj +1 )](α0 ± ik)
(A.22)
Using these series, we can check directly for Re x > x0 , that U and f ± give a potential and its unique Jost solutions. We define h(±) : h(±) = e±ikx f (±) .
(A.23)
The Schrödinger equation for h(±) is now dh± d2 ± (ν; k; x) = U (ν; x)h± (ν; k; x). h (ν; k; x) ∓ 2ik (A.24) dx 2 dx Substituting, expressions (A.21) and (A.22) in the above, we see, after some algebra, that for x > x0 , (A.24) is satisfied if the following algebraic identity holds n n 2 r r−1
αj αj − αr αj . (αn + αn+1 ) = (A.25) n=0
j =0
j =0
j =0
62
N. N. KHURI
But this is equivalent to n n r−1 r 2 r αn αj + (αn+1 ) αj = αj . n=0
j =0
n=0
j =0
(A.26)
j =0
This last equation is an identity and can be proved by induction. The Schrödinger equation is thus valid for all x > x0 . But again using analytic continuation, now in x we easily see that it must hold for all x 0. All the terms in (A.24) are analytic in the half plane Re x 0. Acknowledgements The author wishes to thank James Liu and H. C. Ren for untiring help in checking much of the algebraic manipulations in this paper including the use of Mathematica to produce Tables I and II and carry out other numerical work. This work was supported in part by the U.S. Department of Energy under grant number DOE91ER40651 TaskB. References 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Dyson, F. J.: J. Math. Phys. 3 (1962), 140. Montgomery, H. L.: In: Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, RI, 1973, pp. 181–193. Berry, M. V.: ‘Riemann’s Zeta Function: a Model of Quantum Chaos,’ Lecture Notes in Phys. 262, Springer, New York, 1986. Chadan, K.: private communication, see also K. Chadan and M. Musette, C.R. Acad. Sci. Paris (2) 316 (1993), 1–6. In this paper an example is given with some important properties of the zeta function demonstrated. Meetz, K.: J. Math. Phys. 3 (1962), 690. Gelfand, I. M. and Levitan, B. M.: Izvest. Akad. Nauk. SSSR Ser. Matem. 15 (1951), 309. Marchenko, V. A.: Dokl. Akad. Nauk SSSR 104 (1955), 695. [Math. Rev. 17 (1956), 740]. Martin, A.: Nuovo Cimento 19 (1961), 1257. Jost, R.: Helv. Physica Acta 20 (1947), 256. Levinson, N.: Kgl. Danske Videnskab. Selskab, Math.-fys. Medd. 25(9) (1949). Bargmann, V.: Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 961. Jost, R. and Pais, A.: Phys. Rev. 82 (1951), 840. Chadan, K. and Sabatier, R. C.: Inverse Problems in Quantum Scattering Theory, 2nd edn, Springer, New York, 1989. Faddeev, L. D.: J. Math. Phys. 4 (1963), 72. Blankenbecler, R., Goldberger, M. L., Khuri, N. N. and Treiman, S. B.: Ann. of Phys. 10 (1960), 62. Regge, T.: Nuovo Cimento 14(5) (1959), 951. Martin, A.: Nuovo Cimento 14 (1959), 403. Titchmarsh, E. C.: The Theory of the Riemann Zeta-function, 2nd edn, revised by D. R. HeathBrown, Oxford Univ. Press, Oxford, 1986. Gross, D. J. and Kayser, B. J.: Phys. Rev. 152 (1966), 1441. Cornille, H.: J. Math. Phys. 8 (1967), 2268.
INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS
21.
22. 23.
63
Khuri, N. N.: Inverse scattering revisited: explicit solution of the Marchenko–Martin method, In: S. Ciulli, F. Scheck and W. Thirring (eds), Rigorous Methods in Particle Physics, SpringerVerlag, Berlin, 1990, pp. 77–97. Bargmann, V.: Rev. Modern Phys. 21 (1949), 488. Dyson, F. J.: In: E. Lieb, B. Simon and A. S. Wightman (eds), Studies in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 1976, pp. 151–167.
Mathematical Physics, Analysis and Geometry 5: 65–76, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Algebras of Operators on Holomorphic Functions and Applications M. BEN CHROUDA and H. OUERDIANE Department of Mathematics, Faculty of Sciences of Tunis, Université de Tunis El Manar, 1060 Tunis, Tunisia. e-mail:
[email protected] (Received: 6 March 2001; in final form: 17 August 2001) Abstract. We develop the theory of operators defined on a space of holomorphic functions. First, we characterize such operators by a suitable space of holomorphic functions. Next, we show that every operator is a limit of a sequence of convolution and multiplication operators. Finally, we define the exponential of an operator which permits us to solve some quantum stochastic differential equations. Mathematics Subject Classifications (2000): primary 60H40; secondary 46A32, 46F25, 46G20. Key words: symbols of operators, infinite dimensional holomorphy, convolution product of operators, quantum stochastic differential equations.
1. Introduction Let N be a complex nuclear Fréchet space. Assume that its topology is defined by an increasing family of Hilbertian norms {|.|p , p ∈ N}. Then N is represented as N = p∈N Np , where for p ∈ N the space Np is the completion of N with respect to the norm |.|p . For simplicity, we denote by H the complex Hilbert space N0 and byN−p the dual space of Np , then the dual space N of N is represented as N = p∈N N−p , and it is equipped with the inductive limit topology. We denote by ., . the C-bilinear form on N × N connected to the inner product .|. of H , i.e. z, ξ = ¯z|ξ ,
z ∈ H, ξ ∈ N.
For any n ∈ N we denote by S n N the nth symmetric tensor product of N equipped with the π -topology and by S n Np the nth symmetric Hilbertian tensor product of Np . We will preserve the notation |.|p and |.|−p for the norms on S n Np and S n N−p , respectively. Let n, m ∈ N and 0 k m ∧ n. We denote by ., . k the bilinear map from S n−k N defined by S m N × S n N into S m−k N ⊗ ⊗m ⊗n x , y k := x, y k x ⊗(m−k) ⊗ y ⊗(n−k) , x ∈ N , y ∈ N. The bilinear map ., . k is continuous, then using the density of the vector space generated by {x ⊗m , x ∈ N } in S m N and the density of the vector space generated
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by {y ⊗n , x ∈ N} in S n N, we can extend ., . k to S m N × S n N. Let φm ∈ S m N and ϕn ∈ S n N; then φm , ϕn k is called the right contraction of φm and ϕn of degree k. Let θ be a Young function on R+ , i.e. θ is continuous, convex, increasing function and satisfies lim+∞ θ(x)/x = +∞. We define the conjugate function θ ∗ of θ by θ ∗ (x) := sup(tx − θ(t)).
∀x 0,
(1)
t 0
For a such Young function θ, we denote by Gθ (N) the space of holomorphic functions on N with exponential growth of order θ and of arbitrary type, and by Fθ (N ) the space of holomorphic functions on N with exponential growth of order θ and of minimal type. For every p ∈ Z and m > 0, we denote by exp(Np , θ, m) the space of entire functions f on the complex Hilbert space Np such that nθ,p,m (f ) := sup |f (z)|e−θ(m|z|p ) < +∞. z∈Np
Then the spaces Fθ (N ) and Gθ (N) are represented as exp(N−p , θ, m), Fθ (N ) = p∈N m>0
Gθ (N) =
exp(Np , θ, m),
p∈N m>0
and equipped with the projective limit topology and the inductive limit topology, respectively. Let p ∈ N and m > 0, we define the Hilbert spaces Fθ,m (Np )
n −2 −n 2 , f ∈ S N ; f := θ m |f | < +∞ , = f = (fn )∞ n p θ,p,m n p n=0 n n0
Gθ,m (N−p )
∞ n 2 n 2 (n!θn ) m |φn |−p < +∞ , = φ = (φn )n=0 , φn ∈ S N−p ; φθ,−p,m := n0
where θn = infr>0 eθ(r) /r n , n ∈ N. The sequences θn and θn∗ are connected by the following relation n n ∗ 2n LEMMA 1. For every n ∈ N\{0} we have xe n θn θn e . Proof. We can assume that θ(x) = 0 µ(t) dt where µ is a continuous, increasing function which satisfies lim+∞ µ(x) = +∞ (see [4]). Then θ ∗ (x) = x 0 ω(t) dt, where ω is the inverse function of µ, i.e. µ ◦ ω = ω ◦ µ = id. A direct calculation shows that
θn =
eθ(tn ) tnn
∗
and
θn∗ =
eθ (xn ) , xnn
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ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS
where tn and xn are the solutions of tµ(t) = n and tω(t) = n, respectively, and satisfy tn xn = n. Hence,
n n θ(tn ) θ ∗ (xn ) n ∗ e e n θn θn = tn xn e2n . On the other hand, for every t, x > 0 we have ∗
eθ (t ) eθ (x) et x , (tx)n t n xn
∀n 1.
Then, using the fact that inft >0 Put Fθ (N) =
etx (t x)n
=
en , nn
we obtain en /nn θn θn∗ .
✷
Fθ,m (Np ),
p∈N m>0
Gθ (N ) =
Gθ,m (N−p ).
p∈N m>0
Then the space Fθ (N) equipped with the projective limit topology is a nuclear Fréchet space [4], and Gθ (N ) carries the dual topology of Fθ (N) with respect to the C-bilinear form (., .): n!φn , fn , φ = (φn ) ∈ Gθ (N ), f = (fn ) ∈ Fθ (N). (φ, f ) = n0
For simplicity, we put Fθ (N ) = Fθ ,
Gθ ∗ (N) = Gθ ∗ ,
Fθ (N) = Fθ ,
Gθ (N ) = Gθ
and we denote by Fθ the strong dual of the space Fθ . It was proved in [4] that the Taylor series map S.T yields a topological isomorphism between Fθ (respectively Gθ ∗ ) and Fθ (respectively Gθ ). The nuclear Fréchet space Fθ and its dual Fθ are called the test function space and the distribution space, respectively. The C-bilinear form on Fθ ×Fθ is denoted by ., . . We denote by L(Fθ , Fθ ) the space of continuous linear operators from Fθ into itself, equipped with the topology of bounded convergence. In this paper, we do not restrict ourselves to the theory of Gaussian (white noise) and non-Gaussian analysis studied, for example, in [1, 6, 8, 9] and [10] but we develop a general infinite-dimensional analysis. First, we give a decomposition of convolution operators from Fθ into itself into a sum of holomorphic derivation operators. Second, we establish a topological isomorphism between the Gθ ∗ of holomorphic functions. space L(Fθ , Fθ ) of operators and the space Fθ ⊗
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Next, we develop a new convolution calculus over L(Fθ , Fθ ) and we give sense T n to the expression e := n0 T /n! for some class of operators T . Finally, as an application of this operator theory we solve some linear quantum stochastic differential equations. 2. Some Properties on the Distribution Space Let θ be a Young function. For every ξ ∈ N, the exponential function eξ : z → ez,ξ , z ∈ N belongs to Fθ . Then we define the Laplace transform of a distribution φ ∈ Fθ by φ (ξ ) := φ, eξ ,
ξ ∈ N.
PROPOSITION 1 ([4]). The Laplace transform realizes a topological isomorphism between Fθ and Gθ ∗ . By composition of the Taylor series map with the Laplace transform, we deduce that φ ∈ Fθ if and only if there exists a unique formal series φ = (φn )n0 ∈ Gθ such that ξ ⊗n , φn . φ (ξ ) = n0
Then, the action of the distribution φ on a test function ϕ(z) = given by n!φn , ϕn . φ, ϕ =
n0 z
⊗n
, ϕn is
n0
In particular, for every z ∈ N , the Dirac mass δz defined by δz , ϕ = ϕ(z),
(2)
belongs to Fθ . Moreover, δz coincides with the distribution associated to the formal series ⊗n z . δz := n! n0 Now, we recall some properties of translation operators and convolution product of distributions studied in [2]. Let z ∈ N , the translation operator τ−z is defined by τ−z ϕ(λ) = ϕ(z + λ),
λ ∈ N .
For every z ∈ N , the linear operator τ−z is continuous from Fθ into itself. We define the convolution product of a distribution φ ∈ Fθ with a test function ϕ ∈ Fθ as follows φ ∗ ϕ(z) := φ, τ−z ϕ ,
z ∈ N .
ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS
69
A direct calculation shows that φ ∗ ϕ ∈ Fθ . Let φ1 , φ2 ∈ Fθ , we define the convolution product of φ1 and φ2 , denoted by φ1 ∗ φ2 , by φ1 ∗ φ2 , ϕ := [φ1 ∗ (φ2 ∗ ϕ)](0),
ϕ ∈ Fθ .
Moreover, ∀φ1 , φ2 ∈ Fθ we have φ 1 ∗ φ2 = φ1 φ2 . 3. Convolution Operators In infinite-dimensional complex analysis, a convolution operator on the test space Fθ is a continuous linear operator from Fθ into itself which commutes with translation operators. It was proved in [2, 5] that T is a convolution operator on Fθ if and only if there exists φT ∈ Fθ such that T ϕ = φT ∗ ϕ,
∀ϕ ∈ Fθ .
(3)
Moreover, if the distribution φT is given by z⊗n , ϕn ∈ Fθ , φT = (φm )m0 ∈ Gθ and ϕ(z) = n0
then φT ∗ ϕ(z) =
(n + m)! m0 n0
n!
z⊗n , φm , ϕm+n m .
(4)
In particular, we have φ (ξ )eξ (z). T (eξ )(z) = φT ∗ eξ (z) =
Let θ be a Young function, y ∈ N and ϕ(z) = n0 z⊗n , ϕn ∈ Fθ . We define the holomorphic derivative of ϕ at a point z ∈ N in a direction y by (n + 1)z⊗n , y, ϕn+1 1 . Dy ϕ(z) := n0
LEMMA 2. The operator Dy is continuous from Fθ into itself. Moreover, for every ϕ ∈ Fθ , p ∈ N and m > 0, we have √ Dy ϕθ,p,m mθ1 |y|−py ϕθ,py ∨p, 16m , where py = min{p ∈ N; y ∈ N−p } and py ∨ p = max(py , p). Proof. By definition of the norm .θ,p,m defined on the space Fθ of formal series, we have
1/2 2 −2 −n 2 (n + 1) θn m |y, ϕn+1 1 |p Dy ϕθ,p,m = n0
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M. BEN CHROUDA AND H. OUERDIANE
1/2 2 −2 −n 2 |y|−py (n + 1) θn m |ϕn+1 |p∨py n0
√
m|y|−py
−2 θn+1
n0
√
m|y|−py sup n1
m −n−1 (n + 1)θn+1 2 1/2 2 |ϕn+1 |p∨py 16 22n+2 θn ϕθ,p∨py , 16m .
θn+1 2n+1 θn
Finally, the desired inequality follows immediately using the fact that 2−l−k θl θk ✷ θl+k 2l+k θl θk , ∀l, k ∈ N\{0}. In view of Lemma 2, for each m ∈ N the m-linear operator D defined by D: N × · · · × N → L(Fθ , Fθ ) (y1 , . . . , ym ) → Dy1 . . . Dym is symmetric and continuous, hence, it can be continuously extended to S m N , i.e. → Dφm ∈ L(Fθ , Fθ ). The action of the operator Dφm on a test D: φm ∈ S m N function ϕ(z) = n0 z⊗n , ϕn given by Dφm (ϕ)(z) =
(n + m)! n!
n0
z⊗n , φm , ϕn+m m .
(5)
Then, in view of (3), (4) and (5), we give an expansion of convolution operators in terms of holomorphic derivation operators. convolution operator if and PROPOSITION 2. Let T ∈ L(Fθ , Fθ ), then T is a only if there exists φ = (φm )m0 ∈ Gθ such that T = m0 Dφm . Remark. Let Tφ = equality (3) shows that
m0
Dφm be a convolution operator and n ∈ N. Then
Tφn := Tφ ◦ · · · ◦ Tφ = Tφ∗n .
(6)
n
In particular, φ (ξ ))n eξ (z), Tφn (eξ )(z) = Tφ∗n (eξ )(z) = (
z ∈ N , ξ ∈ N.
4. Symbols of Operators In this section we define the symbol map on the space L(Fθ , Fθ ). Then we give an expansion of such operators in terms of multiplication and derivation operators.
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DEFINITION 1. Let T ∈ L(Fθ , Fθ ), the symbol σ (T ) of the operator T is a C-valued function defined by σ (T )(z, ξ ) := e−z,ξ T (eξ )(z),
z ∈ N , ξ ∈ N.
Similar definitions of symbols have been introduced in various contexts ([7, 10– 12]). In the general theory ([13]), if we take two nuclear Fréchet spaces X and D, then the canonical correspondence T ↔ K T given by T u, v = K T , u ⊗ v ,
u ∈ X, v ∈ D ,
D. In yields a topological isomorphism between the spaces L(X, D) and X ⊗ particular, if we take X = D = Fθ which is a nuclear Fréchet space, then we get L(Fθ , Fθ ) ∼ = Fθ ⊗ Fθ .
(7)
So, the symbol σ (T ) of an operator T can be regarded as the Laplace transform of the kernel K T σ (T )(z, ξ ) = K T (eξ ⊗ δz ),
z ∈ N , ξ ∈ N.
(8)
Moreover, with the help of equalities (2), (7), (8) and Proposition 1 we obtain the following theorem THEOREM 1. The symbol map yields a topological isomorphism between Gθ ∗ . More precisely, we have the following isomorphisms: L(Fθ , Fθ ) and Fθ ⊗ σ Gθ ∗ S.T Gθ , → Fθ ⊗ L(Fθ , Fθ ) → Fθ ⊗ Kl,m , z⊗l ⊗ ξ ⊗m → K = (Kl,m )l,m0 . T → σ (T )(z, ξ ) = l,m
EXAMPLES. (1) Let φm ∈ S m N . Then σ (Dφm )(z, ξ ) = e−z,ξ Dφm (eξ )(z) = e−z,ξ φm , ξ ⊗m ez,ξ = φm , ξ ⊗m .
In particular, the symbol of a convolution operator Tφ = m0 Dφm is given by Dφm (eξ )(z) = φm , ξ ⊗m = φ (ξ ). σ (Tφ )(z, ξ ) = e−z,ξ m0
m0
Hence, the operator Tφ can be expressed in an obvious way by Dφm := φm , D ⊗m = σ (Tφ )(z, D), z ∈ N . Tφ = m0
m0
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(2) Let f ∈ Fθ . We denote by Mf the multiplication operator by the test function f . Its symbol is given by σ (Mf )(z, ξ ) = e−z,ξ (f eξ )(z) = e−z,ξ f (z)eξ (z) = f (z). By the same argument, the multiplication operator is also expressed by Mf = σ (Mf )(z, D). We note that the symbol of a convolution (respectively, multiplication) operator σ (T )(z, ξ ) depends only on ξ (respectively, z). Gθ and assume that K = f ⊗ φ = (fl ⊗ φm )l,m0 . Then the Let K ∈ Fθ ⊗ operator T associated to K (see Theorem 1) satisfies (9) T = Mf Tφ , where f (z) = l0 z⊗l , fl and Tφ is the convolution operator associated to the distribution φ given by φ. Moreover, we have T = Mf Tφ = σ (Mf )(z, D)σ (Tφ )(z, D) = σ (T )(z, D). Gθ , we obtain the following result: Thus, using the density of Fθ ⊗ Gθ in Fθ ⊗ PROPOSITION 3. The vector space generated by operators of type (9) is dense in L(Fθ , Fθ ).
5. Convolution Product of Operators Let T1 , T2 be two operators in L(Fθ , Fθ ); the convolution product of T1 and T2 , denoted by T1 ∗ T2 , is uniquely determined by σ (T1 ∗ T2 ) = σ (T1 )σ (T2 ). If the operators T1 and T2 are of type (9), i.e. T1 = Mf1 Tφ1 and T2 = Mf2 Tφ2 , then T1 ∗ T2 = Mf1 f2 Tφ1 ∗φ2 . In particular, if T = Mf Tφ , then for every n ∈ N we have T ∗n = Mf n Tφ∗n .
(10)
Remark. Let Tφ (resp. Mf ) be a convolution (resp. multiplication) operator. Then for every n ∈ N Tφ∗n = Tφ∗n = Tφn
and
Mf∗n = Mf n = Mfn .
ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS
73
LEMMA 3. Let γ1 , γ2 two Young functions and (Fn ) a sequence belonging to Gγ2 . Then (Fn ) converges in Fγ1 ⊗ Gγ2 if and only if Fγ1 ⊗ Gγ2 . (1) (Fn ) is bounded in Fγ1 ⊗ (2) (Fn ) converges simply. Proof. The proof is based on the nuclearity of the spaces Fγ1 and Gγ2 . A similar proof is established with more details in [3], Theorem 2. ✷ PROPOSITION 4. Let T ∈ L(Fθ , Fθ ); then the operator T ∗n e∗T := n! n0 belongs to L(F(eθ ∗ )∗ , Feθ ). Proof. Let T ∈ L(Fθ , Fθ ) and put Sn =
n T ∗k k=0
k!
.
Geθ to eσ (T ) , from Then, using Lemma 3, we show that σ (Sn) converges in Feθ ⊗ which the assertion follows. ✷ COROLLARY 1. Let T ∈ L(Fθ , Fθ ), and assume that σ (T )(z, ξ ) is a polynomial in z and ξ of degree k and k/(k − 1), respectively, k 2. Then e∗T belongs to L(Fk , Fk ), where Fk is the test space associated to the Young function θ(x) = x k . Let T ∈ L(Fθ , Fθ ) and consider the linear differential equation dE = T E, dt
E(0) = I.
Then the solution is given informally by E(t) = et T ,
t ∈ R.
In the particular case, where T is a convolution or a multiplication operator; the solution E(t) = et T is well defined since eT = e∗T . If T is not a convolution or a multiplication operator then the following theorem gives a sufficient condition on T to insure the existence of its exponential eT . Gθ satisfying Kl,m , Kl ,m k = 0 for every THEOREM 2. Let K = (Kl,m ) ∈ Fθ ⊗ m, l 1, m , l 0 and 1 k m ∧ l and denote by T the operator associated to K (see Theorem 1). Then, T n = T ∗n ,
∀n ∈ N.
Moreover, eT = e∗T ∈ L(F(eθ ∗ )∗ , Feθ ).
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Proof. Using Proposition 3, it will be sufficient to assume that Kl,m = (fl ⊗φm ), i.e. T = Mf Tφ =
Mfl Dφm ,
l,m0
where fl (z) = z⊗l , fl . Assume that fl = η⊗l ,
η∈N
and
φm = y ⊗m ,
y ∈ N .
Then it is easy to see that Dφm Mfl = Mfl Dφm +
m∧l
k!Clk Cmk y, η k Mfl−k Dφm−k ,
k=0
an equality on Fθ . The assumption Kl,m , Kl ,m k = 0 implies that y, η = 0. Then Dφm Mfl = Mfl Dφm .
(11)
Thus, using the density of the vector space generated by {η⊗l , η ∈ N} in the space S l N and the density of the vector space generated by {y ⊗m , y ∈ N } in S m N , we can extend equality (11) to every fl ∈ S l N and φm ∈ S m N such that φm , fl k = 0, ∀1 k l ∧ m. Hence, we obtain Mfl Dφm = Dφm Mfl = Tφ Mf . Mf Tφ = l,m0
l,m0
Using equalities (6) and (10), for every n ∈ N we have T n = (Mf Tφ )n = (Mf )n (Tφ )n = Mf n Tφ∗n = T ∗n . This completes the proof.
✷
Remark. The condition of Theorem 2 is not satisfied by convolution or multiGθ and let T be the operator plication operators. In fact, let K = (Kl,m ) ∈ Fθ ⊗ associated to K. If T is a convolution operator then K = (Kl,m )l,m0 = (K0,m )m0 ∈ Gθ , see Proposition 2. Hence, the right contraction Kl,m , Kl ,m k = 0 with 1 k m ∧ l can not be defined since l = 0. If T is a multiplication operator then K = (Kl,m )l,m0 = (Kl,0 )l0 ∈ Fθ . Thus Kl,m , Kl ,m k = 0 with 1 k m ∧ l can not be defined since m = 0. Gθ which satisfies the Now we give an example of family of kernels K ∈ Fθ ⊗ condition of Theorem 2. EXAMPLE. Let N = S(R) /→ H = L2 (R, dt) /→ N = S (R) and K = S m (S (R)). Assume that there exists Gθ , i.e. Kl,m ∈ S l (S(R))⊗ (Kl,m )l,m0 ∈ Fθ ⊗
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ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS
t ∈ R such that for every l, m ∈ N the support of Kl,m is included in ]−∞, t]l ×]t, +∞[m . Then K satisfies the condition of Theorem 2. Remark. In Theorem 2 we assume that N is a C-vector space of dimension S m N = C. Thus the n 2. However, if N = C then for every m, l 0; S l N ⊗ assumption Kl,m , Kl ,m k = 0 for every m, l 1, m , l 0 and 1 k m∧l is equivalent to Kl,m = 0, ∀l, m ∈ N and the set of operators satisfying the condition of Theorem 2 is reduced to the null operator.
6. Applications to Quantum Stochastic Differential Equations A one-parameter quantum stochastic process with values in L(Fθ , Fθ ) is a family of operators {Et , t ∈ [0, T ]} ⊂ L(Fθ , Fθ ) such that the map t → Et is continuous. For a such quantum process Et we set t E tk , En = n k=0 n n−1
n ∈ N\{0}, t ∈ [0, T ].
Then we prove using Lemma 3 that the sequence (En ) converge in L(Fθ , Fθ ). We denote its limit by t Es ds := lim En in L(Fθ , Fθ ). n→+∞
0
Moreover, we have
t t Es ds = σ (Es ) ds, σ 0
∀t ∈ [0, T ].
0
THEOREM 3. Let t ∈ [0, T ] → f (t) ∈ Fθ and t ∈ [0, T ] → φ(t) ∈ Fθ be two continuous processes and put Lt = Mf (t )Tφ (t). Then the linear differential equation dEt = Mf (t )Et Tφ(t ), E0 = I dt has a unique solution Et ∈ L(F(eθ ∗)∗ , Feθ ) given by
(12)
t
Et = e∗( 0 Ls ds). Proof. Applying the symbol map to Equation (12) we get dσ (Et ) = σ (Lt )σ (Et ), dt t
σ (I ) = 1.
Then σ (Et ) = e 0 σ (Ls ) ds which is equivalent to Et = e∗( conclude by Proposition 4 that Et ∈ L(F(eθ ∗ )∗ , Feθ ).
t 0
Ls ds)
. Finally, we ✷
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THEOREM 4. Let Lt be a quantum stochastic process with values in L(Fθ , Fθ ) such that t
σ Ls ds (z, ξ ) = Kl,m (t), z⊗l ⊗ η⊗m , 0
l,m0
and assume that for every t ∈ [0, T ], m , l 0 and m, l 1 we have Kl,m (t), Kl ,m (t) k = 0,
∀1 k m ∧ l .
Then the following differential equation dE = Lt E, dt
E(0) = I,
(13) t
has a unique solution in L(F(eθ ∗ )∗ , Feθ ) given by E(t) = e
0
Ls ds
.
Acknowledgement We are grateful to the Professor Luis Boutet de Monvel for many stimulating remarks and useful suggestions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Albeverio, S., Daletsky, Yu. L., Kondratiev, Yu. G. and Streit, L.: Non-Gaussian infinite dimensional analysis, J. Funct. Anal. 138 (1996), 311–350. Ben Chrouda, M., Eloued, M. and Ouerdiane, H.: Convolution calculus and applications to stochastic differential equations, To appear in Soochow J. Math. (2001). Ben Chrouda, M., Eloued, M. and Ouerdiane, H.: Quantum stochastic processes and applications, Preprint, 2001. Gannoun, R., Hachaichi, R., Ouerdiane, H. and Rezgui, A.: Un théorème de dualité entre espace de fonctions holomorphes à croissance exponentielle, J. Funct. Anal. 171(1) (2000), 1–14. Gannoun, R., Hachaichi, R., Krée, P. and Ouerdiane, H.: Division de fonction holomorphe a croissance θ-exponentielle, Preprint, BiBos No. E 00-01-04, 2000. Hida, T., Kuo, H.-H., Potthof, J. and Streit, L.: White Noise, An Infinite-Dimentional Calculus, Kluwer Acad. Publ., Dordrecht, 1993. Krée, P. and Raczka, R.: Kernels and symbols of operators in quantum field theory, Ann. Inst. H. Poincaré Sect. A 18(1) (1978), 41–73. Kondratiev, Yu. G., Streit, L., Westerkamp, W. and Yan, J.-A.: Generalized functions in infinite dimensional analysis, Hiroshima Math. J. 28 (1998), 213–260. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Boca Raton, 1996. Obata, N.: White Noise Calculus and Fock Space, Lecture Notes in Math. 1577, Springer, New York, 1994. Obata, N.: Wick product of white noise operators and quantum stochastic differential equations, J. Math. Soc. Japan 51(3) (1999), 613–641. Ouerdiane, H.: Noyaux et symboles d’opérateurs sur des fonctionnelles analytiques gaussiennes, Japan. J. Math. 21(1) (1995), 223–234. Trèves, F.: Topological Vector Space, Distributions and Kernels, Academic Press, New York, 1967.
Mathematical Physics, Analysis and Geometry 5: 77–99, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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On the Gaussian Perceptron at High Temperature MICHEL TALAGRAND Equipe d’Analyse-Tour 46, ESA au CNRS No. 7064, Université Paris VI, 4 Pl. Jussieu, 75230 Paris Cedex 05, France, and Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174, U.S.A. (Received: 9 April 2001; in final form: 25 September 2001) Abstract. For σ = (σi )iN ∈ N = {−1, 1}N , define H (σ ) = −
1 u √ σi gik , N iN kM
where (gik )iN,kM are i.i.d. N(0, 1), and where u is bounded and Borel measurable. When M is a small proportion α of N, we study the system with random Hamiltonian H , at temperature 1. When α is small enough, we prove that the overlap of two configurations taken independently at random for Gibbs’ measure is nearly constant, with a correct estimate of the size of its fluctuations. Mathematics Subject Classifications (2000): Primary: 82D30; secondary: 60D05. Key words: replica-symmetry, pure state, perceptron.
1. Introduction Physicists have developed a remarkable theory of mean field disordered systems [MPV], but the study of these is still in its infancy. The physicists rely upon a number of intuitions or, if one prefers, of heuristic general principles. One of these principles is that ‘at high temperature, the overlap of two configurations chosen independently at random according to Gibbs’ measure is nearly constant’. (The overlap of two configurations is defined after the statement of Theorem 1.1.) Even though this principle emerged from physical experience, the physicists have apparently no qualms to apply it to mathematical objects (such as the one that will be considered here) that are certainly not realistic models for interactions within actual matter. This bold approach seems to yield correct results. This was recently rigorously proved for four of the most popular models (see [T1] for a survey). The case that will be considered here offers a new difficulty (a type of discontinuity). This difficulty appeared serious enough at first sight to have the author doubt that the result should be true. To these doubts, M. Mézard gave in essence the following very interesting answer: ‘But if the system is not in a pure state, what else?’ What else indeed, and whether the physical principles can be supported by a general
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MICHEL TALAGRAND
mathematical principle (rather than by difficult proofs in each special case) are food for further thought. Consider a bounded function u: R → R. Consider i.i.d. N(0, 1) r.v. (gik )iN,kM , that represent the ‘disorder’ of the system, and consider the random Hamiltonian 1 k u √ gi σi . (1.1) HN,M (σ ) = − N iN kM We are interested in the behavior of the system governed by the Hamiltonian (1.1) at inverse temperature 1, that is in the (random Gibbs’) probability measure GN,M on N defined by −1 exp(−HN,M (σ )), GN,M (σ ) = ZN,M
(1.2)
where ZN,M is the normalization factor ZN,M = exp(−HN,M (σ )). σ
(The reason for which we do not consider the usual inverse temperature parameter β is that it can be included in u.) The reason for the name ‘perceptron’ is that if u = β1{xτ } ,
(1.3)
then (hopefully) as β → ∞ the knowledge of GN,M allows to recover information on 1 k gi σi τ , (1.4) card σ ; ∀k M, √ N iN a problem referred to in the neural networks theory as ‘The problem of the capacity of the Gaussian perceptron’. The reason for the term ‘Gaussian’ is that the random variables (gik ) are i.i.d. N(0, 1), while in the usual perceptron they are rather from Bernoulli (P (gik = ±1) = 1/2). The choice of Gaussian r.v. is more natural √ k the point of view of geometry since then in (1.3) the sets {σ ; iN gi σi τ N} are random half-spaces at (nearly) fixed distance from the origin, with a uniformly random direction. The reason why we consider the Gaussian case is simply that this is easier than the Bernoulli case (the results of the present paper are probably true in the Bernoulli case, but we doubt that they are within reach of todays methods). The theory of neural networks is a rich theory full of promises. The problem of the capacity of the perceptron is of fundamental importance in this theory. For the present purpose however, only the geometric formulation we gave is relevant, so we send the reader to [HKP] for a general and readable introduction about neural networks. We also refer the reader to [GD] for the (nonrigorous) approach by the physicists. We will always assume that M is a fixed proportion of N, M = αN, where α > 0.
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
79
In a previous paper [T2], we studied the present problem in the Bernoulli case (but the Gaussian case should be similar) under the additional hypothesis that the first five derivatives of u exist and are bounded, say 5 ⇒ |u() | D .
(1.5)
Then we reached a good understanding of the system provided N N(D ), Lα exp LD 1,
(1.6) (1.7)
where D = sup|u| and where L is a number. It is important that the condition (1.7) does not depend upon D . Unfortunately this result is not useful when u is not five times differentiable and in particular in the case (1.3) (although we could calculate the important quantity limN→∞ N −1 E log ZN,M by approximating u by a differentiable function). The purpose of the present paper is twofold. First, we want to remove conditions (1.5), (1.6), and assume only minimum regularity for u, so as to cover in particular the case (1.3) when β is small enough. Second, we want to obtain the correct rates of convergence, rates that cannot be reached with the previous methods. The reader might think at first that removing a mere smoothness condition is not a big deal, but this has actually required considerable effort, and as a result of these efforts, the methods are not more complicated, but are considerably more powerful than those of [T2]. Moreover, these methods have yielded considerable simplifications of the results previously obtained for other models, as is demonstrated in [T3] for the Sherrington–Kirkpatrick model. Let us explain the basic difficulty. The natural approach is the cavity method, that relates an (N + 1) spin system with an N-spin system. The expert certainly guesses that when attempting this, one meets the quantity
(1.8) u Sk N/N + gk 1/N − u(Sk ) , = kM
where Sk = N −1/2
gik σi ,
N = N + 1
iN
and (gk )kM is a sequence of N(0, 1) r.v. independent of the (gik ). (The nonexpert will of course be explained in detail the variation of the cavity method we will need, for which the difficulties are of the same nature.) When u is three times differentiable, we can write, by a simple computation, that (1.8) is gk 1 2 (g u (Sk ) − Sk u (Sk )) (1.9) √ u (Sk ) + 2N k N kM √ within an error R, where |R| u(3) ∞ / N. Thus, as N → ∞, R vanishes, leaving us with a manageable expression for (1.8). On the other hand, if u(3) ∞
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MICHEL TALAGRAND
does not remain bounded with N, there seems to be no reasonable way to express (1.8), and no reason why it should depend upon u , u only. Yet, for some purposes this is the case. Indeed, if we denote by · averages for Gibbs’ measure, we know how to prove that gk √ u (Sk ) + exp exp N kM
1 2 2 + (u (Sk ) + u (Sk ) − Sk u (Sk ) − u (Sk ) ) , (1.10) 2N √ where means that the error is typically at most L/ N . To appreciate this formula, we observe that by (1.9), is of order 1 for large N when u(3) ∞ < ∞. On the other hand, in the case (1.3) (or of the very small perturbation we √ will consider) is the sum of M terms, each of which having a√chance of order 1/ N to be ±1, so one expects that 2 is typically of order α N. But the right-hand side of (1.10) is order 1; so that in order for (1.10) to hold, rather extraordinary cancelation has to take place. Extraordinary cancelation is indeed the theme of the paper. We now state precisely our results. Throughout the paper, L denotes a number (independent of everything) that need not be the same at each occurrence. We assume throughout the paper that |u| D.
(1.11)
We consider two N(0, 1) variables h, z, and for x ∈ R, 0 < y 1, "(x, y) =
Eh exp u(x + hy) . yE exp u(x + hy)
We consider the systems of equations
ˆ q = Eth2 (z q),
2 √ qˆ = αE" (z q, 1 − q).
(1.12)
(1.13) (1.14)
We leave the reader to check (see [T2] for more details) that if (for a number L large enough) Lα exp LD 1,
(1.15)
then there is a unique solution (q, q) ˆ to (1.13), (1.14). Of course, the meaning of these equations is not so obvious, and the fact that this precise value of q appears naturally is, in a sense, the proof that we are dealing with a subtle and rich situation. THEOREM 1.1. If u satisfies (1.11) and is Borel measurable, then we have under (1.15) that 2 L σ · σ −q (1.16) . E N N
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
81
In (1.16), σ · σ = iN σi σi for σ , σ in N , and the bracket · represents a double integral on N2 with respect to G⊗2 N,M . The quantity σ · σ /N is called the overlap of the two configurations σ , σ , and Theorem 1.1 expresses that the overlap of two configurations chosen independently at random according to Gibbs’ measure is nearly q. The meaning of this condition is not so intuitive either, and we refer the reader to [T3] for a detailed explanation of its fundamental importance in the case of another model, the Sherington–Kirkpatrick model. We will deduce Theorem 1.1 from the following theorem: THEOREM 1.2. There exists a number L with the following property. If u satisfies (1.11), is ten times differentiable, and satisfies ∀ 10,
|u() | exp(N/L),
(1.17)
then (1.16) holds under (1.15). Let us now comment briefly upon the methods of the paper. Our answer to the problem of how to evaluate quantities such as (1.8) is that we do not try to do this. Rather, we use the idea to move along a suitably chosen continuous path from a simple situation to the situation we want to study (Kahane’s principle). The derivatives along the path are studied using integration by parts, on which all the cancelations explained above ultimately rely. In [T2], the cavity argument was broken into a ‘cavity upon N’ part and a ‘cavity upon M part’. It does not seem to be possible to do this here, and both parts of the arguments are combined. Rather, Kahane’s principle has to be used twice. The ‘bottom’ use is the object of Section 2. It is a more powerful version of Lemma 3.2 of [T2]. The ‘top’ use, a kind of cavity argument, is the object of Section 3, that culminates in the proof of Theorem 1.2. The short, final section deduces Theorem 1.1 from Theorem 1.2. 2. Integration by Parts The basic integration by parts principle we will use is that if f is a smooth function of moderate growth, and if g is a centered Gaussian r.v., then E(gf (g)) = Eg 2 Ef (g).
(2.1)
Here is a simple consequence. LEMMA 2.1. Consider a centered Gaussian family g1 , . . . , gm . We assume that for each m, we can write bk, gk , (2.2) g = gk, + k=
where gk, is independent of g1 , . . . , g−1 , g+1 , . . . , gk and where 1 2 ; |bk, | 1. Egk, 4 k=
(2.3)
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MICHEL TALAGRAND
Consider a smooth function F on Rm . Then for each integers r1 , . . . , rm , s1 , . . . , sm we have s E g r1 . . . g rm s ∂ (2.4) sm F (g1 , . . . , gm ) K sup|F |, m 1 1 ∂x1 . . . ∂xm where s = s1 + · · · + sm and where K depends only upon s1 , . . . , sm , r1 , . . . , rm . Proof. The proof goes by induction upon s. Certainly the result is true for s = 0. For the can assume s1 1, and using (2.2) we can write g1 = induction step, we g + k2 bk gk , where k1 |bk | 1, g is independent of g2 , . . . , gm and Eg 2 1/4. We then observe that by (2.1), for any number a, we have E(g(g + a)r1 f (g + a)) = Eg 2 (r1 E((g + a)r1 −1 f (g + a)) + E((g + a)r1 f (g + a)). Using this for a = k2 bk gk at g2 , . . . , gm fixed, for f (x) =
∂ s−1 ∂x1s1 −1 ∂x2s2 . . . ∂xmsm
(2.5)
F (x, g2 , . . . , gm ),
we obtain ∂s F (g1 , . . . , gm )) E(g1r1 . . . gmrm s1 s2 ∂x1 ∂x2 . . . ∂xmsm ∂ s−1 1 r1 rm E (g1 − bk gk )g1 . . . gm s −1 s F (g1 , . . . , gm ) − = Eg 2 ∂x11 ∂x22 . . . ∂xmsm k2 ∂ s−1 r1 −1 r2 rm F (g1 , . . . , gm ) − r1 E g1 g2 . . . gm s −1 s ∂x11 ∂xs 2 . . . ∂xmsm ✷
and this implies the result.
We now present a simple condition that ensures the conditions (2.2), (2.3) of Lemma 2.1. LEMMA 2.2. Assume that the Gaussian r.v. (g )m satisfy 3 ∀ m, Eg2 ; 4
∀ < m, |Eg g |
1 . 4m
(2.6)
Then the conditions of Lemma 1.2 hold. Proof. To prove (2.2) we assume without loss of generality that = 1, and we write bk gk , (2.7) g1 = g + k2
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
83
where g is independent of g2 , . . . , gm . If j 2, we deduce from (2.7) that bj Egj2 = Eg1 gj − bk Egk gj , k2,k=j
so that 3 |b | 4 j
1 1+ |bk | 4m k2
and by summation over j 2, 3 1 |bj | 4 1 + |bk | , 4 j 2
so that
k2
|bj | 12 .
(2.8)
j 2
Now, from (2.7) again, we get bk Eg1 gk Eg12 = Egg1 + k2
Egg1 +
1 1 , |bk | Egg1 + 4m k2 8m
so that Egg1 Eg12 −
1 56 Eg12 8m
and, since Egg1 (Eg 2 )1/2(Eg12 )1/2 , we get Eg 2
5 2 6
Eg12
3 5 2 4 6
14 .
✷
We consider now a finite set J , and a probability measure µ on J . We consider a map f from J m to [−1, 1], and two functions U, V on Rm . We consider a centered Gaussian family (g(j ))j ∈J such that ∀j,
Eg(j )2 34 .
We consider the quantity f (j1 , . . . , jm )U (g(j1 ), . . . , g(jm )) dµ(j1 ) . . . dµ(jm ) . I =E V (g(j1 ), . . . , g(jm )) dµ(j1 ) . . . dµ(jm )
(2.9)
(2.10)
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MICHEL TALAGRAND
LEMMA 2.3. We consider numbers B, C, C > 0. We assume that 1 , B |U | C ,
V
(2.11) (2.12)
∂s F (x1 , . . . , xm ), U (x1 , . . . , xm ) = s1 ∂x1 . . . ∂xmsm
(2.13)
where |F | C.
(2.14)
We set
1 A=µ . (j1 , j2 ); |Eg(j1 )g(j2 )| 8m ⊗
(2.15)
Then we have
1/2 , |I | KB C |f (j1 , . . . , jm )| dµ(j1 ) . . . dµ(jm ) + C A
(2.16)
where K depends only upon m, s1 , . . . sm . Comment. This will be used in situations where C C, but where A is extremely small, so the term C A1/2 will be very small. Proof. We use the Cauchy–Schwarz inequality to write, with obvious short-hand notation, 2 1/2 (∫ f U )2 1/2 B E f U , (2.17) I E ( V )2 using (2.11). Now 2 f (j1 , . . . , jm )f (jm+1 , . . . , j2m ) × E fU × E(U (g(j1 ), . . . , g(jm )) × × U (g(jm+1 ), . . . , g(j2m ))) dµ(j1 ) . . . dµ(j2m ). We write =
+ 61
where
(2.18)
, 6c1
1 . 61 = j1 , . . . , j2m , ∃ < 2m E|g(j )g(j )| 8m
(2.19)
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
85
We observe that µ⊗2m (61 ) m(2m − 1)A,
(2.20)
so that, using (2.12), m(2m − 1)AC 2 . 61
On by
6c1 ,
we use Lemmas 2.1 and 2.2 (with 2m rather than m) to bound the integrand
KC 2 |f (j1 , . . . , jm )| |f (jm+1 , . . . , j2m )| (where K depends only upon m, s1 , . . . , sm ) so that 2 KC 2 , . . . , j )| dµ(j ) . . . dµ(j ) . |f (j 1 m 1 m c
(2.21)
✷
61
3. The Main Estimate In this section, we show how to approximate the quantity (2.10). We consider a number 0 q 1/24m and independent N(0, 1) r.v. z, h1 , . . . hm . We denote by Eh expectation at z given, i.e. in h1 , . . . , hm only. THEOREM 3.1. Assume the conditions of Lemma 2.3, and that, moreover, ∂V C ; |V | C; ∀, m, ∂x 2 2 ∂U ∂ U ∂ V C, C; (3.1) ∂x ∂x ∂x C . ∂x ∂x Then, for some constant K depending only upon m, s1 , . . . , sm , we have I − f (j1 , . . . , jm )dµ(j1 ) . . . dµ(jm ) × √ √ √ √ Eh U (z g + h1 1 − g, . . . , z g + hm 1 − g) ×E √ √ √ √ Eh V (z g + h1 1 − g, . . . , z g + hm 1 − g) 1/2 3 3 2 × f (j1 , . . . , jm ) dµ(j1 ) . . . dµ(jm ) KB C 1/2 2 + × (Eg(j1 )g(j2 ) − q) dµ(j1 ) dµ(j2 ) 1/2
+
(Eg(j )2 − 1)2 dµ(j )
+ KB 3 C 3 A1/2 ,
(3.2)
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MICHEL TALAGRAND
where ⊗2
A=µ
1 . (j, j ); |Eg(j )g(j )| > 24m
(3.3)
Comment. To make this result useful, we will of course choose q to make the right-hand side small. Proof. It will be helpful to consider first the case where µ is replaced by 1 µ = δj , (3.4) R rR r where the points j1 , . . . , jR of J need not be distinct. We set J = {1, . . . , R}, we denote by γ the uniform probability on J . To lighten notation, we write g(r) rather than g(jr ) and f (r1 , . . . , rm ) rather than f (jr1 , . . . , jrm ). We consider f (r1 , . . . , rm )U (g(r1 ), . . . , g(rm )) dγ (r1 ) . . . dγ (rm ) . (3.5) I =E V (g(r1 ), . . . , g(rm )) dγ (r1 ) . . . dγ (rm ) This corresponds to the quantity (2.10) when µ has been replaced by µ . We consider i.i.d. N(0, 1) r.v. h(r), r R, and
√ ξ(r) = z q + h(r) 1 − q. We note that
Eξ(r)ξ(r ) =
q 1
if r = r , if r = r .
(3.6)
For 0 t 1 we consider √ √ ξt (r) = t g(r) + 1 − t ξ(r). We consider the function f (r1 , . . . , rm )U (ξt (r1 ), . . . , ξt (rm )) dγ (r1 ) . . . dγ (rm ) . ψ(t) = E V (ξt (r1 ), . . . , ξt (rm )) dγ (r1 ) . . . dγ (rm ) Thus ψ(1) = I . We write
1
|I − ψ(0)| = |ψ(1) − ψ(0)| 0
|ψ (t)| dt sup |ψ (t)|.
(3.7)
(3.8)
0
This formula bounds the error we make when we approximate I by the simpler quantity ψ(0). (The reason why ψ(0) is simpler than I is that we have replaced the family (g(r)) by the simpler family (ξ(r)).) The rest of the proof consists of estimating the right-hand side of (3.8). Then we will make R !→ ∞ and µ !→ µ to obtain (3.2).
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
To compute ψ , we observe that 1 d 1 1 ξt (r) := ξt (r) = ξ(r) , g(r) − √ √ dt 2 t 1−t
87
(3.9)
so that Eξt (r)ξt (r ) =
1 (Eg(r)g(r ) 2
− Eξ(r)ξ(r ))
:= (r, r ).
(3.10)
With obvious notation, we have 1 ∂U f E ξ (r ) − ψ (t) = D ∂x t m 1 ∂V E ξ (r ) fU − D2 ∂x t
(3.11)
m
for
D=
V =
V (ξt (r1 ), . . . , ξt (rm )) dγ (r1 ) . . . dγ (rm ).
(3.12)
To make sense of (3.11), we integrate by parts, using (3.10), and the following easy generalization of (2.1): E(gf (g1 , . . . , gm )) =
m
E(gg )E
∂f (g1 , . . . , gm ). ∂x
(3.13)
This integration by parts is a cumbersome but straightforward computation, the result of which is that ψ (t) = I + II + III + IV, where
∂ 2U (ξt (r1 ), . . . , ξt (rm ))× ∂x ∂x , m ×(r , r ) dγ (r1 ) . . . dγ (rm ) , 1 ∂U E (ξt (r1 ), . . . , ξt (rm ))× f II = −2 D2 ∂x
I=
1 E D
(3.14)
f
(3.15)
, m
×
∂V (ξt (rm+1 ), . . . , ξt (r2m ))(r , rm+ ) dγ (r1 ) . . . dγ (r2m ) , ∂x
(3.16)
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MICHEL TALAGRAND
III = −
E
, m
×
1 D2
f U (ξt (r1 ), . . . , ξt (rm ))×
∂ 2V (ξt (rm+1 ), . . . , ξt (r2m ))× ∂x ∂x
×(rm+ , rm+ ) dγ (r1 ) . . . dγ (r2m ) , 1 E f U (ξt (r1 ), . . . , ξt (rm ))× IV = 2 D3
(3.17)
, m
∂V ∂V (ξt (rm+1 ), . . . , ξt (r2m )) (ξt (r2m+1 ), . . . , ξt (r3m ))× ∂x ∂x ×(rm+ , r2m+ ) dγ (r1 ) . . . dγ (r3m ) . ×
(3.18)
In these formulas, f = f (r1 , . . . , rm ). Even though these formulas look complicated, the good news is that all these terms are of the same nature as (2.10), provided we replace m by m = 2m or by m = 3m. We will be able to use the bound (2.16), replacing B by B 3 , C by C 3 , C by C 3 , and the quantity A of (2.15) by 1 ⊗2 . (3.19) (r, r ); |Eξt (r)ξt (r )| γ 24m Since for r = r , ξt (r)ξt (r ) = tE(g(r)g(r )) + (1 − t)q and since we assume q 1/24m, the quantity (3.19) is bounded by 1 ⊗2 . (r, r ); |E(g(r)g(r ))| A =γ 24m We also observe that if , 3m, then |f (r1 , . . . , rm )(r , r )| dγ (r1 ) . . . dγ (r3m ) 1/2 2 × f (r1 , . . . , rm ) dγ (r1 ) . . . dγ (rm ) 1/2 2 . × (r , r ) dγ (r ) dγ (r )
(3.20)
Thus, (2.16), (3.14), (3.17) imply that for a constant K depending only on m, s1 , . . . , sm , we have 1/2 × f 2 (r1 , . . . , rm ) dγ (r1 ) . . . dγ (rm ) |ψ (t)| KB 3 C 3
89
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
1/2
×
2 (r1 , r2 ) dγ (r1 ) dγ (r2 ) 1/2
+
2 (r, r) dγ (r)
+
+ KB 3 C 3 A 1/2 .
(3.21)
As µ converges to µ, the right-hand side of (3.21) converges to the right-hand side of (3.2), and I converges to I . Thus, it remains to study how ψ(0) behaves. We have ψ(0) = E(W/S), where (3.22) W = f (r1 , . . . , rm )U (ξ(r1 ), . . . , ξ(rm )) dγ (r1 ) . . . dγ (rm ), (3.23) S = V (ξ(r1 ), . . . , ξ(rm )) dγ (r1 ) . . . dγ (rm ). We consider W = f (r1 , . . . , rm ) dγ (r1 ) . . . dγ (rm )U0 ,
(3.24)
where
√ √ U0 = Eh U (z q + h1 1 − q, . . . , z q + hm 1 − q).
(3.25)
We claim that Eh (W − W )2
K(m) 2 C . R
(3.26)
To see this, one expends the square and uses the fact that Eh (U (ξ(r1 ), . . . , ξ(rm ))) = U0 and Eh (U (ξ(r1 ), . . . , ξ(rm ))U (ξ(rm+1 ), . . . , ξ(r2m ))) = U02 provided all of r1 , . . . , r2m are distinct. We then proceed in a similar manner with S to see that as R → ∞, µ → µ, then ψ(0) approaches the second term of (3.2). ✷
4. Cavity Method (Sort of) We now have the tools to prove Theorem 1.3. Using symmetry between the sites, we have 2 1 2 σ ·σ −q = Ef , (4.1) E N
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MICHEL TALAGRAND
where f =
σ1 · σ2 − q (σN1 σN2 − q). N
Let us define, for a parameter 0 t 1, √ 1 t σN gk k . σi gi + √ Sk,t (σ ) = √ N iN−1 N
(4.2)
(4.3)
There and in the rest of the section we write gk = gNk . Consider a centered Gaussian r.v. Y , independent of all the other r.v. considered previously and such that ˆ EY 2 = q,
(4.4)
where q, qˆ are the solutions of (1.13), (1.14). Then, by (1.13), we have q = Eth2 Y.
(4.5)
We consider the Hamiltonian HN,M,t given by √ u(Sk,t (σ )) + 1 − t σN Y. −HN,M,t (σ ) =
(4.6)
kM
We denote by ·t Gibbs’ measure relative to this Hamiltonian, so that ·1 = ·. We consider ϕ(t) = Ef t ,
(4.7)
so that ϕ(1) = Ef .
(4.8)
It is useful to note that f Et , f t = Et where Et (σ 1 , σ 2 ) = exp
(4.9)
√ (u(Sk,t (σ )) − u(Sk (σ )) + σN 1 − t Y ).
2
We have ϕ(0) = f 0 =
1 1 E(σN1 σN2 − q)2 0 + Ef (σN1 σN2 − q)0 , N N
where f =
1 1 2 σ σ − q. N iN−1 i i
(4.10)
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
91
Looking at the form of HN,M,0 , one sees that f (σN1 σN2 − q)0 = f 0 (th2 Y − q) and that f 0 is independent of Y . Thus by (4.5) we have Ef (σN1 σN2 − q)0 = 0 and by (4.10), |ϕ(0)|
4 . N
(4.11)
We write |ϕ(1) − ϕ(0)| sup |ϕ (t)|.
(4.12)
0
Looking at (4.8), (4.11), (4.12), we see that if we prove that 2 1 2 σ ·σ L 1 −q (4.13) + , ∀t, 0 < t < 1, |ϕ (t)| 2 E N N then we obtain 2 1 2 L σ ·σ −q , E N N which proves Theorem 1.2. So, we turn to the proof of (4.13). Using (4.9), we see that ϕ (t) = I + II + III, for Sk,t (σ ), where, writing Sk,t 1 Egk f σN u (Sk,t )t , I= √ 2 tN 2 kM 1 3 Egk f σN3 u (Sk,t )t , II = − √ tN kM 1 3 2E(Y f σN t ) − E(Y f σN t ) . III = √ 2 1−t 2
To make sense of these expressions, one has to integrate by parts, using (2.1). This is another cumbersome but straightforward computation. We get 1 Ef (u 2 + u )(Sk,t )t + I = 2N 2 kM +
1 1 2 Ef σN1 σN2 u (Sk,t )u (Sk,t )t − N kM
1 3 Ef σN σN3 u (Sk,t )u (Sk,t )t . − N 2 kM
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MICHEL TALAGRAND
By symmetry between replicas, α I = Ef (u 2 + u )(SM,t )t + 2 2 1 2 )u (SM,t )t − + αEf σN1 σN2 u (SM,t 1 3 Ef σN σN3 u (SM,t )u (SM,t )t . −α
(4.14)
2
Similarly, we have 3 )t − II = −αEf (u 2 + u )(SM,t 3 Ef σN σN3 u (SM,t )u (SM,t )t + −α 2 3 4 + 3αEf σN3 σN4 u (SM,t )u (SM,t )t .
(4.15)
Also,
1 2 3 3 4 Ef σN σN t + 3Ef σN σN t . III = −qˆ Ef σN σN t − 2
(4.16)
2
In (4.16), the occurrence of qˆ is of course from (4.4). To bring out the dependence of ·t in SM,t , we introduce the Hamiltonian. √ u(Sk,t (σ )) + 1 − t σN Y (4.17) −HN,M−1,t = kM−1
and we denote by ·t,1 integration with respect to the corresponding Gibbs measure. We have identities such as )t f (u 2 + u )(SM,t
f (u 2 + u )(SM,t ) exp 2 u(SM,t )t,1 = . exp 2 u(SM,t )t,1
(4.18)
We have developed in Section 3 the tools to evaluate the expectation of such a quantity given ·t,1 . Indeed, the right-hand side of (4.18) is of the type (2.10) for m = 2, J = N , µ the Gibbs measure relative to Hamiltonian (4.17), g(σ ) = SM,t (σ ), U (x1 , x2 ) = (∂ 2 /∂x2 )F (x1 , x2 ), F (x1 , x2 ) = V (x1 , x2 ) = exp(u(x1 ) + u(x2 )). Assuming (1.11) and |u() | D
∀ 4,
(4.19)
we see that in Theorem 3.1 we can take B = eLD ,
C = eLD ,
C = D L eLD .
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
93
We note that Eg(σ 1 )g(σ 2 ) =
1 1 2 t σi σi + σi1 σi2 , N iN−1 N
so that 1 2 Eg(σ 1 )g(σ 2 ) − σ · σ 1 N N and
1 2 σ · σ 1 − q + , |Eg(σ )g(σ ) − q| N N 1 |Eg(σ )2 − 1| . N 1
2
Thus (Eg(σ 1 )g(σ 2) − q)2 t,1 2 1 2 2 σ ·σ −q 2 +2 N N t,1 2 1 2 2 σ ·σ −q 2 + eLD , N N t (Eg(σ )2 − 1)2 t,1
1 . N2
(4.20) (4.21)
To control the term A of (3.3), we use the following LEMMA 4.1. If αD L, then if N N0 we have N 1 ⊗2 exp − , σ , σ ; |Eg(σ )g(σ )| Gt 94 L
(4.22)
where Gt is Gibbs’ measure relative to the Hamiltonian (4.17). There is no E in (4.22). This bound is true for all realizations of the disorder. Proof. If N N0 , we have 1 1 ⇒ σi σi 10−2 . |Eg(σ )g(σ )| 96 N iN−1 Let
2 1 −2 σi σi 10 . W = (σ , σ ) ∈ N ; N iN−1
(4.23)
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MICHEL TALAGRAND
We have
exp(−HN,M,t (σ ) − HN,M,t (σ )) ch2 Y exp 2MD card W,
(σ ,σ )∈W
since | kM u(Sk,t (σ ))| MD. Also, exp(−HN,M,t (σ )) 2N ch Y exp(−MD) σ
and thus −2N card W. G⊗2 t (W ) exp(4MD)2
Now, it is well known that 1 Nu2 N+1 σi u 2 exp − card σ ∈ N ; N iN 2 and this implies that
10−4 , card W 22N+1 exp −(N − 1) 2
(4.25)
which, together with (4.24), implies the result.
✷
We are now ready to apply (3.2) to the quantity (3.17). We notice that Lα exp LD 1 implies that q 1/96. We then find, taking expectations in (3.2) that this quantity is √ √ √ √ Eh (u 2 + u )(z q + h 1 − q) exp u(z q + h 1 − q) √ × Ez √ Eh exp u(z q + h 1 − q) (4.26) × Ef t,1 + R, where
2 1/2 1 2 σ ·σ L 2 −q + + Ef t E |R| Le N N t N + LeLD D 3 exp − L 2 1 2 L N σ ·σ LD LD 3 −q + Le D exp − . (4.27) + E Le N N L t LD
We proceed in a similar fashion for all the terms of I and II. We see that by integration by parts, (1.14) means that √ √ √ √ Eh u (z q + h1 1 − q)u (z q + h2 1 − q)E , qˆ = αEz Eh E
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ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
where E = exp
√ u(z q + h 1 − q).
2
We then see that I + II = q(Ef ˆ σN1 σN2 t,1 − 2
Ef σN σN3 t,1 +
2
+
3Ef σN3 σN4 t,1 )
+ αR,
(4.28)
where R is as in (4.23). Since, for a function ξ on Nm , we have ξ exp m u(SM,t (σ ))t,1 , ξ t = exp mu(SM,t (σ ))t,1 we can use again (3.2) to see that in fact Ef σN σN3 t + I + II = qˆ Ef σN1 σN2 t − 2 + 3Ef σN3 σN4 t
2
+ αR
= −III + αR. Thus, we have proved that 2 1 2 1 σ ·σ LD −q + + E |ϕ (t)| Lαe N N t N LD 3 . + Lαe D exp − L
(4.29)
(4.30)
This looks very much like (4.13), except that on the right-hand side we have ·t rather than ·. Quite naturally, we set 2 1 2 σ ·σ −q (4.31) ξ(t) = E N t and we try to compare ξ(t) and ξ(1). We compute ξ (t), which is given by the same expression as ϕ (t), but where now 2 1 2 σ ·σ 1 2 −q . f (σ , σ ) = N To each term of I + II + III we now apply (2.16) rather than (3.2) and we get
N L + LαeLD D 3 exp − . ξ (t) LαeLD ξ(t) + N L
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MICHEL TALAGRAND
Thus, if LαeLD 1, D exp(N/L), this becomes ξ (t) ξ(t) +
L N
and thus ξ(t) Lξ(1) +
L . N
Going back to (4.30), this gives 1 2 2 σ ·σ 1 LD E + −q + |ϕ (t)| Lαe N N N + LαeLD D 3 exp − L and if D exp(N/L), LαeLD 1, this becomes (4.13). We have proved Theorem 1.2.
5. Proof of Theorem 1.1 This proof relies upon an approximation procedure. We will approximate u by a function v such that v satisfies (1.17), and such that the Gibbs’ measures corresponding to u and v respectively are very close in some sense. We consider a parameter a to be chosen later. It is elementary that there exists a function ϕ supported by [−a, a], ϕ 0, such that ϕ(x) dx = 1, (5.1) ∀ 10,
|ϕ ()|
L a +1
.
(5.2)
We define v by ev = ϕ ∗ eu .
(5.3)
Thus, (1.11) implies that |v| D.
(5.4)
Also, ∀ 10,
|v ()|
LeLD . a
(5.5)
ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
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To prove (5.3), we note that v =
ϕ ∗ eu , ϕ ∗ eu
so that (e.g.) |v | ϕ 1 e2D Le2D /a. LEMMA 5.1. If b ∈ R and g is centered Gaussian we have eD a . (5.6) |E(eu(g+b) − ev(g+b))|
Eg 2 Proof. Consider V such that V (x) = eu(x) . Then (ϕ ∗ V ) = ϕ ∗ V = ev . Moreover, since |V | eD , and since the support of ϕ is contained in [−a, a], we have |V − ϕ ∗ V | aeD .
(5.7)
Now, since (V − ϕ ∗ V ) = eu − ev , integration by parts show that Eg 2 E(eu(g+b) − ev(g+b)) = E(g(V (g + b) − ϕ ∗ V (g + b))) so that, using (5.7), we see that Eg 2 |E(eu(g+b) − ev(g+b))| aeD E|g| aeD (Eg 2 )1/2 ✷
and this implies (5.6). We recall the notation Sk (σ ) = N −1/2
iN
gik σi . The key point is as follows.
LEMMA 5.2. Consider a subset A of N . Then 2 u(S (σ )) v(S (σ )) k k − e kM e kM E σ ∈A
Le2MD (card A + a 2 NM 2 (card A)2 ). Proof. The left-hand side of (5.8) is E(B(σ )B(σ )),
(5.9)
σ ,σ ∈A
where
B(σ ) = e
kM
u(Sk (σ ))
−e
kM
v(Sk (σ ))
.
If σ = σ or σ = −σ we use the trivial bound |B(σ )| 2eMD
(5.8)
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MICHEL TALAGRAND
to obtain |E(B(σ )B(σ ))| 4e2MD . Next, we fix σ , σ with σ = σ , σ = −σ . Without loss of generality, we assume σ1 = σ1 , σ2 = −σ2 . To study E(B(σ )B(σ )) we first study E0 (B(σ )B(σ )), where E0 denotes conditional expectation given gik , k M, 3 i N. In other words, we integrate only in gik , k M, i = 1, 2. We note the fact that the 2M variables gk = σ1 g1k + σ2 g2k
and
gk = σ1 g1k + σ2 g2k
are all independent, so that E0 (B(σ )B(σ )) = E0 (B(σ ))E0(B(σ )). Now E0 (B(σ )) =
Xk −
k
Yk ,
(5.10)
(5.11)
k
where Xk = E0 eu(Sk (σ ));
Yk = E0 ev(Sk (σ )) . √ Using Lemma 5.1 with g = gk / N , b = N −1/2 i3 σi gik , we see that |Xk − Yk | aN 1/2 eD . Since |Xk | eD , we see from (5.10) that E0 (B(σ )) aN 1/2 MeMD . Combining with (5.10) this finishes the proof. We write −HN,M,u (σ ) = ZN,M,u =
✷
u(Sk (σ )),
kM
exp(−HN,M,u (σ ))
σ
and similar obvious notation for v. COROLLARY 5.3. For each B > 0, −H (σ ) −H (σ ) (e N,M,u − e N,M,v ) B card A P σ ∈A 1 Le2MD 2 2 + a NM . B2 card A
(5.12)
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ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE
✷
Proof. Use (5.8) and Markov’s inequality.
We now choose a = 2−2rN , where r > 0 is small enough that 21r < 1/L, where L is the constant of (1.17). Then (5.5) shows that for N N(D), v satisfies (1.17) if LαD 1. Next, in (5.12) we use B = exp(−MD − rN). The right-hand side of (5.12) is then bounded by 3rN e 4MD−rN 2 −rN + NM e . (5.13) Le card A Next we observe that ZN,M,u 2N e−MD . If η1 , η2 = ±1, using (5.12) for A = N and for A = {σ ; σ1 = η1 , σ2 = η2 } elementary manipulations then show that E|GN,M,u ({σ1 = η1 , σ2 = η2 }) − GN,M,v ({σ1 = η1 , σ2 = η2 })| LNM 2 e4MD−rN Le−rN/2
(5.14)
provided LαD 1. Writing σ · σ = iN σi σi and using symmetry between sites, we see that E
σ · σ −q N
2 =
N −1 1 + Eσ1 σ2 2 + q 2 − 2qEσ1 2 N N
and combining with (5.14) we get 2 2 σ · σ σ · σ E − qu − qv −E N N u v L|qu − qv | + Le−rN/2 .
(5.15)
We will leave to the reader to check (using (5.6)) that |qu − qv | Le−rN/2 . Inequality (5.15) makes it obvious that Theorem 1.1 follows from Theorem 1.2. References [GD] Gardner, E. and Derrida, B.: Optimal storage properties of neural network models, J. Phys. A 21(1) (1988), 271–284. [HKP] Hertz, J., Krogh, A. and Palmers, R. C.: Introduction to the Theory of Neural Computation, Addison-Wesley, 1991. [MPV] Mézard, M., Parisi, G. and Virasiro, M.: Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. [T1] Talagrand, M.: Huge random structures and mean field models for glasses, In: Proc. Berlin Internat. Congr. Document. Math, Extra Volume I (1988), 507–536. [T2] Talagrand, M.: Intersecting random half spaces: Towards the Derrida–Gardner formula, Ann. Probab. 28 (2001), 725–758. [T3] Talagrand, M.: Mean Field Models for Spin Glasses: A First Course, Saint Flour Summer School in Probability 2000, to appear in Springer Lecture Notes in Math.
Mathematical Physics, Analysis and Geometry 5: 101–123, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
101
Unitary Correlations and the Fejér Kernel Dedicated to Harold Widom on his 70th birthday DANIEL BUMP, PERSI DIACONIS and JOSEPH B. KELLER Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. (Received: 28 March 2002) Abstract. Let M bea unitary matrix with eigenvalues tj , and let f be a function on the unit circle. f (tj ). We derive exact and asymptotic formulae for the covariance of Xf Define Xf (M) = and Xg with respect to the measures |χ(M)|2 dM where dM is Haar measure and χ an irreducible character. The asymptotic results include an analysis of the Fejér kernel which may be of independent interest. Mathematics Subject Classifications (2000): 15A52, 34E05. Key words: random matrix theory, Fejér kernel, unitary group.
1. Introduction Random matrix theory uses the eigenvalues of typical large matrices as models of a wide variety of natural phenomena from nuclear spectra to the zeros of the Riemann zeta function. Surveys of this area may be found in [17, 24] or [9]. We study the eigenvalues of typical unitary matrices M. These are n points tj on T, the unit circle. Sometimes we will write tj = eiθj with 0 θj < 2π . The distribution of the traces of powers of M is studied in [4], the number of eigenvalues in an interval is studied in [26], and the log characteristic polynomial is studied in [13, 14] and [11]. These are all examples of additive functions of M, that is, functions of the form n f (tj ), Xf (M) = j =1
where f : T → C is a function. The limiting behavior of such functionals is studied in the papers cited above as well as in [3, 20, 21], where a variety of central limit theorems are proved. In this paper we study the variance and covariance of Xf and Xg . We are interested not merely in the limiting behavior,but in the behavior for n of moderate size. We will normalize Haar measures T dt and U (n) dM so that the compact groups T and U(n) have volume 1. The convolution of two functions on T is defined by (f ∗ g)(x) = f (t)g(xt −1 ) dt. T
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The Fejér kernel n−1
Kn (t) =
k=−(n−1)
sin( nθ2 )2 |k| k t = , 1− n n sin( θ2 )2
t = eiθ .
(1)
Fejér introduced this kernel to prove that Fourier coefficients determine the function at a point under suitable conditions. See [15] for background and [27], pp. 88–90, for the classical theory. If and are functions on U(n), define the covariance (M)(M) dM − , Cov(, ) = U(n)
where the mean value = (M) dM. U(n)
Let g(t) ˜ = g(t −1 ). THEOREM 1. For f, g ∈ L2 (T), ˜ − n(f ∗ g˜ ∗ Kn )(1). Cov(Xf , Xg ) = n(f ∗ g)(1)
(2)
It is striking that this covariance only depends on f ∗ g. ˜ We will give two proofs of Theorem 1, in Sections 2 and 3. We will see in Section 2 that Theorem 1 is equivalent to an icon of random matrix theory, a formula of Dyson for the pair correlation function. We will generalize Theorem 1 in Theorem 4 below to obtain covariances with respect to the measure |χ(M)|2 dM, where χ is an irreducible character of U(n) by a very similar formula. The Fejér kernel is a Dirac sequence. This means that Kn 0, T Kn (t) dt = 1 and Kn → 0 uniformly on any compact subset of T excluding the point 1. Consequently the sequence of trigonometric polynomials φ ∗ Kn → φ uniformly for continuous functions φ on T. Taking φ = f ∗ g˜ we see that the covariance (2) may be interpreted as the error in the approximation of φ by φ ∗ Kn . and dk are the Fourier coefficients of f and g, so that f (t) = that ck Suppose k ck t and g(t) = dk t k . Recalling that convolution of functions corresponds to multiplication of the Fourier coefficients, we may write (2) in the equivalent form: Cov(Xf , Xg ) = n
∞ k=−∞
ck d−k −
n−1
(n − |k|)ck d−k .
(3)
k=−(n−1)
As an example, we consider the case where f and g are the characteristic functions of two intervals. Thus Xf and Xg count the number of eigenvalues in each interval. Wieand [26] showed that in the limit as n → ∞, these functions are uncorrelated unless the intervals share an endpoint. She found that if they share
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
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a left or right endpoint, then there is a positive limiting correlation, but if the left endpoint of one interval is a right endpoint of the other, then there is a negative limiting correlation. Meanwhile Rains [19] also considered the number of eigenvalues in an interval; he found the complete asymptotic expansion for the variance. It is possible to go from Rains’ asymptotic results on the variance of the number of eigenvalues in an interval I to an asymptotic result on the covariance of the number of eigenvalues in two different intervals I and J . The work involved in this process is not entirely trivial, since the results of [19] would need to be applied to several intervals: if K is a connected component of the symmetric difference I J of I and J , then one needs to know the variances in the number of eigenvalues in the six intervals I , J , K, I K, J K and I (J K) = (I J ) K. By an inclusion exclusion process, one may infer the covariance of the number of eigenvalues in the intervals I and J . We can consider this matter from our point of view using Theorem 1, recovering and extending these results of Wieand and Rains. As an example, let 0 ω 2π , and let f and g be the characteristic functions of the images under θ → eiθ of the intervals [0, ω] and [θ, ω + θ], respectively. We have 1 (1 − e−ikω ), if k = 0, ikθ ck = e dk = 2πik ω , if k = 0. 2π Let φ = f ∗ f˜. Thus φ(eiθ ) =
max(ω − θ, 0) + max(ω + θ − 2π, 0) . 2π
(4)
Because (f ∗ g)(1) ˜ = φ(eiθ )
and
(f ∗ g˜ ∗ Kn )(1) = (φ ∗ Kn )(eiθ ),
Theorem 1 shows that Cov(Xf , Xg ) = n(φ − φ ∗ Kn )(eiθ ), and it is this that we study. The graph of φ(eiθ ) in the case ω = π/2 is shown in Figure 1.
Figure 1. φ(eiθ ) as a function of θ for ω = π/2.
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Figure 2. The function n(φ − φ ∗ Kn ) when n = 42.
The series (3) simplifies to Cov(Xf , Xg ) = n(φ − φ ∗ Kn )(eiθ ) 2 n−1 ω (n − k) iθ − (1 − cos(kω))cos(kθ). = nφ(e ) − n 2π π 2k2 k=1
(5)
This formula is suitable for numerical computation. For n = 42 and ω = π/2, Figure 2 shows the graph of this quantity as a function of θ. Figure 2 displays the covariance between the number of eigenvalues of a random unitary matrix in two given intervals of length π/2 as one of the intervals slides around the circle. We see that the places where (5) has the largest magnitude are the locations of the discontinuities in the derivative φ . We will justify this qualitative observation in Section 4 by an asymptotic analysis of φ − Kn ∗ φ for a function φ which (like this one) is continuous, but whose derivative has jump discontinuities. Let Ci and Si be the cosine and sine integrals, x cos(t) − 1 dt Ci(x) = γ + log(x) + t 0 ∞ cos(t) dt, |arg(x)| < π, (6) = − t x where γ = 0.57721 . . . is Euler’s constant, and ∞ x π sin(t) sin(t) dt = − dt, |arg(x)| < π. (7) Si(x) = t 2 t 0 x The asymptotic expansion of Ci as x → ∞, obtained from the last expression in (6) by integration by parts, is 2! 4! cos(x) 5! 3! sin(x) 1 − 2 + 4 − ··· − 1 − 2 + 4 − · · · . (8) Ci(x) ∼ x x x x2 x x Similarly sin(x) 5! 3! π − 1 − 2 + 4 − ··· − Si(x) ∼ 2 x2 x x 2! 4! cos(x) 1 − 2 + 4 − ··· . (9) − x x x
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
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Figure 3. The function 5 .
If |θ| π , let 2(1 + γ + log(2n)), if θ = 0, n (θ) = 2 Ci(n|θ|) − 2 log sin |θ| + 2 cos(nθ) − nπ |θ| + 2nθ Si(n|θ|), (10) 2 otherwise. By (8) and (9) this function is continuous at θ = 0. We make n into a 2π periodic function. The function n is ‘spiky’, increasingly so as n increases. Indeed, (8) and (9) show that lim n (θ) = −2 log sin(|θ|/2),
n→∞
(11)
n → ∞. The graph of 5 is shown in Figure 3. We can now describe the results of our asymptotic analysis of φ − Kn ∗ φ, which are given in Theorems 9 and 10 below. The most important and interesting contributions to these come from the jump discontinuities in the derivative of φ. Indeed, we will show that if θ → φ(eiθ ) has a jump discontinuity at θ = θ0 , then the asymptotic form of n(φ − Kn ∗ φ)(eiθ ) contains a constant multiple of n (θ − θ0 ). When φ is the function (4), the derivative jumps at θ = 0, ω and 2π − ω, and Theorem 10 shows that Cov(Xf , Xg ) = n(φ − Kn ∗ φ)(θ) 1 1 n (θ) − n (θ − ω) − = 2 2π 4π 2 1 − 2 n (θ − 2π + ω) + O(n−1 ) 4π uniformly in θ. In practice this approximation is quite good. For n = 42, the graph of the approximation is indistinguishable from Figure 2, for its values agree with those of the original function to about five decimal places. The largest error is less than 1.4 × 10−5 .
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Figure 4. The limiting covariance as n → ∞ for Figure 1.
Theorem 1, together with (11) and Theorem 9 show that as n → ∞ the covariance Cov(Xf , Xg ) tends to a limiting distribution which is singular at the discontinuities of the derivative of φ. In the example of Figure 1, (11) shows that the limiting covariance equals
sin θ−π/2 sin θ−3π/2 1
2 2 log
. 2 2
2π sin (θ/2) The graph of this function is shown in Figure 4. We will generalize Theorem 1 by obtaining covariances with respect to the probability measure |χλ (M)|2 dM, where λ is a partition of length n, and χλ is the character of the irreducible representation indexed by λ, defined in (16) below. Denoting this covariance as Covλ , we will show in Theorem 4 that if λ is fixed, there is a Dirac sequence Kn,λ such that Covλ (Xf , Xg ) = n(f ∗ g˜ − f ∗ g˜ ∗ Kn,λ )(1). For example, if λ = (1), that is, the partition (1, 0, 0, . . .), then χλ (M) = tr(M) is just the character of the standard representation of U(n). When n = 42 and φ is the function (4) with ω = π/2, the graph of this covariance is shown in Figure 5. We will show in Theorem 11 that the functions n can be generalized to functions n,λ , giving in a similar way the asymptotic forms of Covλ (Xf , Xg ), at least when f and g are piecewise linear and continuous. We do not work out the exact form of n,λ , though we give the interested reader enough information to do so in the proofs of Theorems 3 and 11. We find that n,λ (θ) − n (θ) is a trigonometric polynomial independent of n. For example, in the case where λ = (1), we find that n,λ (θ) − n (θ) = 2 cos(θ). This result shows that if λ is fixed and n → ∞ the covariance Covλ (Xf , Xg ) tends to a limiting distribution, which is finite except at the discontinuities of the derivative of φ.
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´ KERNEL UNITARY CORRELATIONS AND THE FEJER
Figure 5. The function n(φ − φ ∗ Kn,λ ) when n = 42, λ = (1).
There has been some related work since the circulation of a preliminary version of this paper. Hughes [10] derives related approximations in comparing the variance of the number of eigenvalues and zeta zeros in matching intervals. In the matter of Fejér asymptotics, Pinsky [18] recognized the correction term in Theorem 7 as the Hilbert transform of f . He derives L2 convergence results for a variety of summability kernels and extends his results from the circle to the line. Taylor [23] gives a version of Pinsky’s results for a function on an n-dimensional Riemannian manifold expanded in eigenfunctions of the Laplacian. He further extends our results to uniform asymptotics for piecewise smooth f with a simple jump across a smooth hypersurface. We thank these authors for keeping us informed. 2. Pair Correlation The formula (2) is equivalent to a basic formula of unitary statistics, Dyson’s formula for the correlation function for unitary eigenvalues. The correlation functions were found by Dyson in part III of [5]; Dyson gave a proof and a generalization to other ensembles in [6]; see also [17] and [24] for different proofs and extensions. The m-level correlation Rm (t1 , . . . , tm ) measures the density that t1 , . . . , tm are the eigenvalues of a Haar random unitary matrix. Concretely, if f (t1 , . . . , tm ) is a test function on Tm , then Rm (u1 , . . . , um )f (u1 , . . . , um ) du1 . . . dum Tm ∗ = f (ti1 , . . . , tim ) dM, (12) U(n)
where the sum on the right is over the n!/(n − m)! different m-tuples (i1 , . . . , im ) where the ij are pairwise distinct integers between 1 and n and where t1 , . . . , tn are the eigenvalues of M. Dyson found that Rm (t1 , . . . , tm ) = det(sn (θj − θk ))j,k ,
tj = eiθj ,
(13)
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where
sin(nθ/2)
sn (θ) =
sin(θ/2)
,
n,
θ = 0; θ = 0.
In the case m = 2 (the only case we need), this amounts to R2 (t1 , t2 ) = n2 − nKn (t1 t2−1 ).
(14)
Proof of Theorem 1. If ti are the eigenvalues of M, denote n f (tj )g(tj ). &(M) = j =1
Evidently Xf (M)Xg (M) − &(M) =
f (tj )g(tk ).
j =k
Using (12) we have (Xf (M)Xg (M) − &(M)) dM U(n) R(u1 , u2 )f (u1 )g(u2 ) du1 du2 = 2 T = (n2 − nKn (u1 u−1 2 ))f (u1 )g(u2 ) du1 du2 . T2
The left-hand side here equals Xf (M)Xg (M) dM − n(f ∗ g)(1), ˜ U(n)
and the right side equals Xf Xg − n(f ∗ g˜ ∗ Kn )(1). Comparing these gives (2).
✷
Let λ be a partition, that is, a decreasing sequence λ1 λ2 λ3 · · · of integers such that λj = 0 for j sufficiently large. The largest l such that λl = 0 is called the length l(λ) of λ. If the length of λ is n, let
λ1 +n−1 λ1 +n−1
λ1 +n−1
t t · · · t n 1 2
λ +n−2 λ +n−2
t 2 t2 2 · · · tnλ2 +n−2
1
.. ..
. .
λn
t λn
λn t2 ··· tn 1
n−1 n−1 . (15) sλ (t1 , . . . , tn ) =
t t2 · · · tnn−1
1
t n−2 t n−2 · · · t n−2
1 n 2
.. ..
. .
1 1 ··· 1
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
109
This is the Schur function as defined in [16], (3.1) on p. 30. See [16], volume 2 of [22] and [1] for background on the Schur functions and related representation theory. If t1 , . . . , tn are the eigenvalues of M ∈ U(n), then χλ (M) = sλ (t1 , . . . , tn )
(16)
is the character of an irreducible representation of U(n), denoted sλ (M) in [1]. (This is essentially the Weyl character formula.) With dM Haar measure, |χλ (M)|2 dM defines a probability measure on U(n). We now investigate the m-level correlation function Rm,λ (t1 , . . . , tm ) with respect to the measure |χλ (M)|2 dM. We define this by analogy with (12) by asking that Rm,λ (t1 , . . . , tm )f (t1 , . . . , tm ) dt1 . . . dtm Tm ∗ = f (ti1 , . . . , tim )|χλ (M)|2 dM. (17) U(n)
We will prove a formula generalizing Dyson’s (13) for Rm,λ . To motivate this, rewrite (13) this way: Rm (t1 , . . . , tm ) = det(AA∗ ), where A is the (nonsquare) matrix 1 t1 t12 · · · t1n−1 . . .. . . . . . 2 n−1 1 tm tm · · · tm and A∗ is its conjugate transpose. The Hermitian matrix AA∗ is not equal to the square matrix on the right side of (13), but is conjugate to it by a diagonal matrix. THEOREM 2. We have Rm,λ (t1 , . . . , tn ) = det(Aλ A∗λ ), with
−λ t1 1 . Aλ = .. tm−λ1
t11−λ2 .. . 1−λ2 tm
··· ···
(18)
t1−λn +n−1 .. . . −λn +n−1 tm
Proof. We may prove this formula as follows. First suppose that m = n. Then the matrix Aλ is square, and det(Aλ A∗λ ) = |det(A∗λ )|2 .
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We must show that the function on the right side of (18) satisfies (17). By the Weyl integration formula, if we substitute |det(A∗λ )|2 for Rm,λ , on the left side of (17) we obtain
2
t λ1 t2λ1 ··· tnλ1
1
t λ2 −1 t2λ2 −1 · · · tnλ2 −1
1
.
f (t1 , . . . , tn ) dt1 . . . dtn . ..
. Tn .
.
λn −n+1 λn −n+1
λn −n+1
t1 t2 · · · tn Let F (t1 , . . . , tn ) =
f (tσ (1) , . . . , tσ (n) )
σ ∈Sn
be the symmetrization of f . The Vandermonde identity and the symmetry of the Schur function show that the last expression equals
λ1 +n−1 λ1 +n−1
λ1 +n−1 2
t t · · · t n 1 2
λ +n−2 λ +n−2
t 2 t2 2 · · · tnλ2 +n−2
1
.. ..
. .
λn λn λn
t2 ··· tn t1 1 F (t1 , . . . , tn ) |ti − tj |2 dt1 . . . dtn .
2
n−1 n−1 n−1 n! Tn
t t2 · · · tn i<j
1
t n−2 t n−2 · · · t n−2
1 n 2
. ..
.. .
1 1 ··· 1 Now using (15) and the Weyl integration formula (Goodman and Wallach [8], p. 343), we obtain the right side of (17). Therefore (18) is true when m = n. The case m < n follows by the same argument as in [16], pp. 195–196. The downward induction is based on Theorem 5.2.1 on p. 89 of [16], where in the case at hand the function f (x, y) on T is n (xy −1 )j −1−λj . f (x, y) = j =1
The other ingredient of the proof is the analog of Equation (5.1.2) of [16], which for us is the formula 1 Rn,λ (x1 , . . . , xn ) dxm+1 . . . dxn . Rm,λ (x1 , . . . , xm ) = (n − m)! Tn−m It is straightforward to deduce this from (17) by taking a test function f (t1 , . . . , tn ) ✷ which only depends on t1 , . . . , tm .
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
111
If n l(λ), define
n
2 1
λj −j
t Kn,λ (t) =
.
n j =1 THEOREM 3. (i) The sequence of functions Kn,λ is a Dirac sequence. (ii) There exists polynomials fλ (t) and gλ (t) in t and t −1 such that fλ (t) = fλ (t −1 ) and n(Kn,λ (t) − Kn (t)) = fλ (t) + t n gλ (t) + t −n gλ (t −1 ).
(19)
Proof. It is evident that Kn,λ is positive on T. Writing Kn,λ (t) =
1 λj −λk −j +k t , n j,k
its mean value on T is 1/n times the number of pairs j, k with λj − j = λk − k. Since λj − j is a decreasing sequence, these pairs occur precisely when j = k, so K T n,λ (t) dt = 1. Next we show that if t = 1, then Kn,λ (t) → 0; the convergence is uniform on compact subsets of T − {1}. If l = l(λ) is the length of λ, then
l
2 1
λj 1 − t −(n+1)
−j (20) Kn,λ (t) = (t − 1)t +
. n 1 − t −1 j =1
l
Now j =1 (t λj − 1)t −j is independent of n, and (1 − t −(n+1) )/(1 − t −1 ) is bounded independently of n if t is bounded away from 1. Hence (20) is O(n−1 ). We have established that Kn,λ is a Fejér sequence. It follows from (20) that n(Kn,λ (t) − Kn (t))
2
l l
t λj −j − t −j
(1 − t n+1 ) + t λj −j − t −j + =
1−t j =1
+
j =1
l −λj +j
t
j =1
− tj
1 − t −1
(1 − t −n−1 ).
Noting that (t λj −j − t −j )/(1 − t) is a polynomial in t and t −1 , simplifying this gives (19). ✷ If and are functions on U(n) we will define Covλ (, ) to be the covariance with respect to the measure |χλ (M)|2 dM.
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THEOREM 4. We have ˜ − n(f ∗ g˜ ∗ Kn,λ )(1). Covλ (Xf , Xg ) = n(f ∗ g)(1)
(21)
Proof. If m = 2, we may rewrite Theorem 2 in the form R2,λ (t1 , t2 ) = n2 − nKn,λ (t1 t2−1 ). The proof of Theorem 4 is now the same as that of Theorem 1.
✷
3. Principal Toeplitz Minors In this section we reprove Theorems 1 and 4 by an entirely different method. A classical identity of Heine and Szegö expresses Toeplitz determinants as integrals over the unitary group. See Bump and Diaconis [1] for generalizations and applications of this identity. Here we will derive another generalization of this identity which contains information equivalent to the m-level correlation function of unitary statistics. Let 1 m n. If M is any matrix, let Em (M) denote the sum of the mn principal m × m minors of M. Since this is just the trace of M in the mth exterior power representation of U(n), it is invariant under conjugation, so to compute it we may assume M is diagonal. Thus Em (M) = em (t1 , . . . , tn ), where tj are the eigenvalues of M, and em is the mth elementary symmetric polynomial in n variables. If f is a continuous function on T, we may associate with f a continuous function Uf : U(n) → U(n), namely t1 f (t1 ) −1 −1 .. .. Uf h h , h = h . . tn f (tn ) with h ∈ U(n). Note that this is well defined. If dj are the Fourier coefficients of f , so that f (t) = dj t j , let Tn−1 be the n × n Toeplitz matrix d0 d1 · · · dn−1 d d0 · · · dn−2 −1 . Tn−1 (f ) = . . . . . . d−(n−1) d−(n−2) · · · d0 THEOREM 5. With these notations, Em (Uf (M)) dM = Em (Tn−1 (f )). U(n)
(22)
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
Proof. If m = n, this identity may be written det(Uf (M)) dM = det(Tn−1 (f )),
113
(23)
U(n)
and this is the Heine–Szegö identity. See [1] for a proof and generalizations. The general case of (22) follows by replacing f by 1 + λf in (23), then expanding and ✷ comparing the coefficients of λm . One may deduce Theorem 1 from Theorem 5 by an argument we will give below in the second proof of Theorem 4. To obtain Theorem 4, we need to generalize Theorem 5. λ,µ (f ) be the Toeplitz minor which If λ and µ, are partitions of length n, let Tn−1 is the n × n matrix whose (j, k)th entry is dλj −λk −j +k . It is a minor in a larger Toeplitz matrix. The asymptotics of large Toeplitz minors were studied by Bump and Diaconis [1]. If λ = µ this is a principal minor. These are the ones which we will need. THEOREM 6. We have λ,µ Em (Uf (M))χλ (M)χµ (M) dM = Em (Tn−1 (f )).
(24)
U(n)
Proof. We take λ = µ in Theorem 3 of [1]. The quoted theorem asserts that λ,µ det(Uf (M))χλ (M)χµ (M) dM = det(Tn−1 (f )). U(n)
Proceeding as in the proof of Theorem 5 we obtain (24).
✷
In our applications of Theorem 6 we will take λ = µ. The special case m = 1 of (24) is worth noting: 2 Xf (M)|χλ (M)| dM = n f (t) dt. (25) T
U(n)
Second Proof of Theorem 4. Let dk t k . f (t) = ck t k and g(t) = Let M ∈ U(n) have eigenvalues tj . Then f (tj )g(tk ). Xf (M)Xg (M) = j,k
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We integrate this with respect to the measure |χλ (M)|2 dM, separating the diagonal terms (j = k) from the terms with j = k. The diagonal contribution equals the integral of Xfg (M), which by (25) equals ck d−k . (26) n f (t)g(t) dt = n T
k
We are left with the integral of the off-diagonal terms f (tj )g(tk ) = E2 (Uf +g (M)) − E2 (Uf (M)) − E2 (Ug (M)). j =k
We evaluate this by means of (24). It equals λ,λ λ,λ λ,λ (f + g)) − E2 (Tn−1 (f )) − E2 (Tn−1 (g)). E2 (Tn−1 λ,λ (f + g) has a principal minor of Let 1 k n − 1. If 1 j k n, then Tn−1 the form
cλj −λk −j +k + dλj −λk −j +k
c0 + d0
.
c−(λj −λk −j +k) + d−(λj −λk −j +k) c0 + d0
From this, we subtract the two corresponding minors
cλj −λk −j +k
dλj −λk −j +k d0 c0
+
d−(λj −λk −j +k)
c−(λj −λk −j +k) c0 d0
λ,λ λ,λ (f ) and Tn−1 (g) to obtain of Tn−1
2c0 d0 − cλj −λk −j +k d−(λj −λk −j +k) − c−(λj −λk −j +k) dλj −λk −j +k . Summing these terms gives (cλj −λk −j +k d−(λj −λk −j +k) + (n2 − n)c0 d0 − 1j
= (n2 − n)c0 d0 −
+ c−(λj −λk −j +k) dλj −λk −j +k ) cλj −λk −j +k d−(λj −λk −j +k) . 1j =kn
Adding back the diagonal terms (26) gives Xf (M)Xg (M)|χλ (M)|2 dM U(n)
=n
∞ k=−∞
ck d−k + n(n − 1)c0 d0 −
1j =kn
cλj −λk −j +k d−(λj −λk −j +k) .
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´ KERNEL UNITARY CORRELATIONS AND THE FEJER
By (25) we have Xf = Xf (M)|χλ (M)|2 dM = nc0
and
Xg = nd0 .
U(n)
Thus ∞ ∞ 1 Covλ (Xf , Xg ) = ck d−k − ρl cl d−l , n k=−∞ k=−∞
where ρl = 1 if l = 0; more generally, it is 1/n times the number of pairs (j, k) ✷ with λj − λk − j + k = l. Evidently ρl t l = Kn,λ (t), whence (21).
4. Fejér Asymptotics As we have noted, the convolution of a function f with the Fejér kernel gives a sequence of approximations to f by trigonometric polynomials. We have expressed the covariance of two additive functions on U(n) as the error in such an approximation. The class of functions on T which occurs in Theorem 1 consists of convolutions of pairs of functions, which in the applications might be piecewise smooth with jump discontinuities. The convolution of a pair of such functions is then continuous and piecewise smooth but its derivative can have jump discontinuities. For the analysis of the asymptotics of the convolution of a function f with the Fejér kernel, it will be useful to parametrize the circle by the interval (−π, π ]. We therefore denote kn (x) =
1 − cos(nx) sin2 (nx/2) = , 2 n sin (x/2) 2n sin2 (x/2)
Kn (eix ) = kn (x).
We have π 1 kn (x) dx = 1. 2π −π The convolution of kn with a 2π -periodic function f (θ) is defined by π 1 kn (x)f (θ + x) dx. (f ∗ kn )(θ) = 2π −π We rewrite (29), making use of (28), in the form π 1 kn (x)[f (θ + x) − f (θ)] dx. (f ∗ kn )(θ) = f (θ) + 2π −π
(27)
(28)
(29)
(30)
First we assume that the derivative f (θ) exists. Let R(x, θ; f ) = f (x + θ) − f (θ) − f (θ)x.
(31)
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DANIEL BUMP ET AL.
THEOREM 7. Let f be a 2π -periodic integrable function such that f (θ) exists, and such that R(x, θ; f ) = x 2 φθ (x) where φθ (x) is integrable as a function of x. Then π 1 R(x, θ; f ) dx + o(n−1 ). (32) (f ∗ kn )(θ) = f (θ) + 4π n −π sin2 (x/2) If φθ (x) exists and is integrable then the error term in (32) is O(n2 ). Proof. Using (30), π f (θ) π 1 xkn (x) dx + kn (x)R(x, θ; f ) dx. (f ∗ kn )(θ) = f (θ) + 2π −π 2π −π The first integral on the right vanishes since xkn (x) is odd. By using the second expression in (27), we may write π R(x, θ; f ) 1 dx − (f ∗ kn )(θ) = f (θ) + 4π n −π sin2 (x/2) π x2 1 φθ (x)cos(nx) dx. − 4π n −π sin2 (x/2) By the Riemann–Lebesgue Lemma, the last term is o(n−1 ). If φθ is integrable, then ✷ integration by parts shows that this term is O(n−2 ). The theorem follows. LEMMA 1. We have π 1 1 (1 + γ + log(2n)) + O(n−2 ). xkn (x) dx = 2π 0 nπ
(33)
Proof. Using the second expression in (27) converts the integral in (33) into π 1 4 1 1 − x(1 − cos(nx)) 2 + dx 4π n 0 x sin2 (x/2) (x/2)2 π π 1 1 1 1 − cos(nx) 1 − dx + x dx − = nπ 0 x 4nπ 0 sin2 (x/2) (x/2)2 π 1 1 1 − x cos(nx) dx. − 2 4nπ 0 sin (x/2) (x/2)2 The first integral on the right-hand side is 1 (log nπ + γ − Ci(nπ )), nπ where by (8) we have Ci(nπ ) = O(n−2 ). As for the second integral, π 1 1 1 − log(π/2) 1 − . x dx = 2 2 4nπ 0 nπ sin (x/2) (x/2)
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
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The last integral is O(n−2 ), as can be proved by integration by parts, and putting these together, we obtain the lemma. ✷ Next we assume that f may be discontinuous at θ = θ0 , but that its right and left derivatives f+ (θ0 ) and f− (θ0 ) both exist. Let f (θ0 ) + f+ (θ0 )x + R+ (x, θ0 ; f ), x > 0, f (x + θ0 ) = f (θ0 ) + f− (θ0 )x + R− (x, θ0 ; f ), x < 0. We obtain the asymptotics of (f ∗ kn )(θ) first when θ = θ0 , and later when θ is near to but different from θ0 . Let R+ (x, θ0 ; f ) if x > 0, (34) R(x, θ0 ; f ) = R− (x, θ0 ; f ) if x < 0. THEOREM 8. Let θ = θ0 , where f has a jump discontinuity. Assume that R(x, θ0 ; f ) = x 2 φθ0 (x), where φθ0 (x) is integrable as a function of x. Then f (θ0 ) − f− (θ0 ) (1 + γ + log(2n)) + (f ∗ kn )(θ0 ) = f (θ0 ) + + nπ π R(x, θ0 ; f ) 1 dx + o(n−1 ). + 4π n −π sin2 (x/2)
(35)
Proof. We have
0 1 kn (x)R− (x, θ0 ; f ) dx + (f ∗ kn )(θ0 ) = f (θ0 ) + 2π −π π 1 kn (x)R+ (x, θ0 ; f ) dx + + 2π 0 f+ (θ0 ) π f− (θ0 ) 0 xkn (x) dx + xkn (x) dx + 2π 2π −π 0 π 1 xkn (x) dx + = f (θ0 ) + [f+ (θ0 ) − f− (θ0 )] 2π 0 0 π R− (x, θ0 ; f ) R+ (x, θ0 ; f ) 1 dx + dx − + 4π n −π sin2 (x/2) sin2 (x/2) 0 0 R− (x, θ0 ; f ) 1 cos(nx) dx + − 4π n −π sin2 (x/2) π R+ (x, θ0 ; f ) cos(nx) dx . + sin2 (x/2) 0
The result follows from Lemma 1 and the Riemann–Lebesgue Lemma.
✷
The result of Theorem 7 is valid when f (θ) exists. Its leading term is consistent with Theorem 8, but the subsequent terms differ. Our goal is to obtain a uniform
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DANIEL BUMP ET AL.
expression. Let us now assume that θ is different from θ0 , where θ0 is a discontinuity of f . We will again rely on the Riemann–Lebesgue Lemma, of which we note the following refinement: b LEMMA 2. If φ is an integrable function on [−π, π ], then a cos(nx)φ(x) dx → 0 as n → ∞, uniformly in a and b for −π a b π . Proof. It is easy to see that if χ[a,b] denotes the characteristic function of [a, b] then (a, b) → χ[a,b] φ is a continuous map into L1 ([−π, π ]), so the set of such functions is compact, and the uniformity now follows from the remark to Theorem 2.8 on p. 13 of [12]. ✷ Let 1 K= 4π
π −π
x2 dx = 2.77259 . . . . sin2 (x/2)
LEMMA 3. Assume θ = 0. We have, uniformly in θ: π 1 kn (x) dx 2π θ 1 1 cos nθ − nπ + n Si(nθ) − 12 cot θ2 + O(n−2 ), if θ > 0, 2 θ = 1 1 cos nθ + nπ + n Si(nθ) − 12 cot θ2 + O(n−2 ), if θ < 0, 2 θ 1 2π
1 2π
(37)
π
xkn (x) dx |θ| |θ| |θ| 1 Ci(n|θ|) + cot − log sin + O(n−2 ), = πn 2 2 2
and
(36)
θ
π −π
x 2 kn (x) dx =
K + O(n−2 ). n
(38)
(39)
Proof. Assume that θ > 0. By (27) we have π π cos(nx) 1 1 1 − kn (x) dx = + 2 2π θ 4π n θ sin (x/2) (x/2)2 1 1 dx − 2 + cos(nx) (x/2)2 sin (x/2) θ 1 nπ cos(t) 1 cot − dt + O(n−2 ) = 2π n 2 π nθ t2 θ 1 cos(nπ ) cos(nθ) 1 cot − − + − = 2π n 2 π nπ nθ − Si(nπ ) + Si(nθ) + O(n−2 ).
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
119
Similarly (38) is an even function of θ since xkn (x) is an odd function of x, so we may assume θ > 0 in evaluating it. Then π 1 xkn (x) dx 2π θ π 1 4 1 1 dx − x(1 − cos(nx)) 2 + 2 = 4π n θ x sin (x/2) (x/2)2 1 = (log(π ) − log(θ) − Ci(nπ ) + Ci(nθ)) + πn π 1 1 1 x − dx − + 4π n θ sin2 (x/2) (x/2)2 π 1 1 1 − x cos(nx) dx − 4π n θ sin2 (x/2) (x/2)2 θ θ θ 1 Ci(nθ) − Ci(nπ ) + cot − log sin − = πn 2 2 2 π 1 1 1 − − x cos(nx) dx 4π n θ sin2 (x/2) (x/2)2 θ θ θ 1 Ci(nθ) + cot − log sin + O(n−2 ). = πn 2 2 2 Finally, the difference between the left and right sides of (39) is 1 4π n
π −π
x2 cos(nx) dx = O(n−2 ) sin2 (x/2)
since the integrand is regular at its endpoints.
✷
We will make use of the following function: 1 (x + π )2 , −2π x 0; H (x) = 21 (x − π )2 , 0 x 2π. 2 We note that this function satisfies H (x + 2π ) = H (x) when x and x + 2π are both in its range, so H has an extension to a 2π periodic function. Its derivative has a discontinuity at 0, which is unique modulo 2π . LEMMA 4. We have, recalling n (θ) from (10), (H ∗ kn )(θ) = H (θ) +
1 K − n (θ) + O(n−2 ). 2n n
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DANIEL BUMP ET AL.
Proof. We have, using the fact that kn is even, and Lemma 3: (H ∗ kn )(θ) =
1 4π
−θ
(x + θ + π )2 kn (x) dx +
−π π 1 + (x + θ − π )2 kn (x) dx 4π −θ π π 1 x 2 kn (x) dx − xkn (x) dx + = 4π −π θ π π 1 1 1 2 2 kn (x) dx + kn (x) dx − + 2 (θ + π ) 2π θ 2π −θ π π 1 1 kn (x) dx + kn (x) dx − πθ 2π θ 2π −θ 2 θ K 1 2 2 + 2 (θ + π ) − Ci(nθ) − log sin + = 2n n 2 + cos(nθ) + nθ Si(nθ) + O(n−2 ).
Adding π θ to the second term and subtracting it from the third gives K 1 + H (θ) − n (θ) + O(n−2 ). 2n n
✷
THEOREM 9. Suppose that f has jump discontinuities in its derivative at θi , and that these are the only discontinuities in f modulo 2π . Let αi =
1 (f (θi ) − f− (θi )). 2π +
Assume that φθ (x) = x −2 R(x, θ; f ) is integrable on (−π, π ), where R(x, θ; f ) is defined by (31), or by (34) if θ is a θi . Then with n as in (10), we have (f ∗ kn )(θ) = f (θ) + +
1 4π n
αi n−1 n (θ − θi ) +
i π −π
R(x, θ; f ) dx + o(n−1 ). 2 sin (x/2)
(40)
If θ → φθ is a continuous map of [−π, π ] to L1 ([−π, π ]), then (40) is uniform in θ. If φθ exists and is integrable, the error in (40) is O(n−2 ). In applying this theorem, note that as we move θ around the interval, we want to keep |θi − θ| π . This means that representatives θi are chosen differently
´ KERNEL UNITARY CORRELATIONS AND THE FEJER
121
depending on the location of θ. We’ve done this implicitly by defining n (θ) by (10) when |θ| π , and extending it to a 2π periodic function. Proof. Let f0 = f + i αi Hi , where Hi (x) = H (x−θi ). The function f0 is continuous and has a continuous first derivative, so we may apply Theorem 7. We have π 1 R(x, θ; f0 ) (41) dx + o(n−1 ). (f0 ∗ kn )(θ) − f0 (θ) = 4π n −π sin2 (x/2) If φθ (x) exists is integrable, then Theorem 7 further asserts that the error is O(n−2 ). This estimate may be shown to be uniform in θ along the lines of Lemma 2. By Lemma 4, the left side of (41) equals π 1 f (x + θ)kn (x) dx − f (θ) + 2π −π αi K − αi n−1 n (θ − θi ) + O(n−2 ). + 2n i i We check easily that for the quadratic functions Hi we have R(x, θ; Hi ) = 12 x 2 independent of θ and θi , so using (36) we have π π αi K 1 R(x, θ; f0 ) R(x, θ; f ) 1 dx = dx + . (42) 2 2 4π n −π sin (x/2) 4π n −π sin (x/2) 2n i ✷
Comparing, we obtain (40). In an important special case, the result can be made more explicit.
THEOREM 10. Let f be a continuous, piecewise linear 2π periodic function, and let θi and αi be as in Theorem 9. Then αi n−1 n (θ − θi ) + O(n−2 ) (f ∗ kn )(θ) = f (θ) + i
uniformly in θ. Proof. It is easy to see that φθ exists and is integrable. The theorem will thus follow from Theorem 9 provided we show that π R(x, θ; f ) 1 dx = 0. (43) 4π −π sin2 (x/2) Since f+ (θi ) = f− (θi+1 ), we have i αi = 0. Furthermore, on each interval (θi , θi+1 ) the function f0 defined in the proof of Theorem 9 is polynomial of degree 2, and the coefficient of x 2 is 12 i αi = 0, so f0 is piecewise linear with continuous derivative and 2π periodic; therefore f0 is constant. Thus ✷ R(x, θ; f0 ) = 0, and (43) now follows from (42). Let λ be a partition, and let kn,λ (θ) = Kn,λ (eiθ ).
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THEOREM 11. There exists a function n,λ (θ) such that if f is a piecewise linear and continuous 2π periodic function, then with αi as in Theorems 9 and 10, we have αi n−1 n,λ (θ − θi ) + O(n−2 ). (f ∗ kn,λ )(θ) = f (θ) + i
a trigonometric polynomial, and is independent of n. The function n,λ − n is Proof. Let f0 = f + i αi Hi , where Hi (θ) = H (θ − θi ) as in the proof of Theorems 9 and 10. It was shown in the proof of Theorem 10 that f0 is constant. By Theorem 10, we have (f ∗ kn,λ − f )(θ) αi Hi ∗ (kn,λ − kn )(θ) + O(n−2 ) = αi n−1 n (θ − θi ) + f0 − = αi n−1 n (θ − θi ) − αi Hi ∗ (kn,λ − kn )(θ) + O(n−2 ), (44) i
i
since f0 is constant and kn,λ − kn has mean value 0. By Theorem 3(ii), there exist polynomials fλ and gλ in t and t −1 such that (kn,λ − kn )(θ) =
1 (fλ (eiθ ) + einθ gλ (eiθ ) + e−inθ gλ (e−iθ )). n
Moreover, f (t) = f (t −1 ), so the latter expression may be written as an even trigonometric polynomial – a finite linear combination of functions cosk (θ) = cos(kθ). Substituting this into the right-hand side of (44), the convolution may be worked out using 1 cosk (θ − θi ). k2 Thus fλ contributes n−1 αi G(θ − θi ), where G is a fixed trigonometric polynomial, while gλ contributes terms of order O(n−2 ) which may be discarded. The theorem follows. ✷ (Hi ∗ cosk )(θ) =
References 1. 2. 3. 4.
Bump, D. and Diaconis, P.: Toeplitz minors, J. Combin. Theory Ser. A 97 (2002), 252–271. Coram, M. and Diaconis, P.: New tests of the correspondence between unitary eigenvalues and the zeros of Riemann’s zeta function, to appear in Ann. Statist. Diaconis, P. and Evans, S.: Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), 2615–2633. Diaconis, P. and Shahshahani, M.: On the eigenvalues of random matrices, In: Studies in Applied Probability, a special volume of J. Appl. Probab. A 31 (1994), 49–62.
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Dyson, F.: Statistical theory of the energy levels of complex systems, I, II, III, J. Math. Phys. 3 (1962), 140–156, 157–165, 166–175. Dyson, F.: Correlations between eigenvalues of a random matrix, Comm. Math. Phys. 19 (1970), 235–250. Fejér, L.: Untersuchungen über Fouriershe Reihen, Math. Ann. 58 (1904), 501–569. Goodman, R. and Wallach, N.: Representations and Invariants of the Classical Groups, Cambridge Univ. Press, 1998. Hejhal, D., Friedman, J., Gutzwiller, M. and Odlyzko, A. (eds): Emerging Applications of Number Theory, Springer-Verlag, New York, 1999. Hughes, C.: On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, Dissertation, University of Bristol, 2001. Hughes, C., Keating, J. and O’Connell, N.: On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), 429–451. Katznelson, Y.: An Introduction to Harmonic Analysis, 2nd edn, Dover, New York, 1976. Keating, J. and Snaith, N.: Random matrix theory and ζ ( 12 + it), Comm. Math. Phys. 214 (2000), 57–89. Keating, J. and Snaith, N.: Random matrix theory and L-functions at s = 12 , Comm. Math. Phys. 214 (2000), 91–110. Körner, T.: Fourier Analysis, Cambridge Univ. Press, 1988. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn, Oxford Univ. Press, 1995. Mehta, M.: Random Matrices, 2nd edn, Academic Press, New York, 1991. Pinsky, M.: Fejér asymptotics and the Hilbert transform, Preprint, Department of Math., Northwestern Univ., 17 April 2001. To appear in Amer. Math. Soc. Contemp. Math. Ser. (A. Seeger et al. (eds)). Rains, E.: High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107(2) (1997), 219–241. Soshnikov, A.: Level spacings distribution for large random matrices: Gaussian fluctuations, Ann. of Math. (2) 148 (1998), 573–617. Soshnikov, A.: Central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), 1353–1370. Stanley, R.: Enumerative Combinatorics, Cambridge Univ. Press, 1986, 1997, 1999. Taylor, M.: Multi-dimensional Fejér kernel asymptotics, Preprint, Dept. of Math., Univ. North Carolina, Chapel Hill, 2001. Tracy, C. and Widom, H.: Introduction to random matrices, In: Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992), Lecture Notes in Phys. 424, Springer-Verlag, New York, 1993, pp. 103–130. Tracy, C. and Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices, J. Statist. Phys. 92 (1998), 809–835. Wieand, K.: Eigenvalue distributions of random matrices in the permutation group and compact Lie groups, Harvard PhD Dissertation, 1998. Zygmund, A.: Trigonometric Series, 2nd edn, Cambridge Univ. Press, 1959.
Mathematical Physics, Analysis and Geometry 5: 125–143, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Trajectories Joining Two Submanifolds under the Action of Gravitational and Electromagnetic Fields on Static Spacetimes ROSSELLA BARTOLO1, and ANNA GERMINARIO2, 1 Dipartimento Interuniversitario di Matematica, Politecnico di Bari, Via E. Orabona, 4,
70125 Bari, Italy 2 Dipartimento Interuniversitario di Matematica, Università degli Studi di Bari, Via E. Orabona, 4, 70125 Bari, Italy (Received: 27 September 2001; accepted in final form: 14 March 2002) Abstract. In this paper we present existence and multiplicity results for orthogonal trajectories joining two submanifolds under the action of gravitational and electromagnetic fields on static spacetimes. These trajectories are critical points of unbounded functionals and they can be found by using a variant of the saddle point theorem and the relative category theory. Mathematics Subject Classifications (2000): 58E10, 58E05, 53C50. Key words: Lorentzian manifold, normal trajectory, saddle-points, relative category.
1. Introduction A pair (L, g) is called a Lorentzian manifold if L is a connected finite-dimensional smooth manifold with dim L 2 and g is a Lorentzian metric on L, that is g is a smooth, symmetric, two covariant tensor field such that, for any z ∈ L, the bilinear form g(z)[·, ·] induced on Tz L is nondegenerate and of index ν(g) = 1. The points of L are called events. A Lorentzian manifold (L, g) is called (standard) stationary if L is a product manifold L = M × R,
M any C 3 -connected manifold
and g can be written as ζ, ζ L = ξ, ξ + δ(x), ξ τ + δ(x), ξ τ − β(x)τ τ for any z = (x, t) ∈ L,
ζ = (ξ, τ ),
ζ = (ξ , τ ) ∈ Tz L = Tx M × R,
Work supported by MURST (ex 40% and 60% research funds). Work supported by MURST (ex 40% and 60% research funds).
(1)
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where ·, · , δ and β are, respectively, a Riemannian metric on M, a smooth vector field and a smooth scalar field on M. When δ ≡ 0, L is called (standard) static. We refer to [21] and [23] for all the background material assumed in this paper. Let us consider a smooth stationary vector field A on L, that is ∂t A1 (z) = ∂t A2 (z) = 0, thus A(z) = A(x, t) = A(x) = (A1 (x), A2 (x)),
∀z = (x, t) ∈ L.
In some recent papers, the existence and the multiplicity of trajectories (under the action of A) joining two events in L has been studied. Namely, fixed two events z, w ∈ L, the trajectories joining them satisfy the Euler–Lagrange equation associated to the functional introduced in [6] 1 1 1 γ˙ , γ˙ L ds + A (γ ), γ˙ L ds (2) F (γ ) = 2 0 0 on (z, w; L) = γ ∈ H 1 ([0, 1], L) | γ (0) = z, γ (1) = w , that is Ds γ˙ = ((A (γ ))∗ − A (γ ))[γ˙ ],
(3)
where A is the differential of the vector field A and (A (z))∗ denotes, for any z ∈ L, the adjoint operator of A (z) on Tz L with respect to ·, · L . This problem has been studied in [2] and [12] on complete stationary Lorentzian manifolds, in [3] on open subsets of stationary Lorentzian manifolds, and in [1] in a more general setting. It is clear that this problem generalizes the geodesic connectedness one (see, e.g., [7, 16]). In this paper we extend the results in [2] and [12]. Indeed, we shall look for orthogonal trajectories under a vectorial potential joining two given submanifolds of a stationary Lorentzian manifold L. DEFINITION 1.1. Let S, be two submanifolds of L. A curve γ : [0, 1] → L is called orthogonal trajectory (under the action of A) joining S to if (i) γ satisfies (3), γ (0) ∈ S, γ (1) ∈ , (ii) γ˙ (0) ∈ Tγ (0) S ⊥ , γ˙ (1) ∈ Tγ (1) ⊥ . This problem has been studied when A ≡ 0 in [19] and [9, 10], respectively, on stationary and on orthogonal splitting Lorentzian manifolds. For the sake of simplicity, we shall deal with static Lorentzian manifolds, although our results hold also for stationary Lorentzian manifolds under some additional assumptions on the coefficient δ (see (1)).
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Let P and Q be two submanifolds of M and let us set S1 = P × {0} ,
S2 = Q × {T },
T ∈ R,
S3 = Q × R.
(4)
We shall prove existence and multiplicity results for orthogonal trajectories joining, respectively, S1 to S2 and S1 to S3 . It can be easily proved (see Proposition 2.1) that, if A is orthogonal to Si , i = 1, 2, 3, that is A(z), ζ L = 0,
∀z ∈ S1 ∪ Si , ζ ∈ Tz (S1 ∪ Si ), i = 2, 3,
(5)
then the orthogonal trajectories joining S1 to Si (i = 2, 3) are the critical points of F at (2) on a suitable Hilbert manifold (see Section 2). Before stating our main results, we recall that a vector ζ ∈ Tz L is called timelike (respectively lightlike; spacelike) if ζ, ζ L < 0 (respectively, ζ, ζ L = 0, ζ = 0; ζ, ζ L > 0 or ζ = 0). We remark that Equation (3) has a prime integral, in fact d γ˙ , γ˙ L = 2Ds γ˙ , γ˙ L = (A (γ ))∗ [γ˙ ] − A (γ )[γ˙ ], γ˙ L = 0, ds hence if γ : [0, 1] → L is a trajectory, there exists Eγ ∈ R such that γ˙ (s), γ˙ (s) L = Eγ ,
∀s ∈ [0, 1].
(6)
Therefore a trajectory γ is said to be timelike, lightlike or spacelike according to the causal character of γ˙ . Let us assume that there exist η, b ∈ R such that 0 < η β(x) b,
∀x ∈ M;
(7)
there exist a1 , a2 ∈ R such that |A1 (x)| a1
and
0 A2 (x) a2 ,
∀x ∈ M;
P and Q are closed submanifolds of M such that P or Q is compact; P and Q are disjoint.
(8) (9) (10)
Remark 1.2. A Gauge transformation does not modify the set of the critical points of the functional F . Indeed adding to A any irrotational vector field Y independent of t, say Y = (∇V (x), a) with V ∈ C 2 (M, R) and a ∈ R, the critical points of the corresponding functional still satisfy (3) if (5) holds. Thus, it is enough that A + Y satisfies (8) for such Y (in particular, it suffices that A2 is bounded from below). The following theorems concern, respectively, the existence and the multiplicity of normal trajectories joining S1 to S2 and they will be proved in Section 3. We could consider S = P × {t }, t ∈ R, however, as the metric is stationary, there is not loss 1 0 0 of generality if we assume t0 = 0.
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THEOREM 1.3. Let L = M ×R be a static Lorentzian manifold with M complete. Assume that (5), (7), (8), (9), (10) hold. Then an orthogonal trajectory joining S1 to S2 exists. Moreover, T > 0 exists such that for any T ∈ R with |T | > T there is an orthogonal timelike trajectory joining S1 to S2 . THEOREM 1.4. Let the assumptions of Theorem 1.3 hold. If M is not contractible in itself and P , Q are both contractible in M, then (i) there exists a sequence {γm} of (spacelike) trajectories joining S1 to S2 such that limm→+∞ Eγm = +∞; (ii) denoted by N(T , S1 , S2 ) the number of the timelike orthogonal trajectories joining S1 to S2 , it results lim|T |→+∞ N(T , S1 , S2 ) = +∞. The previous results about spacelike trajectories in Theorems 1.3 and 1.4 have only a geometrical meaning, while the results concerning timelike trajectories have also a physical interpretation. Indeed, the Lorentz world-force law which determinates the motion of relativistic particles γ submitted to an electromagnetic field, is the Euler–Lagrange equation related to the action functional s1 1 s1 −γ˙ , γ˙ L ds + q A (γ ), γ˙ L ds, S(γ ) = −m0 c 2 s0 s0 where m0 is the rest mass of the particle, q is its charge, c is the speed of light (see [23, p. 88]). In [6], it is proved that for timelike trajectories the search of critical points of S is equivalent to that of the critical points of F . In particular, for Eγ < 0, the inertial mass turns out to be a constant of the motion, which is determined by the initial conditions and also the equality between the inertial and gravitational mass can be deduced (see [6]). Remark 1.5. We point out that if L is a static Lorentzian manifold and we replace (7) in Theorems 1.3 and 1.4 by there exist b ∈ R such that 0 < β(x) b ∀x ∈ M, we still get the existence of a trajectory, but we are not able to find timelike trajectories. The following theorems concern, respectively, the existence and the multiplicity of normal trajectories joining S1 to S3 and they will be proved in Section 4. THEOREM 1.6. Let the assumptions of Theorem 1.3 hold. Then there exists a (spacelike) orthogonal trajectory joining S1 to S3 . THEOREM 1.7. Let the assumptions of Theorem 1.4 hold. Then there exists a sequence {γm } of (spacelike) trajectories joining S1 to S3 . Remark 1.8. We point out that if γ = (x, t) is a normal trajectory joining S1 to S3 , as Tγ (1) S3 = Tx(1) Q × R, necessarily t˙(1) = 0, then, from (6) and (1), γ has to be spacelike.
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Remark 1.9. Evaluating the Fréchet differential of F it is clear that if P and Q are not disjoint and (5) holds, then for any x¯ ∈ P ∩ Q the curve γ¯ = (x, ¯ 0) is a trivial (lightlike) trajectory. Thus, assumption (10) is needed in Theorem 1.3 if T = 0 in order to prove the existence of a nontrivial normal trajectory, while, if T = 0, it is necessary only to prove that in some cases the normal trajectory is spacelike (see Remark 3.9). Moreover, if (10) does not hold, γ¯ is always a trivial trajectory joining S1 to S3 . Clearly, in the multiplicity results of Theorems 1.4 and 1.7 assumption (10) is not needed since there exist infinitely many nontrivial trajectories. We have already pointed out that normal trajectories joining S1 to Si , i = 2, 3, are the critical points of the functional F on suitable Hilbert manifolds. We remark that F is unbounded both from above and from below, also modulo compact perturbations, hence the search of its critical points is not trivial. Nevertheless, since the coefficients of the metric (1) do not depend on the variable t, it is possible to prove a variational principle (see Proposition 3.1 and [7]) which reduces our problem to the study of a functional depending only on the ‘spatial’ component. • If we look for normal trajectories γ = (x, t) joining S1 to S2 , as t (1) = T is fixed, the classical Ljusternik–Schnirelmann critical point theory can be applied (see Section 3): indeed the new functional is bounded from below if (7) and (8) hold, and satisfies the well-known Palais–Smale condition (see Definition 3.3). • If we look for normal trajectories γ = (x, t) joining S1 to S3 , as t (1) freely varies in R, the new functional is still unbounded both from below and from above, hence we shall use a different approach. Thanks to the stationarity of the metric, the functional F satisfies the Palais–Smale condition. Then we shall introduce a Galerkin approximation scheme in the variable t, and, by a variant of the Rabinowitz saddle point theorem, we shall find a critical point of F (i.e. a normal trajectory joining S1 to S3 ). In order to get multiplicity results, we shall use the relative category for unbounded functionals (see [13, 15, 25]). Remark 1.10. Plainly, if P and Q reduce respectively to {p} and {q}, then we obtain the results in [2] and [12] for trajectories under a vectorial potential joining two fixed events in L.
2. The Functional Setting Hereafter we shall assume that M is a submanifold of RN for N sufficiently large (see [20]), thus H 1 ([0, 1], L) = z ∈ H 1 ([0, 1], RN+1 ) | z([0, 1]) ⊂ L ,
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where H 1 ([0, 1], RN ) ≡ H 1,2 ([0, 1], RN ) = y ∈ L2 ([0, 1], RN ) | y is absolutely continuous, y˙ ∈ L2 ([0, 1], RN ) . We shall denote by · the usual norm on H 1 ([0, 1], RN ) and by · 2 the usual norm on L2 ([0, 1], RN ). Let us set for i = 2, 3 0(S1 , Si ; L) = z ∈ H 1 ([0, 1], L) | z(0) ∈ S1 , z(1) ∈ Si , then, for any z ∈ 0(S1 , Si ; L), i = 2, 3, Tz 0(S1 , Si ; L) = ζ ∈ Tz H 1 ([0, 1], L) | ζ(0) ∈ Tz(0)S1 , ζ(1) ∈ Tz(1) Si . By using standard arguments (see, e.g., [18]) we can prove the following proposition: PROPOSITION 2.1. Let γ ∈ 0(S1 , Si ; L), i = 2, 3 and assume that (5) holds. Then γ is a critical point of F if and only if it is an orthogonal trajectory joining S1 and Si , i = 2, 3. By Proposition 2.1, the orthogonal trajectories joining S1 to S2 are the critical points of F on ZT := 0(S1 , S2 ; L) = (P , Q; M) × H 1 (0, T ), where (P , Q; M) = x ∈ H 1 ([0, 1], M) | x(0) ∈ P , x(1) ∈ Q is a smooth submanifold of H 1 ([0, 1], M) (see [18]) and H 1 (0, T ) = t ∈ H 1 ([0, 1], R) | t (0) = 0, t (1) = T . For any z = (x, t) ∈ ZT , it results that Tz ZT = Tx (P , Q; M) × H01 ([0, 1], R), where Tx (P , Q; M) = ξ ∈ Tx H 1 ([0, 1], M) | ξ(0) ∈ Tx(0)P , ξ(1) ∈ Tx(1) Q and H01 ([0, 1], R) = τ ∈ H 1 ([0, 1], R) | τ (0) = 0 = τ (1) .
TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION
131
Remark 2.2. If γ = (x, t) is a trajectory joining S1 to S2 , (ii) of Definition 1.1 and (5) can be respectively written as x(0) ∈ P , t (0) = 0, x(1) ∈ Q, x(0) ˙ ∈ Tx(0)P ⊥ , x(1) ˙ ∈ Tx(1) Q⊥ ,
t (1) = T ,
A1 (x), ξ = 0 ∀x ∈ P ∪ Q ∀ξ ∈ Tx P ∪ Tx Q. On the other hand, by Proposition 2.1, the orthogonal trajectories joining S1 to S3 are the critical points of F on Z := 0(S1 , S3 ; L) = (P , Q; M) × W, where W = t ∈ H 1 ([0, 1], R) | t (0) = 0 . By virtue of the Poincaré inequality, the space W can be equipped with the norm equivalent to · t eq = t˙ 2 .
(11)
We remark that W = H01 ([0, 1], R) ⊕ Rj[0,1]
with j[0,1] : s ∈ [0, 1] → s ∈ R.
For any z = (x, t) ∈ Z, it results that Tz Z = Tx (P , Q; M) × W . Remark 2.3. If γ = (x, t) is a trajectory joining S1 to S3 , (ii) of Definition 1.1 can be written as x(0) ∈ P , t (0) = 0, x(1) ∈ Q, x(1) ˙ ∈ Tx(1) Q⊥ , x(0) ˙ ∈ Tx(0)P ⊥ ,
t˙(1) = 0.
3. Proof of Theorems 1.3 and 1.4 As pointed out in Section 2, normal trajectories joining S1 to S2 are the critical points of FT := F on ZT . We have already observed that, as for the geodesic problem on Lorentzian manifolds (see [5, 7]), the functional FT is strongly indefinite. We can overcome such difficulty by a slight variant of the variational principle in [2] which extends the one proved in [7] and which reduces the study of the orthogonal trajectories joining S1 to S2 to the search of the critical points of a suitable functional, defined only on (P , Q; M), which is bounded from below under our assumptions. Indeed, the following proposition holds. PROPOSITION 3.1. Let γ = (x, t) ∈ ZT . The following propositions are equivalent:
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ROSSELLA BARTOLO AND ANNA GERMINARIO
(a) γ is a critical point of FT ; (b) (i) x ∈ (P , Q; M) is a critical point of the C 2 functional JT defined on (P , Q; M) by 1 1 1 x, ˙ x ˙ ds + A1 (x), x ˙ ds + JT (x) = 2 0 0 1 2 1 T + 0 A2 (x) ds 1 1 2 , (12) β(x)A2 (x) ds − + 1 1 2 0 2 ds 0 β(x) (ii) t ∈ H 1 (0, T ) is the solution of the following Cauchy problem: H (x) − A2 (x), t (0) = 0, (13) t˙ = β(x) where 1 T + 0 A2 (x) ds . (14) H (x) = 1 1 ds 0 β(x) Moreover, if (a) or (b) is true, FT (γ ) = JT (x). Remark 3.2. From (7), (8), (12) and the Hölder inequality we get, for any x ∈ (P , Q; M), 2
a22 T 2 1 JT (x) 2 x + + T a2 , ˙ 2 − a1 x ˙ 2−b (15) 2 2 hence JT is bounded from below. In the remainder of this section we shall denote by X a C 2 Hilbert manifold endowed with a Riemannian metric. We shall prove Theorem 1.4 by using the Ljusternik–Schnirelmann category theory for functionals bounded from below. Let us recall some definitions and results (see, e.g., [24]). DEFINITION 3.3. Let f ∈ C 1 (X, R); f satisfies the Palais–Smale condition if every sequence {ym } such that {f (ym )} is bounded
(16)
lim f (ym ) ∗ = 0
(17)
and m→+∞
contains a converging subsequence (where · ∗ is the norm induced on the cotangent bundle by the Riemannian metric on X). DEFINITION 3.4. Let A be a subspace of X. The category of A in X, denoted by catX A, is the minimum number of closed and contractible subsets of X covering A (possibly +∞). We shall write cat X = catX X.
TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION
133
THEOREM 3.5. Let f ∈ C 1 (X, R) be a functional bounded from below, satisfying the Palais–Smale condition and let X be complete. Then f has at least cat X critical points. Moreover, if cat X = +∞, there exists a sequence {ym } of critical points of f such that limm→+∞ f (ym ) = +∞. We shall obtain multiplicity results thanks to Theorem 3.5 and the following theorem (see [11, 14]). THEOREM 3.6. Let M be a noncontractible in itself C 3 Riemannian manifold. Let P and Q be two submanifolds of M both contractible in M. Then there exists a sequence {Km } of compact subsets of (P , Q; M) such that lim cat(P ,Q;M) Km = +∞.
m→+∞
In order to prove the Palais–Smale condition, we recall the following lemma whose proof is essentially contained in [5] (see also [19]). LEMMA 3.7. Assume that P and Q are two closed submanifold of a complete Riemannian manifold M. Let {xm } be a sequence in (P , Q; M) weakly converging to a x ∈ H 1 ([0, 1], RN ). Then x ∈ (P , Q; M) and there exist two sequences {ξm } and {νm } in H 1 ([0, 1], RN ) such that xm − x = ξm + νm , ξm ∈ Txm (P , Q; M), ξm → 0 weakly in H 1 ([0, 1], RN ), νm → 0 strongly in H 1 ([0, 1], RN ). PROPOSITION 3.8. The functional JT (see (12)) satisfies the Palais–Smale condition. Proof. Let {xm } be a Palais–Smale sequence. By Remark 3.2, we get that { x˙m 2 } is bounded.
(18)
Assumption (9) and (18) imply that {xm } is bounded in H 1 ([0, 1], RN ). Hence, x ∈ H 1 ([0, 1], RN ) exists such that, up to a subsequence, xm → x
uniformly.
(19)
By Lemma, 3.7, x ∈ (P , Q; M) since P and Q are both closed in M. From (17), (12), and (14), we easily get
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ROSSELLA BARTOLO AND ANNA GERMINARIO
JT (xm )[ξm ]
1 ˙ = x˙m , ξm ds + A1 (xm )[ξm ], x˙m ds + 0 0 1 1 1 + A1 (xm ), ξ˙m ds + ∇β(xm ), ξm A22 (xm ) ds + 2 0 0 1 β(xm )A2 (xm )∇A2 (xm ), ξm ds − + 0 1 ∇A2 (xm ), ξm ds − −H (xm ) 0 1 1 2 ∇β(xm ), ξm − H (xm ) = o(1), (20) 2 β 2 (xm ) 0 1
where o(1) denotes an infinitesimal sequence and {ξm} is as in Lemma 3.7. From (18), the regularity of β, A1 , A2 , the uniform convergence of {ξm } to 0 and from (20) we get 1 x˙m , ξ˙m ds. (21) o(1) = 0
From (21) and Lemma 3.7, we obtain 1 x˙m , x˙m − x ˙ ds o(1) =
(22)
0
and since x˙m → x˙
weakly in L2 ([0, 1], RN ),
we have
1
o(1) =
1
x, ˙ x˙m − x ˙ ds +
0
x˙m − x, ˙ x˙m − x ˙ ds,
(23)
0
and then x˙m → x˙
strongly in L2 ([0, 1], RN ).
(24)
As L∞ ([0, 1], RN ) is embedded in L2 ([0, 1], RN ), from (19) we have xm → x
strongly in L2 ([0, 1], RN ).
(25)
From (24) and (25) we deduce that xm → x
strongly in H 1 ([0, 1], RN )
and the proof is complete.
✷
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Proof of Theorem 1.3. By Remark 3.2 and Proposition 3.8 we get the existence of a minimum point x of JT , that is an orthogonal trajectory joining S1 and S2 (see Propositions 3.1 and 2.1). As x is a minimum point for JT , it results cT := JT (x) JT (y),
∀y ∈ (P , Q; M).
Therefore, from (12), for a fixed y ∈ (P , Q; M) it results cT c1 − 12 ηT 2 for a suitable c1 > 0. Thus, set γ = (x, t) (see Proposition 3.1), from (6), (13) and (14) we get 1 1 1 Eγ = γ˙ , γ˙ L = cT − A (γ ), γ˙ L ds 2 2 0 1 1 |x| ˙ ds (26) c2 + c3 T − ηT 2 + a1 2 0 for suitable c2 , c3 > 0. By the Young inequality a1 x ˙ 2 14 x ˙ 22 + 4a12 ,
(27)
Equation (15), the Hölder inequality, and (26) we get 1 E c2 + c3 T − 12 ηT 2 + a1 K1 + K2 T + K3 T 2 2 γ for suitable K1 , K2 , K3 , thus the theorem is proved.
✷
Remark 3.9. If (10) holds, from (15) it is easy to see that, as dist(P , Q) > 0, if 14 dist(P , Q) − 4a12 > 0, then for |T | small enough the trajectory found in Theorem 1.3 is spacelike. Proof of Theorem 1.4. By the assumption made, Theorems 3.6, 3.5 and Proposition 3.8, we get the existence of a sequence {xm } of critical points of JT such that limm→+∞ JT (xm ) = +∞, thus by Proposition 3.1 we get the existence of a sequence {γm } of critical points of FT such that lim FT (γm ) = +∞.
(28)
m→+∞
From (6) we have for any m ∈ N 1 1 E = F (γ ) − A (γm ), γ˙m L ds. T m 2 γm 0
Standard calculations show that 1 E 2 γm
FT (γm ) − a1 x˙m 2 − ba2
b (T + a2 ) + a2 , η
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ROSSELLA BARTOLO AND ANNA GERMINARIO
hence from (27) and (15) it follows that
1 E 2 γm
FT (γm ) − a1 FT (γm ) +
16a12
T 2 a22 + + T a2 + 4b 2 2
−
b −ba2 (T + a2 ) + a2 η and from (28), (i) of Theorem 1.4 is proved. Next we shall prove that for any m ∈ N there exists T (m) > 0 such that for any T ∈ R, |T | > T (m), it results that N(T , S1 , S2 ) m. By Theorem 3.6 for any fixed m ∈ N there exists a compact subset Km of (P , Q; M) such that cat(P ,Q;M) Km m. Set cp = inf sup JT (x), B∈0p x∈B
p = 1, . . . , m,
where
0p = B ⊂ (P , Q; M) |
cat
(P ,Q;M)
Bp
(clearly 0p is not empty for any p = 1, . . . , m). Remark that the numbers cp , p = 1, . . . , m, are well defined, therefore there exist at least m critical points of JT corresponding to m critical points γ1 , . . . , γm of FT with critical values cp , p = 1, . . . , m. For any p = 1, . . . , m it results 1 1 Eγ = cp − A (γp ), γ˙p L ds 2 p 0 and, as Km is compact, cp = J (xp ) sup JT (x) c1 − 12 ηT 2 x∈Km
for a suitable c1 > 0. Then, reasoning as in the proof of Theorem 1.3, we easily get ✷ that Eγp < 0 for |T | large enough. 4. Proof of Theorems 1.6 and 1.7 We have already pointed out that the normal trajectories joining S1 to S3 are the critical points of F on Z (see Section 2). As the metric ·, · L does not depend on the time component, the functional F verifies the well-known Palais–Smale condition on Z. Indeed the following proposition holds. PROPOSITION 4.1. F satisfies the Palais–Smale condition on Z. Proof. Let {γm = (xm , tm )} ⊂ Z be a sequence satisfying (16) and (17). Set τm = tm ∈ W ≡ Ttm W . From (17) we get, in particular, the existence of an infinitesimal sequence {:m } such that :m t˙m 2 = F (γm )[(0, tm )] 1 2 ˙ β(xm )tm ds − =− 0
1 0
β(xm )A2 (xm )t˙m ds.
TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION
137
Thus from (16) and (8) we get :m t˙m 2 + 12 x˙m 22 − a1 x˙m 2 c for a suitable c ∈ R. Thus { t˙m 2 } is bounded
(29)
{ x˙m 2 } is bounded.
(30)
and
From (9) and (30) it follows that { xm } is bounded. Hence, from (11) and (29), it follows that {γm = (xm , tm )} is bounded in Z, thus there exists γ = (x, t) ∈ Z (in fact P and Q are closed) such that γm → γ
weakly in H 1 ([0, 1], RN+1 ).
(31)
Let us show that γm → γ
strongly in H 1 ([0, 1], RN+1 ).
Set τm = t − tm ∈ W , we have that τm → 0
weakly in H 1 ([0, 1], R).
(32)
From (17) F (γm )[(ξm , τm )] = o(1),
(33)
where ξm is as in Lemma 3.7 and o(1) denotes an infinitesimal sequence. From (29) and (30), the regularity of β, A1 , A2 and the uniform convergence of {ξm } and {τm } to 0, we get from (33) 1 1 ˙ x˙m , ξm ds − β(xm )t˙m τ˙m ds + 0
0
1
+
A1 (xm ), ξ˙m ds −
0
1
β(xm )A2 (xm )τ˙m ds = o(1).
0
Then, as tm = t − τm , from (32) we get 1 1 ˙ x˙m , ξm ds + β(xm )τ˙m2 ds = o(1), 0
0
therefore 1
0 1 0
x˙m , ξ˙m ds = o(1),
(34)
β(xm )τ˙m2 ds = o(1).
(35)
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ROSSELLA BARTOLO AND ANNA GERMINARIO
From (34) we can reason as in the proof of Proposition 3.8 obtaining xm → x
strongly in H 1 ([0, 1], RN )
and from (35), (32) we have τm → 0
strongly in H 1 ([0, 1], R) ✷
and the proof is complete.
Due to the indefiniteness of the metric ·, · L the functional F is unbounded on infinite-dimensional linear manifolds. Following [5], we shall introduce a finitedimensional Galerkin approximation scheme. Let us set, for any k ∈ N, Wk = Hk ⊕ Rj[0,1] ,
where Hk = span{sin πps | p = 1, . . . , k}
(see also Section 2). Our next aim is to find, for any k ∈ N, a critical point of F restricted to Zk = (P , Q; M) × Wk . Remark that by the same proof of Proposition 4.1, for any k ∈ N Fk := F|Zk satisfies the Palais–Smale condition. We shall use the following variant of the well known saddle point theorem (see [4, 22]). THEOREM 4.2. Let X be a complete Riemannian manifold, H a separable Hilbert space, Y a linear subspace of H with orthonormal basis {ym }, h ∈ H . Set W = Y + h,
Z = X × W,
Wk = span{yp | p = 1, . . . , k} + h
and
Zk = X × Wk
for any k ∈ N, S = {(x, y¯ + h) ∈ Z | x ∈ X}, y¯ ∈ Y, Q(R) = {(x, ¯ w) ∈ Z | w − h − y ¯ R},
x¯ ∈ X, R > 0.
Let f ∈ C (Z, R), assume that fk := f|Zk satisfies the Palais–Smale condition and that there exists R > 0 such that 1
sup f (Q(R)) < +∞,
sup f (∂Q(R)) < inf f (S).
Then, for any k ∈ N there exists a critical point of fk corresponding to a critical level ck such that inf f (S) ck sup f (Q(R)), ck = inf sup fk (g(x)), g∈0k x∈Qk (R)
where 0k = {g ∈ C(Zk , Zk ) | g(x) = x ∀x ∈ ∂Qk (R)}, ¯ w) ∈ Zk | w − h − y ¯ R}. Qk (R) = {(x,
139
TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION
PROPOSITION 4.3. For any k ∈ N there exists a critical point of Fk = F|Zk . Proof. Let us set S = {(x, j[0,1] ) ∈ Z | x ∈ (P , Q; M)}, Q(R) = {(y, t) ∈ Z | t − j[ 0,1] eq R},
(36) R>0
(see (11)), where y ∈ (P , Q; M) is fixed. From (7) and (8) it results that F (z) − 12 b − ba2 − 12 a12 ,
∀z = (x, j[0,1] ) ∈ S.
(37)
Again from (7) and (8), for any z = (y, t) ∈ Q(R), we easily get F (z) c − 12 η t 2eq + ba2 t eq
(38)
for a suitable c, therefore sup F (Q(R)) < +∞. We remark that |R − 1| t eq R + 1,
∀z = (y, t) ∈ ∂Q(R),
then from (38) we have for suitable c1 , c2 > 0 F (z) c1 − 12 ηR 2 + c2 R,
∀z ∈ ∂Q(R).
From (37), (38) and for R large enough it is sup F (∂Q(R)) < inf F (S). Hence by Theorem 4.2 we get, for any k ∈ N, the existence of a critical point γk of Fk such that inf F (S) Fk (γk ) sup F (Q(R)).
(39)
Proof of Theorem 1.6. By Proposition 4.3, we have the existence of a sequence {γk } ⊂ Z such that for any k ∈ N γk is a critical point of Fk and such that (39) holds. We shall prove that {γk } contains a subsequence converging in Z to a critical point γ of F , concluding the proof. Indeed, reasoning as in the proof of Proposition 4.1, we get the boundedness of the sequence {γk }, therefore there exists γ = (x, t) ∈ Z such that, up to a subsequence, γk → γ weakly in Z. Let Pk denote the orthogonal projection operator of W onto Wk for any k ∈ N. Set τk = Pk (t) − tk ,
for any k ∈ N.
Then τk → 0,
weakly in H 1 ([0, 1], R)
and Pk (t) → t,
strongly in H 1 ([0, 1], R).
Again, reasoning as in the proof of Proposition 4.1, we have that γk → γ strongly in Z. The same arguments used in [5, Lemma 3.4] show that γ is a critical point of F . ✷ We recall the notion of relative category (see [13, 15, 25]).
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ROSSELLA BARTOLO AND ANNA GERMINARIO
DEFINITION 4.4. Let Y, W be two closed subsets of a topological space X. The category of W in X relative to Y , denoted by catX,Y W , is the minimum number m (possibily +∞) such that there exist m + 1 closed and contractible subsets W0 , . . . , Wm , covering W and m + 1 functions fi ∈ C([0, 1] × Wi , X), i = 0, . . . , m, such that fi (0, w) = w, ∀w ∈ Wi , i = 0, . . . , m, f0 (1, w) ∈ Y, w ∈ W0 , f0 (s, y) ∈ Y, y ∈ W0 ∩ Y, s ∈ [0, 1], fi (1, w) = wi , w ∈ Wi , for some wi ∈ X, i = 1, . . . , m. In order to prove Theorem 1.7 we shall use the following multiplicity result (see [8] for the proof). THEOREM 4.5. Let X be a C 2 complete Riemannian manifold modelled on a Hilbert space and f ∈ C 1 (X, R). Assume that there exist two subsets C and D of X such that C is a closed strong deformation retract of X \ D and that inf f (z) > sup f (z),
z∈D
cat X > 0.
X,C
z∈C
Assume that f satisfies the Palais–Smale condition. Then f has at least catX,C X critical points in X with critical levels c inf f (D). Moreover, if catX,C X = +∞, then there exists a sequence {zm } of critical points of f such that lim f (zm ) = sup f (z).
m→+∞
z∈X
In order to apply Theorem 4.5 we need the following lemma. LEMMA 4.6. Let η, b as in (7) and a1 , a2 as in (8). Then, there exists h ∈ ˙ 2 ) it C(R+ , R+ ) such that for any z = (x, t) ∈ Zk satisfying t − j[0,1] eq = h( x results F (z) −b − 2ba2 − a12 . Proof. For any z = (x, t) ∈ Zk it results that F (z) x ˙ 22 − 12 η t 2eq + 12 a12 + ba2 t eq , therefore, as | t − j[0,1] eq − 1| t eq t − j[0,1] eq + 1,
∀t ∈ W,
we get F (z) x ˙ 22 − 12 η +
a12 + ba2 + (ba2 + η) t − j[0,1] eq − 12 η t − j[0,1] 2eq . (40) 2
Setting ba2 +1+ h(s) = η the lemma is proved.
b2 a22 8ba2 + 2s 2 + 3a12 + 2b , + η2 η
s 0, ✷
TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION
141
Let us set, for any k ∈ N, ˙ 2 )}. Ck = {z = (x, t) ∈ Zk | t − j[0,1] eq = h( x
(41)
Clearly, by Lemma 4.6 and (37), it follows that sup Fk (Ck ) < inf Fk (S),
(42)
where S is as in (36). Remark 4.7. For any k ∈ N the following results hold (see [10, Lemmas 7.3, 7.4] for the proof). (i) the set Ck at (41) is a closed strong deformation retract of Zk \ S; (ii) if M is a 1-connected Riemannian manifold then for any m ∈ N there exists a compact subset Km of Zk such that catZk ,Ck Km m. In the proof of Theorem 1.7 we can assume that M is a 1-connected Riemannian manifold. Indeed, if the fundamental group π1 (M) is not trivial and finite, (ii) holds (it suffices to consider the universal covering of M). On the other hand, if π1 (M) is not finite we can find a critical point of F on each connected component. Proof of Theorem 1.7. By virtue of Remark 4.7 and (42), Theorem 4.5 can be applied to each Fk , k ∈ N. Theorefore, for any k ∈ N we get the existence of a sequence {γmk } of critical points of Fk such that inf Fk (S) Fk (γmk ),
∀m ∈ N,
and, as in our case catZk ,Ck Zk = +∞ (see, e.g., [10, Theorem 3.10]), limm→+∞ Fk (γmk ) = +∞. We remark that the critical levels in Theorem 4.5 are characterized by Fk (γmk ) = inf sup Fk (z), k B∈0m z∈B
∀m ∈ N,
where
0mk = B ∈ Zk | B is closed, cat B m . Zk ,Ck
Fix m ∈ N; for any k ∈ N, z ∈ Zk it results that ˙ 22 + 12 a12 + ba2 + (ba2 + η)h( x ˙ 2) Fk (z) x (see (40)) hence, reasoning as in [10, Lemma 7.4] it results for a suitable cm > 0: Fk (γmk ) cm ,
∀k ∈ N.
(43)
Now fix c > 0. There exists m = m(c) ∈ N, independent of k ∈ N, such that for any m ∈ N, m m and B ∈ 0mk B ∩ (Ac × {j[0,1] }) = ∅,
(44)
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ROSSELLA BARTOLO AND ANNA GERMINARIO
where
1 Ac = x ∈ (P , Q; M) | 2
1
x, ˙ x ˙ ds c .
0
Indeed, if B ∈ 0mk is such that B ∩(Ac ×{j[0,1]}) is empty, then by (i) of Remark 4.7, it can be proved that cat
(P ,Q;M)
where
Ac cat Km m,
(45)
Zk ,Ck
1 A = x ∈ (P , Q; M) | 2
1
c
x, ˙ x ˙ ds c
0
(see [10] for the details). Remark that, as M is complete and (9) holds, the Riemannian action functional satisfies the Palais–Smale condition (see, e.g., [17]). Therefore cat(P ,Q;M) Ac is finite and from (45), for m large enough (44) holds. From (44) and (37) it follows that Fk (γmk ) c − 12 (b − 2ba2 − a12 ),
∀m m.
(46)
From (43) and (46), reasoning as in the proof of Theorem 1.6, we get, for any m m, the existence of a critical point γm of F satisfying c − 12 (b − 2ba2 − a12 ) F (γm ) cm . As c is arbitrary, we get lim F (γm ) = +∞.
m→+∞
References 1.
2. 3. 4. 5. 6. 7.
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8. 9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
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Candela, A., Giannoni, F. and Masiello, A.: Multiple critical points for indefinite functionals and applications, J. Differential Equations 155 (1999), 203–230. Candela, A. and Salvatore, A.: Light rays joining two submanifolds in space-times, J. Geom. Phys. 22 (1997), 281–297. Candela, A., Masiello, A. and Salvatore, A.: Existence and multiplicity of normal geodesics in Lorentzian manifolds, J. Geom. Anal. 10 (2000), 591–619. Canino, A.: On p-convex sets and geodesics, J. Differential Equations 75 (1988), 118–157. Caponio, E. and Masiello, A.: Trajectories for relativistic particles under the action of an electromagnetic force in a stationary space-time, Nonlinear Anal. 50 (2002), 71–89. Fadell, E.: Lectures in cohomological index theories of G-spaces with applications to critical point theory, Raccolta di seminari, Universitá della Calabria, 1985. Fadell, E. and Husseini, S.: Category of loop spaces of open subsets in Euclidean space, Nonlinear Anal. 17 (1991), 1153–1161. Fournier, G. and Willem, M.: Relative category and the calculus of variations, In: H. Berestycki, J. M. Coron and I. Ekeland (eds), Proc. ‘Variational Methods’, Birkhäuser, Basel, 1990, pp. 95–104. Giannoni, F. and Masiello, A.: On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal. 101(2) (1991), 340–369. Grove, K.: Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics, J. Differential Geom. 8 (1973), 207–223. Klingenberg, W.: Riemannian Geometry, De Gruyter, Berlin, 1982. Molina, J.: Alcune applicazioni della Teoria di Morse a varietá di Lorentz, PhD thesis, Univ. Pisa, 1996. Nash, J.: The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20–63. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983. Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. Math. 65, Amer. Math. Soc., Providence, 1984. Sachs, R. K. and Wu, H.: General Relativity for Mathematicians, Springer, New York, 1977. Schwartz, J. T.: Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. Szulkin, A.: A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal. 15 (1990), 725–739.
Mathematical Physics, Analysis and Geometry 5: 145–182, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in Rd LECH ZIELINSKI LMPA, Université du Littoral, B.P. 699, 62228 Calais Cedex, France. e-mail:
[email protected] and IMJ, Mathématiques, case 7012, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France (Received: 11 October 2001; in final form: 29 March 2002) Abstract. Let A = A0 + v(x) where A0 is a second-order uniformly elliptic self-adjoint operator in Rd and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A0 are Hölder continuous, we study the asymptotic behaviour of the counting function N (A, λ) (λ → ∞) with the remainder estimates depending on the regularity hypotheses. Our strongest regularity hypotheses involve Lipschitz continuity and give the remainder estimate N (A, λ)O(λ−µ ), where µ may take an arbitrary value strictly smaller than the best possible value known in the smooth case. In particular, our results are obtained without any hypothesis on critical points of the potential. Mathematics Subject Classification (2000): 35P20. Key words: spectral asymptotics, Weyl formula, Schrödinger operator, elliptic operator, pseudodifferential operator.
1. Introduction This paper is devoted to a study of a self-adjoint operator in L2 (Rd ), A = A0 + V ,
(1.1)
where A0 = − or more generally A0 is a second-order differential operator uniformly elliptic on Rd and V is the operator of multiplication by a polynomially growing function v. The operator A is bounded from below, its spectrum is discrete and we are interested in the asymptotic behaviour of the associated counting function N (A, λ), defined as the number of eigenvalues (counted with their multiplicities) smaller than λ. Numerous works (cf., e.g., [1, 6, 26, 28] and references therein) show that very weak hypotheses on the potential v are sufficient to establish the following asymptotic formula: −d dx dξ, N (A, λ) ∼ (2π ) a(x,ξ )<λ
where λ → ∞ and a(x, ξ ) = |ξ |2 + v(x) in the case A0 = − .
146
LECH ZIELINSKI
In this paper, we consider estimates of the form N (A, λ) − (2π )−d dx dξ Cλ−µ N (A, λ)
(1.2)
a(x,ξ )<λ
with a certain µ > 0, which are considered, e.g., in [5, 6, 10, 11] (cf. also [18, 22–24, 34] containing similar results for boundary value problems). Concerning the smooth case, we remark that the most precise spectral asymptotics have been obtained by means of microlocal analysis (cf., e.g., [12–14, 16, 17]) and usually one can find an optimal value of µ which cannot be improved in (1.2) (e.g., the explicit computation of eigenvalues for the harmonic oscillator shows that (1.2) holds if and only if µ 1). Concerning the problem we want to study, the estimates (1.2) with optimal values of µ have been proved in particular in [17, 25, 31] and we will present the corresponding results in details in the second part of this section. We note also that estimates (1.2) with smaller values of µ allow us to treat more general smooth operators in Rd by other methods in [2, 4, 15, 21, 27, 29]. The aim of this paper is to establish estimates (1.2) using Hölder continuity hypotheses in the spirit of [32, 34], i.e. the operator of multiplication by a Hölder continuous function is replaced by a suitable pseudodifferential operator and the new problem can be investigated by means of a microlocal analysis. Similarly as in [30] and [34], our assumptions allow us to establish (1.2) with µ by taking an arbitrary value strictly smaller than the optimal value in the smooth situation. The principal difficulty comes from the fact that a covering by conical neighbourhoods used in the elliptic case considered in [34] must be refined similarly as in the paper of A. Mohamed [25]. ± We note that minor changes (e.g., the support of the Fourier transform of fn,λ −1 −1 from Section 3 should be included in the interval [−can−1 h−1 n ; can hn ]) allow to develop the approach of [35] and to establish (1.2) with the optimal value of µ under slightly stronger regularity hypotheses. However, for simplicity, this paper will not treat the case of the optimal value of µ. 1.1. GENERAL DEFINITION OF A We denote x = (1+|x|2 )1/2 and assume that v is a measurable function satisfying x c v(x) C x C
(1.3)
for certain constants C, c > 0. We denote Dj = −i∂/∂xj and formally write Dj (aj,k (x)Dk ), A0 = 1j,kd
where aj,k = ak,j ∈ L∞ (Rd ) are such that aj,k (x)ξj ξk c|ξ |2 a0 (x, ξ ) := 1j,kd
holds for a certain c > 0.
(1.4)
147
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
These hypotheses allow us to define the self-adjoint operator A = A0 + V via the positive quadratic form given by the formula ∂ϕ(x) ∂ψ(x) A[ϕ, ψ] = aj,k (x) + v(x)ϕ(x)ψ(x) dx ∂xj ∂xk Rd 1j,kd
for ϕ, ψ ∈ C0∞ (Rd ) and clearly the resolvent of A is compact. 1.2. REGULARITY HYPOTHESES We fix 0 < r 1, 0 < ρ 1 − (r/2) and assume |x − y| c v(x)ρ ⇒ C −1 v(x) v(y) Cv(x), |x − y| 1 ⇒ |v(x) − v(y)| C|x − y|r v(x)1−ρ , |x − y| 1 ⇒ |aj,k (x) − aj,k (y)| C|x − y|r v(x)−ρ
(1.5) (1.6) (1.7)
for some C, c > 0. THEOREM 1.1. We assume the hypotheses (1.3)–(1.7) and denote a(x, ξ ) := a0 (x, ξ ) + v(x), h(x, ξ ) := v(x)−ρ ξ −r ,
(1.8) (1.9)
where ρ , r are arbitrary positive numbers satisfying ρ < ρ and r < r. If the constant C¯ > 0 is large enough, then we have the estimate N (A, λ) − (2π )−d dx dξ C¯ dx dξ. a<λ
¯ ¯ a(1−Ch)<λ
(1.10)
1.3. COMMENTS We describe some consequences of Theorem 1.1. First of all, we note that performing the integration with respect to ξ , we find dx dξ = (λ − v(x))d/2 ω(x) dx, (1.11) a(x,ξ )<λ
where
v(x)<λ
ω(x) =
dξ. a0 (x,ξ )<1
Therefore, introducing the function dx, V(λ) := v(x)<λ
(1.12)
148
LECH ZIELINSKI
we find that a<λ dx dξ C0 λd/2V(λ) holds for a certain constant C0 > 0 and it is easy to see that the additional assumption (cf. [28]) V(2λ) CV(λ) implies C0−1 λd/2 V(λ)
(1.13)
dx dξ C0 λd/2V(λ).
(1.14)
a<λ
Moreover, we have the following proposition: PROPOSITION 1.2. Assume that (1.13) holds and the dimension d 3. Then for every C¯ > 0, one can find C˜ > 0 such that dx dξ ¯ ¯ a(1−Ch)<λ
Remark 1.3. Similar but more complicated estimates are described in Section 9 in the case of the dimension d = 1 and 2. Let us discuss the important special case when the potential satisfies x m v(x) C x m ,
(1.16)
with certain constants C, m > 0. It is easy to see that (1.16) ensures the estimates C −1 λd/m V(λ) C0 λd/m, 0 v(x)−ρ dx C0 λ−ρ +d/m , v(x)<λ
with a certain C0 > 0, hence (1.10) with (1.15) imply −d N (A, λ) − (2π ) ˜ −µ+d(1/2+1/m) dx dξ Cλ
(1.17)
with µ = (r /2) + ρ and due to (1.14) −1 d(1/2+1/m) dx dξ C0 λd(1/2+1/m). C0 λ
(1.18)
a<λ
a<λ
Of course, (1.18) still holds if we put N (A, λ) instead of a<λ dx dξ , hence the estimate (1.17) can be also written in the form (1.2). The smooth situation with v satisfying (1.16) was considered by H. Tamura [30, 31], who proved the estimate (1.17) with the optimal value µ = 1/2 + 1/m assuming that |∂ α v(x)| Cα x m−|α| , |∂ α aj,k (x)| Cα x −|α| , x · ∇v(x) c|x|m , for |x| > c−1
(1.19) (1.20)
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
149
hold with certain constants Cα , c > 0. The same estimate follows from the paper of A. Mohamed [25], who considered a general microhyperbolic condition |∇v(x)| c|x|m−1 ,
for |x| > c−1
(1.20 )
instead of (1.20). Finally, the theory developed in the book of V. Ivrii [17] gives the possibility of skipping the microhyperbolic condition (1.20 ) and can be also applied to study problems with irregular coefficients as described in [18]. Since the assumptions (1.16), (1.19), imply (1.5), (1.6), (1.7) with r = 1, ρ = 1/m, we find that Theorem 1.1 gives the estimates (1.17) for every µ strictly smaller than the optimal value 1/2 + 1/m. The plan of the paper is detailed in Section 2 and we end this introduction by noting the possibility of considering irregular potentials. More precisely, let A be as in Theorem 1.1 and A˜ = A + V˜ = A0 + V + V˜ ,
(1.21)
where V˜ is the operator of multiplication by the real-valued measurable function v˜ satisfying |v(x)| ˜ Cv(x)1−κ
(1.22)
for certain 0 < κ 1 and C > 0. Since (1.22) implies v 1−κ A1−κ in the sense of quadratic forms (cf. [7]), we have A− CA1−κ A˜ A+ CA1−κ and the min-max principle (cf. [26]) implies ˜ λ) N (A − CA1−κ , λ). N (A + CA1−κ , λ) N (A,
(1.23)
Then it is easy to find λ0 large enough to ensure N (A + CA1−κ , λ) N (A, λ − 2Cλ1−κ ), N (A − CA1−κ , λ) N (A, λ + 2Cλ1−κ ), for λ λ0 and the asymptotic behaviour of N (A, λ ± 2Cλ1−κ ) can be described by using (1.10) with λ ± 2Cλ1−κ instead of λ. It is possible to consider other conditions on v˜ ensuring a decomposition v˜ = v1 + O(v 1−κ ) with v1 satisfying the hypotheses analogical to (1.5), (1.6) (cf., e.g., [34]) and we refer to [3] concerning the question of estimating the error term λ(1−2Cλ−κ )
(2.1)
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LECH ZIELINSKI
in the sense of quadratic forms. Then the min-max principle (cf. [26]) gives the inequalities between the corresponding counting functions N (P+ , λ) N (A, λ) N (P− , λ)
(2.2)
and it suffices to investigate the asymptotic behavior of N (P± , λ). Below in this section, we give the details concerning P± and a Weyl formula for the class of operators of this type is formulated in Theorem 2.1. The proof of Theorem 2.1 is based on a microlocal analysis and its first step is described in Section 3. In this step we replace the Weyl formula of Theorem 2.1 by suitable microlocal estimates formulated in Theorems 3.1 and 3.2. To obtain microlocal estimates of Theorems 3.1 and 3.2, we need to construct a suitable approximation of e−it P± . This construction is presented in Section 4 and it can be seen as a version of constructing a parabolic parametrix if t is imaginary negative (cf. [19]). In Section 5, we begin the investigation of the error that appears when the approximation replaces e−it P± . In particular, simple commutations of pseudodifferential operators with the unitary group e−it P± allow us to observe the fact that modulo negligible errors in our setting the microsupport is conserved by the group. In Section 6, we consider the commutator of e−it P± with xj (the operator of multiplication by the j th coordinate) and in Section 7 we end the proof of Theorem 2.1, using the results of Section 6 in a reasoning similar to the integration by parts described in Lemma 4.4. In Section 8, we present a construction of operators P± , completing the proof of Theorem 1.1 and Proposition 1.2 is proved in Section 9. 2.1. DEFINITION OF A CLASS OF WEIGHT FUNCTIONS Our approach is based on a pseudodifferential calculus with symbols estimates involving some special weight functions like |ξ |2 + v(x) for instance. For x, ξ ∈ Rd and c > 0 we introduce the sets Bx− (c) = {y ∈ Rd : |x − y| < cv(x)ρ },
(2.3)
Bξ+ (c) = {η ∈ Rd : |ξ − η| < c ξ },
(2.3 )
and the metric g˜x,ξ : Rd × Rd → ]0; ∞[ given by the formula g˜ x,ξ (y, η) = v(x)−2ρ |y|2 + ξ −2 |η|2 .
(2.4)
˜ will denote the set of continuous functions m: R × R → ]0; ∞[ Then Mc (g) satisfying d
C −1 (1 + |x| + |ξ |)−C m(x, ξ ) C(1 + |x| + |ξ |)C , m(x, ξ ) C (y, η) ∈ Bx− (c) × Bξ+ (c) ⇒ C −1 m(y, η) for a certain constant C > 0. We will prove the following theorem:
d
(2.5) (2.6)
151
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
˜ be THEOREM 2.1. Let us fix c, ¯ c, σ > 0 and define h by (1.9). Let p ∈ Mc¯ (g) such that p(x, ξ ) c(1 + |x| + |ξ |)c , |∂xβ ∂ξα p(x, ξ )| Cα,β p(x, ξ )h(x, ξ )1+σ |β| ξ |β|−|α|
(2.7) (2.8)
for every α ∈ Nd , β ∈ Nd \ {0}. We set p(x, ˜ ξ ) = ξ |∇ξ p(x, ξ )| + h(x, ξ )p(x, ξ )
(2.9)
and assume that p˜ ∈ Mc¯ (g). ˜ We assume, moreover, that ¯ × Bξ+ (c) ¯ ⇒ 4|∇η p(y, η) − ∇ξ p(x, ξ )| p(x, ˜ ξ ) ξ −1 , (2.10) (y, η) ∈ Bx− (c) the estimate ˜ ξ ) ξ −|α| |∂ξα p(x, ξ )| Cα p(x,
(2.10 )
holds for every α ∈ Nd \ {0} and p˜ Cp for some C > 0. Let p W (x, D) denote the Weyl quantization of p, i.e. the operator acting on ϕ ∈ C0∞ (Rd ) according to the formula W x+y −d i(x−y)ξ , ξ ϕ(y) dy dξ. p e p (x, D)ϕ (x) = (2π ) 2 Then its closure in L2 (Rd ) defines the self-adjoint operator P with discrete spectrum and its counting function N (P , λ) satisfies N (P , λ) − (2π )−d ¯ dx dξ C dx dξ (2.11) p<λ
¯ ¯ p(1−Ch)<λ
for a certain constant C¯ > 0. Then Theorem 1.1 follows from Theorem 2.1 and THEOREM 2.2. Assume that the hypotheses of Theorem 1.1 hold. Then it is possible to find p+ , p− ∈ C ∞ (Rd × Rd ) such that p = p± satisfy the hypotheses of Theorem 2.1 with c¯ > 0 and σ > 0 small enough. Moreover, (2.1) holds with P± W being the closure of p± (x, D) in L2 (Rd ) and |∂ξα (p± − a)(x, ξ )| Cα a(x, ξ )h(x, ξ )1+σ ξ −|α| holds for every α ∈ Nd . Indeed, the condition (2.12) implies a(1 − C0 h) p± a(1 + C0 h),
(2.12)
152
LECH ZIELINSKI
hence using a < λ < p± ⇒ a < λ < a(1 + C0 h) and p± < λ < a ⇒ a(1 − C0 h) < λ < a, we find
dx dξ − p± <λ
a<λ
dx dξ
dx dξ.
Moreover, for every C > 0, we can find C¯ > 0 such that dx dξ dx dξ, p± (1−Ch)<λ
(2.13)
a(1−C0 h)<λ
¯ ¯ a(1−Ch)<λ
(2.14)
hence it is clear that (2.2) and (2.11) with p = p± imply (1.10). 2.2. REMARKS CONCERNING WEIGHT FUNCTIONS It is easy to check ˜ ⇒ m + m, ˜ mm, ˜ m/m ˜ ∈ Mc (g). ˜ m, m ˜ ∈ Mc (g)
(2.15)
Further on, we assume c¯ < 1, which ensures the fact that m(x, ξ ) = ξ s belongs ˜ for every s ∈ R. Moreover, due to (1.5), we can assume that c¯ > 0 is to Mc¯ (g) ˜ In particular, h and 1 + |ξ |2 + v(x) belong to small enough to ensure v ∈ Mc¯ (g). ˜ and since there exist constants C, c > 0 such that Mc¯ (g) c(1 + |ξ |2 + v(x)) a(x, ξ ) C(1 + |ξ |2 + v(x)),
(2.16)
˜ it is clear that a ∈ Mc¯ (g). The next crucial point of our approach is the fact that c|ξ | |∇ξ a(x, ξ )| C|ξ |
(2.17)
holds for some C, c > 0. Indeed, to obtain the left inequality (2.17) it suffices to use the homogeneity of a0 with respect to ξ , writing c|ξ |2 2a0 (x, ξ ) = ξ · ∇ξ a0 (x, ξ ) |ξ ||∇ξ a0 (x, ξ )|. The properties of a imply the following useful fact: in order to find suitable p± defining P± in (2.1), it suffices to guarantee the estimates (2.8) and (2.12). Indeed, other hypotheses on p made in Theorem 2.1 follow thanks to
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
153
LEMMA 2.3. Assume that the hypotheses of Theorem 1.1 hold. Assume that p satisfies the estimates (2.8) for every α ∈ Nd , β ∈ Nd \ {0} and |∂ξα (p − a)(x, ξ )| Cα a(x, ξ )h(x, ξ )1+σ ξ −|α|
(2.18)
˜ and the formula (2.9) defines p˜ ∈ Mc¯ (g) ˜ such for every α ∈ Nd . Then p ∈ Mc¯ (g) that p˜ Cp, (2.10) and (2.10 ) hold for every α ∈ Nd \ {0}. Proof. Let p satisfy (2.18) and (2.8) for α ∈ Nd , β ∈ Nd \ {0}. Step 1. We check that p ∈ Mc¯ (g). ˜ Due to |p − a| C0 ah1+σ there is C¯ > 0 such that |x| + |ξ | C¯ ⇒ C0 h(x, ξ ) 1/2 ⇒ 1/2 (p/a)(x, ξ ) 2. By continuity of the quotient there is C > 0 such that C −1 < a/p < C and ˜ ⇒ p ∈ Mc¯ (g). ˜ a ∈ Mc¯ (g) ˜ p˜ Cp. Since C −1 < Step 2. We define p˜ by (2.9) and check that p˜ ∈ Mc¯ (g), a/p < C and ξ |∇ξ (p − a)| C1 hp, introducing a(x, ˜ ξ ) := ξ |∇ξ a(x, ξ )| + (ha)(x, ξ ),
(2.19)
we obtain ˜ a˜ ξ |∇ξ p| + (C + C1 )hp C2 p,
(2.20)
˜ p˜ ξ |∇ξ a| + (C + C1 )hp C2 a.
(2.20 )
However, due to (2.17) and ha 1, we can find C > 0 such that C −1 (1 + |ξ |2 + ha) a˜ C(1 + |ξ |2 + ha), hence applying (2.20), (2.20 ) we can find C¯ > 0 such that ¯ + |ξ |2 + ha). C¯ −1 (1 + |ξ |2 + ha) p˜ C(1
(2.21)
˜ (2.21) ensures p˜ ∈ Mc¯ (g). ˜ Finally a˜ Ca ⇒ Since 1 + |ξ |2 + ha ∈ Mc¯ (g), p˜ C p. Step 3. We check that (2.10 ) holds for every α ∈ Nd \ {0}. Since (2.21) implies ˜ ξ ), ξ 2 C¯ p(x,
(2.21 )
it is clear that for α ∈ Nd \ {0} we have ˜ ξ ) ξ −|α| |∂ξα a(x, ξ )| C ξ 2−|α| C C¯ p(x,
(2.22)
and using (2.18) with hp p, ˜ we obtain ˜ |∂ξα (p − a)| Cα ξ −|α| h1+σ p Cα ξ −|α| hσ p. Therefore combining (2.22) and (2.23) we obtain (2.10 ).
(2.23)
154
LECH ZIELINSKI
Step 4. We show that (2.10) holds if c¯ > 0 is small enough. Further on we assume y ∈ Bx (c). ¯ Then (1.7) implies |y − x| 1 ⇒ |aj,k (y) − aj,k (x)| C c¯r v(x)(r−1)ρ C c¯r .
(2.24)
Next we check that ¯ ⇒ |aj,k (y) − aj,k (x)| C(1 + |y − x|)v(x)−ρ . y ∈ Bx (c)
(2.25)
Indeed, it suffices to choose n ∈ N such that |y − x| n 1 + |y − x| and note that (1.5)–(1.7) allow us to estimate |aj,k (y) − aj,k (x)| by n
|aj,k (y + k(x − y)/n) − aj,k (y + (k − 1)(x − y)/n)| Cnv(x)−ρ .
k=1
¯ we have Using (2.25), we can affirm that under the hypothesis y ∈ Bx (c) |y − x| 1 ⇒ |aj,k (y) − aj,k (x)| 2C|y − x|v(x)−ρ 2C c¯ 2C c¯r
(2.26)
and, combining (2.24), (2.26), (2.21 ), we obtain |∇ξ a(y, ξ ) − ∇ξ a(x, ξ )| C˜ c¯r |ξ | C˜ C¯ c¯r p(x, ˜ ξ ) ξ −1 .
(2.27)
¯ then If, moreover, η ∈ Bξ+ (c), ˜ − ξ | C˜ c ξ ¯ C˜ C¯ c¯r p(x, ˜ ξ ) ξ −1 (2.28) |∇η a(x, η) − ∇ξ a(x, ξ )| C|η ¯ we obtain and, choosing c¯r 1/(16C˜ C), ¯ × Bξ+ (c) ¯ ⇒ |∇η a(y, η) − ∇ξ a(x, ξ )| 18 p(x, ˜ ξ ) ξ −1 . (2.29) (y, η) ∈ Bx− (c) 1 for |x| + |ξ | C0 and If C0 > 0 is large enough to ensure Cα h(x, ξ )σ 16 |α| = 1 in the right-hand side of (2.23), then (2.29) ensures (2.10) for |x| + |ξ | C0 . It remains to note that the region {|x| + |ξ | C0 } is compact, hence ✷ (2.10) holds for all (x, ξ ) ∈ R2d if c¯ > 0 small enough.
3. Microlocal Partition of Unity To begin we precise some details of our notations. We write R∗ = R\{0}, N∗ = N\{0} and, for Z ⊂ Rk (k ∈ N∗ ), we denote by 1Z the characteristic function of Z defined on Rk . We denote by B(L2 (Rd )) the algebra of bounded operators on L2 (Rd ) and ||·|| is the norm of B(L2 (Rd )). If Q is of trace class on L2 (Rd ), then its trace norm ||Q||tr = tr(QQ∗ )1/2 , where Q∗ denotes the adjoint of Q. Moreover b(x, D) denotes the standard pseudodifferential operator acting according to the formula −d ˆ ) dξ, eixξ b(x, ξ )ϕ(ξ (b(x, D)ϕ)(x) = (2π )
155
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
where ϕˆ is the Fourier transform of ϕ ∈ C0∞ (Rd ). Due to (1.5) the metrics g˜ x− (y) = v(x)−2ρ |y|2 and g˜ξ+ (η) = ξ −2 |η|2 are slowly varying in the sense of the Definition 18.4.1 in [16]. Thus Lemma 18.4.4 in [16] ensures that for any c¯ > 0 small enough one can choose a sequence {x(k)} ¯ k∈N∗ (respectively {ξ¯ (k)}k∈N∗ ) of points in Rd giving a covering of Rd by the family − + ¯ k∈N∗ (respectively {Bξ(k) ¯ k∈N∗ ) satisfying {Bx(k) ¯ (c)} ¯ (c)} ∞
1B −
x(k) ¯
(c) ¯ (x) Nc¯
and
k=1
∞
1B +¯
ξ (k)
(c) ¯ (ξ )
Nc¯
(3.1)
k=1
for a certain Nc¯ ∈ N. Moreover, we can find real valued functions θk− ∈ − + ¯ θk+ ∈ C0∞ (Bξ(k) ¯ satisfying C0∞ (Bx(k) ¯ (c/2)) ¯ (c/2)), ∞
(θk− )2
=
k=1 |∂ α θk− (x)|
∞
(θk+ )2 = 1,
k=1
Cα v(x)−ρ|α| ,
|∂ α θk+ (ξ )| Cα ξ −|α| .
(3.2)
− + + Let 5− k denote the operator of multiplication by θk and 5k = θk (D). Then
I=
∞ ∞ k− =1 k+ =1
− 2 + 5+ k+ (5k− ) 5k+ =
∞
L∗n Ln ,
(3.2 )
n=1
5+ = where n → (kn− , kn+ ) is a fixed bijection N∗ → N∗ × N∗ and Ln = 5− kn− kn+ ln (x, D) with ln (x, ξ ) = θk−− (x)θk++ (ξ ). n
n
Let λ ∈ R. Since 1]−∞, λ[ denotes the characteristic function of the interval ]−∞, λ[, the corresponding spectral projector of P can be written as 1]−∞, λ[ (P ) and due to (3.2 ) and the trace cyclicity, N (P , λ) = tr 1]−∞, λ[ (P ) =
∞
tr Ln 1]−∞, λ[ (P ) L∗n .
(3.3)
n=1
Expression (3.3) allows us to replace Theorem 2.1 by THEOREM 3.1. Let N0 ∈ N. Then one can find a constant C = C(N0 ) such that for every n ∈ N∗ one has the estimate 2 tr Ln 1]−∞; λ[ (P )L∗ − (2π )−d l (x, ξ ) dx dξ n n p<λ N0 1Bn (c) (3.4) C ¯ (x, ξ ) dx dξ + Chn , λ−pn hn
156
LECH ZIELINSKI
where we have denoted pn := p(x(k ¯ n− ), ξ¯ (kn+ )),
hn := h(x(k ¯ n− ), ξ¯ (kn+ )), − ¯ := Bx(k (c) ¯ × Bξ+ ¯ Bn (c) ¯ (k + ) (c). ¯ −) n
n
Let us check that Theorem 3.1 implies Theorem 2.1. Due to (3.3) and 1, it suffices to show that for N0 ∈ N large enough, the quantity ∞ ∞ 0 1Bn (c)¯ (x, ξ ) dx dξ + hN n n=1
λ−pn hn
∞
2 n=1 ln
=
(3.5)
n=1
can be estimated by the right-hand side of (2.11). ˜ and mn = m(x(k ¯ n− ), ξ¯ (kn+ )), then If m ∈ Mc¯ (g) ¯ ⇒ C −1 mn m(x, ξ ) Cmn . (x, ξ ) ∈ Bn (c)
(3.6)
¯ we have the implication In particular for (x, ξ ) ∈ Bn (c) λ − pn hn < p(x, ξ ) < λ + pn hn ⇒ p(1 − Ch)(x, ξ ) < λ < p(1 + Ch)(x, ξ ) and due to (3.1) we find the desired estimate of the first sum of (3.5). ¯ c0 > 0 and To estimate the second sum of (3.5), we note that vol(Bn (c)) ∞ ∞ 0 0 hN c0−1 hN ¯ C h(x, ξ )N0 dx dξ < ∞ n n vol(Bn (c)) n=1
n=1
if N0 is large enough. 3.1. APPROXIMATION OF THE CHARACTERISTIC FUNCTION Let f+ ∈ C0∞ (]0; 1[) and f− ∈ C0∞ (]−1; 0[) be such that f± 0. For λ ∈ R and n ∈ N∗ we define ∞ τ − λ dτ ± (s) := f± . fn,λ hn pn hn pn s
1]−∞,λ[
∞
−∞
f± = 1 and
(3.7)
Then we have − + 1]−∞, λ[ fn,λ 1]−∞, λ+hn pn [ . 1]−∞, λ−hn pn [ fn,λ
(3.8)
± satisfies Moreover, for k 1, the kth derivative of fn,λ ± (k) ) ⊂ [λ − hn pn ; λ + hn pn ], supp(fn,λ ± (k) −k ) | Ck h−k |(fn,λ n pn .
(3.9(k))
± ± denote the inverse Fourier transform of fn,λ . Then Let fˇn,λ ± (t) = ieit λ fˇ± (hn pn t), t fˇn,λ
where fˇ± ∈ S(R) is the inverse Fourier transform of f ± .
(3.10)
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
157
± ⊂ {0} and for every N ∈ N we have Due to (3.10), sing supp fˇn,λ ± (t)| CN hn pn t −N . |t fˇn,λ
(3.11)
The inequalities (3.8) imply − + (P )L∗n tr Ln 1]−∞, λ] (P )L∗n tr Ln fn,λ (P )L∗n tr Ln fn,λ
(3.12)
and it suffices to prove that (3.4) holds with tr Ln 1]−∞, λ] (P )L∗n replaced by ∞ ± ± ∗ dt fˇn,λ (t) tr Ln e−it P L∗n , (3.13) tr Ln fn,λ (P )Ln = −∞
where the integral in a neighbourhood of 0 can be considered as the distribution ± acting on the smooth function t → tr Ln e−it P L∗n . fˇn,λ To recover the behaviour of tr Ln e−it P L∗n we will construct a sequence of operators {QN¯ ,n (t)}N¯ ∈N being a suitable approximation of Ln e−it P . More precisely: instead of Theorem 3.1 it suffices to prove Theorem 3.2. THEOREM 3.2. There exists a sequence of operators {QN¯ ,n (t)}N¯ ∈N such that ∞ ± −it P ∗ ∗ 0 ˇ dt fn,λ (t) tr Ln e Ln − tr QN¯ ,n (t)Ln ChN (3.14) n , −∞ ∞ ± dt fˇn,λ (t) tr QN¯ ,n (t)L∗n − (2π )−d ln (x, ξ )2 dx dξ −∞ p<λ 1Bn (c) (3.15) C ¯ (x, ξ ) dx dξ λ−pn hn
hold if N¯ = N¯ (N0 ) ∈ N and C = C(N¯ , N0 ) > 0 are large enough. Further on we denote [0; t] = {st ∈ R : 0 s 1} and V = {(n, t, τ ) : n ∈ N∗ , t ∈ R∗ , τ ∈ [0; t]}.
(3.16)
Introducing d Q˜ N¯ ,n (t) = QN¯ ,n (t) + iQN¯ ,n (t)P dt and assuming QN¯ ,n (0) = Ln , we can formally write t d QN¯ ,n (t − τ )e−iτ P dτ Ln e−it P − QN¯ ,n (t) = dτ 0 t dτ Q˜ N¯ ,n (t − τ )e−iτ P , =−
(3.17)
0
hence in order to obtain the estimates (3.14), it suffices to prove the existence of a constant C = C(N¯ , N0 ) (independent of (n, t, τ )) such that tr Q˜ N¯ ,n (t − τ )e−iτ P L∗ ChN0 hn pn t C (3.18) n
holds for (n, t, τ ) ∈ V.
n
158
LECH ZIELINSKI
The construction of QN,n ¯ (t) presented in the next section uses pseudodifferential operators and the end of this section is devoted to a description of suitable classes of symbols. 3.2. CLASSES OF SYMBOLS We consider Rd × Rd equipped with the metric gx,ξ (y, η) = h(x, ξ )2σ ξ 2 |y|2 + ξ −2 |η|2 . ˜ then S(m, g) denotes the class of functions b ∈ C ∞ (Rd × Rd ) If m ∈ Mc¯ (g) satisfying |∂xβ ∂ξα b(x, ξ )| Cα,β m(x, ξ )h(x, ξ )σ |β| ξ |β|−|α|
(3.19)
for every α, β ∈ Nd . We recall some basic properties of S(m, g) described in Ch. 18 of [16]: PROPOSITION 3.3. Let b ∈ S(m, g) and b˜ ∈ S(m, ˜ g). Then bb˜ ∈ S(mm, ˜ g),
|b| m ⇒ 1/b ∈ S(1/m, g)
(3.20)
˜ g) such that and there exist b0 ∈ S(m, g), b b˜ ∈ S(mm, bW (x, D) = b0 (x, D),
˜ W (x, D). bW (x, D)b˜ W (x, D) = (b b)
(3.21)
Moreover, ∂xj ∂ξj b ∈ S(m , g) (j = 1, . . . , d) ⇒ b − b0 ∈ S(m , g), ˜ , g), ∂ξj b ∈ S(m , g), ∂xj b ∈ S(m , g), ∂ξj b˜ ∈ S(m
(3.22)
˜ + m m ˜ , g). (3.23) ˜ , g) (j = 1, . . . , d) ⇒ b b˜ − bb˜ ∈ S(m m ∂xj b˜ ∈ S(m We remark that the inequalities (2.8), (2.18) still hold if their left-hand sides contain p + r with r ∈ S(h1+σ p, g) instead of p + r. In particular we have COROLLARY 3.4. If p satisfies (2.8), then using (3.22) with b = p, and m = h1+σ p, we find p0 such that p W (x, D) = p0 (x, D)
and
p − p0 ∈ S(h1+σ p, g).
(3.24)
Moreover, the estimates (2.8), (2.18) hold with p0 instead of p in the left-hand side of each inequality. In Section 4 we consider sequences of symbols {bn }n∈N∗ and later on the families of symbols {bν }ν∈V . We adopt the convention that the notation bn ∈ S(m, g) or bν ∈ S(m, g) always means that the estimates (3.19) hold with bn or bν instead of b with constants Cα,β independent of n ∈ N∗ or ν ∈ V.
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
159
4. Construction of the Approximation In this section N¯ ∈ N is fixed and we describe the construction of QN¯ ,n (t) = e−itp qN¯ ,n (t) (x, D)
(4.25)
such that (3.15) holds. The proof of (3.14) will be given in Sections 5–7. For N = 0, . . . , N¯ we are going to consider 0 t k qk,n (x, ξ ), (4.2(N)) qN,n (t)(x, ξ ) = qN,n (t, x, ξ ) = 0kN 0 = ln and for k 1 we have where qN,n (0) = q0,n 0 qk,n ∈ S(hp(hσ/2p) ˜ k−1 , g),
0 supp qk,n ⊂ Bn (c/2). ¯
(4.3(k))
We construct qN,n (t) by induction with respect to N and its final step corresponding to N = N¯ gives qN¯ ,n (t) to be used in (4.1). For a smooth function (t, x, ξ ) → b(t, x, ξ ) ∈ C we denote (−i)|α| ∂ξa b(t)∂xα p0 /α!, PN¯ b(t) = |α|N¯
where p0 is given by (3.24). PROPOSITION 4.1. Let N = 0, . . . , N¯ . Then we can find qN,n (t) satisfying 0 satisfying (4.3(k)) for k = 1, . . . , N and (4.2(N)) with qN,n (0) = ln , qk,n 0 (t) (4.4(N)) (∂t + iPN¯ ) e−itp qN,n (t) = e−itp q˜N,n holds with
0 (t)(x, ξ ) = q˜N,n
0 t k q˜N,k,n (x, ξ ),
(4.5(N))
NkN+N¯ 0 ∈ S(hp(hσ/2p) ˜ k , g) q˜N,k,n
for k = N, . . . , N + N¯
(4.6(N))
0 ⊂ Bn (c/2). ¯ and supp q˜N,k,n Proof. We introduce the notation P˜N¯ q(t) = eitp (∂t + iPN¯ ) e−itp q(t) .
If q(t, x, ξ ) = q0 (x, ξ ) is independent of t, then t k q˜k0 P˜N¯ q(t) = 0kN¯
(4.7)
160
LECH ZIELINSKI
with q˜00 = i(p¯ 0 − p)q0 − q˜k0 =
(−i)|α|+1 ∂ξα (q0 ∂xα p0 )/α!,
1|α|N¯
cα0 ,...,αk ∂ξα0 (q0 ∂xα0 +···+αk p0 )∂ξα1 p . . . ∂ξαk p
¯ (k = 1, . . . , N).
|α0 +···+αk |N¯ αj !=0 if j!=0
Using the fact that the estimates (2.8) still hold with p0 instead of p, we find q0 ∈ S(m, g) ⇒ ∂ξα0 (q0 ∂xα p0 ) ∈ S(mph1+σ |α| ξ |α|−|α0 | , g)
(4.8)
if α != 0. Moreover, the estimates (2.10 ) imply ∂ξα1 p . . . ∂ξαk p ∈ S(p˜ k ξ −|α1 |−···−|αk | , g)
(4.9)
if α1 , . . . , αk != 0. Combining (4.8), (4.9) with α = α0 + · · · + αk ⇒ |α| max{k, 1} in the expression of q˜k0 we find q0 ∈ S(m, g) ⇒ q˜k0 ∈ S(mph1+σ max{k, 1} p˜ k , g)
(4.10)
(where in the case k = 0 we use, moreover, (3.24)). Therefore in the case N = 0 when we take q(t) = ln ∈ S(1, g) in (4.7), we 0 of the form (4.5(0)) and (4.10) with m = 1 imply (4.6(0)), obtain (4.4(0)) with q˜0,n i.e. Proposition 4.1 holds for N = 0. Further on, we assume that the statement of Proposition 4.1 holds for a given N N¯ − 1 and we prove that it still holds for N + 1 instead of N. Using the induction hypotheses (4.4(N)), (4.5(N)) to express P˜N¯ qN,n (t) we find P˜N¯ qN+1,n (t) 0 ) + P˜N¯ qN,n (t) = P˜N¯ (t N+1 qN+1,n N+1 0 0 0 + q˜N,N,n + P˜N¯ qN+1,n +t = t N (N + 1)qN+1,n
0 t k q˜N,k,n .
N+1kN+N¯
In order to obtain (4.5(N + 1)), it suffices to cancel the term with t N taking 0 0 = −q˜N,N,n /(N + 1) qN+1,n
(4.11)
0 ∈ S(hp(hσ/2p) ˜ N , g) by the induction hypothesis (4.6(N)). and q˜N,N,n Let us introduce the following notation: 0 0 t k S(m(k), g) ⇐⇒ an (t) = t k ak,n with ak,n ∈ S(m(k), g). an (t) ∈ k∈K
k∈K
Then using ph p˜ and max{k, 1} (k + 1)/2, we can write (4.7), (4.10) in the following form: t k S m(hσ/2 p) ˜ k+1 , g . (4.12) q(t) = q0 ∈ S(m, g) ⇒ P˜N¯ q(t) ∈ 0kN¯
161
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
˜ N , we find Since (4.11) gives (4.3(N + 1)), applying (4.12) with m = ph(hσ/2p) 0 t N+1 P˜N¯ qN,n+1 ∈ t N+1+k S(ph(hσ/2p) ˜ N+k+1 , g) 0kN¯
and (4.6(N + 1)) follows.
✷
PROPOSITION 4.2. Let QN¯ ,n (t) be defined by (4.1) with qN¯ ,n given by (4.2(N¯ )) 0 0 ¯ Then (3.15) holds. = ln and qk,n satisfy (4.3(k)) for k = 1, . . . , N. where qn,0 ∞ ∗ d d Proof. For k ∈ N, n ∈ N , q ∈ C0 (R × R ) and λ ∈ R we denote ∞ ± dt fˇn,λ (t)t k Jt (q), (4.13) Nk,n (q, λ) = −∞
where Jt (q) = (2π )
−d
e−itp(x,ξ ) q(x, ξ ) dx dξ.
(4.14)
± ± (k) (t) is the Fourier inverse of i k (fn,λ ) , changing the order of inteSince t k fˇn,λ grals (4.13) and (4.14), we find ± (k) −d ) p(x, ξ ) q(x, ξ ) dx dξ. (4.15) i k (fn,λ Nk,n (q, λ) = (2π )
Since (e−itp q)(x, D)L∗n has the integral kernel −d e−itp(x,ξ )q(x, ξ )ln (y, ξ ) dξ, (x, y) → (2π ) we have tr L∗n (e−itp q)(x, D) = Jt (qln ) and ∞ ± 0 dt fˇn,λ (t) tr QN¯ ,n (t)L∗n = Nk,n (qk,n ln , λ). −∞
(4.16)
0kN¯
0 = ln , hence (4.15) and (3.8) give For k = 0, we have q0,n 0 −d 2 N0,n (q ln , λ) − (2π ) ln (x, ξ ) dx dξ 0,n p<λ ± −d 2 fn,λ − 1]−∞,λ[ p(x, ξ ) ln (x, ξ ) dx dξ = (2π ) 1Bn (c) ¯ (x, ξ ) dx dξ. λ−hn pn
It remains to show that the estimate |Nk,n (qn , λ)| C¯ λ−hn pn
1Bn (c)¯ (x, ξ ) dx dξ
(4.17(k))
162
LECH ZIELINSKI
The expression (4.15) allows us to write the obvious inequality ± (k) |Nk,n (qn , λ)| (fn,λ ) p(x, ξ ) |qn (x, ξ )| dx dξ.
(4.18(k))
In the case k = 1, the above inequality leads to the following corollary: ¯ then |qn | COROLLARY 4.3. If qn ∈ S(ph, g) are such that supp qn ⊂ Bn (c), Chn pn and using (3.9(1)) in (4.18(1)), we obtain (4.17(1)). In particular, (4.17(1)) 0 ln . holds with qn = q1,n 0 In order to prove (4.17(k)) for qn = qk,n ln , k 2, let χ ∈ C0∞ (]−2; 2[) be such that χ = 1 on [−1; 1] and for s > 0 let
χs (x, ξ ) = χ(m(x, ξ )/s 2 ) with m = ξ 2 |∇ξ p|2 h−2 p −2 .
(4.19)
We also consider χ˜s := 1 − χs and remark that using 1 + m ∈ S(1 + m, g), it is easy to check that χs , χ˜s ∈ S(1, g). Moreover (x, ξ ) ∈ supp χs ⇒ p(x, ˜ ξ ) = ((1 + m1/2 )hp)(x, ξ ) (1 + 2s)(hp)(x, ξ ).
(4.20)
0 ln χs | Ck hkn pnk and using (3.9(k)) in (4.18(k)) we find that Therefore |qk,n 0 ln χs , k 2. (4.17(k)) holds with qn = qk,n 0 ln χ˜s , k 2. Thus it remains to show that (4.17(k)) holds with qn = qk,n To obtain this result, it suffices to show
˜ g) such that LEMMA 4.4. If qn ∈ S(m, g), then we can find q˜n ∈ S(m/p, supp q˜n ⊂ supp qn and tJt (qn χ˜s ) = Jt (q˜n χ˜ s/2 ).
(4.21)
Indeed, iterating the assertion of Lemma 4.4, we can write ¯
0 ln χ˜s ) = t k−1 Jt (q1,k,n χ˜s/2 ) = · · · = t k−k Jt (qk,k,n χ˜ s/2k¯ ) t k Jt (qk,n ¯
(4.22)
¯
∈ S(php˜ k−1−k , g) with supp qk,k,n ⊂ Bn (c/2). ¯ Thus using (4.22) for some qk,k,n ¯ ¯ with k¯ = k − 1 we obtain 0 ln χ˜s , λ) = N1,n (qn , λ) Nk,n (qk,n
¯ and (4.17(k)) holds due to Corollary 4.3. with qn ∈ S(hp, g), supp qn ⊂ Bn (c/2), Proof of Lemma 4.4. Using ξ 2 |∇ξ p|2 ∈ S(p˜ 2 , g) and ˜ ξ )2 2s 2 (hp)(x, ξ )2 + 2 ξ 2 |∇ξ p(x, ξ )|2 s 2 p(x, 2(s 2 + 1) ξ 2 |∇ξ p(x, ξ )|2
163
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
for (x, ξ ) ∈ supp χ˜s , we obtain χ˜s ξ −2 |∇ξ p|−2 ∈ S(p˜ −2 , g) due to (3.20) and combining with ξ 2 ∂ξj p ∈ S( ξ p, ˜ g), we can define ˜ g). χ˜ j,s := χ˜s ∂ξj p|∇ξ p|−2 ∈ S( ξ /p, Writing qn χ˜s = dj=1 qn χ˜j,s ∂ξj p and integrating by parts, we find −d i∂ξj e−itp(x,ξ ) (qn χ˜j,s )(x, ξ ) dx dξ tJt (qn χ˜j,s ∂ξj p) = (2π )
(4.23)
= −iJt (∂ξj (qn χ˜ j,s )), ˜ g) and supp χ˜j,s ∩ which completes the proof due to ∂ξj (qn χ˜j,s ) ∈ S(m/p, ✷ supp χs/2 = ∅.
5. Preliminary Remarks about the Approximation Error In this section we begin a study of tr Q˜ N¯ ,n (t − τ )e−iτ P L∗n with the purpose of establishing (3.18), which implies (3.14). To abbreviate notations, we denote the elements of V by the letter ν, adopting the following convention: if the index ν appears in a formula simultaneously with a letter n, t or τ , then ν = (n, t, τ ). Moreover, the notation sν = s˜ν + O(mn,t ) means that |sn,t,τ − s˜n,t,τ | Cmn,t holds with a constant C > 0 independent of ν = (n, t, τ ) ∈ V. Assume that Qν = (e−itp qν )(x, D) with qν ∈ S(m, g),
supp qν ⊂ Bn (c) ¯
and
Yν ∈ B(L2 (Rd ))
for ν ∈ V.
Then Lemma 9.1 allows us to estimate k+C0 0 %Yν %, |t k tr Qν e−iτ P Yν | |t|k %Qν %tr %Yν % Cmn h−C n hn pn t
(5.1)
where mn are as in (3.6). The family {Yν }ν∈V ⊂ B(L2 (Rd )) will be called negligible, if for every N ∈ N, we can find C(N) > 0 such that C(N) . (5.2) %Yν % = O hN n hn pn t If N , k ∈ N and {Yν }ν∈V is negligible, then (5.1) allows to find C(k, N ) > 0 such that C(k,N ) . (5.3) t k tr Qν e−iτ P Yν = O hN n hn pn t PROPOSITION 5.1. Let c > c/2 ¯ and consider bν ∈ S(m, g) satisfying supp bν ∩ Bn (c) = ∅. Then the family Rν = bνW (x, D)e−iτ P L∗n is negligible.
(5.4)
164
LECH ZIELINSKI
Before giving the proof of this result we describe its consequences. We assume c¯ > c > c > c/2 ¯ and note that the method of Ch. 18.4 in [16] (cf. − ˜− the beginning of Section 3) allows us to find θ˜k− ∈ C0∞ (Bx(k) ¯ (c)) such that θk = 1 − on Bx(k) ¯ (c ) and |∂ α θ˜n− (x)| Cα v(x)−ρ|α| . ˜− ˜− If 5 k denotes the operator of multiplication by θk , then Proposition 5.1 ensures − − ˜ − )e−iτ P L∗n and P (I − 5 ˜ − )e−iτ P L∗n are negligible, hence for every that (I − 5 kn kn N ∈ N, we can find C(N) > 0 such that −iτ P ∗ C(N) ˜ N,n ˜ −− e−iτ P L∗n + O hN Ln = tr Q˜ N¯ ,n (t − τ )5 . (5.5) tr Q ¯ (t − τ )e n hn pn t kn
˜ −− instead of Q˜ N¯ ,n (t − τ ) Further on, we prefer using the operators Q˜ N¯ ,n (t − τ )5 kn and for this purpose we introduce new notations. 5.1. SYMBOLS DEPENDING ON (x, ξ, y) ∈ Rd × Rd × Rd If qν ∈ C0∞ (Rd × Rd × Rd ), then Op(ei(τ −t )p qν ) will denote the integral operator with the kernel −d ei(x−y)ξ +i(τ −t )p(x,ξ )qν (x, ξ, y) dξ. (5.6) (x, y) → Kν (q, x, y) = (2π ) For c > 0 and n ∈ N∗ , we will denote − B˜ n (c) := B − − (c) × B + ¯ + ×B x(k ¯ n)
x(k ¯ n− )
ξ (kn )
(c)
˜ ˜ we will write qν ∈ S(m) if and only if there is c < c¯ such that and, for m ∈ Mc¯ (g), ∞ ˜ qν ∈ C0 (Bn (c)) and β ∂ξα qν (x, ξ, y)| Cα,β mn hσn |β| ξ |β|−|α| |∂x,y
holds for every α ∈ Nd , β ∈ N2d , where mn are as in (3.6). These notations will be used below to express the approximation error in a particular form, similar to (4.13), (4.14), (4.16). ˜ −− e−iτp L∗n 5.2. EXPRESSIONS OF tr Q˜ N,n ¯ (t − τ )5 k n
We have p W (x, D) = p0 (x, D) = p0 (x, D)∗ and
QN¯ ,n (t)p0 (x, D)∗ = Op e−itp q˜N¯ ,n,t
with q˜N¯ ,n,t (x, ξ, y) = qN¯ ,n (t)(x, ξ )p0 (y, ξ ).
165
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
Writing Taylor’s development of q˜N¯ ,n,t (x, ξ, ·) in x and applying standard integrations by parts in the integrals of the form (5.6) based on the equality (x − y)α ei(x−y)ξ = i |α| ∂ξα ei(x−y)ξ , we find
Q˜ N¯ ,n (t) = Op e−itp (q˜N0¯ ,n + q˜N1¯ ,n )(t) ,
where q˜N0¯ ,n (t) are as in Proposition 4.1 and the remainder term of Taylor’s development of order N¯ gives 1 ¯ ds(1 − s)N q˜N¯ ,n (t, s, x, ξ, y), (5.7) q˜N1¯ ,n (t, x, ξ, y) = eitp(x,ξ )(N¯ + 1) 0
with q˜N¯ ,n (t, s, x, ξ, y) (−i)|α| ∂ξα qN¯ ,n (t)e−itp (x, ξ )∂xα p0 (x + s(y − x), ξ ) /α!. = |α|=N¯ +1
We can write q˜N0¯ ,n (t − τ, x, ξ )θ˜k−− (y) = n
=
N¯ k2N¯
(t − τ )k q˜N0¯ ,k,n (x, ξ )θ˜k−− (y) n
t k q˜N0¯ ,k,ν (x, ξ, y),
N¯ k2N¯
where ν = (n, t, τ ) according to our convention and τ k 0 0 q˜N¯ ,k,n (x, ξ )θ˜k−− (y). q˜N¯ ,k,ν (x, ξ, y) = 1 − n t ¯
Nσ/2 k ˜ p˜ ) and using the form of qN0¯ ,n (t) in (5.7), we find It is clear that q˜N0¯ ,k,ν ∈ S(ph a similar expression t k q˜N1¯ ,k,ν (x, ξ, y) q˜N1¯ ,n (t − τ )(x, ξ, y)θ˜k−− (y) = n
0k2N¯
¯ Nσ/2 ˜ p˜ k ). with q˜N1¯ ,k,ν ∈ S(ph
˜ then the quantity If Yν ∈ B(L2 (Rd )) and qν ∈ S(m), Jν (q, Y ) := tr Op(ei(τ −t )p qν )e−iτ P Yν
(5.8)
is well defined due to Lemma 9.1 and, analogically to (5.1), we have k+C0 0 %Yν %, |t k Jν (q, Y )| Cmn h−C n hn pn t
where mn are as in (3.6). Using the above notation, we can state the following conclusion:
(5.9)
166
LECH ZIELINSKI ¯
Nσ/2 k ˜ COROLLARY 5.2. There exist q˜N¯ ,k,ν ∈ S(ph p˜ ) such that ˜ N¯ ,n (t − τ )5 ˜ −− e−iτ P L∗n = t k Jν (q˜N¯ ,k , Y ) tr Q k n
(5.10)
0k2N¯
holds with Yν = L∗n . We will complete the proof of Theorem 2.1 by showing the following proposition: ˜ PROPOSITION 5.3. Assume N0 , k ∈ N∗ , Yν = L∗n and qν ∈ S(m). Then we can find k0 ∈ N∗ and C(N0 ) > 0 such that C(N0 ) 0 t k Jν (q, Y ) = Jν (qk¯ , Yk¯ ) + O hN (5.11) n hn pn t ¯ 0 1kk
holds for certain families of symbols {qk,ν ¯ }ν∈V and operators {Yk,ν ¯ }ν∈V satisfying C(N0 ) ˜ p˜ −k ), %Yk,ν (5.12) qk,ν ¯ ∈ S(m ¯ % = O hn pn t for k¯ = 1, . . . , k0 . ¯
Indeed, using (5.10) and Proposition 5.3 with qν = q˜N¯ ,k,ν , m = phNσ/2 p˜ k , we can write ˜ N¯ ,n (t − τ )5 ˜ −− e−iτ P L∗n tr Q kn N ¯ 0 ) C(N,N 0 = Jν (qN¯ ,k¯ , YN, , ¯ k¯ ) + O hn hn pn t ¯ ¯ 1kk( N)
with ¯ Nσ/2 ˜ ), qN¯ ,k,ν ¯ ∈ S(ph
¯ 0) C(N,N %YN, , ¯ k,ν ¯ % = O hn pn t
hence choosing N¯ = N¯ (N0 ) large enough, we can ensure N ¯ N¯ ,N0 ) C( 0 , Jν (qN¯ ,k¯ , YN, ¯ k¯ ) = O hn hn pn t due to (5.9). This proves (3.18), completing the proof of Theorems 3.2 and 2.1. Proof of Proposition 5.1. The method of Ch. 18.4 in [16] allows us to find the symbols ln0 (x, ξ ) = θk0−− (x)θk0++ (ξ ) n
n
satisfying ∂xj ln0 ∈ S(v(x)−ρ , g), ln0 ∈ S(1, g), ¯ ln0 = 1 on Bn (c ) with c > c > c/2.
supp ln0 ⊂ Bn (c),
(5.13)
167
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
We set L0n = ln0 (x, D) and note that %bνW (x, D)L0n % = O(hN n ) for every N ∈ N 0 holds due to supp bν ∩ supp ln = ∅ (cf. Theorems 18.5.4 and 18.6.3 in [16]). Therefore, it suffices to show that k (5.14(k)) %(I − L0n )e−iτ P L∗n % = O hkσ n hn pn τ holds for every k ∈ N if ln0 satisfies (5.13). Obviously (5.14(k)) holds for k = 0 and we will prove the general statement by induction with respect to k ∈ N. Further on, we assume that σ > 0 is small enough to ensure ρ (1 + σ ) ρ and r (1 + σ ) 1, implying v(x)−ρ ξ h(x, ξ )1+σ . Then ∂xj p ∈ S( ξ ph1+σ , g), ∂ξj p ∈ S( ξ −1 p, g),
∂ξj ln0 ∈ S( ξ −1 , g), ∂xj ln0 ∈ S(v(x)−ρ , g) ⊂ S( ξ h1+σ , g)
and (3.23) ensures [I − L0n , P ] = −[L0n , P ] = l˜nW (x, D)
with l˜n ∈ S(ph1+σ , g).
(5.15)
Moreover, we can find ∂xj ln1 ∈ S(v(x)−ρ , g), ln1 ∈ S(1, g), ¯ ln1 = 1 on Bn (c ) with c > c > c/2
supp ln1 ⊂ Bn (c ),
(5.16)
and setting L1n = ln1 (x, D), we have supp l˜n ∩ supp ln1 = ∅, hence %l˜nW (x, D)L1n % = O(hN n)
for every N ∈ N.
(5.17)
Since %(I − L0n )L∗n % = O(hN n ) for every N ∈ N and
(I − L0n )e−iτ P L∗n = e−iτ P (I − L0n )L∗n + I − L0n , e−iτ P L∗n ,
(5.18)
it remains to estimate the norm of 1
dsτ ei(s−1)τ P [I − L0n , P ]e−isτ P L∗n I − L0n , e−iτ P L∗n =
(5.19)
0
and due to (5.17) it suffices to estimate |τ |%l˜nW (x, D)(I − L1n )e−isτ P L∗n %.
(5.20)
(cf. Theorem 18.6.3 in [16]) and using However, we have %l˜nW (x, D)% Cpn h1+σ n the induction hypothesis, we can assume that (5.14(k)) holds with L1n instead of L0n and c instead of c . Thus, the quantity (5.20) can be estimated by kσ k (k+1)σ hn pn τ k+1 , C|τ |pn h1+σ n hn hn pn τ Chn
completing the proof of (5.14(k + 1)).
✷
168
LECH ZIELINSKI
6. Auxiliary Commutator Formulas 6.1. NOTATIONS We will write bν ∈ Sn (m, g) if and only if bν ∈ S(m, g) and there exists c0 < c, ¯ ln0 ∈ S(1, g) satisfying supp ln0 ⊂ Bn (c0 ) and (1 − ln0 )bν ∈ S(hN , g) for every N ∈ ˜ b b˜ ∈ Sn (mm, ˜ g) if bν ∈ Sn (m, g), N. Then Theorem 18.5.4 in [16] ensures bb, ˜bν ∈ S(m, ˜ S(m m ˜ g) and (3.23) still holds with b, b, ˜ + m m ˜ , g) replaced by ˜ +m m ˜ , g). Moreover, bν ∈ Sn (m, g) implies bν , b˜ν , Sn (m m |∂xβ ∂ξα bν (x, ξ )| Cα,β mn hσn |β| ξ |β|−|α| ,
(6.1)
where mn are as in (3.6) and Theorem 8.6.3 [16] ensures %bνW (x, D)% Cmn ,
%(I − ln0 W (x, D))bνW (x, D)% = O(hN n)
(6.2)
for every N ∈ N. We introduce the following formal notation: Y (τ, B) := e−iτ P Beiτ P .
(6.3)
˜ We write Let {Yν }ν∈V be a family of bounded operators and let m ∈ Mc¯ (g). Yν ∈ Y(m) if and only if there exist N ∈ N, C0 > 0, the weights m(k, k ) ∈ ˜ the symbols bk,k ,ν ∈ Sn (m(k, k ), g) and functions sk,k : [0; 1]N → R, Mc¯ (g), sk,ν : [0; 1]N → C, satisfying N
m(k, k ) m,
|sk,k (w)| C0 ,
|sk,ν (w)| C0 hn pn t C0 ,
k =1
for k, k = 1, . . . , N and N dwsk,ν (w)Y (sk,1 (w)τ, Bk,1,ν ) . . . Y (sk,N (w)τ, Bk,N,ν ) + Rν , (6.4) Yν = k=1
[0; 1]N
W where Bk,k ,ν = bk,k ,ν (x, D) and the family {Rν }ν∈V is negligible. Taking sk,ν (w) = sk,k (w) = 1, we can forget the integration with respect to w, hence
Y (τ, Bk,1,ν ) . . . Y (τ, Bk,N,ν ) ∈ Y(m) and, more generally, ˜ ⇒ Yν Y˜ν ∈ Y(mm), ˜ Yν ∈ Y(m), Y˜ν ∈ Y(m) C Yν ∈ Y(m) ⇒ %Yν % Cmn hn pn t .
6.2. REFORMULATION OF PROPOSITION In Section 7 we will prove
5.3
(6.5) (6.6)
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
˜ Y0,ν ∈ Y(1) and N0 ∈ N. Then one can PROPOSITION 6.1. Assume q0,ν ∈ S(m), ∗ find k0 ∈ N and C(N0 ) > 0 such that (Jν (qk¯ , Yk¯ ) + tJν (q−k¯ , Y−k¯ ))+ tJν (q0 , Y0 ) = ¯ 0 1kk
C(N0 ) 0 + O hN n hn pn t
(6.7)
holds with certain symbols q±k,ν ¯ and operators Y±k,ν ¯ satisfying ˜ qk,ν p), ˜ ¯ ∈ S(m/
σ ˜ q−k,ν ), ¯ ∈ S(mh
Y±k,ν ¯ ∈ Y(1)
(6.8)
for k¯ = 1, . . . , k0 . It is easy to see that Proposition 6.1 implies Proposition 5.3. Indeed, first of all we note that the assertion of Proposition 6.1 can be applied to express tJν (q−k¯ , Y−k¯ ), k¯ = 1, . . . , k0 and iterating this procedure N times, we find the expression of Nσ ˜ ), k¯ = 1, . . . , kN . tJν (q0 , Y0 ) in the form (6.7) with new symbols q−k,ν ¯ ∈ S(mh Thus, for N = N(N0 ) large enough, all terms tJν (q−k¯ , Y−k¯ ), k¯ = 1, . . . , kN , beC(N0 ) 0 ), i.e. the assertion of Proposition 6.1 holds with q−k¯ = 0 come O(hN n hn pn t for k¯ 1. This proves Proposition 5.3 in the case k = 1 and it is clear that the general case follows after k iterations. In the remaining part of this section, we describe the properties of Yν ∈ Y(m) needed in the proof of Proposition 6.1. More precisely, we consider the commutator of Yν with the operator of multiplication by the j th coordinate, denoted by xj . LEMMA 6.2. Assume Yν ∈ Y(m). Then there exist Yν+ ∈ Y( ξ −1 m),
Yν− ∈ Y( ξ −1 ph ˜ σ m)
(6.9)
such that [Yν , xj ] = Yν+ + τ Yν− . W Proof. Let Bk,k ,ν = bk,k ,ν (x, D) with bk,k ,ν ∈ Sn (m(k, k ), g). If we know that + − [Y (τ, Bk,k ,ν ), xj ] = Yk,k ,ν + τ Yk,k ,ν
(6.10)
holds with + −1 m(k, k )), Yk,k ,ν ∈ Y( ξ
− −1 Yk,k ph ˜ σ m(k, k )), ,ν ∈ Y( ξ
then succesively commuting xj with Y (sk,k (w)τ, Bk,k ,ν ), k = 1, . . . , N, we obtain easily the general statement of Lemma 6.2. To begin we write [Y (τ, Bk,k ,ν ), xj ] = e−iτ P [Bk,k ,ν , Y (−τ, xj )]eiτ P
(6.11)
and denote Pj := [iP , xj ] = ∂ξj p W (x, D) = ∂ξj p0 (x, D).
(6.12)
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LECH ZIELINSKI
Then we can write the Taylor formula Y (−τ, xj ) = xj − τ ∂τ Y (0, xj ) + τ
1
2 0
ds(1 − s)∂τ2 Y (−sτ, xj )
= xj + τ Pj + τ e−iτ P Y (τ, [iτ P , Pj ])eiτ P , where
1
Y (τ, B) = τ
(6.13)
ds(1 − s)Y ((−s − 1)τ, B).
0
Using (6.13) we can express the commutator (6.11) in the form Y (τ, [Bk,k ,ν , xj ]) + τ Y (τ, [Bk,k ,ν , Pj ]) + τ [Y (τ, Bk,k ,ν ), Y (τ, [iτ P , Pj ]) (6.14) W and since [Bk,k ,ν , xj ] = ∂ξj bk,k ,ν (x, D), it is clear that the first term of (6.14) is −1 ˜ σ m(k, k )) is a consein Y( ξ m(k, k )). Then Y (τ, [Bk,k ,ν , Pj ]) ∈ Y( ξ −1 ph quence of
˜ σ m(k, k ), g), bk,k ,ν ∂ξj p − ∂ξj p bk,k ,ν ∈ Sn ( ξ −1 ph which follows from (3.23) due to ∂xj bk,k ,ν ∈ Sn ( ξ hσ m(k, k ), g),
∂ξj ∂ξj p ∈ S( ξ −2 p, ˜ g),
∂ξj bk,k ,ν ∈ Sn ( ξ −1 m(k, k ), g),
∂xj ∂ξj p ∈ S(ph1+σ , g) ⊂ S(ph ˜ σ , g)
(we use hp p˜ in the last inclusion). Moreover, using ∂xj p ∈ S( ξ h1+σ p, g),
∂ξj p ∈ S( ξ −1 p, ˜ g),
we find that (3.23) ensures ˜ 1+σ p, g). p ∂ξj p − ∂ξj p p ∈ S( ξ −1 ph
(6.15)
Introducing ln0 ∈ S(1, g) such that (I − L0n )Bk,k ,ν is negligible with L0n = ln0 (x, D) ¯ we can write and supp ln0 ⊂ Bn (c0 ) with c0 < c, Y (τ, Bk,k ,ν )Y (τ, [Pj , iτ P ]) = Y (τ, Bk,k ,ν )Y (τ, L0n [Pj , iτ P ]) + Rk,k ,ν
(6.16)
with {Rk,k ,ν }ν∈V negligible. However, using hn pn τ −1 |τ |ln0 ∈ Sn (h−1 p −1 , g) ˜ σ , g) such that and (6.15), we find pj,ν ∈ Sn ( ξ −1 ph W (x, D), hn pn τ −1 τ L0n [Pj , iP ] = pj,ν
˜ σ ). Therefore the right-hand side of (6.16) hence Y (τ, L0n [Pj , iτ P ]) ∈ Y( ξ −1 ph ˜ σ m(k, k )) and Y (τ, [Pj , iτ P ])Y (τ, Bk,k ,ν ) belongs to the belongs to Y( ξ −1 ph same class, i.e. (6.14) gives the desired decomposition (6.10). ✷
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
COROLLARY 6.3. Let Y0,ν ∈ Y(1). If Pj = [iP , xj ], then one has + − τ Pj e−iτ P Y0,ν = [e−iτ P Y0,ν , xj ] + e−iτ P (Y0,ν + τ Y0,ν )
(6.17)
+ − ∈ Y( ξ −1 ) and Y0,ν ∈ Y( ξ −1 ph ˜ σ ). with some Y0,ν −iτ P Proof. Indeed, using (6.13) to express [e , xj ] and applying Lemma 6.2 with Yν = Y0,ν we can write
[e−iτ P Y0,ν , xj ] = [e−iτ P , xj ]Y0,ν + e−iτ P [Y0,ν , xj ] = −τ Pj e−iτ P Y0,ν − τ e−iτ P Y (τ, [Pj , iτ P ])Y0,ν + e−iτ P (Yν+ + τ Yν− ). It remains to remark that the reasoning of the proof of Lemma 6.2 ensures the fact + − = Yν+ and Y0,ν = −Y (τ, [Pj , iτ P ])Y0,ν + Yν− belong to the indicated that Y0,ν classes. ✷
7. End of the Proof of Theorem 2.1 Throughout this section, we use the following notation: Qν = Op(ei(τ −t )p qν )
˜ m/p). with qν ∈ S( ξ ˜
(7.1)
We adopt the convention that the symbol (x, ξ ) → p(x, ξ ) can be considered as ˜ by the formula (x, ξ, y) → p(x, ξ ), allowing us to define qν ∂ξj p ∈ S(m) (qν ∂ξj p)(x, ξ, y) = qν (x, ξ, y)∂ξj p(x, ξ ).
(7.2)
LEMMA 7.1. If Qν and qν ∂ξj p are as in (7.1), (7.2), then [Qν , xj ] = (t − τ ) Op(ei(τ −t )p qν ∂ξj p) + Op(ei(τ −t )p i∂ξj qν ).
(7.3)
Proof. Since the integral kernel of [Qν , xj ] is −d (yj − xj )ei(x−y)ξ +i(τ −t )p(x,ξ )qν (x, ξ, y) dξ (x, y) &→ (2π )
(7.4)
and (yj − xj )ei(x−y)ξ = −i∂ξj ei(x−y)ξ , the integration by parts allows us to write (7.4) in the form −d ei(x−y)ξ +i(τ −t )p(x,ξ )((t − τ )∂ξj pqν + i∂ξj qν )(x, ξ, y) dξ, (x, y) &→ (2π ) which gives (7.3).
✷
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LECH ZIELINSKI
The computation of the composition kernel gives Op(ei(τ −t )p qν )bν (x, D)∗ = Op(ei(τ −t )p (qν • bν )) with (qν • bν )(x, ξ, y) = (2π )
−d
(7.5)
˜ ˜ −ξ) qν (x, ξ, y)b ˜ ν (y, ξ˜ ) ei(y−y)(ξ
and the usual Taylor development with integrations by parts give qν • bν = qν •N bν + rN ,ν , with
(qν •N bν )(x, ξ, y) = rN ,ν =
(7.6)
(−i)|α| ∂yα qν (x, ξ, y)∂ξα bν (y, ξ )/α!,
|α|
(qα,ν (z) • ∂ξα bν )|z=y ,
|α|=N
1
qα,ν (z)(x, ξ, y) = 0
(7.7)
∂yα qν (x, ξ, z
i N N ds . + s(y − z)) α!
Then, for an arbitrary N ∈ N, we can find N = N (N) such that the family of operators RN ,ν = Op(ei(τ −t )p rN ,ν ) satisfies C(N) . %RN ,ν % CN hN n hn pn t
(7.8)
Denoting (qν b¯νB )(x, ξ, y) = qν (x, ξ, y)bν (y, ξ ), we can write ˜ m), ˜ mm ˜ bν ∈ S(m , g) ⇒ qν •N bν ∈ S( ˜ ), qν ∈ S( ˜ mm ˜ hσ ). qν •N bν − qν b¯νB ∈ S( ˜ m/p), ˜ ∂ξj p0 ∈ S( ξ −1 p, ˜ g), then If, in particular, qν ∈ S( ξ Qν Pj = Op(ei(τ −t )p (qν • ∂ξj p0 )), ˜ qν •N ∂ξj p0 ∈ S(m), B ˜ σ m). qν •N ∂ξj p0 − qν ∂ξj p0 ∈ S(h
˜ g), Since (3.20) ensures ∂ξj p − ∂ξj p0 ∈ S( ξ −1 h1+σ p, g) ⊂ S( ξ −1 hσ p, ˜ σ m). qN− ,ν := qν •N ∂ξj p0 − qν ∂ξj p B ∈ S(h For j = 1, . . . , d, we introduce τ τ ∂ξj p(x, ξ ) + ∂ξj p(y, ξ ). p(j ),ν (x, ξ, y) := 1 − t t
(7.9)
(7.10)
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
˜ m/p), ˜ Y0,ν ∈ Y(1) and N0 ∈ N. Then we can PROPOSITION 7.2. Let qν ∈ S( ξ find C(N0 ) > 0 such that C(N0 ) 0 , (7.11) (Jν (qk , Yk ) + tJν (q−k , Y−k )) + O hN tJν (qp(j ) , Y0 ) = n hn pn t 1k2
where for k = 1 and 2 we have ˜ qk,ν ∈ S(m/ p), ˜
σ ˜ q−k,ν ∈ S(mh ),
Yk,ν ∈ Y(1),
Y−k,ν ∈ Y(1). (7.12)
Proof. By definition of p(j ),ν , we have t Op(ei(τ −t )p qν p(j ),ν ) = (t − τ ) Op(ei(τ −t )p qν ∂ξj p) + τ Op(ei(τ −t )p qν ∂ξj p B )
(7.13)
and applying (7.3), (7.9), we can write the above expression as [Qν , xj ] + Qν τ Pj + Op(ei(τ −t )p (τ qN− ,ν − i∂ξj qν )) + τ RN ,ν with RN ,ν satisfying (7.8). Thus tJν (qp(j ) , Y0 ) can be written as tr[Qν , xj ]e−iτ P Y0,ν + tr Qν τ Pj e−iτ P Y0,ν + C(N0 ) 0 ). + Jν (−i∂ξj q, Y0 ) + τ Jν (qN− , Y0 ) + O(hN n hn pn t
(7.14)
Due to Corollary 6.3, the sum of two first terms in (7.14) equals tr[Qν , xj ]e−iτ P Y0,ν + tr Qν [e−iτ P Y0,ν , xj ] + Jν (q, Y0+ ) + τ Jν (q, Y0− ) = tr[Qν e−iτ P Y0,ν , xj ] + Jν (q, Y0+ ) + τ Jν (q, Y0− ) (7.15) = Jν (q, Y0+ ) + τ Jν (q, Y0− ). Thus, we obtain (7.11) with q1,ν = i∂ξj qν ,
τ − q , Y1,ν = Y−1,ν = Y0,ν , t N ,ν + Y2,ν = ξ¯ (kn+ ) Y0,ν ,
q−1,ν =
−1 q2,ν = ξ¯ (kn+ ) qν ,
−1 q−2,ν = ξ¯ (kn+ ) p˜ n hσn qν ,
− Y−2,ν = ξ¯ (kn+ ) p˜ n−1 h−σ n Y0,ν ,
¯ n− ), ξ¯ (kn+ )). where, according to our convention, p˜n = p˜n (x(k
✷
Proof of Proposition 6.1. Let χs , χ˜ s be as in the proof of Corollary 4.3, then −1 ˜ ˜ g) ⇒ q˜ν := q0,ν χs h−1 ˜ g) q0,ν ∈ S(m, n pn ∈ S(m/p,
and ˜ Y˜ ) tJν (q0 χs , Y0 ) = Jν (q,
with Y˜ν = hn pn tY0,ν ∈ Y(1).
Therefore, it suffices to prove the statement of Proposition 6.1 with q0,ν χ˜s instead of q0,ν .
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LECH ZIELINSKI
Further on, we assume s = 1 and note that for (x, ξ ) ∈ supp χ˜ 1 we have ˜ ξ )/2 |∇ξ p(x, ξ )| 2 ξ −1 p(x, ˜ ξ) ξ −1 p(x,
(7.16)
and, due to (2.10), for (x, ξ, y) ∈ supp q0,ν χ˜1 , we have
d 1/2 2 |∇ξ p(x, ξ )| − p(j ),ν j =1
|∇ξ p(x, ξ ) − ∇ξ p(y, ξ )| ˜ ξ )/4 |∇ξ p(x, ξ )|/2, ξ −1 p(x, which implies 1/2
d 2 p(j ),ν |∇ξ p(x, ξ )|/2 ξ −1 p(x, ˜ ξ )/4.
(7.17)
j =1
Using (7.16), (7.17), we can write q0,ν χ˜1 =
q(j ),ν = q0,ν χ˜1 p(j ),ν
d
d
j =1
q(j ),ν p(j ),ν with
−1 2 p(j ),ν
˜ m/p, ∈ S( ξ ˜ g)
j =1
and we complete the proof applying Proposition 7.2 with qν = q(j ),ν .
✷
8. Proof of Theorem 2.2 8.1. THE FAMILY OF MOLLIFYING FUNCTIONS Let γ , γ˜ ∈ C0∞ ({x ∈ Rd : |x| < c}) ¯ be such that γ (x) dx = γ˜ (x) dx = 1, γ˜ 0 and γ˜ (x) c0 > 0 for x ∈ supp γ . For s > 0, we denote γ˜s (x) := s −d γ˜ (x/s). γs (x) := s −d γ (x/s), We note that γs (x) dx = γ˜s (x) dx = 1 and α γs (x)| Cα s −|α| γ˜s (x). |∂x,s
(8.1)
8.1.1. Mollifying v Let 0 < ε (ρ − ρ )/2, r /r < δ < 1 and define −δ
ε ¯ ξ , ξ , sk (ξ , ξ ) = v(x(k))
(8.2)
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
where k ∈ N,
ξ , ξ ∈ Rd
and
ξ , ξ := (1 + |ξ |2 + |ξ |2 )1/2 .
It is easy to check that
|∂ξα ∂ξα sk (ξ , ξ )| Cα ,α sk (ξ , ξ ) ξ, ξ
−|α |−|α|
(8.3)
.
Using the partition of unity of Section 3, we set vk := vθk−
and
v˜k (ξ , x, ξ ) := (vk ∗ γsk (ξ ,ξ ) )(x).
(8.4)
ε Similarly as (2.25), we check that |x − y| cv(x) ¯ implies
|v(x) − v(y)| C max{|x − y|r , 1 + |x − y|}v(x)1−ρ 2C|x − y|r v(x)1−ρ+(1−r)ε . Using (8.5) to estimate
(8.5)
v˜k (ξ , x, ξ ) − vk (x) =
(vk (y) − vk (x))γsk (ξ ,ξ ) (x − y) dy,
we find |v˜k (ξ , x, ξ ) − vk (x)|
|x − y|r |γsk (ξ ,ξ ) (x − y)| dy 1−ρ+(1−r)ε r sk (ξ , ξ ) |y|r |γ (y)| dy = Cv(x(k)) ¯ 1−ρ+(1−r)ε Cv(x(k)) ¯
C v(x)1−ρ+ε ξ , ξ −δr . β If |α | + |α| + |β| 1, then ∂ξα ∂ξα ∂x γsk (ξ ,ξ ) (x − y) dy = 0 and α α β ∂ξ ∂ξ ∂x v˜k (ξ , x, ξ ) = (vk (y) − vk (x))∂ξα ∂ξα ∂xβ γsk (ξ ,ξ ) (x − y) dy.
(8.6)
(8.7)
However, using (8.1) and (8.3), we can find
|∂ξα ∂ξα ∂xβ γsk (ξ ,ξ ) (x − y)| Cα ,α,β γ˜sk (ξ ,ξ ) (x − y)sk (ξ , ξ )
−|β|
−|α |−|α|
ξ , ξ
and, estimating the integral (8.7), we find −|α |−|α|
|∂ξα ∂ξα ∂xβ v˜k (ξ , x, ξ )| Cα ,α,β v(x)1−ρ+ε sk (ξ , ξ )r−|β| ξ , ξ
.
(8.8)
8.2. ESTIMATES OF PSEUDODIFFERENTIAL OPERATORS We fix σ > 0 small enough to ensure (1 + σ )r < rδ,
(1 + σ )ρ ρ − ε,
σ r 1 − δ,
σρ ε.
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LECH ZIELINSKI
These conditions on σ ensure
v(x)ε−ρ ξ −rδ v(x)−(1+σ )ρ ξ −(1+σ )r = h(x, ξ )1+σ , (8.9) −ε δ−1 −σρ −σ r σ v(x) ξ = h(x, ξ ) . (8.10) v(x) ξ ∞ We define V˜ = k=0 V˜k , where V˜k has the kernel given by the oscillatory integral −2d ei(x−x )ξ +i(x −y)ξ v˜k (ξ , x, ξ ) dξ dx dξ. (x, y) → (2π ) Using Theorem 2.1 of Kumano-Go and Nagase [20], we note that the estimates α α s/2 ∂ ∂ ξ (vk (x) − v˜k (ξ , x, ξ )) ξ s/2 ξ
ξ
(s−rδ)/2−|α |
1−ρ+ε Cα ,α v(x(k)) ¯ ξ
ξ (s−rδ)/2−|α|
with s = (1 + σ )r < rδ ensure 1−ρ+ε % D (1+σ )r /2 (Vk − V˜k ) D (1+σ )r /2 % Cv(x(k)) ¯ ,
(8.11)
where Vk denotes the operator of multiplication by vk . The estimate (8.11) can be written as the inequality (in the sense of quadratic forms)
1−ρ+ε ¯ D −(1+σ )r . ±(Vk − V˜k ) Cv(x(k))
(8.12)
˜− ˜ ˜− ˜− ˜− ˜− ˜k0 (x, D). Then 5 Let 5 k be as in Section 5 and consider 5k Vk 5k = v k V k 5k = V k and standard symbol expansions (cf. [19, 20]) give the approximation of v˜k0 by i |α| ∂xα ∂ξα v˜k (ξ , x, ξ )|ξ =ξ /α!, (8.13) |α|N
hence (8.6), (8.8), (8.9), (8.10) imply |v˜k0 (x, ξ ) − vk (x)| Cv(x)1−ρ+ε ξ −rδ Cv(x)h(x, ξ )1+σ , |∂xβ ∂ξα v˜k0 (x, ξ )| Cα,β v(x)1−ρ+ε ξ −rδ (v(x)−ε ξ δ−1 )|β| ξ |β|−|α| Cα,β v(x)h(x, ξ )1+σ (1+|β|) ξ |β|−|α| ,
(8.14) (8.15)
if |α| + |β| 1. Let Aj,j ,k denote the operator of multiplication by the function aj,j ,k := θk− aj,j and similarly as before, let a˜ j,j ,k (ξ , x, ξ ) := (aj,j ,k ∗ γsk (ξ ,ξ ) )(x).
(8.16)
Reasoning as before, we find the estimates −rδ
|aj,j ,k (x) − a˜ j,j ,k (ξ , x, ξ )| Cv(x)ε−ρ ξ , ξ α ξ
(8.17)
, δ(|β|−r)−|α |−|α|
|∂ ∂ξα ∂xβ a˜ j,j ,k (ξ , x, ξ )| Cα ,α,β v(x)ε(1−|β|)−ρ ξ , ξ
, (8.18)
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
if |α | + |α| + |β| 1. Using [20] similarly as before, we find ε−ρ % D (1+σ )r /2 (Aj,j ,k − A˜ j,j ,k ) D (1+σ )r /2 % Cv(x(k)) ¯ ,
(8.19)
if the operator A˜ j,j ,k has the kernel (x, y) → (2π )−d ei(x−x )ξ +i(x −y)ξ a˜ j,j ,k (ξ , x, ξ ) dξ dx dξ and (8.19) implies ˜ ( (Aj,j ,k − Aj,j ,k )Dj ϕ, Dj ϕ) 1j,j d
ε−ρ ( D 2−(σ +r ) ϕ, ϕ). (8.20) Cv(x(k)) ¯ ˜ ˜ − ˜ k,0 (x, D) with ˜− Introducing A˜ k,0 = 1j,j d Dj A˜ j,j ,k Dj , we find 5 k Ak,0 5k = a a˜ k,0 satisfying
|∂ξα (a˜ k,0 − a0 )(x, ξ )| Cα v(x)ε−ρ ξ 2−δr−|α| , |∂ξα ∂xβ a˜ k,0 (x, ξ )|
Cα,β h(x, ξ )
1+σ |β|
ξ
2+|β|−|α|
(8.21) (8.22)
,
where a0 is given by (1.4), α ∈ Nd , β ∈ Nd \ {0}. If C > 0 is large enough, then (8.12), (8.20) ensure (2.1) with P± =
∞
˜ ˜ ˜− ˜− 5 k (A0,k + Vk ± CRk )5k ,
k=1
1−(1+σ )ρ −(1+σ )ρ ¯ D −(1+σ )r + v(x(k)) ¯ D 2−(1+σ )r Rk = v(x(k)) W 1+σ ˜− ˜− and R = ∞ , g), hence p = p± k=1 5k Rk 5k = r (x, D) holds with r ∈ S(ah satisfy the hypotheses of Theorem 2.1 due to Lemma 2.3 and remark at the end of Section 3.
9. Appendix LEMMA 9.1. (a) Let Qν = (e−itp qν )(x, D) with qν ∈ S(m, g) such that supp qν ⊂ ¯ If C > 0 is large enough, then Bn (c). C %Qν %tr mC,ν := Cmn h−C n hn pn t .
(9.1)
˜ then the estimate (9.1) still holds. (b) If Qν = Op(e−itp qν ) with qν ∈ S(m), Proof. For k ∈ N let Fk (x, ξ ) = x 2k ξ 2k . If k > d, then Fk (x, D)−1 is of trace class and %Qν %tr %Fk (x, D)−1 %tr %Fk (x, D)Qν %.
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LECH ZIELINSKI
A direct calculus gives Fk (x, D)Qν = Op(e−itp )qk,ν with |qk,ν | Ck m|t|2k p 2k Fk and for C > 0 large enough, we obtain |qk,ν | mC,ν . Using the same kernel expression (5.6) in cases (a) and (b), we complete the proof due to Schwarz lemma 1/2 −itp %Op(e qk,ν )% suppy |Kν (q, x, y)| dx suppx |Kν (q, x, y)| dy
mC,ν vol(Bn (c)) ¯ C mC,ν h−C n .
✷
Proof of Proposition 1.2. Our aim is to estimate the volume of G(λ) = {(x, ξ ) ∈ R2d : a(1 − C0 h)(x, ξ ) < λ < a(1 + C0 h)(x, ξ ), |x| + |ξ | > C1 }. We assume C1 > 0 large enough to ensure λ/2 a(x, ξ ) 2λ
for (x, ξ ) ∈ G(λ).
Case 1. We consider (x, ξ ) ∈ G(λ) satisfying v(x) a0 (x, ξ ). Then λ/2 a(x, ξ ) 2a0 (x, ξ ) C|ξ |2
(9.2)
ensures
(ah)(x, ξ ) 2a0 (x, ξ ) ξ −r v(x)−ρ C a0 (x, ξ )λ−r /2 v(x)−ρ
(9.3)
and (x, ξ ) ∈ G(λ) implies
a0 (x, ξ )(1 − C0 λ−r /2 v(x)−ρ )
λ − v(x) a0 (x, ξ )(1 + C0 λ−r /2 v(x)−ρ ).
(9.4)
If λ λ0 with λ0 > 0 large enough, then (9.4) implies λ − v(x) 0 and λ− (x) a0 (x, ξ ) λ+ (x)
(9.5)
with
λ± (x) = (λ − v(x))(1 ± 2C0 λ−r /2 v(x)−ρ ).
(9.6)
Indeed, if λ± (x) are given by (9.6), then
λ− (x) (λ − v(x))(1 + C0 λ−r /2 v(x)−ρ )−1 (λ − v(x))(1 − C0 λ−r /2 v(x)−ρ )−1 λ+ (x), implying (9.5). Since for d 2 we have dξ = ω(x)(λ+ (x)d/2 − λ− (x)d/2 ) λ− (x)
C(λ+ (x) − λ− (x))λ+ (x)d/2−1
(9.7)
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
and λ± (x) Cλ, we have dξ C v(x)−ρ λd/2−r /2 .
(9.8)
λ− (x)
Case 2. We note that for a given C¯ > 0 we have 1−ρ dξ = C¯ λ 2+r d
¯ 1−ρ |ξ |2+r Cλ
(9.9)
and d − r 1 − ρ − ρ . d 2 + r 2
d 3 ⇒
Case 3. We consider (x, ξ ) ∈ G(λ) satisfying a0 (x, ξ ) v(x). Then
λ/2 a(x, ξ ) 2v(x) ⇒ (ah)(x, ξ ) Cλ1−ρ a0 (x, ξ )−r /2 ,
(9.10)
hence (x, ξ ) ∈ G(λ) implies
a0 (x, ξ ) − C1 λ1−ρ a0 (x, ξ )−r /2 < λ − v(x) < a0 (x, ξ ) + C1 λ1−ρ a0 (x, ξ )−r /2 .
(9.11)
Due to (9.9), it remains to consider the region where
a0 (x, ξ )1+r /2 c|ξ |2+r 3C1 λ1−ρ
(9.12)
and the condition (9.12) ensures
λ − v(x) > a0 (x, ξ ) − C1 λ1−ρ a0 (x, ξ )−r /2 2C1 λ1−ρ a0 (x, ξ )−r /2 , λ − v(x) < a0 (x, ξ ) + C1 λ1−ρ a0 (x, ξ )−r /2 a0 (x, ξ ) + (λ − v(x))/2, hence 2a0 (x, ξ ) λ − v(x). Due to the last inequality, (9.11) implies (9.5) with
λ± (x) = λ − v(x) ± 2C1 λ1−ρ (λ − v(x))−r /2 .
(9.13)
Let us check that λ+ (x) C(λ − v(x)).
(9.14)
Indeed, using once more (9.10) and (9.11), we can write
λ − v(x) > a0 (x, ξ ) − C1 λ1−ρ a0 (x, ξ )−r /2 a0 (x, ξ )/2, hence 1 C λ1−ρ 2 1
1+r /2 > a0 (x, ξ )1+r /2 2(λ − v(x))
and (9.14) follows.
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LECH ZIELINSKI
Using (9.7) with (9.13) we find dξ Cλ1−ρ (λ − v(x))−r /2 λ+ (x)d/2−1 λ− (x)
C λ1−ρ (λ − v(x))(d−r )/2−1 C λ(d−r )/2−ρ . To complete the proof of Proposition 1.2, it remains to perform the integration with respect to x in the region v(x) 2λ. ✷
9.1. CASES d = 1 AND d = 2 Using (9.9) to estimate the integral in the left-hand side of (1.15) in the region ¯ 1−ρ , we obtain an additional term V(λ)O(λ(1−ρ )d/(2+r ) ) in v(x) < λ, |ξ |2+r Cλ the right-hand side of (1.15). However, sometimes we can obtain a better estimate if there exists κ ∈ [0; 1] such that 1 τ λ/2 ⇒ dx Cτ V(λ)λ−κ . (9.15) λ−τ
Indeed, using (9.15) instead of (9.9), we find the error estimate 1−ρ −κ+ 1−ρ (d− r2 ) ˜ 2+r ξ −r /2 dξ CV(λ)λ . CV(λ)λ1−ρ −κ
¯ 1−ρ |ξ |2+r Cλ
Acknowledgement This paper would not be written without aid of Janina Dawczynska and it is dedicated to her memory. References 1. 2. 3. 4. 5. 6.
Birman, M. and Solomyak, M.: Asymptotic behaviour of spectrum of differential equations, J. Soviet. Math. 12 (1979), 247–282. Boimatov, K.: Spectral asymptotics of pseudodifferential operators, Soviet Math. Dokl. 42(2) (1990), 196–200. Buzano, E.: Some remarks on the Weyl asymptotics by the approximate spectral projection method, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2000), 775–792. Dencker, N.: The Weyl calculus with locally temperate metrics and weights, Ark. Mat. 24 (1986), 59–79. Edmunds, D. E. and Evans, W. D.: On the distribution of eigenvalues of Schrödinger operators, Arch. Rational Mech. Anal. 89 (1985), 135–167. Edmunds, D. E. and Evans, W. D.: Spectral Theory and Differential Operators, Oxford Math. Monogr., Oxford, 1989.
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
7. 8. 9. 10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
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Faris, W. G.: Self-adjoint Operators, Lecture Notes in Math. 433, Springer-Verlag, New York, 1975. Feigin, V. I.: The asymptotic distribution of the eigenvalues of pseudodifferential operators in Rn , Math. USSR-Sb. 28 (1976), 533–552. Feigin, V. I.: Sharp estimates of the remainder in the spectral asymptotics for pseudodifferential operators in Rn , Functional Anal. Appl. 16 (1982), 88–89. Fleckinger, J. and Lapidus, M.: Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), 329–356. Fleckinger, J. and Lapidus, M.: Schrödinger operators with indefinite weight functions: asymptotics of eigenvalues with remainder estimates, Differential Integral Equations 7 (1994), 1389–1418. Guillemin, V. and Sternberg, S.: Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915–955. Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque 112 (1984). Helffer, B. and Robert, D.: Propriétés asymptotiques du spectre d’opérateurs pseudodifférentiels sur Rn , Comm. Partial Differential Equations 7 (1982), 795–881. Hörmander, L.: On the asymptotic distribution of the pseudodifferential operators in Rn , Ark. Mat. 17 (1979), 297–313. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols 1–4, SpringerVerlag, New York, 1983, 1985. Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998. Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients, Internat. Math. Res. Notices (2000), 1155–1166. Kumano-Go, H.: Pseudodifferential Operators, MIT Press, Cambridge, MA, 1981. Kumano-Go, H. and Nagase, M.: Pseudodifferential operators with non-regular symbols and applications, Funkcial. Ekvac. 21 (1978), 151–192. Levendorskii, S. Z.: Asymptotic Distribution of Eigenvalues of Differential Operators, Math. Appl., Kluwer Acad. Publ., Dordrecht, 1990. Métivier, G.: Valeurs propres des problèmes aux limites irréguliers, Bull. Soc. Math. France Mem. 51–52 (1977), 125–219. Miyazaki, Y.: A sharp asymptotic remainder estimate for the eigenvalues of operators associated with strongly elliptic sesquilinear forms, Japan. J. Math. 15 (1989), 65–97. Miyazaki, Y.: The eigenvalue distribution of elliptic operators with Hölder continuous coefficients, Osaka J. Math. 28 (1991), 935–973; Part 2, Osaka J. Math. 30 (1993), 267–302. Mohamed, A.: Comportement asymptotique avec estimation du reste, des valeurs propres d’une classe d’opérateurs pseudo-différentiels sur Rn , Math. Nachr. 140 (1989), 127–186. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vols I–IV, Academic Press, New York, 1972, 1975, 1979. Robert, D.: Propriétés spectrales d’opérateurs pseudo-différentiels, Comm. Partial Differential Equations 3 (1978), 755–826. Rozenblyum, G. V.: Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSRSb. 22 (1974), 349–371. Shubin, M. A. and Tulovskii, V. A.: On the asymptotic distribution of eigenvalues of pseudodifferential operators in Rn , Math. USSR-Sb. 21 (1973), 565–573. Tamura, H.: Asymptotic formula with remainder estimates for eigenvalues of Schrödinger operators, Comm. Partial Differential Equations 7 (1982), 1–54. Tamura, H.: Asymptotic formula with sharp remainder estimates for eigenvalues of elliptic operators of second order, Duke Math. J. 49 (1982), 87–119. Zielinski, L.: Asymptotic behaviour of eigenvalues of differential operators with irregular coefficients on a compact manifold, C.R. Acad. Sci. Paris Sér. I Math. 310 (1990), 563–568.
182 33. 34. 35.
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Zielinski, L.: Asymptotic distribution of eigenvalues for elliptic boundary value problems, Asymptotic Anal. 16 (1998), 181–201. Zielinski, L.: Asymptotic distribution of eigenvalues of some elliptic operators with intermediate remainder estimates, Asymptotic Anal. 17 (1998), 93–120. Zielinski, L.: Sharp spectral asymptotics and Weyl formula for elliptic operators with nonsmooth coefficients, Math. Phys. Anal. Geom. 2 (1999), 291–321; Part 2, Colloq. Math. 92 (2002), 1–18.
Mathematical Physics, Analysis and Geometry 5: 183–200, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
183
Heat Kernel Expansions on the Integers F. ALBERTO GRÜNBAUM and PLAMEN ILIEV Department of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A. e-mail: {grunbaum, iliev}@math.berkeley.edu (Received: 27 December 2001; in final form: 21 April 2002) Abstract. In the case of the heat equation ut = uxx +V u on the real line, there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula. We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L0 . We show if L denotes the result of applying a finite number of Darboux transformations to L0 then the fundamental solution of ut = Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument. Mathematics Subject Classifications (2000): 35Q58, 37K20, 58J72. Key words: heat kernel expansions, Toda lattice hierarchy, Darboux transformations.
1. Heat Kernel Expansions The subject of heat kernel expansions on Riemannian manifolds with or without boundaries cuts across a number of branches of mathematics and serves as an interesting playground for a whole array of interactions with physics. It suffices to mention, for instance, the work of Kac [18] on the issue of recovering the shape of a drum from its pure tones, as well as the suggestion by McKean and Singer [20] that one should be able to find a ‘heat equation proof’ of the index theorem. For an updated account, see [8, 23]. In the simpler case of the whole real line, the fundamental solution of the equation ut = uxx + V (x)u,
u(x, 0) = δy (x)
admits an asymptotic expansion valid for small t and x close to y, in the form (x−y)2 ∞ e− 4t 1+ Hn (x, y)t n . u(x, y, t) ∼ √ 4π t n=1 When V is taken to be a potential such that L = (d/dx)2 + V belongs to a rank one bispectral ring, then something remarkable happens, namely this expansion gives rise to an exact formula consisting of a finite number of terms and valid for
184
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all x, y as well as all t. For a few examples of this see [15], as well as references in [3]. See also [24] where the author points to [21] for useful connections between the heat equation and KdV invariants. For a general discussion of the bispectral problem see [10] and [27] and for a sample of problems touching upon this area see [4–7, 9, 11, 13]. In the case above the potentials V s are the rational solutions of the KdV equation decaying at infinity, see [2]. They can be obtained by a finite number of applications of the Darboux process starting from the trivial potential V = 0, see [1]. The remarks above rest on the basic fact that the Darboux process maps operators of the form d2 /dx 2 + V into themselves. Following some preliminary explorations in [14] we would like to use this ‘soliton technology’ as a tool for the discovery of the general form of a heat kernel expansion on the integers. In this case, there seems to be no general theory predicting even the existence of asymptotic expansions. If one replaces the real line by the integers and looks for the fundamental solution of ut (n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t) ≡ L0 u(n, t) with u(n, 0) = δnm , one obtains, in terms of the Bessel function In (t) of imaginary argument, the well-known result u(n, t) = e−2t In−m (2t). A nice reference for this is [12]. In the context of l 2 (Z) it is simplest to study L0 by Fourier methods and obtain for the fundamental solution of the heat equation the expression dx e−2t −1 et (x+x ) x n−m . (1.1) 2π i x One can replace the second difference operator L0 above by an appropriate perturbation of it and look at the corresponding heat equation and its fundamental solution. The purpose of this paper is to describe the result when the ordinary second difference operator is subject to a finite number of Darboux factorization steps. In this case the spectrum is a finite interval and the factorization can be performed at each end (see (2.1)). Formula (1.1) above will be modified properly, in (5.3), when L0 is subject to a finite number of applications of the Darboux process. We close this introduction with the simplest nontrivial example and then we describe the organization of the paper. Let us write the operator L0 in the form L0 = − 2 Id + −1 , where stands for the customary shift operator, acting on functions of a discrete variable n ∈ Z by f (n) = f (n + 1). We can factorize L0 as L0 = P0 Q0 ,
(1.2)
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HEAT KERNEL EXPANSIONS ON THE INTEGERS
where the operators P0 and Q0 are given by P0 = Id −
τn−1 (δ) −1 τn (δ)
and
Q0 = −
τn+1 (δ) Id, τn (δ)
(1.3)
with τn (δ) = n+δ. Applying one Darboux step with parameter δ to the operator L0 amounts to producing a new operator L1,0 by exchanging the order of the factors in (1.2), i.e. τn+1 (δ)τn−1 (δ) −1 1 Id + L1,0 = Q0 P0 = − 2 + . (1.4) τn (δ)τn+1 (δ) τn (δ)2 The fundamental solution to ut = L1,0 u
(1.5)
is given by u(n, m, t) =
e−2t [τm τn+1 In−m (2t) − tIn−m (2t) − tIn−m+1 (2t)]. τm+1 τn
(1.6)
Indeed, it is obvious that u(n, m, 0) = δnm , and Equation (1.5) can be verified using well known identities satisfied by the Bessel functions (see (4.2a) and (4.2b)). It is important to point out that, when the real line is replaced by the integers, the operators obtained from L0 by the Darboux process are no longer of the form L0 plus a potential. The operators L, obtained from L0 by a finite number of applications of the Darboux process at the ends of the spectrum, have been recently determined, see [16, 17]. These operators belong to a rank-one commutative ring AV of difference operators with unicursal spectral curve. The common eigenfunction pn (x) to all operators of AV , with spectral parameter x, is also an eigenfunction to a rank-one commutative ring of differential operators in x with solitonic spectral curve, i.e. we have a difference-differential bispectral situation. Moreover, the functions pn (x) satisfy an orthogonality relation on the circle. These results will be summarized in the next section and put to use in later ones. The ring of operators AV is also discussed in Section 2 and certain operators in it are exhibited in Section 3. In Section 4 we collect some properties of Bessel functions and then all of this is used in Section 5 to obtain our main result, Theorem 5.1. We close the paper with one example to illustrate all the steps of the proof of our main result what covers all cases when L belongs to a rank one bispectral ring. 2. Rank-One Bispectral Second-Order Difference Operators Denote by and , respectively, the customary shift and difference operators, acting on functions of a discrete variable n ∈ Z by f (n) = f (n + 1)
and
f (n) = f (n + 1) − f (n) = ( − Id)f (n).
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The formal adjoint to an operator aj (n)j X= j
is defined to be X∗ = −j · aj (n) = aj (n − j )−j . j
j
In this section we describe the operator LR,S obtained by successive Darboux transformations from the operator L0 = − 2 Id + −1 . The symmetric operator L0 admits (as d2 /dx 2 did in the case of the real line) a unique selfadjoint extension in l 2 (Z) (‘limit point’ case of Weyl’s classification at both end points). Its spectrum is the interval (−4, 0). The steps of the Darboux transformations are as follows L0 = P0 Q0 L1,0 = Q0 P0 = P1 Q1 · · · LR,0 = QR−1 PR−1 , LR,0 + 4 Id = PR QR LR,1 + 4 Id = QR PR = PR+1 QR+1 · · · LR,S + 4 Id = QR+S−1 PR+S−1 . (2.1) At each step we perform a lower-upper factorization, as we did in (1.3), of the corresponding operator and then we produce a new operator by interchanging the factors. The operator LR,S is obtained by performing R Darboux steps at one end of the spectrum of L0 and S steps at the other end. For details of the exposition that follows we refer the reader to [17]. Let e(k, λ) be the linear functional acting on a function g(z) by the formula
e(k, λ), g = g (k) (λ),
λ ∈ C, k 0.
We shall denote by Exp(n; r, z) the exponential function ∞ ri zi , Exp(n; r, z) = (1 + z)n exp i=1
where r = (r1 , r2 , . . .). Let us introduce also the functions Sjε (n; r) =
1
e(j, ε − 1), Exp(n; r, z), j!
(2.2)
and write 1 φj (n; r) = S2j −1 (n + j − 1; r)
and
−1 ψj (n; r) = S2j −1 (n + j − 1; r).
(2.3)
When ε = 1, the functions Sj1 (n; r) are a shifted version of the classical elementary ∞ j j Schur polynomials, defined by exp( ∞ j =1 rj z ) = j =0 Sj (r)z : Sj1 (n; r) = Sj (r1 + n, r2 − n/2, r3 + n/3, . . .).
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HEAT KERNEL EXPANSIONS ON THE INTEGERS
For ε = −1, the functions Sj−1 (n; r) are of the form
∞ Sj−1 (n; r) = (a polynomial in n, r1 , r2 , . . .) × (−1)n exp (−2)j rj . j =1
Finally, let us define τ (n; r) = Wr (φ1 (n; r), . . . , φR (n; r), ψ1 (n; r), . . . , ψS (n; r)) × ∞ nS i ri (−2) , × (−1) exp −S
(2.4)
i=1
where Wr denotes the discrete Wronskian with respect to the variable n: Wr (f1 (n), f2 (n), . . . , fk (n)) = det(i−1 fj (n))1i,j k . The purpose of the exponential factor in (2.4) is to cancel the exponential factor that comes out from the functions ψj (n; r), 1 j S. With this normalization, the function τ (n; r) becomes a quasipolynomial in the variables n, r1 , r2 , . . ., i.e. in general, τ (n, r) depends on infinitely many variables, but there exists a positive integer N, such that τ (n, r) is a polynomial in every variable of degree at most N. The operator LR,S in (2.1) can be expressed in terms of the function τ (n, r) via the formula ∂ τ (n + 1; r) Id + log LR,S = + −2 + ∂r1 τ (n; r) τ (n − 1; r)τ (n + 1; r) −1 . (2.5) + τ (n; r)2 From (2.4) it follows immediately that τ (n, r) is an adelic tau function+ of the KP hierarchy defined by R + S one-point conditions, with R conditions at the point zero and S conditions at the point −2. The reason why the end points of the spectrum of L0 , 0 and −4, are replaced by 0 and −2 can be traced back to (2.2) and (2.3). After a suitable linear change of time variables r1 , r2 , . . . the tridiagonal operator LR,S solves the Toda lattice hierarchy, i.e. ∂L = [(Lj )+ , L], ∂rj
j = 1, 2, . . . ,
where (Lj )+ denotes the positive difference part++ of the operator Lj , and {rj }∞ j =1 are related to {rj }∞ j =1 via linear transformation of the form rj =
rj
+
∞
cj k rk ,
k=j +1 + By an adelic tau function we mean a tau function built from a plane belonging to Wilson’s adelic Grassmannian [27] via the construction in [16]. ++ Equivalently, if we think of Lj as an infinite matrix, (Lj ) is the upper part of this matrix, + including the main diagonal.
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see [16] for explicit formulas. The wave function w(n; r, z) and the adjoint wave function w ∗ (n; r, z) are defined by τ (n; r − [z−1 ]) Exp(n; r, z) τ (n; r) ∞ wj (n; r)z−j Exp(n; r, z) = 1+
w(n; t, z) =
(2.6)
j =1
and τ (n; r + [z−1 ]) Exp−1 (n; r, z) τ (n; r) ∞ ∗ −j wj (n; r)z Exp−1 (n; r, z), = 1+
w ∗ (n; r, z) =
(2.7)
j =1
where [z] = (z, z2/2, z3 /3, . . .). The wave operator W (n; r) and the adjoint wave operator (W −1 )∗ can be written in the form W (n; r) = 1 +
∞
wj (n; r)−j = Q( − Id)−R ( + Id)−S ,
(2.8)
j =1
(W −1 )∗ (n; r) = 1 +
∞
wj∗ (n + 1; r)∗ −j
j =1 ∗
= P (∗ − Id)−R (∗ + Id)−S ,
(2.9)
where P and Q are finite-band forward difference operators of order R + S satisfying P Q = ( − Id)2R ( + Id)2S ,
(2.10)
see [17, Theorem 3.3]. The tau function τ (n; r) can be viewed as an infinite sequence (indexed by n) of tau functions of the standard KP hierarchy, associated with a flag of nested subspaces V:
· · · ⊂ Vn+1 ⊂ Vn ⊂ Vn−1 ⊂ · · · ,
(2.11)
with Vn = Span{w(n; 0, z), w(n + 1; 0, z), w(n + 2; 0, z), . . .}.
(2.12)
The plane V0 belongs to Wilson’s adelic Grassmannian [27]. Let x = z + 1, ∞ rj (x − 1)j , (2.13) pn (x) = W (n; r)x n = w(n, r, x − 1) exp − j =1
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HEAT KERNEL EXPANSIONS ON THE INTEGERS
and similarly pn∗ (x) = W ∗ (n − 1; r)x −n = w ∗ (n, r, x − 1) exp
∞
rj (x − 1)j .
(2.14)
j =1
From (2.1) it follows easily that pn (x) are eigenfunctions of the operator LR,S with eigenvalue x − 2 + x −1 , i.e. LR,S pn (x) = (x − 2 + x −1 )pn (x).
(2.15)
Moreover, pn (x) are also eigenfunctions of a differential operator in x (with coefficients independent of n) and thus provide a difference-differential analog of the rank-one solutions (the KdV family) of the bispectral problem considered by Duistermaat and Grünbaum in [10]. Below we discuss the spectral curve and the common eigenfunction of the maximal commutative ring of difference operators AV containing LR,S . The correspondence between commutative rings of difference operators and curves was studied by van Moerbeke and Mumford [26] and Krichever [19] (see also [22] where the case of singular curves was treated very completely). Let us introduce the ring AV of Laurent polynomials in x that preserve the flag V: AV = {f (x) ∈ C[x, x −1 ] : f (x)Vn ⊂ Vn+k , for some k ∈ Z and ∀n ∈ Z}.
(2.16)
For each f (x) ∈ AV there exists a finite band operator Lf , such that Lf pn (x) = f (x)pn (x). 2 j Moreover, if f (x) = m j =m1 aj x with am1 = 0 and am2 = 0, then the operator Lf has support [m1 , m2 ], i.e. Lf =
m2
bj (n)j .
j =m1
We can think of Lf as an operator obtained by a Darboux transformation from the constant coefficient operator f (). The ring AV is generated by the functions+ 1 1 (x − 1)2R+1 (x + 1)2S+1 1 x+ and v = R+S+1 w= , (2.17) 2 x 2 x R+S+1 i.e. AV = C[w, v]. Thus the operator LR,S constructed above belongs to a maximal rank-one commutative ring of difference operators AV isomorphic to the ring of Laurent polynomials AV . The spectral curve of AV is Spec(AV ):
v 2 = (w − 1)2R+1 (w + 1)2S+1 .
+ Notice that these generators differ slightly from the ones chosen in [17].
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The operators Q and P ∗ in formulas (2.8) and (2.9), can be expressed in terms of the functions {φi (n; r)}Ri=1 and {ψj (n; r)}Sj=1 as Q f (n) =
Wr (φ1 (n; r), . . . , φR (n; r), ψ1 (n; r), . . . , ψS (n; r), f (n)) Wr (φ1 (n; r), . . . , φR (n; r), ψ1 (n; r), . . . , ψS (n; r))
(2.18)
and P ∗ f (n) =
Wr∗ (φ1∗ (n; r), . . . , φR∗ (n; r), ψ1∗ (n; r), . . . , ψS∗ (n; r), f (n)) , (2.19) Wr∗ (φ1∗ (n; r), . . . , φR∗ (n; r), ψ1∗ (n; r), . . . , ψS∗ (n; r))
with φi∗ (n; t) = φi (n + R + S; t),
1 i R,
and ψj∗ (n; t) = ψj (n + R + S; t),
1 j S.
Using these explicit formulas for Q and P , one can check that pn (x −1 ) =
τ (n + 1; r) ∗ xpn+1 (x). τ (n; r)
(2.20)
Finally, (see [17, Theorem 5.2]) one can prove the following orthogonality relation: τ (n + 1; t) dx 1 = δnm , ∀n, m ∈ Z, pn (x)pm (x −1 ) (2.21) 2π i C x τ (n; t) where C is any positively oriented simple closed contour surrounding the origin, not passing through the points x = ±1. Indeed, pn (x) are rational functions on the Riemann sphere with poles only at x = 0, ±1, ∞. The spectral curve Spec(AV ) has cusps at ±1, hence using the fact that the residue of a regular differential at a cusp is always zero (see [25]), we deduce that ∗ (x) dx = 0. resx=±1 pn (x)pm
The proof of (2.21) now follows from the discrete Kadomtsev–Petviashvili bilinear identities and the relation between the wave and the adjoint wave functions in (2.20). 3. Certain Laurent Polynomials in AV The main result in this section is Proposition 3.1 below which allows us to construct some ‘universal’ Laurent polynomials in AV . This result is crucial for the proof of Theorem 5.1. As we shall see in Section 5, applying Proposition 3.1, we can reduce the infinite sum in the computation of the fundamental solution of (5.1)–(5.2) to a finite sum, modulo some identities among the Bessel functions to be discussed in the next section.
191
HEAT KERNEL EXPANSIONS ON THE INTEGERS
PROPOSITION 3.1. Let T be max(R, S), and let s0 , s1 , . . . , sT be distinct nonzero integers, such that sj ≡ sk (mod 2) and sj + sk = 0, for 0 j, k T . Then Fs0 ,...,sT (x) =
T k=0
sk
x sk ∈ AV . 2 2 j =k (sk − sj )
(3.1)
The proof of Proposition 3.1 is based on two simple lemmas related to the Lagrange interpolation polynomial and the Chebyshev polynomials of the second kind. LEMMA 3.2. Let q(n) be an odd polynomial in n of degree at most 2T − 1, and let s0 , s1 , . . . , sT be distinct positive numbers. Then T k=0
sk
q(sk ) = 0. 2 2 j =k (sk − sj )
(3.2)
Proof of Lemma 3.2. Consider the Lagrange interpolation polynomial for q(n) at the nodes ±s0 , ±s1 , . . . , ±sT . Since deg q(n) 2T − 1, the Lagrange polynomial must be identically equal to q(n), i.e. we have T
T n + sk n2 − sj2 n − sk n2 − sj2 q(sk ) + q(−s ) = q(n). k 2sk j =k sk2 − sj2 −2sk j =k sk2 − sj2 k=0 k=0
Hence, the coefficient of n2T +1 on left-hand side must be zero, which gives (3.2). ✷ The Chebyshev polynomials of the second kind are defined by the following three term recurrence relation 2wUn (w) = Un+1 (w) + Un−1 (w),
n = 1, 2, . . . ,
(3.3)
with U0 (w) = 1 and U1 (w) = 2w. From the last formula one sees immediately that Un (w) is an even/odd polynomial if n is an even/odd positive integer, respectively. The Chebyshev polynomials of the second kind can be obtained from the Jacobi polynomials by taking α = β = 1/2: −n, n + 2 1 − w Un (w) = (n + 1)2 F1 3 2 2 n k 2k ((n + 1)2 − j 2 ) (w − 1)k . (3.4) = (n + 1) (2k + 1)! k=0 j =1 These explicit expressions for Un (w) will be needed below.
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F. ALBERTO GRÜNBAUM AND PLAMEN ILIEV
LEMMA 3.3. If s0 , s1 , . . . , sT are distinct positive integers, then the polynomial Qs0 ,...,sT (w) =
T k=0
Us −1 (w)
k 2 sk j =k (sk − sj2 )
(3.5)
is divisible by (w − 1)T . Moreover if s0 ≡ s1 ≡ · · · ≡ sT (mod 2), then (w 2 − 1)T /Qs0 ,...,sT (w).
(3.6)
Proof. From (3.4) we see that (k) Un−1 (1) =
k 2k k! n (n2 − j 2 ), (2k + 1)! j =1
(k) (1) is an odd polynomial in n of degree 2k +1. From Lemma 3.2 it follows i.e. Un−1 that
Q(k) s0 ,...,sT (1) = 0,
k = 0, 1, . . . , T − 1,
hence (w − 1)T /Qs0 ,...,sT (w). If s0 ≡ s1 ≡ · · · ≡ sT (mod 2), then Qs0 ,...,sT (w) is either even or odd polynomial, which proves (3.6). ✷ Proof of Proposition 3.1. Notice that Fs0 ,...,sk−1 ,sk ,sk+1 ,...,sT (x) − Fs0 ,...,sk−1 ,−sk ,sk+1 ,...,sT (x) 1 (x sk + x −sk ) ∈ AV .
= sk j =k (sk2 − sj2 ) Thus, without any restriction, we may assume that s0 , . . . , sT are positive integers. We can rewrite the first equation in (2.17) as x 2 = 2wx − 1. From this relation, one can easily see by induction that x k = Uk−1 (w)x − Uk−2 (w),
for k = 1, 2, . . . ,
(3.7)
with the understanding that U−1 (w) = 0. Using (3.7), we can write the function in (3.1) in the form Fs0 ,...,sT (x) = Qs0 ,...,sT (w)x +
T k=0
Us −2 (w) ,
k 2 sk j =k (sk − sj2 )
(3.8)
where Qs0 ,...,sT (w) is the polynomial defined by (3.5). Clearly, the sum in (3.8) belongs to AV , so it remains to show that Qs0 ,...,sT (w)x ∈ AV .
(3.9)
HEAT KERNEL EXPANSIONS ON THE INTEGERS
193
Using (2.17) we can write v in terms of w and x as v = (w − 1)R (w + 1)S (x − w), which gives (w − 1)R (w + 1)S x = v + w(w − 1)R (w + 1)S ∈ AV . Therefore, if T max(R, S), we have (w 2 − 1)T x ∈ AV . The proof now follows from Lemma 3.3.
(3.10) ✷
4. Some Identities Satisfied by the Bessel Functions The Bessel functions of imaginary argument are defined by the generating function −1 Ik (t)x k = et (x+x )/2 , (4.1) k∈Z
whence Ik (t) = I−k (t), and Ik (−t) = (−1)k Ik (t). Differentiating (4.1) with respect to x and t, one gets kIk (t) =
t (Ik−1 (t) − Ik+1 (t)) 2
(4.2a)
Ik (t) =
(Ik−1 (t) + Ik+1 (t)) ,
(4.2b)
and 1 2
respectively. Similarly, one can show that Ik (t) satisfy the modified Bessel equation (t 2 ∂t2 + t∂t − (t 2 + k 2 ))Ik (t) = 0,
(4.3)
where ∂t = d/dt. The main result in this section is Proposition 4.1 below, which says that if q(j ) is an odd polynomial of j , then the sum q(j )Ij (t), j>k j odd/even
can be written as a finite linear combination of Bessel functions of the form αj (t)Ij (t), finitely many j ’s where each coefficient αj (t) is a polynomial in t. Let us define a sequence of polynomials αjn (t) for j ∈ Z and n = 0, 1, . . . as follows:
t t/2 if j = 0, (4.4) αj0 (t) = δj 0 = 0 if j = 0 2
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F. ALBERTO GRÜNBAUM AND PLAMEN ILIEV
and αjn+1 (t)
=
(t 2 ∂t2 +
+
t∂t )αjn (t)
t 2 (αjn+1 (t) + αjn−1 (t)) + + t ∂t + 2
t2 n (α (t) − 2αjn (t) + αjn−2 (t)). 4 j +2
(4.5)
n (t). MoreFrom the symmetry of the defining relation, it follows that αjn (t) = α−j over, it is clear that
/ [−2n, 2n], αjn = 0 for j ∈
(4.6)
n n (t) = α2n (t) = t 2n+1 /22n+1 . For arbitrary j ∈ [−2n, 2n], αjn (t) is a and α−2n polynomial in t of degree at most 2n + 1.
PROPOSITION 4.1. Let k and n be integers, n 0. Then
j
2n+1
Ij (t) =
j>k j≡k+1 (mod 2)
n αs−k (t)Is (t)
=
2n+k
n αs−k (t)Is (t).
(4.7)
s=−2n+k
s∈Z
Proof by induction. For n = 0 we have to show that t j Ij (t) = Ik (t), 2 j>k
(4.8)
j≡k+1 (mod 2)
which easily follows from (4.2a). Assume that (4.7) holds for some n. Then, using the modified Bessel equation (4.3) we obtain ∞
(k + 2l + 1)2n+3 Ik+2l+1 (t)
l=0
=
(t 2 ∂t2
∞ + t∂t − t ) (k + 2l + 1)2n+1 Ik+2l+1 (t)
=
(t 2 ∂t2
+ t∂t − t )
2
l=0
2
n αs−k (t)Is (t),
s∈Z
which shows that (4.7) holds for n + 1, upon using (4.2b).
✷
5. Heat Kernel Expansions on the Integers In the case of the real line, the solution of the heat equation is not unique unless the class of solutions satisfies a condition of the form 2
|u(x, t)| c1 ec2 x ,
c1 , c2 0.
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HEAT KERNEL EXPANSIONS ON THE INTEGERS
When one says that the Gaussian kernel is the fundamental solution of the heat equation one (implicitly) assumes that one is considering functions of moderate growth at infinity. Denote by u(n, m, t) the solution of ut = LR,S u, u|t =0 = δnm ,
(5.1) (5.2)
where LR,S is the second-order difference operator (acting on functions of a discrete variable n ∈ Z) constructed in Section 2. We make the same implicit assumption here when we say that u(n, m, t) is the fundamental solution. From (2.15) and (2.21) and elementary spectral theory, it follows that dx τ (m) e−2t −1 (5.3) et (x+x ) pn (x)pm (x −1 ) , u(n, m, t) = τ (m + 1) 2π i C x where, for simplicity, we have omitted the dependence on the parameters r = (r1 , r2 , . . .). Using (2.20) we can write also the fundamental solution in terms of the wave and the adjoint wave functions as e−2t −1 ∗ u(n, m, t) = et (x+x ) pn (x)pm+1 (x) dx. (5.4) 2π i C THEOREM 5.1. The solution of (5.1) with initial condition (5.2) can be written in the form βj (n, m, t)Ij (2t), (5.5) u(n, m, t) = e−2t finitely many j ’s where βj (n, m, t) are polynomials in t of degree at most 2T − 1, with T = max(R, S). Proof. From (2.20) it follows that u(n, m, t) =
τ (m)τ (n + 1) u(m, n, t). τ (m + 1)τ (n)
Thus, we can assume that k = n − m 0. Around x = 0, we have the expansion τ (m + 1) pn (x)pm (x −1 ) ∗ = pn (x)pm+1 (x) x τ (m) x n−m−1 (τ (n + 1)τ (m) + O(x)), = τ (n)τ (m) which shows that dx =0 resx=0 x j pn (x)pm (x −1 ) x
for j 1 − k.
(5.6)
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For ; = 1, 2 and i = 0, 1, . . . , T − 1 denote qk+;+2i (j ) =
T j j 2 − (k + ; + 2l)2 . k + ; + 2i l=1 (k + ; + 2i)2 − (k + ; + 2l)2
(5.7)
l=i
qk+;+2i (j ) is an odd polynomial in j of degree 2T − 1. We have −1 et (x+x ) = Ij (2t)x j + Ik (2t)x −k + j 1−k
+
Ij (2t) x
j >k+2T
+
2 T −1 ;=1 i=0
−j
−
T −1
qk+;+2i (j )x
i=0
(5.8a)
−(k+;+2i)
+
(5.8b)
qk+;+2i (j )Ij (2t) x −(k+;+2i) ,
(5.8c)
jk+;+2i j≡k+; (mod 2)
where in the second sum, for every fixed j > k + 2T , we choose ; = 1 or 2, so that j ≡ k + ; (mod 2). Denote by fj (x) the Laurent polynomial of x in the second sum in (5.8), i.e. fj (x) = x −j −
T −1
qk+;+2i (j )x −(k+;+2i) .
i=0
Notice that fj (x) = −j
T
(j 2 − (k + ; + 2l)2 )Fs0 ,s1 ,...,sT (x),
l=1
where Fs0 ,s1 ,...,sT (x) is the function defined by (3.1) with si = −(k + ; + 2i), i = 0, 1, . . . , T − 1 and sT = −j . Thus, by Proposition 3.1, fj (x) ∈ AV , and therefore there exists a difference operator Lfj with support [−j, −(k + ;)], satisfying fj (x)pn (x) = Lfj pn (x). Since n − (k + ;) = m − ; < m we see that dx = 0. (5.9) fj (x)pn (x)pm (x −1 ) x C From (5.3), (5.6) and (5.9) it follows that the infinite sums in (5.8a) and (5.8b) do not ‘contribute’ to the fundamental solution u(n, m, t). Finally, the infinite sum in (5.8c) can be rewritten as a finite linear combination of Bessel functions with polynomial coefficients, according to Proposition 4.1, which completes the proof. ✷ We shall illustrate all steps of the proof by considering the case R = S = 1.
197
HEAT KERNEL EXPANSIONS ON THE INTEGERS
EXAMPLE 5.2. Let R = S = 1. From (2.2) and (2.3) one computes φ1 (n; r) = S11 (n; r) = n + r1 ; ψ1 (n; r) = S1−1 (n; r) = −n +
∞
∞ (−2)j rj . (−1)n exp
(−2)j −1 j rj
j =1
j =1
∞
Denote for simplicity α = r1 and β = j =2 (−2)j −1 j rj . The tau function is given by formula (2.4): n+α −n + α + β . τ (n) = (5.10) 1 2n + 1 − 2(α + β) The second-order difference operator L1,1 is given by formula (2.5), with ∂/∂r1 = ∂/∂α. From (2.6), (2.8), (2.13) and (2.18) we get n+α −n + α + β 1 n x . n + 1 + α n + 1 − α − β x (5.11) pn (x) = τ (n)(x 2 − 1) n + 2 + α −n − 2 + α + β x 2 From the last formula one can easily deduce that near x = 0 we can expand pn (x)pm (x −1 ) as ∞ τ (n + 1) + γj x j , (5.12) x n−m τ (n) j =1 where γj = −
4 [(m − α − β + 1)(n − α − β + 1)(n − m + j ) + τ (n)τ (m) + (−1)j (m + α + 1)(n + α + 1)(n − m + j )]. (5.13)
From (5.13) and (4.8) one can see that indeed u(n, m, t) is a finite linear combination of Bessel functions. Below, we shall illustrate how this can be seen following the proof of Theorem 5.1 using just the first few coefficients in (5.12). Proposition 3.1 tells us that ∀s, l = 0, such that s ≡ l (mod 2) xl xs − ∈ AV . s l −1
Following (5.8), we can write et (x+x ) as −1 Ij (2t)x j + Ik (2t)x −k + et (x+x ) = j 1−k
+
j>k+2 j≡k+1 (mod 2)
Ij (2t) x
−j
j −(k+1) x − + k+1
(5.14a) (5.14b)
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+
Ij (2t) x
−j
j>k+2 j≡k (mod 2)
1 + k+1 1 + k+2
j −(k+2) − x + k+2
(5.14c)
j Ij (2t) x −(k+1) +
j>k j≡k+1 (mod 2)
(5.14d)
j Ij (2t) x −(k+2) ,
(5.14e)
j>k+1 j≡k (mod 2)
where k = n − m. Using (4.8), we can write the sums in (5.14d) and (5.14e) as j Ij (2t) = tIk (2t) and j Ij (2t) = tIk+1 (2t). j>k j≡k+1 (mod 2)
j>k+1 j≡k (mod 2)
Thus (5.14) can be rewritten as −1 Ij (2t)x j + et (x+x ) = j 1−k
+
Ij (2t) x
j>k+2 j≡k+1 (mod 2)
+
j>k+2 j≡k (mod 2)
+x
−k
(5.15a)
Ij (2t) x
−j
−j
j −(k+1) x − + k+1
j −(k+2) x − + k+2
(5.15b)
(5.15c)
t t −1 −2 Ik (2t)x + Ik+1 (2t)x Ik (2t) + . (5.15d) k+1 k+2
The sums in (5.15a), (5.15b) and (5.15c) ‘do not contribute’ to the integral (5.3) (see the proof of Theorem 5.1). Thus u(n, m, t)
t t −1 −2 Ik (2t)x + Ik+1 (2t)x × = e k+1 k+2 τ (m) pn (x)pm (x −1 ) × τ (m + 1) x k+1 τ (n + 1) γ1 γ2 τ (m) Ik (2t) + tIk (2t) + tIk+1 (2t) , (5.16) = e−2t τ (m + 1) τ (n) k+1 k+2 −2t
resx=0 Ik (2t) +
where γ1 and γ2 are the coefficients in the expansion (5.12). Using (5.13) we get the following explicit formula for u(n, m, t) u(n, m, t) =
e−2t [τ (m)τ (n + 1)In−m (2t) + τ (m + 1)τ (n)
HEAT KERNEL EXPANSIONS ON THE INTEGERS
+ 4t (β + 2α)(n + m − β + 2)In−m (2t) − − 4t (2mn − βn + 2n − βm + 2m + β 2 + + 2αβ − 2β + 2α 2 + 2)In−m+1 (2t)].
199
(5.17)
If we put δ = α + 1/2 and let β → ∞ we get u(n, m, t) =
e−2t [τm τn+1 In−m (2t) − tIn−m (2t) − tIn−m+1 (2t)], τm+1 τn
where τn = n + δ. Notice that this is exactly formula (1.6) for the fundamental solution computed in the introduction (R = 1, S = 0). The referee has raised the interesting possibility of using the formula t t AB sBA e =1+A e ds B 0
to get an alternative proof of Theorem 5.1. This remains a challenging problem. Acknowledgements We thank Henry P. McKean and the referee for suggestions that led to an improved version of this paper. References 1. 2.
3. 4.
5. 6. 7. 8. 9. 10.
Adler, M. and Moser, J.: On a class of polynomials connected with the Korteweg–de Vries equation, Comm. Math. Phys. 61 (1978), 1–30. Airault, H., McKean, H. P. and Moser, J.: Rational and elliptic solutions of the Korteweg– de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95– 148. Avramidi, I. G. and Schimming, R.: Heat kernel coefficients for the matrix Schrödinger operator, J. Math. Phys. 36 (1995), 5042–5054. Berest, Y.: Huygens principle and the bispectral problem, In: The Bispectral Problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998, pp. 11–30. Berest, Y. and Kasman, A.: D-modules and Darboux transformations, Lett. Math. Phys. 43 (1998), 279–294. Berest, Y. and Veselov, A. P.: The Huygens principle and integrability, Uspekhi Mat. Nauk 49 (1994), 7–78, transl. in Russian Math. Surveys 49 (1994), 5–77. Berest, Y. and Wilson, G.: Classification of rings of differential operators on affine curves, Internat. Math. Res. Notices 2 (1999), 105–109. Berline, N., Getzler, E. and Vergne, M.: Heat Kernels and Dirac Operators, Grundlehren Math. Wiss. 298, Springer-Verlag, Berlin, 1992. Chalykh, O. A., Feigin, M. V. and Veselov, A. P.: Multidimensional Baker–Akhiezer functions and Huygens’ principle, Comm. Math. Phys. 206 (1999), 533–566. Duistermaat, J. J. and Grünbaum, F. A.: Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240.
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Felder, G., Markov, Y., Tarasov, V. and Varchenko, A.: Differential equations compatible with KZ equations, Math. Phys. Anal. Geom. 3 (2000), 139–177. Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1990. Granovskii, Ya. I., Lutzenko, I. M. and Zhedanov, A. S.: Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Phys. 217 (1992), 1–20. Grünbaum, F. A.: The bispectral problem: an overview, In: J. Bustoz et al. (eds), Special Functions 2000: Current Perspective and Future Directions, 2001, pp. 129–140. Grünbaum, F. A.: Some bispectral musings, In: The Bispectral Problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998, pp. 11–30. Haine, L. and Iliev, P.: Commutative rings of difference operators and an adelic flag manifold, Internat. Math. Res. Notices 6 (2000), 281–323. Haine, L. and Iliev, P.: A rational analogue of the Krall polynomials, In: Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000), J. Phys. A: Math. Gen. 34 (2001), 2445–2457. Kac, M.: Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1–23. Krichever, I. M.: Algebraic curves and non-linear difference equations, Uspekhi Mat. Nauk 33 (1978), 215–216, transl. in Russian Math. Surveys 33 (1978), 255–256. McKean, H. P. and Singer, I.: Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43–69. McKean, H. P. and van Moerbeke, P.: The spectrum of Hill’s equation, Invent. Math. 30 (1975), 217–274. Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg–de Vries equation and related non-linear equations, In: M. Nagata (ed.), Proceedings of International Symposium on Algebraic Geometry (Kyoto 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 115–153. Rosenberg, S.: The Laplacian on a Riemannian Manifold. An Introduction to Analysis on Manifolds, London Math. Soc. Stud. Texts 31, Cambridge Univ. Press, Cambridge, 1997. Schimming, R.: An explicit expression for the Korteweg–de Vries hierarchy, Z. Anal. Anwendungen 7 (1988), 203–214. Serre, J.-P.: Groupes algébriques et corps de classes, Hermann, Paris, 1959. van Moerbeke, P. and Mumford, D.: The spectrum of difference operators and algebraic curves, Acta Math. 143 (1979), 93–154. Wilson, G.: Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177–204.
Mathematical Physics, Analysis and Geometry 5: 201–241, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Classification of Gauge Orbit Types for SU(n)-Gauge Theories G. RUDOLPH1, M. SCHMIDT1 and I. P. VOLOBUEV2
1 Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany. e-mail: [email protected], [email protected] 2 Nuclear Physics Institute, Moscow State University, 119899 Moscow, Russia
(Received: 15 February 2001; in final form: 4 March 2002) Abstract. A method for determining the orbit types of the action of the group of gauge transformations on the space of connections for gauge theories with gauge group SU(n) in spacetime dimension d 4 is presented. The method is based on the one-to-one correspondence between orbit types and holonomy-induced reductions of the underlying principal SU(n)-bundle. It is shown that the orbit types are labelled by certain cohomology elements of spacetime satisfying two relations. Thus, for every principal SU(n)-bundle the corresponding stratification of the gauge orbit space can be explicitly determined. As an application, a criterion characterizing kinematical nodes for physical states in Yang–Mills theory with the Chern–Simons term proposed by Asorey et al. is discussed. Mathematics Subject Classifications (2000): 53C05, 53C80. Key words: classification, gauge orbit space, nongeneric strata, orbit types, quantum nodes, stratification.
1. Introduction One of the basic principles of modern theoretical physics is the principle of local gauge invariance. Its application to the theory of particle interactions gave rise to the standard model, which proved to be a success from both the theoretical and phenomenological points of view. The most impressive results of the model were obtained within the perturbation theory, which works well for high energy processes. On the other hand, the low energy hadron physics, in particular, the quark confinement, turns out to be dominated by nonperturbative effects, for which there is no rigorous theoretical explanation yet. The application of geometrical methods to non-Abelian gauge theories revealed their rich geometrical and topological properties. In particular, it showed that the configuration space of such theories, which is the space of gauge group orbits in the space of connections, may have a highly nontrivial structure. In general, the orbit space possesses not only orbits of the so-called principal type, but also orbits of other types, which may give rise to singularities of the configuration space. This stratified structure of the gauge orbit space is believed to be of importance for
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both the classical and quantum properties of non-Abelian gauge theories in the nonperturbative approach, and it has been intensively studied in recent years. Let us discuss some aspects indicating its physical relevance. First, studying the geometry and topology of the generic (principal) stratum, one gets a deeper understanding of the Gribov ambiguity and of anomalies in terms of index theorems. In particular, one gets anomalies of the purely topological type, which cannot be seen by perturbative quantum field theory. These are well known results from the eighties. Moreover, there are partial results and conjectures concerning the relevance of nongeneric strata. First of all, nongeneric gauge orbits affect the classical motion on the orbit space due to boundary conditions and, in this way, may produce nontrivial contributions to the path integral. They may also lead to localization of certain quantum states, as it was suggested by finite-dimensional examples [10]. Further, the gauge field configurations belonging to nongeneric orbits can possess a magnetic charge, i.e., they can be considered as a kind of magnetic monopole configurations, which are responsible for quark confinement. This picture was found in three-dimensional gauge systems [3], and it is conjectured that it can hold for four-dimensional theories as well [4]. Finally, it was suggested in [16] that nongeneric strata may lead to additional anomalies. Most of the problems mentioned here are still awaiting a systematic investigation. In a series of papers, we are going to make a new step in this direction. We give a complete solution to the problem of determining the strata that are present in the gauge orbit space for SU(n) gauge theories in compact Euclidean spacetime of dimension d = 2, 3, 4. Our analysis is based on the results of a paper by Kondracki and Rogulski [23], where it was shown that the gauge orbit space is a stratified topological space in the ordinary sense (cf. [22] and references therein). Moreover, these authors found an interesting relation between orbit types and certain bundle reductions, which we are going to use. We also refer to [14] for the discussion of a very simple, but instructive special example (orbit types of SU(2)-gauge theory on S4 ). The paper is organized as follows. In Section 2 we introduce the basic notions related to the action of the group of gauge transformations on the space of connections, state the definitions of stabilizer and orbit type and recall basic results concerning the stratification structure of the gauge orbit space. In Section 3 we introduce holonomy-induced bundle reductions and establish their connection with orbit types. As a tool for determining such bundle reductions, we introduce the notions of a Howe subgroup and a Howe subbundle. Section 4 is devoted to the study of the Howe subgroups of SU(n). In Section 5 we give a classification of the Howe subbundles of SU(n)-bundles for spacetime dimension d 4. In Section 6 we prove that any Howe subbundle of SU(n)-bundles is holonomy-induced. In Section 7 we implement the equivalence relation of Howe subbundles due to the action of SU(n). As an example, in Section 8 we determine the orbit types for gauge group SU(2). Finally, in Section 9 we discuss an application to Chern–Simons theory in
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2 + 1 dimensions. In two subsequent papers, we shall investigate the natural partial ordering on the set of orbit types and the structure of another, coarser stratification (see [24]) obtained by first factorizing with respect to the so-called pointed gauge group and then by the structure group. 2. Gauge Orbit Types and Stratification We consider a fixed topological sector of a gauge theory with gauge group G on a Riemannian manifold M. Within the geometrical setting, it means that we are given a smooth right principal fibre bundle P with structure group G over M. G is assumed to be a compact connected Lie group and M is assumed to be compact, connected, and orientable. Denote the sets of connection forms and gauge transformations of P of Sobolev class W k by Ak and Gk , respectively. For generalities on Sobolev spaces of crosssections in fibre bundles, see [28]. Provided 2k > dim M, Ak is an affine Hilbert space and Gk+1 is a Hilbert Lie group acting smoothly from the right on Ak [23, 26, 32]. We shall even assume that 2k > dim M + 2. Then, by the Sobolev Embedding Theorem, connection forms are of class C 1 and, therefore, have continuous curvature. If we view elements of Gk+1 as G-space morphisms P → G, the action of g ∈ Gk+1 on A ∈ Ak is given by A(g) = g −1 Ag + g −1 dg.
(1)
Let M k denote the quotient topological space Ak /Gk+1 . This space represents the configuration space of our gauge theory. For this to make sense, M k should not depend essentially on the purely tech nical parameter k. Indeed, let k > k. Then one has natural embeddings Gk +1 → Gk+1 and Ak → Ak . As a consequence of the first, the latter projects to a map ϕ: M k → M k . Since the image of Ak in Ak is dense, so is ϕ(M k ) in M k . To see (g) that ϕ is injective, let A1 , A2 ∈ Ak and g ∈ Gk+1 such that A2 = A1 . Then (1) implies dg = gA2 − A1 g.
(2)
Due to 2k > 2k > dim M, by the multiplication rule for Sobolev functions, the right-hand side of (2) is of class W k+1 . Then g is of class W k+2 . This can be iterated until the right-hand side is of class W k . Hence, g ∈ Gk +1 , so that A1 and A2 are representatives of the same element of M k . This shows that M k can be identified with a dense subset of M k . Another question is whether the stratification structure of M k , which will be discussed in a moment, depends on k. Fortunately, the answer to this question is negative, see Theorem 3.3. In general, the orbit space of a smooth Lie group action does not admit a smooth manifold structure. The best one can expect is that it admits a stratification. For the notion of stratification of a topological space, see [22] or [23, § 4.4]. For the gauge orbit space M k , a stratification was constructed in [23], using a method which is
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known from compact Lie group actions on completely regular spaces [7]. In order to explain this, let us recall the notions of stabilizer and orbit type. The stabilizer, or isotropy subgroup, of A ∈ Ak is the subgroup = {g ∈ Gk+1 | A(g) = A} Gk+1 A of Gk+1 . It has the following transformation property: For any A ∈ Ak and g ∈ Gk+1 , = g −1 Gk+1 Gk+1 A g. A(g) Thus, there exists a natural map, called a type map, assigning to each element of M k the conjugacy class in Gk+1 made up by the stabilizers of its representatives in Ak . Let k denote the image of this map. The elements of k are called orbit types. The set k carries a natural partial ordering: τ τ iff there are representatives Gk+1 A k+1 of τ and Gk+1 ⊇ Gk+1 A of τ such that GA A . Note that this definition is consistent with [7] but not with [23] and several other authors who define it just inversely. As was shown in [23], the subsets Mτk ⊆ M k , consisting of gauge orbits of type τ , can be equipped with a smooth Hilbert manifold structure and the family {Mτk | τ ∈ k } is a stratification of M k . Accordingly, the manifolds Mτk are called strata. In particular, Mτk , Mk = τ ∈ k
where for any τ ∈ k , Mτk is open and dense in τ τ Mτk . Similarly to the case of compact Lie groups, there exists a maximal orbit type τ0 , called the principal orbit type. Since the corresponding stratum Mτk0 is open and dense in M k , τ0 and Mτk0 are also called generic orbit type and generic stratum, respectively. The above considerations show that the set k , together with its natural partial ordering, carries the information about which strata occur and how they are patched together. To conclude, let us remark that instead of using Sobolev techniques, one can also stick to smooth connection forms and gauge transformations. Then one obtains essentially analogous results about the stratification of the corresponding gauge orbit space where, roughly speaking, one has to replace ‘Hilbert manifold’ and ‘Hilbert Lie group’ by ‘tame Fréchet manifold’ and ‘tame Fréchet Lie group’, see [1, 2]. 3. Correspondence between Orbit Types and Bundle Reductions In this section, let p0 ∈ P be fixed. For A ∈ Ak , let HA and PA denote the holonomy group and holonomy subbundle, respectively, of A based at p0 . We assume 2k > dim M + 2. Then, by the Sobolev Embedding Theorem, A is of
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class C 1 so that PA is a bundle reduction of P of class C 2 . For any g ∈ Gk+1 , let ϑg denote the associated vertical automorphism of P , given by ϑg (p) = p · g(p),
∀p ∈ P .
(3)
For H ⊆ G, let CG (H ) denote the centralizer in G. We abbreviate C2G (H ) = CG (CG (H )). Note that H ⊆ C2G (H ). Let A ∈ Ak . Since the elements of Gk+1 map A-horizontal paths in P to AA horizontal paths, they are constant on PA . Conversely, any gauge transformation which is constant on PA leaves A invariant. Thus, for any g ∈ Gk+1 one has ⇐⇒ g|PA is constant. g ∈ Gk+1 A
(4)
This suggests characterizing orbit types by certain classes of bundle reductions of P . These will be constructed now. For any subgroup S ⊆ Gk+1 , define a subset (S) ⊆ P by (S) = {p ∈ P | g(p) = g(p0 ) ∀g ∈ S}.
(5)
LEMMA 3.1. 2 (a) For any A ∈ Ak , (Gk+1 A ) = PA · CG (HA ). k+1 k+1 = Gk+1 (b) Let A, A ∈ Ak , then (GA ) = (Gk+1 A ) implies GA A . (c) Let g ∈ Gk+1 . For any subgroup S ⊆ Gk+1 , (gSg −1) = ϑg ((S)) · g(p0 )−1 .
Remark. According to (a), if the subgroup S is the stabilizer of a connection A, then (S) is a bundle reduction of P . In [23], the image PA · C2G (HA ) is called the evolution bundle generated by A. Proof. (a) Let A ∈ Ak . Recall that PA has a structure group HA . Hence, in view of (4), the equivariance property of gauge transformations implies {g(p0 ) | g ∈ Gk+1 A } = CG (HA ).
(6)
Thus, by equivariance again, ⇒ g|PA ·C2 (HA ) is constant. g ∈ Gk+1 A
(7)
G
This shows PA · C2G (HA ) ⊆ (Gk+1 A ). Conversely, let p ∈ P such that g(p) = k+1 g(p0 ) for all g ∈ GA . There exists a ∈ G such that p · a −1 ∈ PA . Due to (4), g(p0 ) = g(p · a −1 ) = ag(p)a −1 = ag(p0 )a −1 ,
∀g ∈ Gk+1 A .
Due to (6), then a ∈ C2G (HA ). Hence, p = (p · a −1 ) · a ∈ PA · C2G (HA ). (b) Let A, A ∈ Ak be given. For any g ∈ Gk+1 , we have ⇐⇒ g|(Gk+1) is constant. g ∈ Gk+1 A A
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Here implication from left to right is due to (7) and assertion (a), the inverse implication follows from PA ⊆ (Gk+1 A ) and (4). Since a similar characterization holds , the assertion follows. for Gk+1 A (c) Let p ∈ P , h ∈ S. Using (3) we compute g(p0 )−1 g(p)h(p)g(p)−1g(p0 ) = h(ϑg −1 (p) · g(p0 )). This allows us to write down the following chain of equivalences: p ∈ (gSg −1) ⇐⇒ g(p)h(p)g(p)−1 = g(p0 )h(p0 )g(p0 )−1 ⇐⇒ g(p0 )−1 g(p)h(p)g(p)−1g(p0 ) = h(p0 ) ⇐⇒ h(ϑg −1 (p) · g(p0 )) = h(p0 ) ∀h ∈ S ⇐⇒ ϑg −1 (p) · g(p0 ) ∈ (S). This proves assertion (c).
∀h ∈ S ∀h ∈ S
✷
DEFINITION 3.2. A bundle reduction Q ⊆ P will be called holonomy-induced ˜ ⊆ P to a subgroup H˜ such that of class C r iff there exists a connected reduction Q ˜ · C2G (H˜ ). Q=Q
(8)
Let Red∗ (P ) denote the set of isomorphy classes of holonomy-induced bundle reductions of P of class C 0 , factorized by the action of the structure group G. We equip Red∗ (P ) with the following natural partial ordering: η η iff there exist representatives Q of η and Q of η such that Q ⊆ Q . It is evident that in the definition of Red∗ (P ), continuity could be replaced by any differentiability class. THEOREM 3.3. Let M be compact, dim M 2. Then the assignment induces, by passing to quotients, an order-preserving bijection from k onto Red∗ (P ). Proof. Let τ ∈ k and choose a representative S ⊆ Gk+1 . There exists A ∈ k A such that S = Gk+1 A . According to Lemma 3.1(a), (S) can be obtained by extending the bundle reduction PA ⊆ P to the structure group C2G (HA ). Since PA is of class C 0 , so is (S). Since PA is connected, (S) is holonomy-induced of class C 0 . According to Lemma 3.1(c), if S is conjugate in Gk+1 to some S , (S) and (S ) are conjugate under the actions of Gk+1 and G, then, since gauge transformations from Gk+1 are continuous, (S) and (S ) are C 0 -isomorphic. Thus, projects to a map from k to Red∗ (P ). To check that this map is surjective, let Q ⊆ P be a holonomy-induced bundle reduction of P of class C 0 . Let Q˜ ⊆ Q be a connected bundle reduction of P of class C 0 , with a structure group H˜ , such that (8) holds. Due to well-known ˜ and Q are of class C ∞ . smoothing theorems [17, Ch. I, §4], we may assume that Q ˜ Since M is compact and dim M 2, Moreover, up to the action of G, p0 ∈ Q. ∞ ˜ Q carries a C -connection with holonomy group H˜ [21, Ch. II, Thm. 8.2]. This
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˜ and HA = H˜ . connection prolongs to a unique (smooth) A ∈ Ak obeying PA = Q k+1 2 ˜ ˜ Then Lemma 3.1(a) and (8) imply (GA ) = Q · CG (H ) = Q. This proves surjectivity. To show that the projected map is injective, let τ, τ ∈ k . Choose representatives S, S and assume that (S ) and (S) · a are C 0 -isomorphic, for some a ∈ G. Then there exists a continuous gauge transformation g such that (S ) = ϑg ((S) · a).
(9)
LEMMA 3.4. Let A ∈ Ak and let Q ⊆ P be a bundle reduction of class C ∞ . If there exists a continuous gauge transformation h of P such that (Gk+1 A ) = ϑh (Q),
(10)
then h may be chosen from Gk+1 . Before proving the lemma, let us assume that it holds and finish the arguments. Again, due to smoothing theorems, (S) is C 0 -isomorphic to some bundle reduction Q ⊆ P of class C ∞ , i.e., there exists a continous gauge transformation h such that (S) = ϑh (Q). Due to Lemma 3.4, we can choose h ∈ Gk+1 . Moreover, due to (9), (S ) = ϑgh (Q · a). By application of Lemma 3.4 again, we can achieve gh ∈ Gk+1 . This shows that we may assume, from the beginning, g ∈ Gk+1 . Now consider (9). Since p0 ∈ (S),
ϑg (p0 ) · a = p0 · (g(p0 )a) ∈ (S ).
Since also p0 ∈ (S ), g(p0 )a is an element of the structure group of (S ). Then (S ) · (a −1 g(p0 )−1 ) = (S ), so that (9) and Lemma 3.1(c) yield (S ) = ϑg ((S)) · g(p0 )−1 = (gSg −1 ). Due to Lemma 3.1(b), then S = gSg −1 . This proves injectivity. Proof of Lemma 3.4. Let A and Q be given. Under the assumption that (10) holds, (Gk+1 A ) and Q have the same structure group H . There exist an open covering {Ui } and local trivializations k+1 ξi : Ui × H → (Gk+1 A )|Ui of (GA )
and
ηi : Ui × H → Q|Ui of Q.
These define local trivializations ξ˜i , η˜ i : Ui × G → P |Ui of P over {Ui }. Here ηi , η˜ i are of class C ∞ . As for ξi and ξ˜i , we note that (Gk+1 A ) contains the holonomy k bundle PA . Since A is of class W , PA admits local cross-sections of class W k+1 (cf. the proof of Lemma 1 in [21, Ch. II, §7.1]). Hence, ξi and ξ˜i may be chosen from the class W k+1 . Due to (10), the family {ϑh ◦ ηi } defines a local trivialization of class C 0 of 0 (Gk+1 A ) over {Ui }. Hence, there exists a vertical automorphism ϑ of class C of (Gk+1 A ) such that ξi = ϑ ◦ ϑh ◦ ηi , ∀i. By equivariant prolongation, ϑ defines
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a unique gauge transformation h of P of class C 0 . Since ϑh leaves (Gk+1 A ) k+1 invariant, (GA ) = ϑh h (Q). Thus, by possibly redefining h we may assume that h = 1, i.e., that ϑ is trivial. Then (11) ξ˜i = ϑh ◦ η˜i , ∀i. As we shall argue now, (11) implies h ∈ Gk+1 . By definition, h ∈ Gk+1 iff the local representatives hi = h ◦ η˜ i ◦ ι are of class W k+1 . Here ι denotes the embedding Ui → Ui × G, x → (x, 1). Using η˜ i (x, hi (x)) = ϑh ◦ η˜ i (x, 1),
∀x ∈ Ui ,
we find that hi = pr2 ◦ η˜ i−1 ◦ ϑh ◦ η˜ i ◦ ι, where pr2 is the canonical projection Ui × G → G. Using (11) we obtain hi = pr2 ◦ η˜ i−1 ◦ ξ˜i ◦ ι, ∀i. Here ξ˜i is of class W k+1 and the other maps are of class C ∞ . Thus, according to the composition rules of Sobolev mappings, hi is of class W k+1 . It follows h ∈ Gk+1 . This proves Lemma 3.4 and, therefore, Theorem 3.3. ✷ Remarks. (1) As an important consequence of Theorem 3.3, k does not depend on k. (2) Theorem 3.3 also shows that the notion of holonomy-induced bundle reduction may be viewed as an abstract version of the notion of evolution subbundle generated by a connection, introduced in [23]. (3) General arguments show that Red∗ (P ) is countable, see [23, §4.2]. Hence, so is k . Countability of k is a necessary condition for this set to define a stratification in the sense of [22]. It was first stated in Theorem 4.2.1 in [23]. In fact, the proof of this theorem already contains most of the arguments needed to prove Theorem 3.3. Unfortunately, although in the proof of Theorem 4.2.1 the authors used that isomorphy of evolution subbundles implies conjugacy under the action of Gk+1 , they did not give an argument for that. Such an argument is provided by our Lemma 3.4. (4) The geometric ideas behind the proof of Theorem 3.3 are also contained in [15, §2]. However, a rigorous proof was not given there. In view of Theorem 3.3, we are left with the problem of determining the set Red∗ (P ) together with its partial ordering. To begin with, we make the following observation. By construction, the structure group of a holonomy-induced reduction of P has the form H = CG (H˜ ), for some H˜ ⊆ H . Such subgroups are known as Howe subgroups in the literature, cf. [27]. They can equivalently be characterized by the property H = C2G (H ). DEFINITION 3.5. Howe subbundle.
A reduction of P to a Howe subgroup of G will be called a
As remarked above, the class of Howe subbundles of P contains the class of holonomy-induced reductions of P . Thus, we are lead to the following programme:
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PROGRAMME Step 1. Determination of the Howe subgroups of G. Since G-action on bundle reductions conjugates the structure group, classification up to conjugacy is sufficient. Step 2. Determination of the Howe subbundles of P up to isomorphy. Step 3. Specification of the Howe subbundles which are holonomy-induced. Step 4. Factorization by G-action. Step 5. Determination of the natural partial ordering. In this paper, we perform steps 1–4 for the group G = SU(n). The determination of the natural partial ordering can be found in [37]. 4. The Howe Subgroups of SU(n) Let Howe(SU(n)) denote the set of conjugacy classes of Howe subgroups of SU(n). In order to derive Howe(SU(n)), we consider SU(n) as a subset of Mn (C), the associative algebra of complex (n × n)-matrices. In the literature, it is customary to consider, instead of Howe subgroups, reductive Howe dual pairs. A Howe dual pair is an ordered pair of subgroups (H1 , H2 ) of G such that H1 = CG (H2 ),
H2 = CG (H1 ).
The assignment H → (H, CG (H )) defines a one-to-one relation between Howe subgroups and Howe dual pairs. The pair is called reductive iff its members are reductive. In our case, this condition is automatically satisfied because SU(n) is compact and Howe subgroups are always closed. Reductive Howe dual pairs play an important role in the representation theory of Lie groups, cf. [18]. Although for SU(n) it is not necessary to go into the details of the classification theory of reductive Howe dual pairs, we note that there exist, essentially, two methods. One applies to the isometry groups of Hermitian spaces and uses the theory of Hermitian forms [27, 29, 31]. The other method applies to complex semisimple Lie algebras and uses root space techniques [30]. Let K(n) denote the collection of pairs of sequences (of equal length) of positive integers J = (k, m) = ((k1 , . . . , kr ), (m1 , . . . , mr )),
r = 1, 2, 3, . . . , n,
which obey k·m=
r
ki mi = n.
(12)
i=1
For a given element J = (k, m) of K(n), let g denote the greatest common divisor r ) by g mi = mi , ∀i. Moreover, for of the members of m. Define m = ( m1 , . . . , m
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any permutation σ of r elements, define σ J = (σ k, σ m). Any J ∈ K(n) generates a decomposition Cn = (Ck1 ⊗ Cm1 ) ⊕ · · · ⊕ (Ckr ⊗ Cmr )
(13)
and an associated injective homomorphism Mk1 (C) × · · · × Mkr (C) → Mn (C) (D1 , . . . , Dr ) → (D1 ⊗ 1m1 ) ⊕ · · · ⊕ (Dr ⊗ 1mr ).
(14)
We denote the image of this homomorphism by MJ (C), its intersection with U(n) by U(J ) and its intersection with SU(n) by SU(J ). Note that U(J ) is the image of the restriction of (14) to U(k1 ) × · · · × U(kr ) ⊆ Mk1 (C) × · · · × Mkr (C). LEMMA 4.1. A subgroup of U(n) (resp. SU(n)) is Howe if and only if it is conjugate, under the action of SU(n) by inner automorphisms, to U(J ) (resp. SU(J )) for some J ∈ K(n). Proof. We give only the proof for SU(n). Let H be a Howe subgroup of SU(n). Then H = CSU(n) (K) = CMn (C) (K) ∩ SU(n) for some subgroup K ⊆ SU(n). Since K is ∗-invariant, so is M := CMn (C) (K). Since M also contains the unit matrix, it is a unital ∗-subalgebra (or von Neumann algebra) of Mn (C). Thus, as a basic fact, M is conjugate under SU(n)-action to MJ (C), for some J . Then H is conjugate in SU(n) to MJ (C) ∩ SU(n) = SU(J ). Conversely, let J ∈ K(n). It suffices to show that SU(J ) is Howe. Consider the centralizer M := CMn (C) (MJ (C)). Since MJ (C) is a unital ∗-subalgebra, so is M . In particular, M is spanned by the subset M˜ = M ∩ SU(n) (which is a sub˜ = group of SU(n)). Moreover, the Double Commutant Theorem yields CMn (C) (M) MJ (C). Thus, we obtain CSU(n) (M˜ ) = CMn (C) (M˜ ) ∩ SU(n) = CMn (C)(M ) ∩ SU(n) = MJ (C) ∩ SU(n) = SU(J ). This shows that SU(J ) is Howe.
✷
LEMMA 4.2. Let J, J ∈ K(n). Then SU(J ) and SU(J ) are conjugate under the action of SU(n) by inner automorphisms if and only if there exists a permutation σ such that J = σ J . Proof. It suffices to check the assertion for the subalgebras MJ (C) and MJ (C) of Mn (C). If a permutation σ exists, there exists T ∈ SU(n) mapping the factors Cki ⊗ Cmi of the decomposition (13), defined by J , identically onto the factors Ckσ (i) ⊗ Cmσ (i) of the decomposition defined by J . Then MJ (C) = T −1 MJ (C)T . Conversely, if MJ (C) = T −1 MJ (C)T for some T ∈ SU(n), then MJ (C) and MJ (C) are isomorphic. Hence, k = σ k for some permutation σ . Since T is an isomorphism of the representations J
Mk1 (C) × · · · × Mkr (C) −→ Mn (C)
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and J
σ
Mk1 (C) × · · · × Mkr (C) −→ Mk1 (C) × · · · × Mkr (C) −→ Mn (C), where J , J indicate the respective embeddings (14), it does not change the multi✷ plicities of the irreducible factors. Thus, m = σ m. It follows J = σ J . As a consequence of Lemma 4.2, we introduce an equivalence relation on the ˆ denote the set of set K(n): J ∼ J iff J = σ J for some permutation σ . Let K(n) equivalence classes. ˆ THEOREM 4.3. The assignment J → SU(J ) induces a bijection from K(n) onto Howe(SU(n)). Proof. According to Lemma 4.1, the assignment J → SU(J ) induces a surjecˆ tive map K(n) → Howe(SU(n)). Due to Lemma 4.2, this map projects to K(n) and the projected map is injective. ✷ This concludes the classification of Howe subgroups of SU(n), i.e., Step 1 of our programme. 5. The Howe Subbundles of SU(n)-Bundles In this section, let P be a principal SU(n)-bundle over M, dim M 4. We are going to derive the Howe subbundles of P up to isomorphy. As we have seen above, we can restrict attention to the structure groups SU(J ), J ∈ K(n). Thus, let J ∈ K(n) be fixed. Let Bun(M, SU(J )) denote the set of isomorphism classes of principal SU(J )-bundles over M (where principal bundle isomorphisms are assumed to commute with the structure group action and to project to the identical map on the base space). Moreover, let Red(P , SU(J )) denote the set of isomorphism classes of reductions of P to the subgroup SU(J ) ⊆ SU(n). We shall first derive a description of Bun(M, SU(J )) in terms of suitable characteristic classes and then give a characterization of the subset Red(P , SU(J )). The classification of Bun(M, SU(J )) will involve the construction of the Postnikov tower of the classifying space BSU(J ) up to level 5. For the convenience of the reader, the basics of this method will be briefly explained below. Note that in the sequel maps of topological spaces are always assumed to be continuous, without explicitly stating this. 5.1. PRELIMINARIES Universal Bundles and Classifying Spaces. Let G be a Lie group. As a basic fact in bundle theory, there exists a so-called universal G-bundle G → EG → BG with the following property: For any CW complex (hence, in particular, any manifold) X the assignment [X, BG] −→ Bun(X, G),
f → f ∗ EG,
(15)
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is a bijection [19]. Here [·, ·] means the set of homotopy classes of maps and f ∗ denotes the pull-back of bundles. Both EG and BG can be realized as CW complexes. They are unique up to homotopy equivalence. BG is called the classifying space of G. The homotopy class of maps X → BG associated to P ∈ Bun(X, G) by virtue of (15) is called the classifying map of P . We denote it by fP . Note that a principal G-bundle is universal iff its total space is contractible. As a consequence, the exact homotopy sequence of fibre spaces [8] implies for the homotopy groups πi (G) ∼ = πi+1 (BG),
i = 0, 1, 2, . . . .
(16)
Associated Principal Bundles Defined by Homomorphisms. Let ϕ: G → G be a Lie group homomorphism and let P ∈ Bun(X, G). By virtue of the action G × G → G ,
(a, a ) → ϕ(a)a ,
G becomes a left G-space and we have an associated bundle P [ϕ] = P ×G G . P [ϕ] can be viewed as a principal bundle in an obvious way. One has the natural bundle morphism ψ: P → P [ϕ] ,
p → [(p, 1G )].
(17)
It obeys ψ(p · a) = ψ(p) · ϕ(a) and projects to the identical map on X. In the special case where ϕ is a Lie subgroup embedding, (17) is an embedding of P onto a reduction of P [ϕ] to the subgroup (G, ϕ) of G . Then P [ϕ] is the extension of P by G . In this case, if no confusion about ϕ can arise, we shall often write P [G ] instead of P [ϕ] . Classifying Maps Associated to Homomorphisms. Again, let ϕ: G → G be a homomorphism. One can associate to ϕ a map Bϕ: BG → BG which is defined as the classifying map of the principal G -bundle (EG)[ϕ] associated to the universal G-bundle EG. It has the following functorial property: For ϕ: G → G and ϕ : G → G there holds B(ψ ◦ ϕ) = Bψ ◦ Bϕ.
(18)
Using Bϕ, the classifying map of P [ϕ] can be expressed through that of P : fP [ϕ] = Bϕ ◦ fP .
(19)
We note that in the special case where ϕ is a normal Lie subgroup embedding, Bϕ is a principal bundle Bϕ
G /G → BG −→ BG .
(20)
The classifying map of this bundle is Bp [6], where p: G → G /G is the natural projection.
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CLASSIFICATION OF GAUGE ORBIT TYPES
Characteristic Classes. Let G be a Lie group. Consider the cohomology ring H ∗ (BG, π ) of the classifying space with values in some Abelian group π . For any P ∈ Bun(X, G), the homomorphism (fP )∗ , induced on cohomology, maps H ∗ (BG, π ) to H ∗ (X, π ). Therefore, given γ ∈ H ∗ (BG, π ), one can define a map χγ : Bun(X, G) → H ∗ (X, π ),
P → (fP )∗ γ .
(21)
This is called the characteristic class for G-bundles over X defined by γ . By construction, one has the following universal property of characteristic classes: Let f : X → X be a map and let P ∈ Bun(X , G). Then χγ (f ∗ P ) = f ∗ χγ (P ).
(22)
Observe that if two bundles are isomorphic then their images under arbitrary characteristic classes coincide, whereas the converse, in general, does not hold. This is due to the fact that characteristic classes can control maps X → BG only on the level of the homomorphisms induced on cohomology. In general, the latter do not give sufficient information on the homotopy properties of the maps. In certain cases, however, they do. For example, such cases are obtained by specifying G to be U(1) or discrete, or by restricting X in dimension. In these cases there exist sets of characteristic classes which classify Bun(X, G). Eilenberg–MacLane Spaces. Let π be a group and n a positive integer. An arcwise connected CW complex X is called an Eilenberg–MacLane space of type K(π, n) iff πn (X) = π and πi (X) = 0 for i = n. Eilenberg–MacLane spaces exist for any choice of π and n, provided π is commutative for n 2. They are unique up to homotopy equivalence. The simplest example of an Eilenberg–MacLane space is the 1-sphere S1 , which is of type K(Z, 1). Two further examples, K(Z, 2) and K(Zg , 1), are briefly discussed in the Appendix. Note that Eilenberg–MacLane spaces are, apart from very special examples, infinite dimensional. Assume π to be commutative also in the case n = 1. Due to the Universal Coefficient Theorem, Hom(Hn (K(π, n)), π ) is isomorphic to a subgroup of H n (K(π, n), π ). Due to the Hurewicz Theorem, Hn (K(π, n)) ∼ = πn (K(π, n)) = n π . It follows that H (K(π, n), π ) contains elements which correspond to isomorphisms Hn (K(π, n)) → π . Such elements are called characteristic. If γ ∈ H n (K(π, n), π ) is characteristic then for any CW complex X, the map [X, K(π, n)] → H n (X, π ),
f → f ∗ γ ,
(23)
is a bijection [8, §VII.12]. In this sense, Eilenberg–MacLane spaces provide a link between homotopy properties and cohomology. Path-Loop Fibration. Let X be an arcwise connected topological space. Consider the path-loop fibration over X, 0(X) → P (X) −→ X. Here 0(X) and P (X) denote the loop space and the path space of X, respectively (both based at some
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G. RUDOLPH ET AL.
point x0 ∈ X). Since P (X) is contractible, the exact homotopy sequence induced by the path-loop fibration implies πi (0(X)) ∼ = πi+1 (X), i = 0, 1, 2, . . . . Thus, 0(K(π, n + 1)) = K(π, n), ∀n, and the path-loop fibration over K(π, n + 1) reads K(π, n) → P (K(π, n + 1)) → K(π, n + 1).
(24)
Postnikov Tower. A map f : X → X of topological spaces is called an n-equivalence iff the homomorphism induced on homotopy groups f∗ : πi (X) → πi (X ) is an isomorphism for i < n and surjective for i = n. Let f : X → X be an n-equivalence and let Y be a CW complex. Then the map [Y, X] → [Y, X ], g → f ◦ g, is bijective for dim Y < n and surjective for dim Y = n [8, Ch. VII, Cor. 11.13]. A CW complex Y is called simple iff it is arcwise connected and the natural action of π1 (Y ) on πi (Y ) is trivial for all i 1. The following theorem describes how a simple CW complex can be approximated by n-equivalent spaces constructed from Eilenberg–MacLane spaces. THEOREM 5.1. Let Y be a simple CW complex. There exist: (a) a sequence of CW complexes Yn and principal fibrations qn
K(πn (Y ), n) → Yn+1 −→ Yn ,
n = 1, 2, 3, . . . ,
(25)
induced by maps θn : Yn → K(πn (Y ), n + 1), (b) a sequence of n-equivalences yn : Y → Yn , n = 1, 2, 3, . . . , such that Y1 = ∗ (one point space) and qn ◦ yn+1 = yn for all n. Proof. The assumption that Y be simple implies that the constant map Y → ∗ is a simple map (see [8, Ch. VII, Def. 13.4] for a definition of the latter). Thus, the assertion is a consequence of a more general theorem about simple maps given in [8, Ch. VII, Thm. 13.7]. ✷ Remarks. (1) The sequence of spaces and maps (Yn , yn , qn ), n = 1, 2, 3, . . . , is called Postnikov tower, or Postnikov system, or Postnikov decomposition of Y . (2) For the principal fibrations (25) to be induced by a map θn : Yn → K(πn (Y ), n + 1) means that they are given as pull-back of the path-loop fibration (24) over K(πn (Y ), n + 1). Strategy. We wish to classify Bun(M, SU(J )) by means of characteristic classes. For that purpose, we have to find out whether this is possible and which characteristic classes are necessary for classification. We start from the general classification result Bun(M, SU(J )) = [M, BSU(J )]. In general, [M, BSU(J )] is hard to handle and it cannot be expected to be classified by characteristic classes. However, Theorem 5.1 allows us to successively construct n-equivalent approximations BSU(J )n , up to n = 5, starting from BSU(J )1 = ∗. Thus, if we assume
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CLASSIFICATION OF GAUGE ORBIT TYPES
dim M 4, [M, BSU(J )] = [M, BSU(J )5 ] and the explicit form of BSU(J ) allows us to determine the kind of characteristic classes which are necessary to classify Bun(M, SU(J )). Finally, we shall construct these classes explicitly. We remark that the strategy described is usual in dealing with bundle classification problems, see, for instance, [5, 36]. Now let us turn to the construction of BSU(J5 ). First of all, we need information about the low-dimensional homotopy groups of SU(J ). 5.2. THE HOMOTOPY GROUPS OF SU(J ) For a positive integer a, denote the embedding Za → U(1) by ja and the endomorphism of U(1) mapping z → za by pa . Let jJ and iJ denote the natural embeddings ) : U(J ) → SU(J ) → U(J ) and U(J ) → U(n), respectively. Finally, let prU(J i U(ki ) denote the natural projections. Recall that SU(J ) = ker(detU(n) ◦ iJ ). Let D ∈ U(J ). Writing D = (D1 ⊗ ) (D) ∈ U(ki ), we have 1m1 ) ⊕ · · · ⊕ (Dr ⊗ 1mr ), where Di = prU(J i r r pmi ◦ detU(ki ) (Di ) = pg pmi ◦ detU(ki ) (Di ) . detU(n) ◦ iJ (D) = i=1
i=1
Thus, we can decompose detU(n) ◦ iJ = pg ◦ λJ ,
(26)
where λJ : U(J ) → U(1) is defined by λJ (D) =
r
) pmi ◦ detU(ki ) ◦ prU(J (D), i
∀D ∈ U(J ).
(27)
i=1
Due to (26), the restriction of λJ to the subgroup SU(J ) takes values in ker pg = jg (Zg ). Hence, we can define λSJ : SU(J ) → Zg by requiring λJ ◦ jJ = jg ◦ λSJ .
(28)
In the following lemma, let (SU(J ))0 denote the arcwise connected component of the identity. Note that it is also a connected component. LEMMA 5.2. The homomorphism λSJ projects to an isomorphism SU(J )/ (SU(J ))0 → Zg . Proof. Consider the homomorphism λSJ : SU(J ) → Zg . The target space being discrete, λSJ must be constant on connected components. Hence, (SU(J ))0 ⊆ ker λSJ , so that λSJ projects to a homomorphism SU(J )/(SU(J ))0 → Zg . The latter is surjective, because λSJ is surjective. To prove injectivity, we show ker λSJ ⊆ ) ◦ jJ (D). Define (SU(J ))0 . Let D ∈ ker λSJ and denote Di = prU(J i ϕ: U(1r ) → U(1),
1 r (z1 , . . . , zr ) → z1m · · · zrm .
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G. RUDOLPH ET AL.
Then λSJ (D) = ϕ(detU(k1 ) D1 , . . . , detU(kr ) Dr ). By assumption, (detU(k1 ) D1 , . . . , detU(kr ) Dr ) ∈ ker ϕ. Since the exponents defining ϕ have greatest common divisor 1, ker ϕ is connected. Thus, there exists a path (γ1 (t), . . . , γr (t)) in ker ϕ running from (detU(k1 ) D1 , . . . , detU(kr ) Dr ) to (1, . . . , 1). For each i = 1, . . . , r, define a path Gi (t) in U(ki ) as follows: First, go from Di to (detU(ki ) Di ) ⊕ 1ki −1 , keeping the determinant constant, thus using connectedness of SU(ki ). Next, use the path γi (t) ⊕ 1ki −1 to get to 1ki . By construction, the image of (G1 (t), . . . , Gr (t)) under the embedding (14) is a path in SU(J ) connecting D with 1n . This proves ✷ ker λSJ ⊆ (SU(J ))0 . THEOREM 5.3. The homotopy groups of SU(J ) are π0 (SU(J )) ∼ = Zg ,
π1 (SU(J )) ∼ = Z⊕(r−1)
and πi (SU(J )) ∼ = πi (U(k1 )) ⊕ · · · ⊕ πi (U(kr ))
for i > 1.
In particular, π1 (SU(J )) and π3 (SU(J )) are torsion-free. Proof. The group π0 (SU(J )) = SU(J )/(SU(J ))0 is given by Lemma 5.2. For i > 1, the assertion follows from the exact homotopy sequence induced by the bundle SU(J ) → U(J ) → U(1) with projection ϕ = detU(n) ◦ iJ . For i = 1, consider the following portion of this sequence: ϕ∗
π2 (U(1)) → π1 (SU(J )) → π1 (U(J )) → π1 (U(1)) → π0 (SU(J )) → π0 (U(J )) →Z → Zg → 0. 0 → π1 (SU(J )) → Z⊕r One has Z⊕r / ker(ϕ∗ ) ∼ = im (ϕ∗ ). Exactness implies ker(ϕ∗ ) ∼ = π1 (SU(J )) and
im (ϕ∗ ) = gZ ∼ = Z.
It follows π1 (SU(J )) ∼ = Z⊕(r−1), as asserted.
✷
5.3. THE POSTNIKOV TOWER OF BSU(J ) UP TO LEVEL
5
Let r ∗ denote the number of indices i for which ki > 1. THEOREM 5.4. The fifth level of the Postnikov tower of BSU(J ) is given by (BSU(J ))5 = K(Zg , 1) ×
r−1 j =1
∗
K(Z, 2) ×
r j =1
K(Z, 4).
(29)
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CLASSIFICATION OF GAUGE ORBIT TYPES
Proof. First, we check that BSU(J ) is a simple space. To see this, note that any inner automorphism of SU(J ) is generated by an element of (SU(J ))0 , hence is homotopic to the identity automorphism. Consequently, the natural action of π0 (SU(J )) on πi−1 (SU(J )), i = 1, 2, 3, . . . , induced by inner automorphisms, is trivial. Since the natural isomorphisms πi−1 (SU(J )) ∼ = πi (BSU(J )) transform this action into that of π1 (BSU(J )) on πi (BSU(J )), the latter is trivial, too. Thus, we can apply Theorem 5.1 to construct the Postnikov tower of BSU(J ). According to Theorem 5.3, the relevant homotopy groups are π1 (BSU(J )) = Zg , π3 (BSU(J )) = 0,
π2 (BSU(J )) = Z⊕(r−1) , ∗
π4 (BSU(J )) = Z⊕r .
(30)
Moreover, we shall need that H ∗ (K(Z, 2), Z) is torsion-free and that H 2i+1 (K(Z, 2), Z) = 0, H 2i+1 (K(Zg , 1), Z) = 0,
(31)
i = 0, 1, 2, . . . ,
see Appendix. We start with (BSU(J ))1 = ∗. (BSU(J ))2 : Being a fibration over (BSU(J ))1 , (BSU(J ))2 must coincide with the fibre: (BSU(J ))2 = K(Zg , 1).
(32)
(BSU(J ))3 : In view of (32) and (30), (BSU(J ))3 is the total space of a fibration q2
K(Z⊕(r−1) , 2) → (BSU(J ))3 −→ K(Zg , 1)
(33)
induced from the path-loop fibration over K(Z⊕(r−1) , 3) by some map θ : K(Zg , 1) → K(Z⊕(r−1) , 3). Note that K(Z⊕(r−1) , n) = r−1 2 j =1 K(Z, n), ∀n. Then, due to (23), [K(Zg , 1), K(Z
⊕(r−1)
, 3)] =
r−1
H 3 (K(Zg , 1), Z).
i=1
Here the right-hand side is trivial by (31). Hence, θ2 is homotopic to a constant map, so that the fibration (33) is trivial. It follows that (BSU(J ))3 = K(Zg , 1) ×
r−1
K(Z, 2).
(34)
j =1
(BSU(J ))4 : In view of (30), (BSU(J ))4 is given by a fibration over (BSU(J ))3 with fibre K(0, 3) = ∗. Hence, it just coincides with the base space.
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G. RUDOLPH ET AL.
(BSU(J ))5 : This is the total space of a fibration ∗
q4
K(Z⊕r , 4) → (BSU(J ))5 → (BSU(J ))3 ,
(35)
which is induced by a map θ4 : (BSU(J ))3 → K(Z to the case of θ2 ,
⊕r ∗
, 5). Similarly
∗
[(BSU(J ))3 , K(Z
⊕r ∗
, 5)] =
r
H 5 ((BSU(J ))3 ).
(36)
i=1
Now consider (33). Since H ∗ (K(Z, 2), Z) is torsion-free, we can apply the Künneth Theorem for cohomology [25, Ch. XIII, Cor. 11.3] to write H 5 ((BSU(J ))3 ) as a sum over tensor products H j (K(Zg , 1), Z) ⊗ H j1 (K(Z, 2), Z) ⊗ · · · ⊗ H jr−1 (K(Z, 2), Z), where j + j1 + · · · + jr−1 = 5. Due to this constraint, each summand contains a tensor factor of odd degree, hence is trivial by (31). Then (36) is trivial, and so is the fibration (35). This proves the assertion. ✷ The fact that (BSU(J ))5 is a direct product of Eilenberg–MacLane spaces immediately yields the following corollary. COROLLARY 5.5. Let J ∈ K(n) and dim M 4. Let P , P ∈ Bun(M, SU(J )). Assume that for any characteristic class α defined by an element of H 1 (BSU(J ), Zg ), H 2 (BSU(J ), Z), or H 4 (BSU(J ), Z) there holds α(P ) = α(P ). Then P and P are isomorphic. Proof. Let pr1 , pr21 , . . . , pr2r -1 , and pr41 , . . . , pr4r ∗ denote the natural projections of the direct product (29) onto its factors. Let γ1 , γ2 , and γ4 be characteristic elements of H 1 (K(Zg , 1), Zg ), H 2 (K(Z, 2), Z), and H 4 (K(Z, 4), Z), respectively. Consider the map ϕ: [M, BSU(J )] → [M, (BSU(J ))5 ] ∗
r−1 r → [M, K(Zg , 1)] × [M, K(Z, 2)] × [M, K(Z, 4)] i=1
→ H 1 (M, Zg ) ×
r−1 i=1
H 2 (M, Z) ×
i=1 r∗
H 4 (M, Z),
i=1
that takes f to ∗
∗ ∗ r (f ∗ (pr1 ◦ y5 )∗ γ1 , {f ∗ (pr2i ◦ y5 )∗ γ2 }r−1 i=1 , {f (pr4i ◦ y5 ) γ4 }i=1 ).
Here y5 : BSU(J ) → (BSU(J ))5 is the 5-equivalence provided by Theorem 5.1. According to Theorem 5.4, the second step of ϕ and, therefore, the whole map, is a bijection.
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CLASSIFICATION OF GAUGE ORBIT TYPES
Now let P , P ∈ Bun(M, SU(J )) as proposed in the corollary. Then, by assumption, the homomorphisms (fP )∗ and (fP )∗ , induced on H 1 (BSU(J ), Zg ), H 2 (BSU(J ), Z), and H 4 (BSU(J ), Z), coincide. This implies ϕ(fP ) = ϕ(fP ). ✷ Hence, fP and fP are homotopic. This proves the corollary. We remark that, of course, the cohomology elements (pr1 ◦ y5 )∗ γ1 , (pr4i ◦ y5 )∗ γ4 ,
(pr2i ◦ y5 )∗ γ2 , i = 1, . . . , r ∗
i = 1, . . . , r − 1,
and
define a set of characteristic classes which classifies Bun(M, SU(J )). These classes are independent and surjective. However, they are hard to handle, because we do not know the homomorphism y5∗ explicitly. Therefore, we prefer to work with characteristic classes defined by some natural generators of the cohomology groups in question. The price we have to pay for this is that the classes so constructed are subject to a relation and that we have to determine their image explicitly. 5.4. GENERATORS FOR H ∗(BSU(J ), Z) Instead of generators for H 2 (BSU(J ), Z) and H 4 (BSU(J ), Z) only, we can construct generators for the whole of H ∗ (BSU(J ), Z) without any additional effort. Consider the homomorphisms ) BprU(J i
BjJ
BSU(J ) −→ BU(J ) −→ BU(ki ). Recall that H ∗ (BU(k), Z) is generated freely over Z by the elements (2j )
γUk ∈ H 2j (BU(k), Z),
j = 1, . . . , k, (2j )
see [6]. We assume that the signs of the γU(ki ) are chosen in such a way that for the (2j ) (2j ) canonical embedding ϕ: U(k) → U(l), k l, one has ϕ ∗ γU(l) = γU(k) , 0 j k. (2j ) Then the characteristic class defined by γU(k) is the j th Chern class of U(k)-bundles over M. We denote (2) (2k) + · · · + γU(k) . γU(k) = 1 + γU(k)
(37) (2j )
Of course, γU(k) defines the total Chern class. The generators γU(ki ) define elements ) ∗ ) γU(ki ) , γ˜J,i = (B prU(J i (2j )
(2j ) γJ,i
(2j )
= (BjJ )
∗
(38)
) ∗ (2j ) (B prU(J ) γU(ki ) i
(39)
of H 2j (BU(J ), Z) and H 2j (BSU(J ), Z), respectively. We denote (2) (2ki ) + · · · + γ˜J,i , γ˜J,i = 1 + γ˜J,i
i = 1, . . . , r,
(40)
(2) (2ki ) + · · · + γJ,i , γJ,i = 1 + γJ,i
i = 1, . . . , r,
(41)
as well as γ˜J = (γ˜J,1 , . . . , γ˜J,r ) and γJ = (γJ,1 , . . . , γJ,r ).
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G. RUDOLPH ET AL.
LEMMA 5.6. H ∗ (BU(J ), Z) is generated freely over Z by γ˜J,i , j = 1, . . . , ki , i = 1, . . . , r. Proof. Using the isomorphism (2j )
dr
U(J ) −→
r
U(J )
U(J )
−→
i=1
r
U(ki ),
i=1
where dr denotes r-fold diagonal embedding, the assertion follows from the fact that H ∗ (
σ∗
=c1 (η)
(BjJ )∗
σ∗
=c1 (η)
−→ H 1 (BSU(J ), Z) −→ H 0 (BU(J ), Z) −→ H 2 (BU(J ), Z) −→ H 2 (BSU(J ), Z) −→ H 1 (BU(J ), Z) −→ H 3 (BU(J ), Z) −→ · · · . (42)
(On the level of differential forms, the homomorphism σ ∗ is given by integration over the fibre.) If η was trivial, we would have π1 (BSU(J )) ∼ = π1 (BU(J ) × U(1)) ∼ = Z, which would contradict Theorem 5.3. Hence, η is nontrivial, so that c1 (η) = 0. Due to Lemma 5.6, H ∗ (BU(J ), Z) has no zero divisors. It follows that multiplication by c1 (η) is an injective operation on H ∗ (BU(J ), Z). Then exactness of the Gysin sequence (42) implies that the homomorphism σ ∗ is trivial and, ✷ therefore, (BjJ )∗ is surjective. Lemmas 5.6 and 5.7 yield the following corollary: COROLLARY 5.8. H ∗ (BSU(J ), Z) is generated over Z by γJ,i , j = 1, . . . , ki , i = 1, . . . , r. (2j )
(2) of H ∗ (BSU(J ), Z) are subject to a relation. Since Remark. The generators γJ,i this relation turns out to be a consequence of a more fundamental relation which will be derived below, it does not play a role in the sequel.
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CLASSIFICATION OF GAUGE ORBIT TYPES
5.5. GENERATOR FOR H 1 (BSU(J ), Zg ) Consider the homomorphism λSJ : SU(J ) → Zg and the induced homomorphism (BλSJ )∗ : H 1 (BZg , Zg ) −→ H 1 (BSU(J ), Zg ).
(43)
Due to Lemma 5.2, λSJ ∗ : π0 (SU(J )) → π0 (Zg ) is an isomorphism. Hence, so is (BλSJ )∗ : π1 (BSU(J )) → π1 (BZg ). Then the Hurewicz and Universal Coefficient Theorems imply that (43) is an isomorphism. Thus, generators of H 1 (BSU(J ), Zg ) can be obtained as the images of generators of H 1 (BZg , Zg ) under (BλSJ )∗ . Note that H 1 (BZg , Zg ) ∼ = Zg , see Appendix. To choose a generator, we recall that the short exact sequence µg
@g
0 −→ Z −→ Z −→ Zg −→ 0,
(44)
where µg denotes multiplication by g and @g reduction modulo g, induces a long exact sequence of coefficient homomorphisms, see [8, §IV.5], βg
µg
@g
βg
µg
· · · −→ H i (·, Z) −→ H i (·, Z) −→ H i (·, Zg ) −→ H i+1 (·, Z) −→ · · · .
(45)
The connecting homomorphism βg is usually called Bockstein homomorphism. LEMMA 5.9. There exists a unique element δg ∈ H 1 (BZg , Zg ) such that (2) . βg (δg ) = (Bjg )∗ γU(1)
(46)
It is a generator of H 1 (BZg , Zg ). (2) are elements of Proof. First we notice that both βg (δg ) and (Bjg )∗ γU(1) H 2 (BZg , Zg ) so that Equation (46) makes sense. Consider the following portion of the exact sequence (45): @g
βg
µg
H 1 (BZg , Z) −→ H 1 (BZg , Zg ) −→ H 2 (BZg , Z) −→ H 2 (BZg , Z) −→ Zg −→ Zg . 0 −→ Zg (See Appendix for the cohomology groups.) Since µg is trivial here, βg is an (2) . In order to check isomorphism. Hence, we can define δg = βg−1 ◦ (Bjg )∗ γU(1) that this is a generator, consider J ◦ = ((1), (g)) ∈ K(g). Observe that Zg ∼ = SU(J ◦ ), U(1) ∼ = U(J ◦ ), and that jg corresponds to jJ ◦ : SU(J ◦ ) → U(J ◦ ). Then Lemma 5.7 implies that (Bjg )∗ is surjective. Thus, H 2 (BZg , Z) is generated by (2) and, therefore, H 1 (BZg , Zg ) is generated by δg . ✷ (Bjg )∗ γU(1) We define δJ = (BλSJ )∗ δg . COROLLARY 5.10. H 1 (BSU(J ), Zg ) is generated by δJ , where βg (δJ ) = (2) . (BλSJ )∗ (Bjg )∗ γU(1)
222
G. RUDOLPH ET AL.
5.6. THE RELATION BETWEEN GENERATORS Since βg maps H 1 (BSU(J ), Zg ) to H 2 (BSU(J ), Z), it establishes a relation between the generators of these groups. This will be derived now. For any topological space X, let H0even (X, Z) denote the subset of H even (X, Z) consisting of elements of the form 1 + α (2) + α (4) + · · · . It is a semigroup w.r.t. the cup product. Thus, for any finite sequence of nonnegative integers a = (a1 , . . . , as ), we can define a polynomial function Ea :
s
(α1 , . . . , αs ) → α1a1 = . . . = αsas , (47)
H0even (X, Z) → H0even (X, Z),
i=1
where powers are taken w.r.t. the cup product. By straightforward computation, for the components of 2nd and 4th degree one obtains Ea(2) (α1 , . . . , αs ) Ea(4) (α1 , . . . , αs )
= =
s i=1 s
ai αi(2), ai αi(4)
i=1
+
s
(48)
s ai (ai − 1) (2) αi = αi(2) + + 2 i=1
ai aj αi(2) = αj(2).
(49)
i<j =2
As an immediate consequence of (48), for any l ∈ Z, (2) El(2) a = lEa .
(50)
LEMMA 5.11. The following two formulae hold: (BiJ )∗ γU(n) = Em (γ˜J ), (BλJ )
∗
(2) γU(1)
=
(51)
(2) Em (γ˜J ).
(52)
Proof. First, consider (51). We decompose iJ as follows: dr
iJ : U(J ) −→
U(J )
U(J ) −→
i
i
U(ki ) −→
mi
(U(ki ) × · · · × U(ki ))
i
j
−→ U(n). Here dr , dmi denote r-fold and mi -fold diagonal embedding, respectively, and j stands for the natural (blockwise) embedding. Since we have chosen the generators (2j ) γU(k) for different k in a consistent way, m1
mr
(Bj )∗ γU(n) = (γU(k1 ) × · · · × γU(k1 ) ) × · · · × (γU(kr ) × · · · × γU(kr ) ).
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CLASSIFICATION OF GAUGE ORBIT TYPES
Using this we obtain ∗
(BiJ ) γU(n) =
dr∗
◦
i
= dr∗ ◦
) B prU(J i
∗
∗ ◦ dmi ◦ (Bj )∗ γU(n) i
∗
∗ U(J ) B pri ◦ dmi ×
i
i m1
mr
× (γU(k1 ) × · · · × γU(k1 ) ) × · · · × (γU(kr ) × · · · × γU(kr ) )
∗ U(J ) m1 mr ∗ B pri (γU(k × · · · × γU(k ) = dr ◦ r) 1)
i m1 mr × · · · × γ˜J,r ) = dr∗ (γ˜J,1 m1 mr = γ˜J,1 = . . . = γ˜J,r ,
hence (51). Now consider (52). Due to (26), (2) (2) = (BiJ )∗ (B detU(n) )∗ γU(1) . (BλJ )∗ (Bpg )∗ γU(1)
Inserting (2) (2) = γU(n) (B detU(n) )∗ γU(1)
and
(2) (2) (Bpg )∗ γU(1) = gγU(1)
and using (51) and (50), we obtain (2) (2) (2) = Em (γ˜J ) = gEm g(BλJ )∗ γU(1) (γ˜J ).
Since this relation holds in H 2 (BU(J ), Z) which is free Abelian, it implies (52). ✷ (2) THEOREM 5.12. There holds the relation βg (δJ ) = Em (γJ ). Proof. We compute (2) βg (δJ ) = (BλSJ )∗ (Bjg )∗ γU(1) ∗
= (BjJ ) (BλJ )
∗
(2) γU(1)
(2) = (BjJ )∗ Em (γ˜J )
=
by Corollary 5.10 by (28)
by (52)
(2) Em (γJ ).
✷
5.7. CHARACTERISTIC CLASSES FOR SU(J )- BUNDLES (2j )
Using the cohomology elements γJ,i and δJ constructed above, we define the following characteristic classes for SU(J )-bundles over a manifold M: αJ,i : Bun(M, SU(J )) → H0even (M, Z), ξJ : Bun(M, SU(J )) → H 1 (M, Zg ),
Q → (fQ )∗ γJ,i , Q → (fQ )∗ δJ .
i = 1, . . . , r, (53) (54)
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Sorted by degree, (2) (2ki ) αJ,i (Q) = 1 + αJ,i (Q) + · · · + αJ,i (Q),
where αJ,i (Q) = (fQ )∗ γJ,i . Moreover, we introduce the notation αJ (Q) = (αJ,1 (Q), . . . , αJ,r (Q)). Then (2j )
(2j )
αJ (Q) = (fQ )∗ γJ
(55)
and αJ can be viewed as a map from Bun(M, SU(J )) to the set H (J )(M, Z)
= (α1 , . . . , αr ) ∈
r i=1
(2j ) H0even (M, Z) αi = 0 for j > ki .
(56)
By construction, the relation which holds for γJ and δJ carries over to the characteristic classes αJ and ξJ . From Theorem 5.12 we infer (2) Em (αJ (Q)) = βg (ξJ (Q)),
∀Q ∈ Bun(M, SU(J )).
(57)
In order to derive expressions for αJ and ξJ in terms of the ordinary characteristic classes for U(ki )-bundles and Zg -bundles, let Q ∈ Bun(M, SU (J )). There are two kinds of principal bundles associated in a natural way to Q: The U(ki )-bundles U(J ) S Q[pri ◦jJ ] , i = 1, . . . , r, and the Zg -bundle Q[λJ ] . For the first ones, using (19) and (39), we compute U(J )
c(Q[pri
◦jJ ]
) = (f
Q
U(J ) [pr ◦jJ ] i
) ∗ )∗ γU(ki ) = (fQ )∗ ◦ (BjJ )∗ ◦ (B prU(J ) γU(ki ) i
= (fQ )∗ γJ,i , so that U(J )
αJ,i (Q) = c(Q[pri
◦jJ ]
),
i = 1, . . . , r.
(58)
As for the second one, let χg denote the characteristic class for Zg -bundles over M defined by the generator δg ∈ H 1 (BZg , Zg ), i.e., χg (R) = (fR )∗ δg , ∀R ∈ Bun(M, Zg ). Then (19) yields χg (Q[λJ ] ) = (f S
Q
[λS J]
)∗ δg = (fQ )∗ ◦ (BλSJ )∗ δg = (fQ )∗ δJ .
Consequently, ξJ (Q) = χg (Q[λJ ] ). S
5.8. CLASSIFICATION OF SU(J )- BUNDLES We denote (2) K(M, J ) = {(α, ξ ) ∈ H (J ) (M, Z) × H 1 (M, Zg ) | Em (α) = βg (ξ )}.
(59)
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CLASSIFICATION OF GAUGE ORBIT TYPES
THEOREM 5.13. Let M be a manifold, dim M 4, and let J ∈ K(n). Then the characteristic classes αJ and ξJ define a bijection from Bun(M, SU(J )) onto K(M, J ). Proof. The map is injective by Corollary 5.5. In the following lemma we prove that it is also surjective. ✷ LEMMA 5.14. Let M be a manifold, dim M 4, and let J ∈ K(n). Let (α, ξ ) ∈ K(M, J ). Then there exists Q ∈ Bun(M, SU(J )) such that αJ (Q) = α and ξJ (Q) = ξ . Proof. We give a construction of Q in terms of U(ki ) and Zg -bundles. There exists R ∈ Bun(M, Zg ) such that χg (R) = ξ . Due to dim M 4, there exist also ˜ = Q1 ×M Qi ∈ Bun(M, U(ki )) such that c(Qi ) = αi , i = 1, . . . , r. Define Q · · · ×M Qr (Whitney, or fibre, product). By identifying U(k1 ) × · · · × U(kr ) with ˜ becomes a U(J )-bundle. Then U(J ), Q Q˜ [pri
U(J )
]
∼ = Qi ,
i = 1, . . . , r.
(60)
Consider the U(1)-bundles Q˜ [λJ ] and R [jg ] associated to Q˜ and R, respectively. Assume, for a moment, that they are isomorphic. Then R is a reduction of Q˜ [λJ ] with structure group Zg . Let Q denote the pre-image of R under the natural bundle ˜ with structure group ˜ → Q˜ [λJ ] , see (17). This is a reduction of Q morphism Q being the pre-image of Zg under λJ , i.e., with structure group SU(J ). Using (58), ˜ and (60), for i = 1, . . . , r, Q[jJ ] = Q, U(J )
αJ,i (Q) = c(Q[pri
◦jJ ]
U(J )
) = c((Q[jJ ] )[pri
]
˜ [pri ) = c(Q
U(J )
]
) = c(Qi ) = αi .
∼ R. Thus, (59) yields ξJ (Q) = Moreover, by construction of Q, Q[λJ ] = χg (R) = ξ . It remains to prove Q˜ [λJ ] ∼ = R [jg ] . Using (19), (52), (38) and (60) we find S
(2) c1 (Q˜ [λJ ] ) = Em (α).
(61)
Using (19) and (46) we get c1 (R [jg ] ) = βg (ξ ).
(62)
Thus, due to (α, ξ ) ∈ K(M, J ), c1 (Q˜ [λJ ] ) = c1 (R [jg ] ). It follows that, indeed, ✷ Q˜ [λJ ] ∼ = R [jg ] . This proves the lemma and, therefore, the theorem.
5.9. CLASSIFICATION OF SU(J )- SUBBUNDLES OF SU(n)- BUNDLES Let P be a principal SU(n)-bundle over a manifold M and let J ∈ K(n). We are going to characterize the subset Red(P , SU(J )) ⊆ Bun(M, SU(J )) in terms of the characteristic classes αJ and ξJ . Recall that for Q ∈ Bun(M, SU(J )), Q[SU(n)] denotes the extension of Q by SU(n).
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LEMMA 5.15. For any Q ∈ Bun(M, SU(J )), c(Q[SU(n)] ) = Em (αJ (Q)). Proof. Note that c(Q[SU(n)] ) = c(Q[U(n)] ) = c(Q[iJ ◦jJ ] ). Hence, using (19), (51), ✷ (39) and (55) we obtain c(Q[SU(n)] ) = Em (αJ (Q)). We define K(P , J ) = {(α, ξ ) ∈ K(M, J ) | Em (α) = c(P )}. THEOREM 5.16. Let P be a principal SU(n)-bundle over a manifold M, dim M 4, and let J ∈ K(n). Then the characteristic classes αJ , ξJ define a bijection from Red(P , SU(J )) onto K(P , J ). Proof. Let Q ∈ Bun(M, SU(J )). Then (αJ (Q), ξJ (Q)) ∈ K(M, J ). Lemma 5.15 implies that (αJ (Q), ξJ (Q)) ∈ K(P , J ) if and only if c(Q[SU(n)] ) = c(P ). Due to dim M 4, the latter is equivalent to Q[SU(n)] ∼ = P , i.e., to Q ∈ Red(P , SU(J )). ✷ (2) (α) = 0 The equation Em (α) = c(P ) actually contains the two equations Em (4) and Em (α) = c2 (P ). However, under the assumption that (α, ξ ) ∈ K(M, J ), the first one is redundant, because then, due to (50), (2) (2) (α) = gEm Em (α) = gβg (ξ ) = 0.
Thus, the relevant equations are (2) Em (α) = βg (ξ ),
(63)
(4) (α) = c2 (P ), Em
(64)
where α ∈ H (J )(M, Z), ξ ∈ H 1 (M, Zg ). The set of solutions of Equation (63) yields K(M, J ), hence Bun(M, SU(J )). The set of solutions of both Equations (63) and (64) yields K(P , J ) and, therefore, Red(P , SU(J )). This concludes the classification of Howe subbundles of P , i.e., Step 2 of our programme.
5.10. EXAMPLES We are going to determine K(P , J ) for several choices of J and for base manifolds M = S4 , S2 × S2 , T4 , and L3p × S1 . Here L3p denotes the three-dimensional lens space which is defined to be the quotient of the restriction of the natural action of U(1) on the sphere S3 ⊂ C2 to the subgroup Zp . (Note that there exist more general lens spaces in three dimensions.) Preliminary Remarks. Due to compactness and orientability, H 4 (M, Z) ∼ = Z. Let us derive the Bockstein homomorphism βg : H 1 (M, Zg ) → H 2 (M, Z). Since for products of spheres the integer-valued second cohomology is torsion-free, βg is trivial here. For M = L3p × S1 , we use the Universal Coefficient Theorem
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CLASSIFICATION OF GAUGE ORBIT TYPES
to compute the necessary cohomology groups of L3p from its singular homology which can be found in most textbooks, see, for example, [35, §II.7.7]: H 1 (L3p , Z) = 0,
H 2 (L3p , Z) = Zp ,
H 1 (L3p , Zg ) = Z!p,g" .
(65)
Here !p, g" denotes the greatest common divisor of p and g. Let 1L3p ;Zg , γL(1) 3 ;Z , g p
0 3 1 3 2 3 and γL(2) 3 ;Z be generators of H (Lp , Zg ), H (Lp , Zg ), and H (Lp , Z), respectively. p
0 1 1 1 Moreover, we choose generators 1S1 and γS(1) 1 of H (S , Z) and H (S , Z). According to the Künneth Theorem for cohomology,
H 1 (L3p × S1 , Zg ) = Z!p,g" ⊕ Zg ,
H 2 (L3p × S1 , Z) = Zp ,
(66)
(2) ×γS(1) with generators γL(1) 3 ;Z ×1S1 and 1L3 1 in degree 1 and γL3 ;Z ×1S1 in degree 2. p ;Zg p g p βg acts on the generators as (1) βg (γL(1) 3 ;Z × 1S1 ) = βg (γL3 ;Z ) × 1S1 , g g p
βg (1L3p ;Zg ×
p
γS(2) 1 )
= βg (1L3p ;Zg ) × γS(2) 1 .
The second term vanishes due to (65). The left one can be deduced, using (65), from the following portion of the exact sequence (45) @g
βg
H 1 (L3p , Z) −→ H 1 (L3p , Zg ) −→ H 2 (L3p , Z). Thus, up to a redefinition of the generator γL(1) 3 ;Z , g p
βg (γL(1) 3 ;Z × 1S1 ) = g p
p γ (2) × 1S1 , 3 !p, g" Lp ;Z
βg (1L3p ;Zg × γS(1) 1 ) = 0.
(67)
Finally, we note that L3p is orientable, hence H 4 (L3p ×S1 , Z) is torsion-free. In view of (66), this implies that the cup product is trivial in second degree. Now let us discuss some special choices for J . For brevity, we write J in the form J = (k1 , . . . , kr |m1 , . . . , mr ). J = (1|n) ∈ K(n). Here SU(J ) = Zn , the center of SU(n). Moreover, g = n. Variables are ξ ∈ H 1 (M, Zn ) and α = 1 + α (2), α (2) ∈ H 2 (M, Z). The system of Equations (63) and (64) reads α (2) = βn (ξ ),
n(n − 1) (2) α = α (2) = c2 (P ). 2
Note that we have used (48) and (49). The first equation yields n α (2) = 0, so that the second one requires c2 (P ) = 0. Hence, K(P , J ) is nonempty iff P is trivial and is then parametrized by ξ . This coincides with what is known about Zn -subbundles of SU(n)-bundles.
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G. RUDOLPH ET AL.
J = (n|1) ∈ K(n). Here SU(J ) = SU(n), the whole group. Due to g = 1, the only variable is α = 1 + α (2) + α (4), where α (2j ) ∈ H 2j (M, Z), j = 1, 2. The system of Equations (63) and (64) is α (2) = 0,
α (4) = c2 (P ).
Thus, K(P , J ) consists of P itself. J = (1, 1|2, 2) ∈ K(4).
One can check that SU(J ) has connected components
{diag(z, z, z−1 , z−1 ) | z ∈ U(1)},
{diag(z, z, −z−1, −z−1 ) | z ∈ U(1)}.
It is, therefore, isomorphic to U(1) × Z2 . Variables are ξ ∈ H 1 (M, Z2 ) and αi = 1 + αi(2), αi(2) ∈ H 2 (M, Z), i = 1, 2. The system of equations under consideration is α1(2) + α2(2) = β2 (ξ ),
α1(2) = α1(2) + α2(2) = α2(2) + 4α1(2) = α2(2) = c2 (P ).
Since products including β2 (ξ ) vanish, by eliminating α2(2) we obtain −2α1(2) = α1(2) = c2 (P ).
(68)
For base manifold M = S4 , H 2 (M, Z) = 0. Hence, K(P , J ) is nonempty iff c2 (P ) = 0, in which case it consists of the (necessarily trivial) U(1) × Z2 -bundle over S4 . For M = L3p × S1 , in case c2 (P ) = 0, K(P , J ) is parametrized by ξ ∈ H 1 (M, Zg ) ∼ = Z!2,p" ⊕ Z2
and
α1(2) ∈ H 2 (M, Z) ∼ = Zp .
Otherwise it is empty. 2 2 For M = S2 × S2 , H 1 (M, Zg ) = 0. Let γS(2) 2 be a generator of H (S , Z). Then (2) (2) H 2 (M, Z) ∼ = Z ⊕ Z is generated by γS2 × 1S2 and 1S2 × γS2 , whereas H 4 (M, Z) (2) is generated by γS(2) 2 × γS2 . Writing (2) α1(2) = aγS(2) 2 × 1S2 + b1S2 × γS2
(69)
with a, b ∈ Z, Equation (68) becomes (2) −4abγS(2) 2 × γS2 = c2 (P ).
If c2 (P ) = 0, there are two series of solutions: a = 0 and b ∈ Z as well as a ∈ Z (2) and b = 0. Here K(P , J ) is infinite. If c2 (P ) = 4lγS(2) 2 × γS2 for some l = 0, then a = q and b = −l/q, where q runs through the (positive and negative) divisors of l. Hence, in this case, the cardinality of K(P , J ) is twice the number of divisors of l. If c2 (P ) is not divisible by 4 then K(P , J ) is empty. Finally, for M = T4 one has H 1 (M, Z2 ) ∼ = Z⊕4 2
and
H 2 (M, Z) ∼ = Z⊕6 .
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CLASSIFICATION OF GAUGE ORBIT TYPES
Moreover, H 2 (M, Z) is generated by elements γT(2) 4 ;ij , 1 i < j 4, where (1) (1) γT(2) 4 ;12 = γS1 × γS1 × 1S1 × 1S1 ,
(1) (1) γT(2) 4 ;13 = γS1 × 1S1 × γS1 × 1S1 ,
etc.,
(1) (1) (1) (1) whereas H 4 (M, Z) is generated by γT(4) 4 = γS1 × γS1 × γS1 × γS1 . One can check (2) (4) γT(2) 4 ;ij = γT4 ;kl = Gij kl γT4 ,
(70)
where Gij kl denotes the totally antisymmetric tensor in four dimensions. Writing α1(2) = aij γT(2) (71) 4 ;ij 1i<j 4
and using (70), Equation (68) yields −4(a12 a34 − a13 a24 + a14 a23 )γT(4) 4 = c2 (P ). Hence, we find that K(P , J ) is again nonempty iff c2 (P ) is divisible by 4, in which case it now has always infinitely many elements. J = (1, 1|2, 3) ∈ K(5). The subgroup SU(J ) of SU(5) consists of matrices of the form diag(z1 , z1 , z2 , z2 , z2 ), where z1 , z2 ∈ U(1) such that z12 z23 = 1. We can parameterize z1 = z3 , z2 = z−2 , z ∈ U(1). Hence, SU(J ) is isomorphic to U(1). Variables are αi = 1 + αi(2), i = 1, 2. The equations to be solved read 2α1(2) + 3α2(2) = 0,
α1(2) = α1(2) + 3α2(2) = α2(2) + 6α1(2) = α2(2) = c2 (P ).
Parametrizing α1(2) = 3η, α2(2) = −2η, where η ∈ H 2 (M, Z), we obtain −15η = η = c2 (P ). The discussion of this equation is analogous to that of Equation (68) above. J = (2, 3|1, 1) ∈ K(5). Here SU(J ) ∼ = S(U(2) × U(3)), the symmetry group of the standard model. In the grand unified SU(5)-model this is the subgroup to which SU(5) is broken by the heavy Higgs field. Moreover, the subgroup SU(J ) is the centralizer of the subgroup SU(1, 1|2, 3) discussed above. Variables are αi = (2j ) 1 + αi(2) + αi(4) , where αi ∈ H 2j (M, Z), i, j = 1, 2. Equations (63) and (64) read α1(2) + α2(2) = 0,
α1(4) + α2(4) + α1(2) = α2(2) = c2 (P ).
Replacing α2(2) = −α1(2) we obtain α2(4) = c2 (P ) − α1(4) + α1(2) = α1(2). Thus, K(P , J ) can be parametrized by α1 (or α2 ), i.e., by the Chern class of one of the factors U(2) or U(3). Due to the important role S(U(2) × U(3)) is playing in symmetry breaking, this has been known for a long time [20].
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J = (2|2). The subgroup SU(J ) of SU(4) consists of matrices D⊕D, D ∈ U(2), (det D)2 = 1. Hence, it has connected components {D ⊕ D | D ∈ SU(2)} and {(iD) ⊕ (iD) | D ∈ SU(2)}. One can check that the map SU(2) × Z4 → SU(J ), (D, a) → e2πia/4 D, induces an isomorphism from (SU(2) × Z4 )/Z2 onto SU(J ). Variables are ξ ∈ H 1 (M, Z2 ) and α = 1 + α (2) + α (4). We have α (2) = β2 (ξ ),
α (2) = α (2) + 2α (4) = c2 (P ).
(72)
The first equation expresses α (2) in terms of ξ . For example, in case M = L3p × S1 , × γS(1) by expanding ξ = ξL γL(1) 3 ;Z × 1S1 + ξS 1L3 1 , Equation (67) implies p g p
α (2) = qξL γL(2) 3 ;Z × 1S1 p
for p = 2q,
α (2) = 0 for p = 2q + 1.
(73)
The second equation becomes 2α (4) = c2 (P ). Thus, K(P , J ) is nonempty iff c2 (P ) is even and is then parametrized by ξ ∈ H 1 (M, Z2 ). Let us point out that for M = L3p × S1 , p even, Equation (73) implies that principal SU(J )-bundles which are nontrivial over the factor L3p have a magnetic charge. This distinguishes SU(J ) from (SU(2)×Z2 ), because a principal (SU(2)× Z2 )-bundle can never have a magnetic charge. To conclude, we remark that Equations (63) and (64) will be studied systematically elsewhere. Let us only note the following. Equation (64) always leads to a diophantine equation. For the base manifolds considered above, the latter is bilinear. For such equations, there exists an algorithm to parameterize the set of solutions [33]. The situation is different, for example, for M = CP2 . Here the equation obtained from (64) is quadratic and, therefore, substantially harder to discuss. 6. Holonomy-Induced Bundle Reductions In the next step of our programme we have to determine which of the Howe subbundles of P are holonomy-induced. LEMMA 6.1. Let H ⊆ H ⊆ SU(n) be Howe subgroups. If dim H = dim H , then H = H . Proof. There exist J, J ∈ K(n) such that H and H are conjugate to SU(J ) and SU(J ), respectively. Consider U(J ) and U(J ). Due to H ⊆ H , there exists D ∈ SU(n) such that D −1 U(J )D ⊆ U(J ). Moreover, by assumption, dim U(J ) = dim SU(J ) + 1 = dim SU(J ) + 1 = dim U(J ). Since U(J ) is connected and D −1 U(J )D is closed in U(J ), equality of dimension implies D −1 U(J )D = U(J ). Then D −1 SU(J )D = D −1 (U(J ) ∩ SU(n))D = (D −1 U(J )D) ∩ SU(n) = U(J ) ∩ ✷ SU(n) = SU(J ). It follows H = H . THEOREM 6.2. Any Howe subbundle of a principal SU(n)-bundle is holonomyinduced.
CLASSIFICATION OF GAUGE ORBIT TYPES
231
Proof. Let P be a principal SU(n)-bundle and let Q be a Howe subbundle of P with structure group H . Denote the structure group of a connected component of Q by H˜ . Since H is Howe, H˜ ⊆ C2SU(n) (H˜ ) ⊆ C2SU(n) (H ) = H . Since H˜ and H have the same dimension, so do C2SU(n) (H˜ ) and H . Then Lemma 6.1 implies ✷ C2SU(n) (H˜ ) = H . Hence the assertion. For the reader who wonders whether there exist Howe subbundles which are not holonomy-induced we give an example. Consider the Lie group SO(3). One checks that the subgroup H = {13 , diag(−1, −1, 1)} is Howe. Thus, the reduction ˜ Q = M × H of M × SO(3) is a Howe subbundle. Any connected reduction Q of Q has the center {13 } as its structure group. Since the center is Howe itself, Q˜ · C2G ({13 }) = Q˜ = Q, i.e., Q is not holonomy-induced. 7. Factorization by SU(n)-Action In Step 4 of our programme to determine Red∗ (P ), we actually have to take the disjoint union of Red(P , H ) over all Howe subgroups H of SU(n) and to factorize by the action of SU(n). Since SU(n)-action on bundle reductions conjugates their structure groups, however, it suffices to take the union only over SU(J ), J ∈ K(n): Red(P , SU(J )). (74) J ∈K(n)
We define K(P ) =
K(P , J ).
(75)
J ∈K(n)
We shall denote the elements of K(P ) by L and write them in the form L = (J ; α, ξ ), where J ∈ K(n) and (α, ξ ) ∈ K(P , J ). Due to Theorem 5.16, the collection of characteristic classes {(αJ , ξJ ) | J ∈ K(n)} defines a bijection from (74) onto K(P ). Now we reverse this bijection: For L ∈ K(P ), L = (J ; α, ξ ), define QL ∈ Red(P , SU(J )) by αJ (QL ) = α,
ξJ (QL ) = ξ.
(76)
LEMMA 7.1. Let L, L ∈ K(P ), where L = (J ; α, ξ ), L = (J ; α , ξ ). There exists D ∈ SU(n) such that QL = QL · D if and only if ξ = ξ and J = σ J , α = σ α for some permutation σ . Proof. We start with some preliminary considerations. For J1 , J2 ∈ K(n), we introduce the notation N(J1 , J2 ) := {D ∈ SU(n) | D −1 SU(J1 )D ⊆ SU(J2 )}. Any D ∈ N(J1 , J2 ), defines embeddings D −1 CD, hD : U(J1 ) → U(J2 ), C → hSD : SU(J1 ) → SU(J2 ), C → D −1 CD.
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Let L, L be given as in the lemma. Assume that J = σ J for some permutation σ and let D ∈ N(J, J ). Then hD and hSD are isomorphisms. One can check that [hS ]
QL · D = QL D . Accordingly, the classifying map of QL · D is f(QL·D) = BhSD ◦ fQL ,
(77)
cf. (19). We decompose hD into a pure permutation of factors and an inner automorphism of U(J ) as follows. There exists a permutation σD of 1, . . . , r such that hD maps the σD (i)th factor of U(J ) isomorphically onto the ith factor of U(J ). Then, in particular, J = σD J
(78)
(note that σD can differ from σ by a permutation which leaves J invariant). Fur) ) ◦ hC = prU(J thermore, there exists C ∈ N(J, J ) such that prU(J i σD (i) , ∀i. Then
) ) ◦ jJ ◦ hSC = prU(J prU(J i σD (i) ◦ jJ .
(79)
We define B = DC −1 . Then BhSD = BhSB ◦BhSC . By construction, B ∈ N(J, J ) and hB is an automorphism of U(J ) which leaves each factor invariant separately. One can check that hB is inner. Then hSB is an inner automorphism of SU(J ) and can, therefore, be generated by an element of the connected component of the identity. It follows BhSB = BidSU(J ) ≡ idBSU(J ), hence BhSD = BhSC , up to homotopy. Thus, (77) becomes f(QL·D) = BhSC ◦ fQL .
(80)
Now we can compute the characteristic classes of QL · D in terms of those of QL . Using (80), (39) and (79) we find: αJ (QL · D) = σD αJ (QL ).
(81)
In a similar way, using the obvious equality λSJ ◦ hSC = λSJ , one can check that ξJ (QL · D) = ξJ (QL ).
(82)
Now let us turn to the proof of the lemma. First, assume that there exists D ∈ SU(n) such that QL = QL ·D. Then D ∈ N(J, J ) and J = σ J for some permutation σ . Thus, we can apply the above argumentation. First of all, (78) holds. Moreover, due to (81), α = αJ (QL ) = αJ (Q · D) = σD αJ (QL ) = σD α. Similarly, using (82), one can check that ξ = ξ . This yields the assertion, where the desired permutation is σD . Conversely, assume that ξ = ξ and α = σ α, J = σ J for some permutation σ . Due to Lemma 4.2, there exists D ∈ N(J, J ).
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Obviously, D can be chosen in such a way that σD = σ . Consider QL · D. Due to (81), αJ (QL · D) = σ αJ (QL ) = σ α = α = αJ (QL ). Similarly, using (82), one can verify ξJ (QL · D) = ξJ (QL ). Consequently, QL · D = QL . ✷ As suggested by Lemma 7.1, we introduce an equivalence relation on K(P ): ˆ ) Write L ∼ L iff ξ = ξ and J = σ J , α = σ α for some permutation σ . Let K(P denote the of equivalence classes. ˆ ) onto THEOREM 7.2. The assignment L → QL induces a bijection from K(P Red∗ (P ). Proof. The assignment L → QL induces a surjective map K(P ) → Red∗ (P ). ˆ ) and the projected map is injective. Due to Lemma 7.1, it projects to K(P ✷ With Theorem 7.2 we have accomplished the determination of Red∗ (P ) and, therefore, of the set of orbit types k . Calculations for the latter can now be ˆ ). performed entirely on the level of the classifying set K(P 8. Example: Gauge Orbit Types for SU(2) In this section, we are going to determine k for an SU(2)-gauge theory over the base manifolds discussed in Subsection 5.10. The set K(2) contains the elements Ja = (1|2),
Jb = (1, 1|1, 1),
Jc = (2|1).
Here SU(Ja ) = center, SU(Jb ) ∼ = U(1) (toral subgroup) and SU(Jc ) = SU(2). The strata of M k corresponding to the elements of K(P , Ja ), K(P , Jb ), K(P , Jc ) are, in the respective order, those with stabilizers isomorphic to SU(2), U(1), and the generic stratum. Accordingly, we shall refer to the first class as SU(2)-strata and to the second class as U(1)-strata. We have K(P ) = K(P , Ja ) ∪ K(P , Jb ) ∪ K(P , Jc ) (disjoint union). As we already know, K(P , Ja ) is parametrized by ξ ∈ H 1 (M, Z2 ) if P is trivial and is empty otherwise. Moreover, K(P , Jc ) consists of P itself. For K(P , Jb ), variables are αi = 1 + αi(2), αi(2) ∈ H 2 (M, Z), i = 1, 2. Equations (63), (64) read α1(2) + α2(2) = 0,
α1(2) = α2(2) = c2 (P ).
Replacing α2(2) we obtain −α1(2) = α1(2) = c2 (P ).
(83)
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Note that here α1(2) is just the first Chern class of a reduction of P to the subgroup U(1). According to this, Equation (83) has been discussed in connection with spontaneous symmetry breaking of SU(2) to U(1), see [20]. Note also that when passing ˆ ), the pairs (α1 , α2 ) and (α2 , α1 ) label the same class of bundle from K(P ) to K(P reductions. Hence, solutions α1(2) of (83) have to be identified with their negative. Let us now consider specific choices for the base manifold M. ˆ ) contains M = S4 . Equation (83) requires c2 (P ) = 0. Thus, if P is trivial, K(P the Z2 -bundle, the U(1)-bundle (both necessarily trivial) and P itself. Accordingly, in M k there exist, besides the generic stratum, an SU(2)-stratum and a U(1)stratum. This is well known and was studied in detail, for instance, in [14]. The authors found that the two nongeneric strata of M k can be parametrized by means of an affine subspace of Ak which is acted upon by the Weyl group of SU(2). In ˆ ) consists of P alone. Accordingly, nongeneric strata do case P is nontrivial, K(P k not exist, i.e., M is already a manifold. M = S2 × S2 . We use the notation of Subsection 5.10. Writing α1(2) in the form (69), Equation (83) becomes (2) −2abγS(2) 2 × γS2 = c2 (P ).
ˆ ) contains the Z2 -bundle, which is trivial, and the Thus, if P is trivial then K(P U(1)-bundles labelled by a 0, b = 0 and a = 0, b 0, i.e., which are trivial over one of the 2-spheres. Accordingly, M k contains an SU(2)-stratum and infinitely (2) ˆ many U(1)-strata. In case c2 (P ) = 2lγS(2) 2 × γS2 , l = 0, K(P ) contains the U(1)bundles with a = q and b = −l/q, where q is a (positive) divisor of m. Hence, here the nongeneric part of M k consists of finitely many U(1)-strata. In case c2 (P ) is odd, M k consists only of the generic stratum. M = T4 .
Writing α1(2) in the form (71) and using (70), Equation (83) reads
−2(a12 a34 − a13 a24 + a14 a23 )γT(4) 4 = c2 (P ).
(84)
Hence, the result is similar to that for the case M = S2 × S2 . The only difference is that, due to H 1 (M, Z2 ) ∼ = Z⊕4 2 , there exist 16 different Z2 -bundles which are all ˆ contained in K(P ) for P being trivial. Accordingly, in this case M k contains 16 SU(2)-strata. Moreover, in case c2 (P ) = 2lγT(4) 4 , l = 0, the number of solutions of (84) is infinite. Therefore, in this case there exist infinitely many U(1)-strata. We remark that pure Yang–Mills theory on T4 has been discussed in [12, 13], where the authors have studied the maximal Abelian gauge for gauge group SU(n). They have found that if P is nontrivial, this gauge fixing is necessarily singular on Dirac strings joining magnetically charged defects in the base manifold. It is an interesting question whether there is a relation between such defects and the nongeneric strata of M k .
CLASSIFICATION OF GAUGE ORBIT TYPES
235
ˆ ) contains M = L3p × S1 . Here (83) requires c2 (P ) = 0. Thus, if P is trivial, K(P 1 3 1 the Z2 -bundles which are labelled by the elements of H (Lp × S , Z2 ), i.e., by Z2 ⊕ Z2 in case p = 0 and even or by Z2 otherwise, as well as the U(1)-bundles, (2) (2) which are labelled by α1(2) ∈ H 2 (L3p × S1 , Z) ∼ = Zp , modulo α1 ∼ −α1 . Accordingly, if p is even, the nongeneric part of M k consists of four SU(2)-strata and (p/2 + 1) U(1)-strata. If p is odd, it consists of two SU(2)-strata and ((p + 1)/2) U(1)-strata. If P is nontrivial, nongeneric strata do not exist. 9. Application: Kinematical Nodes in Yang–Mills Theory with Chern–Simons Term Following [3], we consider Yang–Mills–Chern–Simons theory with gauge group SU(n) in the Hamiltonian approach. The Hamiltonian in Schrödinger representation is given by √ d2 x δ iK µν 2 1 J G Aν + + d2 x hTr(Fµν F µν ). H =− √ Tr 2 M h δAµ 4π 4J M Geometrically, Aµ and Fµν are the local representatives of a connection A and its curvature F in a (necessarily trivial) SU(n)-bundle P over the two-dimensional space M. Physical states are given by functionals ψ on the space Ak of connections in P which obey the Gauss law µ
∇A
δ δAµ (x)
ψ(A) =
iK µν G ∂µ Aν (x)ψ(A). 4π
(85)
Here ∇A denotes the covariant derivative w.r.t. A. In [3] it was shown that if A carries a nontrivial magnetic charge, i.e., if it can be reduced to some reduction of P with nontrivial first Chern class, all physical states obey ψ(A) = 0. Such a connection is called a kinematical node. (Note that there exist also dynamical nodes which differ from state to state.) The authors of [3] argue that nodal gauge field configurations are relevant for the confinement mechanism. In the following, we shall show that being a node is a property of strata. For that purpose, we reformulate the result of [3] in our language. ˆ ), where L = (J ; α, ξ ). THEOREM 9.1. Let A ∈ Ak have orbit type [L] ∈ K(P (2) If αi = 0 for some i then A is a kinematical node, i.e., ψ(A) = 0 for all physical states ψ. Proof. The proof follows the lines of [3]. By assumption, A can be reduced to a connection on QL ∈ Red(P , SU(J )). M being a compact orientable 2-manifold, (2) H 2 (M, Z) ∼ = Z. Hence, αi = ci γ (2), where ci ∈ Z and γ (2) is a generator of ˜ t | t ∈ R} of SU(n) by H 2 (M, Z). We define a 1-parameter subgroup {O
˜ t = exp i c1 t 1k1 ⊗ 1m1 ⊕ · · · ⊕ exp i cr t 1kr ⊗ 1mr . O k1 kr
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Due to (63), (m1 c1 + · · · + mr cr )γ (2) = g( m1 α1(2) + · · · + m r αr(2)) = g βg (ξ ) = 0. Hence, ˜ t = exp{it (m1 c1 + · · · + mr cr )} = 1, det O ˜ t defines a 1-parameter ˜ t ⊆ SU(n), ∀t. Commuting with SU(J ) for all t, O so that O subgroup {Ot | t ∈ R} of Gk+1 by ˜ t, Ot (q) = O
∀q ∈ QL , t ∈ R.
Each element of this subgroup is constant on Q. Hence, so is the generator φ = ˙ O(0):
cr c1 1k ⊗ 1m1 ⊕ · · · ⊕ 1k ⊗ 1mr , ∀q ∈ QL . (86) φ(q) = i k1 1 kr r In particular, ∇A φ = 0.
(87)
According to this, for any state ψ,
δ µ δ µ Tr φ∇A µ ψ(A) = − Tr (∇A φ) µ ψ(A) = 0. δA δA M M For physical states, the Gauss law implies Tr(φ dA)ψ(A) = 0.
(88)
M
Using (87), as well as the structure equation F = dA + 12 [A, A], we obtain Tr(φ dA) = 2 Tr(φF ). M
(89)
M
Since A is reducible to QL , F has block structure (F1 ⊗ 1m1 ) ⊕ · · · ⊕ (Fr ⊗ 1mr ) with Fj being (kj × kj )-matrices. Thus, using (86), Tr(φF ) = i M
r mj j =1
kj
Tr Fj .
cj
(90)
M
Now cj being just the first Chern classes of the elementary factors of the U(k1 ) × · · · × U(kr )-bundle QL[U(J )] , we have Tr Fj = −2π icj , j = 1, . . . , r. M
CLASSIFICATION OF GAUGE ORBIT TYPES
237
Inserting this into (90) and the latter into (89) we obtain r mj Tr(φ dA) = 4π (cj )2 . k j M j =1 Thus, in view of (88), r mj j =1
kj
(cj )2 ψ(A) = 0.
(91)
It follows that if one of the cj is nonzero then ψ(A) = 0 for all physical states ψ, i.e., A is a kinematical node. This proves the theorem. ✷ Remark. Let us compare (91) with Formula (6) in [3]. Define ki = ki mi and mi = 1. Then J = (k , m ) ∈ K(n) and U(J ) ⊆ U(J ). Consider the extension )] ⊆ P . One can check that the elementary factors of this bundle have first Q[U(J L Chern classes ci = mi ci . Inserting ki , li , and ci into (91) one obtains formula (6) )] , rather than that it in [3]. In fact, the authors of [3] use that A is reducible to Q[U(J L is even reducible to QL . This argument being ‘coarser’ than ours, it still suffices to prove that any connection which is reducible to a bundle reduction with nontrivial magnetic charge is a kinematical node. As a consequence of Theorem 9.1, one can speak of nodal and nonnodal strata. This information can be read off directly from the labels of the strata. Let us discuss this in some more detail. Let J ∈ K(n) be given and consider Equation (63) (Equation (64) is trivially satisfied). Variables are ξ ∈ H 1 (M, Zg ) and αi(2) ∈ H 2 (M, Z), i = 1, . . . , r. Since H 2 (M, Z) is torsion-free, βg is trivial. Thus, the equation to be solved is (2) Em (α) = 0.
(92)
Writing αi(2) = ci γ (2) again, (92) becomes r
m i ci = 0.
i=1 ⊕r The set of solutions is a subgroup Gm ⊆ Z . According to Theorem 9.1, the nonnodal strata are parametrized by ξ and the neutral element of Gm , whereas the nodal strata are labelled by ξ and all the other elements of Gm . For example, in the case of SU(2) we obtain the following. For J = (1|2), Gm = {0} ⊆ Z, hence all ⊕2 strata are nonnodal. For J = (1, 1|1, 1), Gm = {(c, −c) | c ∈ Z} ⊆ Z . Since also ξ = 0, each value of c labels one stratum. That corresponding to c = 0 is nonnodal, the others are nodal. For J = (2|1), we have the generic stratum, which is nonnodal.
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10. Summary Starting from a principal SU(n)-bundle P over a compact connected orientable Riemannian 4-manifold M, we have derived a classification of the orbit types of the action of the group of gauge transformations of P on the space of connections in P . Orbit types are known to label the elements of the natural stratification, given by Kondracki and Rogulski [23], of the gauge orbit space associated to P . The interest in this stratification is due to the fact that the role of nongeneric strata in gauge physics is not clarified yet. In order to accomplish the classification, we have utilized that orbit types are in one-to-one correspondence with a certain class of bundle reductions of P (called holonomy-induced), factorized by isomorphy and the natural action of the structure group. We have shown that such classes of bundle reductions are labelled by symbols [(J ; α, ξ )], where J = ((k1 , . . . , kr ), (m1 , . . . , mr )) is a pair of sequences of positive integers obeying r
ki mi = n,
i=1
α = (α1 , . . . , αr ), where αi ∈ H even (M, Z) are admissible values of the Chern class of U(ki )-bundles over M, and ξ ∈ H 1 (M, Zg ) with g being the greatest common divisor of (m1 , . . . , mr ). The cohomology elements αi and ξ are subject to the relations r mi i=1 α1m1
g
αi(2) = βg (ξ ),
= . . . = αrmr = c(P ),
where βg : H 1 (M, Zg ) → H 2 (M, Z) is the connecting (i.e., Bockstein) homomorphism associated to the short exact sequence of coefficient groups 0 → Z → Z → Zg → 0. Furthermore, for any permutation σ of {1, . . . , r}, the symbols [(J ; α, ξ )] and [(σ J ; σ α, ξ )] have to be identified. The result obtained enables one to determine which strata are present in the gauge orbit space, depending on the topology of the base manifold and the topological sector, i.e., the isomorphism class of P . For some examples we have discussed this dependence in detail. We have also shown that our result can be used to reformulate a sufficient condition, on a connection to be a node for all physical states in Yang–Mills Theory with Chern–Simons term, derived in [3]. Our result may be viewed as one more step towards a systematic investigation of the physical effects related to nongeneric strata of the gauge orbit space. We remark that orbit types still carry more information about the stratification structure. Namely, their partial ordering encodes how the strata are patched
CLASSIFICATION OF GAUGE ORBIT TYPES
239
together in order to build up the gauge orbit space (cf. [23, Thm. (4.3.5)]). A derivation of this partial ordering can be found in [37]. Appendix: The Eilenberg–MacLane Spaces K(Z, 2) and K(Zg , 1) In this appendix, we construct a model for each of the Eilenberg–MacLane spaces K(Z, 2) and K(Zg , 1) and derive the integer-valued cohomology of these spaces. Consider the natural free action of U(1) on the sphere S∞ which is induced from the natural action of U(1) on S2n−1 ⊂ Cn . The orbit space of this action is the complex projective space CP∞ . Moreover, by viewing Zg as a subgroup of U(1), this action gives rise to a natural free action of Zg on S∞ . The orbit space of the latter is the lens space L∞ g . By construction, one has principal bundles U(1) → S∞ −→ CP∞, Zg → S∞ −→ L∞ g .
(93) (94)
Due to πi (S∞ ) = 0, ∀i, the exact homotopy sequences induced by (93), (94) yield Z, i = 2, ∞ πi (CP ) = πi−1 (U(1)) = 0, i = 2, πi (L∞ g )
= πi−1 (Zg ) =
Zg , i = 1, 0, i = 2, 3, . . . ,
respectively. As a consequence, CP∞ is a model of K(Z, 2) and L∞ g is a model of K(Zg , 1). In particular, Z, i even, i i ∞ (95) H (K(Z, 2), Z) = H (CP , Z) = 0, i odd, see [8, Ch. VI, Prop. 10.2], and
Z, i = 0, Zg , i = 0, even, , Z) = H i (K(Zg , 1), Z) = H i (L∞ g 0, i= 0, odd,
(96)
see [35, §II.7.7] (and use the Universal Coefficient Theorem). We notice that the vanishing of all homotopy groups of S∞ also implies that the bundles (93) and (94) are universal for U(1) and Zg , respectively. Hence, CP∞ and L∞ g are models of BU(1) and BZg , respectively. For BZg , this has been used in the proof of Lemma 5.9. Acknowledgements The authors would like to thank S. Boller, C. Fleischhack, S. Kolb, and A. Strohmaier for interesting discussions. They are also grateful to T. Friedrich, J. Hilgert,
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and H.-B. Rademacher for useful suggestions, as well as to L. M. Woodward, ˇ C. Isham, and M. Cadek who have been very helpful in providing specific information.
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On the Essential Spectrum of a Class of Singular Matrix Differential Operators. I: Quasiregularity Conditions and Essential Self-adjointness PAVEL KURASOV1 and SERGUEI NABOKO2 1 Dept. of Mathematics, Lund Institute of Technology, Box 118, 221 00 Lund, Sweden.
e-mail: [email protected] 2 Dept. of Mathematical Physics, St. Petersburg Univ., 198904 St. Petersburg, Russia. e-mail: [email protected] (Received: 5 December 2001; in final form: 26 August 2002) Abstract. The essential spectrum of singular matrix differential operator determined by the operator matrix d ρ(x) d + q(x) d β − dx x dx dx m(x) d − βx dx 2 x
is studied. It is proven that the essential spectrum of any self-adjoint operator associated with this expression consists of two branches. One of these branches (called regularity spectrum) can be obtained by approximating the operator by regular operators (with coefficients which are bounded near the origin), the second branch (called singularity spectrum) appears due to singularity of the coefficients. Mathematics Subject Classifications (2000): Primary: 47A10, 76W05; secondary: 34L05, 47B25. Key words: essential spectrum, quasiregularity conditions, Hain–Lüst operator.
1. Introduction Systems of ordinary and partial differential and pseudodifferential equations is a subject of interest for many mathematicians (see [19] and numerous references therein). Matrix ordinary differential operators of mixed order appear in many problems of theoretical physics: hydrodynamics, plasma physics, quantum field theory, and others. Mathematically rigorous treatment of such problems has been carried out by several authors: J. A. Adam, V. Adamyan, J. Descluox, G. Geymonat, G. Grubb, T. Kako, H. Langer, A. E. Lifchitz, R. Mennicken, M. Möller, G. D. Raikov, A. Shkalikov, and others [1, 2, 4, 5, 8–11, 15, 17, 20–23, 28, 29, 31, 37]. Matrix differential operators with singular coefficients are of special interest in plasma physics, for example so-called force operators describing equilibrium state of plasma in toroidal region are exactly of this kind [20]. A more general
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class of 3 × 3 matrix differential operators with singularities was considered by V. Hardt, R. Mennicken, and S. Naboko [17], where a new branch of the essential spectrum determined by the singularity was observed and described. This new branch had been predicted by J. Descloux and G. Geymonat [5]. To study the essential spectrum of the operator, so-called quasiregularity conditions were introduced ([17]). These conditions are necessary and sufficient for the boundedness of the essential spectrum of the singular operator. A different approach to this class of matrix operators satisfying the quasiregularity conditions was developed by M. Faierman, R. Mennicken, and M. Möller [10]. Recently, R. Mennicken, S. Naboko, and Ch. Tretter suggested clarifying approach to study this class of singular operators ([30]). It was discovered that the new branch of the essential spectrum can be characterized as the zero set for the symbol of the asymptotic Hain–Lüst operator introduced in [30]. It should be mentioned that in the new approach, the authors used Proposition A.1 from the current paper. Investigation of the essential spectrum of differential and partial differential operators attracts attention of many scientists ([40, 41, 44]). For example the spectrum of pseudodifferential operators with piecewise continuous symbols has been investigated by S. C. Power [35, 36]. In [14] (Chapter 3), it is shown how to calculate the essential spectrum for pseudodifferential boundary value problems using the principal interior and boundary symbol operators. A new class of matrix differential operators with singular coefficients is introduced and investigated in this paper. This class consists of 2× 2 matrices instead of the 3 × 3 operator matrices studied in [30], which is a formal simplification. (The method elaborated in the paper can be applied to m × m operator matrices.) But all essential features of the problem are still present. Additionally the singularities of the matrix elements are distributed in a different way. We decided to study this class of singular operators in order to illustrate the mechanism of the appearance of the additional branch of essential spectrum using the most explicit example. This helps us to avoid tedious calculations and at the same time preserves the main features of the original problem. For this reason we tried to develop a proper Calkin calculus (see Appendix B), which allows one to justify calculations from [17, 20–22] being incomplete. On the other hand, employment of Calkin calculus makes all calculations transparent and easier. For example, the authors of [4], investigating nonsingular operator matrices, used subtle results from operator theory due to P. E. Sobolevskii [26]. Developing these methods, some new results on Banach space operators were obtained. These results on the spectrum of the sum of three operators are of an abstract nature and can be used in other problems as well. Investigating this problem, we tried to elaborate a new general approach to singular matrix differential operators. We hope to be able to apply this method to most general singular matrix differential operators including partial differential operators. The operator under investigation is determined formally by the following expression:
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
245
d d β d − dx ρ(x) dx + q(x) dx x . (1) m(x) β d − x dx x2 We use this form of the matrix differential operator in order to display explicitly the singularities of three matrix elements at the origin. The most interesting (and complicated) case is when the functions β and m do not vanish at the origin. Therefore, the operator defined by the functions β and m having zeroes at the origin of order 1 and 2 respectively, will be called regular. In this case, all singularities are artificial. The essential spectrum of the corresponding operator can easily be investigated using the methods of [4]. All other operators from the described class will be called singular and we are going to concentrate our attention on the case of singular operators only. It is clear that the matrix symbol does not determine the self-adjoint operator uniquely even in the regular case. The extension theory, of the minimal operator in the regular case has been developed by H. de Snoo [43] and in the case of nonsingular leading matrix coefficient in [38, 39]. Our interest in singular problem is motivated by the new spectral phenomenon which can be observed in this case: the essential spectrum of any selfadjoint operator corresponding to the symbol (1) in L2 [0, 1] ⊕ L2 [0, 1] cannot be described as a limit of the essential spectra of the operators determined by the same symbol in L2 [ , 1] ⊕ L2 [ , 1] as → +0. Such limit determines only a certain part of the essential spectrum of the operator in L2 [0, 1] ⊕ L2 [0, 1]. An additional branch of the essential spectrum appears due to the singularity of the coefficients at the origin. Trivial counterpart of this phenomena is well known for infinite intervals, since for example the essential spectrum of −(d2 /dx 2 ) on a finite interval [−an , bn ] is empty and therefore does not give the essential spectrum of −(d2 /dx 2 ) on the whole line when an , bn → ∞. The phenomenon described in the current note is more sophisticated and is due to rather complicated interplay between the components of the matrix differential operator. On the other hand, the coefficient of the matrix determining the operator have singularities at the boundary points. This new branch of essential spectrum is absent in the case of regular operators, since the limit procedure for the essential spectrum described above gives the correct answer in the regular case. Spectral analysis in the regular case is well known and can be carried out using methods developed in [4, 15]. In what follows, the two branches of the essential spectrum will be called the regularity spectrum and singularity spectrum, respectively. We introduce quasiregularity conditions for the singular operator which guarantees boundedness of the regularity spectrum. The quasiregularity conditions determine a special class of singular matrix differential operators for which we are able to calculate the essential spectra. Note that in many physical applications, i.e. in plasma physics ([20]), these conditions are fulfilled. The singularities of the operator coefficients at the origin play an important role even at the stage of the definition of the self-adjoint operator corresponding to the
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formal expression (1). The indices of the minimal differential operator produced by the singular point are investigated by considering the extension of the minimal operator to the set of functions satisfying certain symmetric boundary condition at the regular point. (In this way the singular x = 0 and regular x = 1 endpoints are treated separately and in different ways.) It is proven that this extended operator has trivial deficiency indices (is essentially self-adjoint) if and only if the quasiregularity conditions are satisfied and β(0) = 0. (The condition β(0) = 0 together with the quasiregularity condition (8) imply for smooth coefficients that m(0) = m (0) = 0 and therefore that the operator L is not singular.) If at least one of the quasiregularity conditions is not satisfied or the function β vanishes at the origin then the deficiency indices of the described extended operator are nontrivial like it is in the regular case. We would like to note that the quasiregularity conditions introduced originally to guarantee boundedness of the regular branch of the essential spectrum play an important role in the investigation of the deficiency indices. (Note that the name quasiregularity conditions has nothing to do with the regularity of the extension problem for the operator. It refers to the essential spectrum only.) After the family of self-adjoint operators corresponding to the formal expression (1) is determined, we discuss the transformation of the operator using the exponential map of the interval [0, 1] onto the half-infinite interval [0, ∞). This map transforms the singular point at the origin to a point at ∞ and enables us to use the standard Fourier transform in L2 (R). So the reason to use this exponential map is pure technical. Since we are interested in the essential spectrum of the corresponding selfadjoint operators, the choice of the boundary conditions in the limit circle case is not important. The difference between the resolvents of any two operators from this family is a finite rank operator. To calculate the essential spectrum of any such selfadjoint operator we use the even stronger fact that the essential spectra of any two self-adjoint operators coincide if the difference between their resolvents is compact (Weyl theorem). We develop a so-called cleaning procedure which enables one to reduce the calculation of the essential spectrum of the complicated matrix differential operator given by (1) to the calculation of the essential spectrum of a certain asymptotic singular operator with real coefficients. The singular coefficients of the asymptotic operator are chosen to have the same singularities as those of the original operator. In other words the asymptotic operator is chosen so that the difference between the resolvents of the original and asymptotic operators is compact. The Hain–Lüst operator can be considered as a regularized determinant of the 2×2 matrix differential operator (1), and it plays a very important rˆole in the cleaning procedure. In the considered case the Hain–Lüst operator is an ordinary (scalar) second order differential operator in L2 (R+ ). We hope that the approach developed in the current paper can be applied to more general operators including arbitrary dimension matrix differential operators and matrix partial differential operators. The method of cleaning of the resolvent modulo compact operators is of general
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
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nature. Several abstract lemmas proven in the present paper can be applied without even minor changes. To calculate the essential spectrum of the asymptotic operator we use the fact that its resolvent is equal to the separable sum of two pseudodifferential operators. We call the sum of two pseudodifferential operators separable if the symbol of one of these two operators depends only on the space variable, and the symbol of the other operator depends only on the momentum variable. Calculation of the essential spectrum of such operators is based on Proposition A.1 from Appendix A. We observe that the essential spectrum of the model operator under consideration coincides with the set of zeroes of the symbol of the asymptotic Hain–Lüst operator. That operator is a modified version of the original Hain–Lüst operator which preserves information on the behavior of the coefficients at the singular point only. This operator has a more simple expression: it is a second-order differential operator with constant coefficients. Unfortunately all information concerning the regularity spectrum disappears during this rectification. This probably general relation between the symbol of the asymptotic Hain–Lüst operator and the singularity spectrum will be investigated in one of the forthcoming publications. The methods developed in this article can easily be extended to include differential operators determined by operator matrices of higher dimension. For example, the case when the coefficient m appearing in (1) is a matrix can easily be investigated. The developed methods can help to study matrix partial differential operators as well. These subjects will be discussed in a future publication. 2. The Minimal Operator Let us consider the linear operator defined by the following operator valued 2 × 2 matrix d d β d ρ(x) + q(x) − dx dx dx x , (2) L := m(x) β d − x dx x2 where the real-valued functions ρ(x), q(x), β(x), and m(x) are continuously differentiable in the closed interval [0, 1] ρ, q, β, m ∈ C 2 [0, 1].
(3)
In addition we suppose that the density function ρ is positive (definite) ρ(x) ρ0 > 0.
(4)
Certainly these conditions on the coefficients are far from being necessary for our analysis, but we assume these conditions in order to avoid unnecessary complications. In this way we are able to present certain new ideas explicitly without getting the most optimal result.
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The operator matrix (2) determines rather complicated matrix differential operator. Indeed in its formal determinant which controls the spectrum of the whole operator the differential order of the formal product of the diagonal elements d d m(x) − ρ(x) + q(x) dx dx x2 coincides with that of the formal product of the antidiagonal elements β d d β − . dx x x dx The same holds true for the orders of the singularities at the origin. These relations can be expressed by the diagrams 2 + 0 = 1 + 1 for the order of differential operators and 0 + 2 = 1 + 1 for the orders of the power-like singularities at the origin. These conditions imply that the nondiagonal coupling cannot be considered as a weak perturbation of the diagonal part of the operator and therefore no existing perturbation theory can be applied to the study of the operator. The aim of this article is to describe new spectral phenomena appearing due to this interplay between the singularities. The operator matrix given by (2) does not determine unique self-adjoint operator in the Hilbert space H = L2 [0, 1] ⊕ L2 [0, 1]. To describe the family of self-adjoint operators corresponding to (2) let us consider the minimal operator Lmin with the domain C0∞ (0, 1) ⊕ C0∞ (0, 1). The operator Lmin is symmetric but is not self-adjoint. Let us keep the same notation for the closure of the operator. Any self-adjoint operator corresponding to the operator matrix (2) is an extension of the minimal operator Lmin . It will be shown in Section 4 that the deficiency indices of Lmin are finite and all self-adjoint extensions of the operator can be described by certain boundary conditions at the end points of the interval [0, 1]. In what follows we are going to consider local boundary conditions only. Such boundary conditions do not connect the boundary values of functions at different end point of the interval. As usual each self-adjoint extension of the operator Lmin is a restriction of the adjoint operator L∗min ≡ Lmax , which is defined by the same operator matrix (2) on the domain of functions from W22 [0, 1] ⊕ W21 [0, 1] ⊂ H satisfying the following two additional conditions ([33]) d d β d ρ(x) u1 + qu1 + u2 ∈ L2 [0, 1]; dx dx dx x m β d u1 + 2 u2 ∈ L2 [0, 1]. − x dx x
−
Since the original operator Lmin has finite deficiency indices, the difference between the resolvents of any two self-adjoint extensions of Lmin is a finite rank operator. Therefore all these self-adjoint operators have just the same essential spectrum by the Weyl theorem [24].
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3. Quasiregularity Conditions Consider an arbitrary self-adjoint extension L of the operator Lmin . The essential spectrum of the operator L will be denoted by σess(L) in what follows. One part of σess (L) can be calculated using the Glazman splitting method (see [3]) already at this stage. Indeed consider the operator L0 ( ) being the restriction of the operator L to the domain
d 0 Dom(L ( )) = F = (f1 , f2 ) ∈ Dom(L) : f1 ( ) = f1 ( ) = f2 ( ) = 0 . dx Consider the following decomposition of the Hilbert space L2 [0, 1] = L2 [0, ] ⊕ L2 [ , 1]. The corresponding decomposition of the Hilbert space H is defined as follows H = H ⊕ H = (L2 [0, ] ⊕ L2 [0, ]) ⊕ (L2 [ , 1] ⊕ L2 [ , 1]). Using this decomposition the operator L0 ( ) can be represented as an orthogonal sum of two symmetric operators acting in H and H respectively. The point x = is regular for the operator matrix (2) and one of the self-adjoint extensions of the operator L0 ( ) is defined by Dirichlet boundary conditions at x = ± . (The fact that the Dirichlet boundary condition at any regular point determines a selfadjoint extension is not trivial for matrix differential operators and has been proven rigorously in [43].) Let us denote this extension by L( ). The difference between the resolvents of the operators L( ) and L is at most a rank 2 operator. Therefore the essential spectra of these two operators coincide. In particular the essential spectrum of the operator L contains the essential spectrum of the operator L( ) restricted to the subspace H = L2 [ , 1] ⊕ L2 [ , 1] σess (L) ⊃ σess (L( )|H ),
∈ (0, 1).
(5)
The restricted operator L( )|H is a regular matrix self-adjoint operator and its essential spectrum can be calculated using the results of [4] (Theorem 4.5) β(x)2 m(x) . (6) − σess (L( )|L2 ( ,1)) = Rangex∈[ ,1] x2 x 2 ρ(x) For any > 0 the essential spectrum of L( )|H fills in a certain finite interval, since the functions m, β, and ρ −1 are finite and therefore bounded on [ , 1]. Since obviously β(x)2 m(x) , (7) σess (L( )|H ) = Rangex∈(0,1] − 2 σess (L) ⊃ x2 x ρ(x) >0 the essential spectrum of L is bounded only if the following quasiregularity conditions hold d (ρm − β 2 )|x=0 = 0. (8) ρm − β 2 |x=0 = 0, dx
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The quasiregularity conditions appeared first in [17] and were also used later in [9, 10]. Note that the function (ρm − β 2 )/x 2 is related to the leading coefficient of the formal determinant of the matrix L (2). The rˆole of the quasiregularity conditions is explained by the following statement based on formula (51) to be proven in Section 8. LEMMA 3.1. Under the assumptions (3) and (4) on the coefficients ρ, β, m, and q the quasiregularity conditions are fulfilled if and only if the essential spectrum of at least one (and, hence, any) self-adjoint extension of Lmin is bounded. Proof. Formula (7) implies that quasiregularity conditions are fulfilled if the essential spectrum for at least one self-adjoint extension of Lmin . Here we used that the coefficients satisfy (3). On the other hand, formula (51) valid for any operator matrix satisfying the quasiregularity conditions implies the boundedness of the essential spectrum for all self-adjoint extensions of Lmin . The lemma is proven, provided formula (51) holds true. ✷ In what follows we are going to call the matrix L quasiregular if the quasiregularity conditions (8) on the coefficients are satisfied. Regular matrices form a subset of quasiregular operator matrices. The subfamily of regular matrices can be characterized by one of the following two additional conditions m(0) = 0
∨
β(0) = 0.
(9)
Really each of these conditions together with the first quasiregularity condition imply the other one. Then the second quasiregularity condition implies that m (0) = 0. Hence, the corresponding matrix is regular, since m(0) = m (0) = β(0) = 0. Therefore we are going to concentrate our attention on the case of quasiregular matrices which are not regular, since the regular matrices have been studied earlier ([43]). 4. Deficiency Indices Self-adjoint extensions of the minimal operator Lmin are investigated in this section. These extensions can be described by certain (generalized) boundary conditions on the functions from the domain of the extended operator. These boundary conditions relates the boundary values at the endpoints x = 0 and x = 1. We restrict our studies to local boundary conditions. The boundary conditions are called local if they do not join together the boundary values at different points. Every self-adjoint extension of the operator Lmin is a certain restriction of the adjoint operator L∗min . To calculate the adjoint operator it is enough to consider the operator Lmin restricted to the set of functions from C0∞ (0, 1) ⊕ C0∞ (0, 1), since the adjoint operator is invariant under closure. One concludes using standard calculations ([33]) that the adjoint operator is determined by the same operator valued matrix (2) on the set of functions satisfying the following five conditions
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
(1) U = (u1 , u2 ) ∈ L2 [0, 1] ⊕ L2 [0, 1]; (2) u1 ∈ W21 ( , 1) for any 0 < < 1; (3) The function ωU (x) := −ρ(x)u 1 (x) +
251 (10) (11)
β(x) u2 (x) x
(12)
is absolutely continuous on [0, 1]; d d β(x) d ωU (x) = −ρ(x) u1 + u2 ∈ L2 [0, 1]; (4) dx dx dx x m β(x) d u1 + 2 u2 ∈ L2 [0, 1]. (5) − x dx x
(13) (14)
The function ωU is called transformed derivative and is well-defined for any function U = (u1 , u2) ),
1 u1 ∈ W2,loc (0, 1) ∩ L2 [0, 1],
u2 ∈ L2 [0, 1].
The transformed derivative appearing in the boundary conditions for the matrix differential operator L plays the same rˆole as the usual derivative for the standard one-dimensional Schrödinger operator. The function ωU corresponding to U ∈ Dom(L∗ ) belongs to W21 (0, 1), since it is absolutely continuous and (13) holds. Let us calculate the sesquilinear boundary form of the adjoint operator. This form can be used to describe all self-adjoint extensions of Lmin as restrictions of the adjoint operator to Lagrangian planes with respect to this form. Let U , V ∈ Dom(L∗min ), then integrating by parts we get L∗min U, V − U, L∗min V
β m β d d −ρu 1 + u2 , v1 + − u1 + 2 u2 , v2 − = dx x x dx x
d β β d m −ρv1 + v2 − u2 , − v1 + 2 v2 − u1 , dx x x dx x τ τ m d β ωU v1 dx + − u 1 + 2 u2 v2 dx − = lim 0,τ 1 dx x x
τ τ m d β u1 ωV dx − u2 − v1 + 2 v2 dx − dx x x τ τ β τ u1 v2 dx − ωU v1 dx − = lim ωU (x)v 1 (x)|x= − 0,τ 1 x The transformed derivative is a generalization of the quasi-derivatives described, for example,
by W. N. Everitt, C. Bennewitz and L. Markus [6, 7].
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− u1 (x)ωV (x)|τx=
τ
+
u 1 ωV
dx +
=
τ
β u2 v1 dx x
lim {ωU (x)v 1 (x)|τx= − u1 (x)ωV (x)|τx= }.
0,τ 1
(15)
Note that the limits in the last formula cannot be always substituted by the limit values of the functions, since the functions u1 and v1 are not necessarily bounded at the origin. On the other hand the limit as τ 1 can be calculated using continuity of all four functions at the regular endpoint x = 1. This boundary form will be used to determine the deficiency indices of the operator Lmin and describe its selfadjoint extensions. This method of using boundary forms to describe self-adjoint extensions of symmetric operators is classical and is well described for example in [3] (vol. 2) and [33]. THEOREM 4.1. The operator Lmin is a symmetric operator in the Hilbert space H with finite equal deficiency indices. (1) If the operator matrix L is singular quasiregular (i.e. quasiregularity conditions are satisfied and m(0) = 0), then the deficiency indices of Lmin are equal to (1, 1) and all self-adjoint extensions of Lmin are described by the standard boundary condition ωU (1) = h1 u1 (1),
h1 ∈ R ∪ {∞}.
(16)
(2) If the operator matrix is regular or is not quasiregular then the deficiency indices of Lmin are equal to (2, 2). The self-adjoint extensions of Lmin are described by pair of boundary conditions using the following alternatives covering all possibilities: (a) If ρ(0)m(0) − β 2 (0) = 0 or β(0) = 0, then the first component u1 of any vector from the domain of the adjoint operator L∗min is continuous on the closed interval [0, 1]. All local self-adjoint extensions of the operator Lmin are described by the standard boundary conditions ωU (1) = h1 u1 (1),
ωU (0) = h0 u1 (0),
h0,1 ∈ R ∪ {∞}.
(17)
(b) If d (ρm − β 2 )(0) = 0, and β(0) = 0, dx then the first component u1 of any vector from the domain of the adjoint operator L∗min admits the asymptotic representation ρ(0)m(0) − β 2 (0) = 0,
u1 (x) = kwU (0) ln x + cU + o(1),
as x → 0,
(18)
The family of all self-adjoint extensions of L min can easily be described using our analysis. The corresponding formulas are not written here only in order to make the presentation more transparent. In the case h = ∞, α = 0, 1 the corresponding boundary condition should be written as α u1 (α) = 0 or cU = 0.
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where k=−
β 2 (0) ρ(0)
d (ρm dx
1 − β 2 )|x=0
and cU is an arbitrary constant depending on U . Then all local self-adjoint extensions of the operator Lmin are described by the nonstandard boundary conditions ωU (1) = h1 u1 (1),
ωU (0) = h0 cU ,
h0,1 ∈ R ∪ {∞}.
(19)
Information concerning the deficiency indices of Lmin and self-adjoint local boundary conditions is collected in Table I. Proof. In order to describe all local boundary conditions the points x = 0 and x = 1 can be considered separately. The point x = 1 is a regular boundary point, since the functions ρ −1 , β/x, m/x 2 are infinitely differentiable in a neighborhood of this point. The symmetric boundary condition at the point x = 1 can be written in the form ωU (1) = h1 u1 (1),
(20)
where h1 ∈ R ∪ ∞ is a real constant parametrizing all symmetric conditions (see [43] and Case C below for details). The extension of the operator Lmin to the set of infinitely differentiable functions with support separated from the origin and satisfying condition (20) at the point x = 1 will be denoted by Lh1 . Let us study the deficiency indices of the operator Lh1 . The operator adjoint to Lh1 is the restriction of L∗min to the set of functions satisfying (20). This operator is defined by the operator matrix with real coefficients, therefore the deficiency Table I. ρ(0)m(0) − β 2 (0) = 0
β(0) = 0 β(0) = 0
ρ(0)m(0) − β 2 (0) = 0 d 2 dx (ρm − β )|x=0 = 0
d 2 dx (ρm − β )|x=0 = 0
A
B
C
indices (2,2) 2 standard b.c. (17) indices (2,2) 2 standard b.c. (17)
indices (2,2) 2 standard b.c. (17) indices (2,2) 2 nonstandard b.c. (19)
indices (2,2) 2 standard b.c. (17) indices (1,1) 1 standard b.c. (16)
The letters A, B, and C refer to the three cases considered in the proof of the theorem.
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indices of Lh1 are equal. Moreover, the differential equation on the deficiency element g λ for any λ ∈ / R [3] is given by d λ β(x) λ d −ρ(x) g1 + g + q(x)g1λ = λg1λ , dx dx x 2 (21) β(x) d λ m(x) λ λ g + 2 g2 = λg2 ; − x dx 1 x and it can be reduced to the following scalar differential equation for the first component β(x) 1 β(x) d λ d ρ(x) + g + q(x)g1λ = λg1λ . (22) − dx x λ − m(x)/x 2 x dx 1 The component g2λ can be calculated from g1λ using the formula g2λ = −
1 β(x) d λ g . 2 λ − m(x)/x x dx 1
Equation (22) is a second-order ordinary differential equation with continuously differentiable coefficients. Since the principle coefficient in this equation for nonreal λ is separated from zero on the interval ( , 1], the solutions are two times continuously differentiable functions (18). Boundary condition (20) implies that the first component satisfies the boundary condition at point x = 1 d λ β 2 (1) g (1) = h1 g1λ (1). (23) − ρ(1) + λ − m(1) dx 1 This condition is nondegenerate, since λ is nonreal. Therefore the subspace of solutions to Equation (21) satisfying condition (20) has dimension 1. But these solutions do not necessarily belong to the Hilbert space H = L2 [0, 1] ⊕ L2 [0, 1]. If the nontrivial solution is from the Hilbert space, g λ ∈ H , then the operator Lh1 is symmetric with deficiency indices (1, 1). Otherwise the operator Lh1 is essentially self-adjoint ([42]). If the principal coefficient of Equation (22) is bounded and separated from zero on the interval [0, 1], then g λ ∈ H and the operator Lh1 has deficiency indices (1,1). The last condition is satisfied if for example m(0) = 0 and ρ(0)m(0) − β 2 (0) = 0, since λ = 0. Complete analysis of Equation (22) can be carried out using WKB method ([34]). We are going instead to analyze the boundary form. Let us study the singular point x = 0 in more detail. We are going to consider the following three possible cases: (A) The first quasiregularity condition (8) is not satisfied. (B) The first quasiregularity condition is satisfied, but the second quasiregularity condition (8) is not satisfied. (C) The quasiregularity conditions (8) are satisfied.
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The case C includes the set of regular operator matrices. Case A. Consider arbitrary cutting function ϕ ∈ C ∞ [0, 1] equal to 1 in a certain neighborhood of the origin and vanishing in a neighborhood of the point x = 1. The function W = (m(0)xϕ(x), β(0)xϕ(x)) obviously belongs to the domain of the adjoint operator L∗h1 , since the support of the function W is separated from the point x = 1 and condition (20) is therefore satisfied. The function W is not identically equal to zero, since the first quasiregularity condition (8) is not satisfied. Consider arbitrary U ∈ Dom (L∗min ). Then formula (15) implies that the limit lim{−ωU ( )w1 ( ) + u1 ( )ωW ( )}
0
exists. Taking into account that ωU is absolutely continuous on the interval [0, 1]; lim 0 w1 ( ) = 0; lim 0 ωW ( ) = −ρ(0)m(0) + β 2 (0) = 0; we conclude that the limit u1 (0) = lim 0 u1 ( ) exists for arbitrary function U ∈ Dom (L∗ ). Hence, the boundary form of the operator L∗h1 is given by L∗h1 U, W − U, L∗h1 W = −ωU (0)w1 (0) + u1 (0)ωW (0), and is not degenerate. The operator L(h1 ) has deficiency indices (1,1), and all symmetric boundary conditions at the point x = 0 are standard ωU (0) = h0 u1 (0).
(24)
Case B. Let us introduce the following notation d (ρ(x)m(x) − β 2 (x))|x=0 = 0. (25) dx In addition we suppose that β(0) = 0. To prove that the boundary form is not degenerate (and hence the deficiency indices of Lh1 are (1, 1)) consider the two vector functions x β(t) dt 1+ (26) F = 0 ρ(t) , x 1 β(t) c0 + dt − ρ(0)β(0) tρ(t) (27) G= . x 1 c0 =
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Multiplying the functions F and G by the scalar function ϕ introduced above one gets functions from the domain of the operator L∗h1 . The fact that these functions satisfy (10), (11), (13), (14) is a result of straightforward calculations. We have ωF ( ) ≡ 0,
lim f1 ( ) = 1,
0
and ωG ( ) = −
c0 ρ( ), β(0)ρ(0)
g1 ( ) =
β(0) (ln ) + cG + o(1). ρ(0)
Hence the boundary form of L∗h1 (h1 ) calculated on ϕF and ϕG is given by L∗h1 ϕG, ϕF − ϕG, L∗h1 ϕF =
c0
= 0. β(0)
Therefore the deficiency indices of Lh1 are equal to (1,1). Let us prove that the asymptotic representation (18) holds for any function V from the domain of the operator adjoint to Lmin . Consider the boundary form of the adjoint operator calculated on the function V and the above introduced function G. The following limits obviously exist ∃ lim[−ωG ( )v¯1 ( ) + g1 ( )ω¯ V ( )] 0
√ c0 + o( ) v¯1 ( ) + = lim − − 0 β(0) √ β(0) ln + cU + o(1) (ω¯ V (0) + o( )) + ρ(0) √ β(0) c0 (1 + o( ))v¯1 ( ) + ω¯V (0) ln . ⇒ ∃ lim 0 β(0) ρ(0)
It follows that (18) holds. The parameters ωU (0) and cU are independent, when U runs over Dom(L∗h1 ). This follows easily from the fact that the function (u1 , u2 ) = (1, 0) belongs to the domain of L∗min . Substituting the asymptotic representation (18) for arbitrary U, V ∈ Dom(Lh1 ) into the boundary form L∗h1 U, V − U, L∗h1 V = lim(−ωU ( )v 1 ( ) + u1 ( )ωV ( )) 0
= −ωU (0)cV + cU ωV (0). Hence all local self-adjoint extensions are described by nonstandard boundary conditions (19). To complete the study of Case B, let β(0) = 0. Consider the function F given by (26) and the function S =
x . 0
Then the boundary form calculated on the vectors
ϕF and ϕS is nondegenerate L∗h1 ϕS, ϕF − ϕS, L∗h1 ϕF = ρ(0) = 0,
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and therefore the operator Lh1 has deficiency indices (1, 1). Let us prove that the component u1 of any vector from the domain of the adjoint operator is continuous in the closed interval. Note that ωS (x) = −ρ(x)
and
s1 (0) = 0.
Consider the boundary form of L∗h1 calculated on ϕS and arbitrary V ∈ Dom (Lh1 ) Lh1 ϕS, V − ϕS, Lh1 V = lim(ρ( )v1 ( ) + ωV ( )) 0
= − lim ρ( )v1 ( ). 0
Since ρ(0) is not equal to zero, the limit lim 0 v1 ( ) exists and therefore selfadjoint boundary conditions can be written in the standard form (17) as in Case A. This completes investigation of Case B. Case C. Suppose in addition that β(0) = 0. It follows that the matrix is singular quasiregular. Consider the vector function 1 β(t) dt − E= , x tρ(t) 1 which belongs to the domain of the adjoint operator L∗min due to quasiregular conditions. Therefore ϕE ∈ Dom(L∗h1 ). Then for any function U ∈ Dom(L∗h1 ) the boundary form is given by L∗h1 U, ϕE − U, L∗h1 ϕE = − lim ωU ( )e1 ( ), 0
since ωE ( ) ≡ 0. Note that e1 diverges to infinity due to our assumption β(0) = 0 v1 ( ) ∼ 0
β(0) ln → ∞. ρ(0)
Since the limit lim 0 ωU ( ) exists it should be equal to zero ωU (0) = 0. Hence taking into account that ωU ∈ W21 [0, 1] one concludes that √ ωU ( ) = o( ).
(28)
(29)
On the other hand, condition (13) implies that x
β x d u1 = u2 − ωU ∈ L2 [0, 1]. dx ρ ρ
It follows from Cauchy inequality that 1 u1 ( ) = O √ .
(30)
(31)
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Formulas (29) and (31) imply that the boundary form is identically equal to zero. Therefore the operator L(h1 ) is essentially self-adjoint in this case. (Note that each function from the domain of arbitrary self-adjoint extension of Lmin automatically satisfies the boundary condition (28) at the singular point.) To accomplish the investigation of Case C, assume β(0) = 0. The first quasiregularity condition (8) implies that m(0) = 0. The second quasiregularity condition (8) implies then that (d/dx)m|x=0 = 0. It follows that point zero is a regular point for the operator matrix L. Therefore the deficiency indices of L(h1 ) are equal to (1, 1) and the local self-adjoint extensions are described by standard boundary conditions ([17, 43]). We have already proven this result. Indeed taking into account that u1 ∈ W21 (0, 1) and that the function ω( ) is absolutely continuous the above mentioned fact follows immediately from (15). This accomplishes the investigation of Case C. The theorem is proven. ✷ COROLLARY 4.1. The theorem implies that the operator Lh1 is essentially selfadjoint if and only if the operator matrix is singular quasiregular. Otherwise it has deficiency indices (1,1). Nonstandard boundary conditions (19) at the singular point described by Theorem 4.1 are similar to the boundary conditions appearing in the studies of onedimensional Schrödinger operator with Coulomb potential γ d2 in L2 (R). − 2 dx x In what follows we are going to study the essential spectrum of the self-adjoint extensions of the operator Lmin . Since the deficiency indices of this operator are always finite, the essential spectrum does not depend on the particular choice of the boundary conditions. The same holds true for nonlocal boundary conditions and therefore our restriction to the case of local boundary conditions can be waived. Therefore in the course of the paper we are going to denote by L some self-adjoint extension of the minimal operator. −
5. Transformation of the Operator In the current section we are going to transform the self-adjoint operator L to another self-adjoint operator acting in the Hilbert space H = L2 [0, ∞)⊕L2 [0, ∞). The reason to carry out this transformation is pure technical – we would like to be able to use Fourier transform. Consider the following change of variables x = e−y ,
dx = −e−y dy = −x dy,
(32)
mapping the interval [0, ∞) onto the interval [0, 1] and the corresponding unitary transformation between the spaces L2 [0, 1] and L2 [0, ∞) ˜ .: ψ(x) %→ ψ(y) = ψ(e−y )e−y/2 .
(33)
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259
The points 0 and ∞ are mapped to 1 and 0, respectively, and the following formula holds ∞ 1 2 ' ψ(x) ' dx = ' ψ(e−x ) '2 e−y dy. 0
0
The inverse transform is given by 1 ˜ ˜ x). %→ ψ(x) = √ ψ(−ln .−1 : ψ(y) x
(34)
To determine the transformed operator denoted by K let us calculate the transformed operator matrix first componentwise K11 : √
1 d d ˜ x) + q(x) √ ψ(−ln x − ρ dx dx x √ 1 d 1 ˜ ψ(−ln x) + 3/2 ψ˜ (−ln x) + q(x)ψ˜ (−ln x) = x − ρ dx 2x 3/2 x √ 1 1 ˜ ψ(−ln x) + 3/2 ψ˜ (−ln x) + = x ρx 2x 3/2 x −3 −1 1 3 ˜ + 5/2 ψ˜ (−ln x) + x) + 3/2 ψ˜ (−ln x) + ρ − 5/2 ψ(−ln 4x 2x x 2x −1 1 + + 3/2 ψ˜ (−ln x) x x
+ q(x)ψ˜ (−ln x) ρ = − 2 ψ˜ (−ln x) + x
ρ 3ρ ρx ρx ˜ ˜ − 2 2 ψ (−ln x) + − ψ(−ln x) + x x 2x 4 x2
+ q(x)ψ˜ (−ln x)
ρx 3ρ ˜ d ρ d ˜ ψ(−ln x) + q(x) + − 2 ψ(−ln x). =− dy x 2 dy 2x 4x K12 :
d β 1 ˜ x) √ ψ(−ln dx x x √ d β ˜ ψ(−ln x) = x dx x 3/2 3β β β ˜ x) x − 2 = − 2 ψ˜ (−ln x) + ψ(−ln x x 2x
√ x
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PAVEL KURASOV AND SERGUEI NABOKO
d β ˜ 2β ˜ βx =− − 3 ψ(−ln ψ(−ln x) − x x) + 2 2 dy x x x β d β ψ˜ + 2 . =− 2 dy x 2x
3β ˜ βx − 2 ψ(−ln x) x 2x
K21 is the conjugated expression to K12 1β β d + . x 2 dy 2 x2 K22 : m/x 2 . Finally the transformed operator matrix will be denoted by K and it is given by ρx 3ρ d ρ d β d β + q(x) + − − + − dy x 2 dy 2x 4x 2 dy x 2 2x 2 A C∗ K = . (35) := C D 1β β d m + x 2 dy 2 x2 x2 To define a self-adjoint operator corresponding to this operator matrix one has to consider first the minimal operator Kmin being the closure of the differential operator given by (35) on the domain of functions from C0∞ [0, ∞) ⊕ C0∞ [0, ∞). Then one has to study the deficiency indices of this operator and describe all its self-adjoint extensions. This analysis is equivalent to the one carried out in the previous section for the operator Lmin . The self-adjoint extensions of the operators Lmin and Kmin are in one-to-one correspondence given by the unitary equivalence (33), (34). Therefore we conclude that the deficiency indices of the operator Kmin are equal and finite ((1, 1) or (2, 2) depending on the properties of the coefficients). Let us denote by K one of the self-adjoint extensions of the minimal operator. The essential spectrum of the operator will be studied. The analysis does not depend on the choice of self-adjoint extension, since the deficiency indices of the minimal operator are finite. It is easier to study pseudodifferential operators on the whole axis instead of the half axis. The reason is that the manifold [0, ∞) has nontrivial boundary and therefore even the momentum operator cannot be defined as a self-adjoint operator in L2 [0, ∞). It appears more convenient for us to study the corresponding problem on the whole real line in order to avoid these nonessential difficulties related to the boundary point y = 0. In this way the problem of studies of the matrix differential operator can be reduced to a certain pure algebraic problem. Consider the Hilbert space H = L2 (R) ⊕ L2 (R). The operator K acting in H can be chosen in such a way that its essential spectrum coincides with the essential spectrum of the operator K. In order to simplify the discussion of the essential spectrum we have to chose special continuation of the operator. However, this program applied to the operator Kmin itself meets some difficulties and it appears more convenient for us to perform
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261
this program on a later stage of the investigation of the operator, namely during the studies of the cleaned resolvent of the operator. 6. Resolvent Matrix and the Hain–Lüst Operator The resolvent of the operator K will be used to study its essential spectrum. The difference between the resolvents of any two self-adjoint extensions of the minimal operator Kmin is a finite rank operator and it follows that the essential spectrum is independent of the chosen self-adjoint extension. In fact it is enough to calculate the resolvent of the operator K on any subspace of finite codimension, for example on the range of the minimal operator Kmin . We are going to consider the resolvent equation (Kmin − µ)−1 F = U, for µ satisfying one of the following two conditions (i) (µ = 0; (ii) µ ∈ R, |µ| ) 1. Formula (36) below shows that resolvent’s denominator T (µ) has no additional singularities outside x = 0 for all nonreal values of the parameter µ. For sufficiently large real µ the same holds true if either m(0) = 0, or m(0) = 0, the quasiregularity conditions (8) hold and sign µ sign m(0+ ) = −1. If the quasiregularity conditions hold then m(0+ ) 0 and the parameter µ can always be chosen to be small negative, µ * −1. For F ∈ R(Kmin) and U ∈ C0∞ [0, ∞) ⊕ C0∞ [0, ∞) the resolvent equation can be written as follows f1 = (A − µ)u1 + C ∗ u2 ,
f2 = Cu1 + (D − µ)u2 .
Using the fact that the operator (D −µ) is invertible for nonreal µ one can calculate u2 from the second equation u2 = (D − µ)−1 f2 − (D − µ)−1 Cu1 and substitute it into the first equation to get f1 = ((A − µ) − C ∗ (D − µ)−1 C)u1 + C ∗ (D − µ)−1 f2 . The last equation can easily be resolved using Hain–Lüst operator, which is analogous to the regularized determinant of the matrix K T (µ) = (A − µI ) − C ∗ (D − µI )−1 C d β2 d ρ −µ+ − 2 =− 2 2 dy x x (m − µx ) dy
3ρ d β2 β2 ρ − x . (36) + q(x) + x − 2 − 2 2x 4x 4x (m − µx 2 ) dx 2x 2 (m − µx 2 )
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Elementary calculations show that under quasiregular conditions (8) both coefficients in the expression above are smooth and bounded. The principle coefficient ρ β2 − x2 x 2 (m − µx 2 ) is uniformly separated from zero. We consider this operator for µ * −1 on the set C0∞ [0, ∞) and use the same notation for its Friedrichs extension described by the Dirichlet boundary condition at the origin. This operator has been introduced in a special case by K. Hain and R. Lüst during the investigation of problems of magnetohydrodynamics. In what follows we are going to show that Hain–Lüst operator plays the key rˆole in the investigation of the essential spectrum. The rˆole of the quasiregularity conditions for the Hain–Lüst operator is explained by the following lemma. LEMMA 6.1. Let µ ∈ / Rangex∈[0,1]((m(x))/x 2 ), then the coefficients of the Hain– Lüst operator (36) f (x) = and
β2 ρ , − x2 x 2 (m − µx 2 )
3ρ d β2 β2 ρx − 2− 2 −x − µ, g(x) = q(x) + 2x 4x 4x (m − µx 2 ) dx 2x 2 (m − µx 2 )
are uniformly bounded functions if and only if the quasiregularity conditions (8) hold. Comment. The condition µ ∈ / Rangex∈[0,1]((m(x))/x 2 ) holds, for example, if the parameter µ either nonreal or µ ∈ R, µ * −1. Proof. Let the quasiregularity conditions (8) be satisfied. Then the coefficient f (x) =
ρm − β 2 − µx 2 x 2 (m − µx 2 )
is uniformly bounded, since by (8) ρ(x)m(x) − β 2 (x) ∼x→0 cx 2 and the factor m − µx 2 is uniformly separated from 0. The function g(x) − q(x) + µ +
f (x) 4
ρ β2 ρx − 2 −x = 2x x 2x 2 (m − µx 2 ) x
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ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
ρ β 2 − ρµ ρµ ρx − 2 −x −x 2 2 2 2x x 2x (m − µx ) x 2x (m − µx 2 ) β 2 − ρµ ρx 2 = −x + µx 2x 2 (m − µx 2 ) x 2x 2 (m − µx 2 ) x
=
x
is also uniformly bounded. On the other hand, the boundedness of the leading coefficient f (x) =
ρm − β 2 − µx 2 x 2 (m − µx 2 )
implies conditions (8) under the assumptions of the lemma. The lemma is proven. ✷ Similar result has been proven for magnetohydrodynamic operator in [17]. The resolvent matrix can be presented by M(µ) ≡ (Kmin − µ)−1 T −1 (µ) = −1 −[(D − µI )
−T −1 (µ)[C ∗ (D − µI )−1 ]
C]T −1 (µ) (D − µI )−1 + [(D − µI )−1 C]T −1 (µ)[C ∗ (D − µI )−1 ]
. (37)
The last expression determines the resolvent of any self-adjoint extension K of the minimal operator Kmin on the subspace R(Kmin ) which has finite codimension. Therefore this resolvent matrix determines the essential spectrum of any self-adjoint extension K. In order to calculate the essential spectrum we are going to consider perturbations of the calculated resolvent by compact operators. This is discussed in the following section. 7. The Asymptotic Hain–Lüst Operator The essential spectra of two operators coincide if the difference between their resolvents is a compact operator. This idea of relatively compactness was used in applications to magnetohydrodynamics by T. Kako [22]. Even if the expression for the resolvent is much more complicated than the one for operator itself we prefer to handle with the resolvent. We are going to simplify the expression for the resolvent step by step using Weyl theorem. We call this procedure cleaning of the resolvent. Therefore we are going to perturb the resolvent operator M(µ) by compact operators in order to simplify it. Our aim is to factorize the pseudodifferential operator M(µ) into a sum of two pseudodifferential operators with symbols depend on the coordinate and momentum, respectively. In our calculations we are going to use the Calkin calculus [13]. We say that any two operators A and B are equal in Calkin algebra if their difference is a compact operator. The following notation for the equivalence relation in Calkin algebra will be used throughout the
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paper: A = ˙ B. Since all operators appearing in the decomposition (37) are in fact pseudodifferential the following notation for the momentum operator will be used 1 d p= . (38) i dy This symbol will denote the differential expression in the first half of this section. The same notation will be used for the symbol of the pseudodifferential operator on the real line in the rest of the paper. Let us introduce the asymptotic Hain–Lüst operator for the generic case m(0) = 0 d2 (39) Tas (µ) = a(µ) − 2 + c(µ) ≡ a(µ)(p 2 + c(µ)), dy where β2 ρ(0) ρ = l0 − µ , a(µ) = lim 2 − 2 2 x→0 x x (m − µx ) m(0) 2 ρ − βm , (40) l0 = lim x→0 x2 µ 1 . c(µ) = − 4 a(µ) The domain of the asymptotic Hain–Lüst coincides with the set of functions from the Sobolev space W22 satisfying the Dirichlet boundary condition at the origin: {ψ ∈ W22 ([0, ∞)), ψ(0) = 0}. We obtain the asymptotic Hain–Lüst operator by substitution the coefficients of the second-order differential Hain–Lüst operator by their limit values at the singular point. It will be shown that the additional branch of essential spectrum of L is determined exactly by the symbol of asymptotic Hain– Lüst operator. To prove that the difference between the inverse Hain–Lüst and inverse asymptotic Hain–Lüst operators is compact we are going to use Lemma B.4. We decided to devote a separate appendix to this lemma which is of special interest in the theory of pseudodifferential operators (see Appendix B, where the proof of this lemma can be found). This lemma implies that the difference of the inverse Hain–Lüst operators is compact T −1 (µ) − Tas−1 (µ) ∈ S∞
(41)
for sufficiently large |µ| to guarantee the invertibility of the both operators. Note that both operator functions −T −1 (µ) and −Tas−1 (µ) are operator valued Herglotz functions ([32]). 8. Cleaning of the Resolvent This section is devoted to the cleaning of the resolvent, which is based on formula (41). The main algebraic tool is Calkin calculus ([13]) and Appendix B.
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
265
Using Calkin algebra and Lemma B.1 formula (41) can be almost rigorously written as follows 1 ˙ ρ . (42) p T −1 (µ)p = β2 − x2 x 2 (m−µx 2 ) In fact to apply Lemma B.1 one needs extra regularizator h – any bounded vanishing at infinity function (see formula (80)). The operator p T −1 (µ)p here is the closure of the bounded operator defined originally on W21 [0, ∞). Let us introduce the function β . (43) b(x, µ) = m − µx 2 Our aim is to find a matrix differential operator equivalent in Calkin algebra to the operator M(µ) given by (37). Using (41) and the fact (the result of straightforward calculations) that the operators C ∗ (D − µI )−1 and (D − µI )−1 C under quasiregular conditions are first order differential operators with bounded smooth coefficients we obtain ip+1/2 1 1 − b(0,µ) a(µ) p 2 +c(µ) a(µ) p 2 +c(µ) M(µ) = ˙ . (44) 2 b(0,µ) −ip+1/2 x −1 −1 ∗ −1 − a(µ) p2 +c(µ) m−µx 2 + [(D − µI ) C]T (µ)[C (D − µI ) ] The expressions (±ip + 1/2)/(p 2 + c(µ)) are considered as bounded operators defined on L2 [0, ∞) by (±ip + 1/2)(p 2 + c(µ))−1 , where (p 2 + c(µ))−1 is the resolvent of the Laplace operator p 2 with the Dirichlet boundary condition at the origin. Substituting expressions for the operators C and D from (35) we get b(0,µ) ip+1/2 1 1 M(µ) = ˙
a(µ) p 2 +c(µ) −ip+1/2 − b(0,µ) a(µ) p 2 +c(µ)
−
x2 m−µx 2
a(µ) p 2 +c(µ)
+ b(x, µ)(−ip + 1/2)T −1 (µ)(ip + 1/2)b(x, µ)
.
Let us concentrate our attention to the element (22). We consider this differential operator on the set W21 [0, ∞). b(x, µ)(−ip + 1/2)T −1 (µ)(ip + 1/2)b(x, µ) = b(x, µ)(−ip + 1/2)Tas−1 (µ)(ip + 1/2)b(x, µ) + + b(x, µ)(−ip + 1/2)T −1 (µ)(Tas (µ) − T (µ))Tas−1 (µ)(ip + 1/2)b(x, µ) p 2 + 1/4 b(x, µ) + = ˙ b(x, µ) a(µ)(p 2 + c(µ)) In fact only the condition m(0) = 0 is used here. This relation follows from the first
quasiregularity condition (8).
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+ b(x, µ)(−ip + 1/2)T ×
−1
d d ρ β2 − a(µ) × (µ) − 2 2 2 dy x x (m − µx ) dy
ip + 1/2 b(x, µ). a(µ)(p 2 + c(µ))
The last equality in Calkin algebra holds due to the following observations: (1) The operator Tas−1 (µ)(ip + 1/2) is bounded. (2) Since the minor terms in both T (µ) and Tas (µ) are bounded functions, Lemma B.3 and (1) imply that the following operator is compact (−ip + 1/2)T −1 (µ) {bounded function tending to 0 at infinity} = ˙ (−ip + 1/2)Tas−1 (µ) {bounded function tending to 0 at infinity} = ˙ 0. To transform the first term the following equality has been used (−ip + 1/2)Tas−1 (µ)(ip + 1/2) = ˙
p 2 + 1/4 . p 2 + c(µ)
Using b(x, µ)
p 2 + 1/4 p 2 + 1/4 b(x, µ) = ˙ b(0, µ) b(0, µ) p 2 + c(µ) p 2 + c(µ)
(b ∈ L∞ [0, ∞) and has limit at ∞, Lemma 6.1 from [17]), we get b(x, µ)(−ip + 1/2)T −1 (µ)(ip + 1/2)b(x, µ) b2 (x, µ) b2 (0, µ) 1/4 − c(µ) + + = ˙ a(µ) a(µ) p 2 + c(µ) ρ β2 − a(µ) × + b(x, µ)(−ip + 1/2)T −1 (µ) −p 2 − 2 x x (m − µx 2 ) ip 2 + p/2 b(x, µ). × a(µ)(p 2 + c(µ)) The operator ip 2 + p/2 b(x, µ) ≡ (ip 2 + p/2)Tas−1 (µ)b/x, µ) a(µ)(p 2 + c(µ)) is bounded. Consider the operator ρ β2 −1 − a(µ) b(x, µ)(−ip + 1/2)T (µ) −p 2 − 2 x x (m − µx 2 ) β2 ρ 1 − a(µ) − 2 = ˙ b(x, µ) ρ β2 x2 x (m − µx 2 ) − 2 2 2 x x (m−µx )
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ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
due to Lemma B.1 and the equality following from (8) β2 ρ − a(µ) |x=0 = 0. − 2 x2 x (m − µx 2 )
(45)
Lemma B.1 could be applied here, since one can easily that the operator ρ β2 −1 − a(µ) b(x, µ)(1/2)T (µ) −p 2 − 2 x x (m − µx 2 ) is compact. Therefore the element (22) is equivalent in Calkin algebra to the following operator b2 (x, µ) b2 (0, µ) 1/4 − c(µ) x2 + − + m − µx 2 a(µ) a(µ) p 2 + c(µ) 1 β2 ρ 1 − a(µ) b(x, µ). − 2 − b(x, µ) ρ 2 2 β2 x x (m − µx ) a(µ) 2 − 2 2 x
x (m−µx )
The following formula for the cleaned resolvent matrix has been obtained M(µ) = ˙
1 1 a(µ) p 2 + c(µ) −
b(0, µ) −ip + 1/2 a(µ) p 2 + c(µ)
−
b(0, µ) ip + 1/2 a(µ) p 2 + c(µ)
x2
b2 (0, µ) 1/4 − c(µ) b2 (x, µ) + + 2 2 2 ρ a(µ) m − µx p + c(µ) − 2 β 2 x2 x (m−µx )
.
(46)
Let us remind that the formal expression 1 1 2 a(µ) p + c(µ) in all four matrix entries denotes the resolvent of the asymptotic Hain–Lüst operator. The last matrix can be written (at least formally) as a sum of two matrices depending on x and p only: M(µ) = ˙ X(x) + P (p), where 0 0 2 2 X(x) = , b (x, µ) x 0 + 2 β ρ m − µx 2 − 2 2 2 x
x (m−µx )
b(0, µ) ip + 1/2 − a(µ) p 2 + c(µ) P (p) = . b(0, µ) −ip + 1/2 b2 (0, µ) 1/4 − c(µ) − a(µ) p 2 + c(µ) a(µ) p 2 + c(µ)
1 1 2 a(µ) p + c(µ)
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In Section 4, to handle pseudodifferential operators, we discussed the extension of all operators to certain operators acting in the Hilbert space H = L2 (R) ⊕ L2 (R) ⊃ L2 [0, ∞) ⊕ L2 [0, ∞). This procedure can easily be carried out for the cleaned resolvent. Let us continue all involved functions b(x(y), µ), ρ(x(y)) and m(x(y)) to the whole real line as even functions of y. Consider the operator generated by the continued matrix symbol X(x(y)) + P (p). This operator is bounded operator defined on the whole Hilbert space H. The essential spectrum of the new operator coincides (without counting multiplicity) with the essential spectrum of the original operator M(µ). Really Glazman’s splitting procedure ([3]) and Weyl theorem on compact perturbations ([24]) imply that the essential spectrum of the new operator coincides with the union of the essential spectra of the two operators generated by the operator matrix on the two half-axes: p2
1 1 1 |L2 (R) = |L2 (−∞,0] ⊕ 2 |L [0,∞) , ˙ 2 + c(µ) p + c(µ) p + c(µ) 2
where p2
1 |L (−∞,0] + c(µ) 2
and
p2
1 |L [0,∞) + c(µ) 2
denote the resolvents of the Laplace operator p 2 on the corresponding semiaxis with the Dirichlet boundary condition at the origin. In the last formula p denotes the momentum operator in the left-hand side and the differential expression in the right one. One can easily prove that the unitary transformation f1 (−y) f1 (y) %→ f2 (y) −f2 (−y) relates the matrix operators generated in the orthogonal decomposition of the Hilbert space H = (L2 (−∞, 0] ⊕ L2 (−∞, 0]) ⊕ (L2 [0, ∞) ⊕ L2 [0, ∞)). Hence, the two operators appearing in this orthogonal decomposition are unitary equivalent and therefore have the same essential spectrum. The problem of calculation of the essential spectrum has been transformed to a pure algebraic problem. 9. Calculation of the Essential Spectrum In order to apply Proposition A.1 from Appendix A let us introduce two matrix operator functions 0 0 (47) Q= ρ(0) 0 m(0)a(µ)
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
and
Y (y) =
0
0
x2 + 0 m − µx 2
b2 (x, µ) ρ x2
−
269
β2 x 2 (m−µx 2 )
ρ(0) . − m(0)a(µ)
(48)
Let us remind the reader that everywhere in the paper x is considered as a function of the variable y, x = e|y| , where we have taken into account the even continuation of all parameters of the matrix for negative values of y. The matrices Q, Y (y) and P (p) satisfy the conditions of Proposition A.1. In addition, the matrix functions Y (y) and P (p) are continuous on the real line and have zero limits at infinity. All matrix functions are depending on the parameter µ. Therefore the essential spectrum of the resolvent operator M(µ) is given by (72) σess (M(µ)) = σess (Q + P) ∪ σess(Q + Y). To calculate the essential spectra of the operators Q + P and Q + Y we use the fact that the determinants of the corresponding matrices Q + P (p) and Q + Y (y) are equal to zero identically. It follows that one of the two eigenvalues of the each matrix is identically zero. Therefore the essential spectra of the operators coincides with the range of the second (nontrivial) eigenvalues when y resp.p runs over the whole real axis. This simple fact is a result of straightforward calculations. The nontrivial eigenvalues coincide with the traces of the corresponding 2 × 2 matrices Q + P (p) and Q + Y (y). The trace of the matrix M(µ) is given by Tr (M(µ)) = Tr (Y (y)) + Tr (P (p)) − Tr (Q) 1 x2 1 + + = a(µ) p 2 + c(µ) m − µx 2 b2 (x, µ) b2 (0, µ) 1/4 − c(µ) + + ρ β2 a(µ) p 2 + c(µ) 2 − 2 x
x
.
(m−µx 2 )
The last expression can be factorized into the sum of three factors Tr(M(µ)) = ϕ(x(y)) + ψ(p) − Tr(Q), Tr Q =
ρ(0) , m(0)a(µ)
where the functions ϕ(x(y)) and ψ(p) tend to zero as y resp.p tend to ∞. The factorization is unique and obvious ϕ(x) =
x2 + m − µx 2
b2 (x, µ) ρ x2
−
β2 x 2 (m−µx 2 )
;
1 b2 (0, µ) 1/4 − c(µ) ρ(0) 1 + + . ψ(p) = a(µ) p 2 + c(µ) a(µ) p 2 + c(µ) m(0)a(µ)
(49)
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Proposition A.1 implies that the essential of the resolvent operator is given by σess (M(µ)) = (Range(ϕ(x)) ∪ Range(ψ(x)) + ϕ(0)).
(50)
Straightforward calculations imply σess (L) = Rangex∈[0,1]
m− x2
β2 ρ
∪
, , ρ(0) ρ(0)
l0 4+
m(0)
l0
(51)
m(0)
where l0 is given by (40). The parameter µ disappears eventually as one can expect. This parameter is pure axillary. We conclude that the essential spectrum of L consists of two parts having different origin. The so-called regularity spectrum ([30]) Rangex∈[0,1]
m−
β2 ρ
x2
is determined by all coefficients of the operator matrix on the whole interval [0, 1]. This part of the spectrum coincides with the limit of the essential spectra of the truncated operators L( ) Rangex∈[0,1]
m−
β2 ρ
x2
=
σess(L( )).
>0
On the contrary the singularity spectrum l0 l0 , ρ(0) ρ(0) 4 + m(0) m(0) is due to the singularity of the operator matrix at the origin and depends on the behavior of the matrix coefficients at the origin only. This part of the essential spectrum is absent for all truncated operators L( ) and cannot be obtained by the limit procedure → 0. This fact explains the name singularity spectrum given in [30]. The appearance of this interval of the essential spectrum generated by the singularity was predicted by J. Descloux and G. Geymonat. Note that the end point l0 /(ρ(0)/m(0)) of the singularity spectrum always belongs to the interval of regularity spectrum, since lim
x→0
m− x2
β2 ρ
=
l0 ρ(0) m(0)
.
Remark. Let us remind that the essential spectrum has been calculated provided m(0) = 0 and the quasiregularity conditions are satisfied. If m(0) = 0, the quasiregularity conditions imply that β(0) = 0 and hence m (0) = 0. No singularity
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
271
appears in the coefficients of the matrix L given by (2). Therefore the operator is regular and its essential spectrum equals to 2
m − βρ (4). Rangex∈[0,1] x2 No singularity spectrum appears in this case. There is another way to describe the singularity spectrum using the roots of the symbol of the asymptotic Hain–Lüst operator, observed first for a different matrix differential operator in [30]. LEMMA 9.1. The singularity spectrum l0 l0 , ρ(0) ρ(0) 4 + m(0) m(0) of the operator L coincides with the set of singular points (roots) of the symbol of the asymptotic Hain–Lüst operator . = {µ ∈ R | ∃p ∈ R ∪ {∞} : a(µ)(p 2 + c(µ)) = 0}. Proof. The set of singular points of the symbol a(µ)(p 2 + c(µ)) coincides with the set . = {µ ∈ R | c(µ) 0}. Formula (40) implies
ρ(0) l0 − µ m(0) 4 .= µ∈R|0 µ l0 l0 , . = ρ(0) ρ(0) 4 + m(0) m(0) Note that p = ∞ formally corresponds to right endpoint of the last interval. The lemma is proven. ✷ In our opinion this connection between the singular set of the symbol of the asymptotic Hain–Lüst operator and the singularity spectrum has general character. Studies in this direction will be continued in one of our forthcoming publications. Remark. We would like to mention that the regularity spectrum 2
m − βρ Rangex∈[0,1] x2 under quasiregularity conditions can be calculated using just the symbol of the Hain–Lüst operator. Really trivial calculations show that the regularity spectrum coincides with the set of real µ for which the principle coefficient of the Hain–Lüst operator degenerates, i.e. equals zero. Roughly speaking this idea has been utilized by physicists K. Hain and R. Lüst ([16]) (see also [12]).
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10. Semiboundedness of the Operator In many applications to physics semibounded operators play very important rˆole. Semiboundedness of the considered operator is related to the quasiregularity conditions. THEOREM 10.1. Suppose that the real valued functions q, β, ρ, m satisfy the following conditions: q ∈ L∞ [0, 1],
β, m, ρ ∈ C 2 [0, 1],
ρ c0 > 0.
(52)
Then the symmetric operator Lmin corresponding to the operator matrix (2) is semibounded if and only if one of the following three conditions is satisfied (1) (m − β 2 /ρ)|x=0 > 0, (2) (m − β 2 /ρ)|x=0 = 0 and (m − β 2 /ρ) |x=0 > 0, (3) (m − β 2 /ρ)|x=0 = 0 and (m − β 2 /ρ) |x=0 = 0 (quasiregularity conditions). COROLLARY 10.1. Under assumptions of Theorem 10.1 the operator Lmin admits self-adjoint extensions. Every such extension L is a semibounded operator if and only if one of the conditions (1)–(3) is satisfied. Proof. Since the coefficients of the matrix L are real valued functions, the deficiency indices of Lmin are equal. On the other hand the equation for the deficiency element is a system of ordinary differential equations. Therefore the set of solutions has finite dimension. Hence the operator Lmin always has finite equal deficiency indices and admits self-adjoint extensions. Theorem 10.1 implies that every such extension is semibounded if and only if one of the three conditions is satisfied (see [3]). ✷ Proof of Theorem 10.1. Without loss of generality one can suppose that q = 0, q 0 since the operator corresponding to the matrix 0 0 is bounded in H and cannot change the semiboundedness of the whole operator Lmin . The theorem will be proven by estimating the quadratic form of Lmin defined on the domain C0∞ [0, 1] ⊕ C0∞ [0, 1] by the following operator matrix d d d β ρ − dx dx dx x . L= m β d − x dx x2
The quadratic form of this operator is
β β m u2 , u 1 − u1 , u2 + 2 u2 , u2 Lmin U, U = ρu 1 , u 1 − x x x
(53)
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
β √ √ 1 −√ ρu 1 ρu1 ρ , = u2 u2 . β m −√ x x ρ
273
(54)
Considering functions with zero second component U = (u1 , 0) we conclude that the operator Lmin is not bounded from above, since the quadratic form coincides with the quadratic form of the operator −(d/dx)ρd/dx in this case. Therefore the operator Lmin is semibounded if and only if it is bounded from below. To get the second necessary condition for the semiboundedness of the operator consider the set of functions with zero first component U = (0, u2 ). The quadratic form is then given by
m Lmin U, U = 2 u2 , u2 . x Hence the operator Lmin is semibounded only if m(0) > 0 or
m(0) = 0 and
m (0) 0.
(55)
Case A. Suppose that the determinant of the matrix β 1 −√ 2 ρ =m− β det β ρ m −√ ρ is negative at point zero (and therefore in a neighborhood of this point as well) m(0) −
β(0)2 < 0. ρ(0)
(56)
It follows that the matrix has precisely one negative eigenvalue λ(x) < 0 for small enough values of x. Let us denote by (α, γ ) the corresponding normalized real eigenvector depending continuously on x in a neighborhood of the origin. Suppose that α(0) = 0. Then the first equation for the eigenvector implies that β(0) = 0 and therefore m(0) < 0 due to (56). This contradicts (55) and therefore α(0) = 0 in a certain neighborhood of the origin due to the continuity of α. Consider arbitrary real function h ∈ C0∞ [0, 1] such that the derivative of h is equal to 1 in the interval (1/4, 1/2) and the family of scaled functions h = √ h(x/ ). The corresponding family of vector functions U = (h , γα ρxh ) is well-defined for sufficiently small . Since λ(x)ρ 2 h dx, Lmin U , U = α2 0
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and 'U ' =
γ 2ρ 2 2 h + 2 x h dx, α 2
2
0
the quotient LU, U /' U '2 tends to −∞ as → 0. Hence the operator Lmin is not semibounded in this case. Case B. Suppose that m(0) −
β(0)2 > 0. ρ(0)
The operator L is semibounded in this case. Indeed the quadratic form can be decomposed as follows β(0) √ √ ρu1 ρu1 1 −√ ρ(0) Lmin U, U = u2 , u2 + β(0) m(0) −√ x x ρ(0) β(0) β 0 −√ √ ρ ρ(0) + β β(0) −√ m − m(0) √ ρ ρ(0)
√ √ ρu1 ρu1 u , u . 2 2 x x
The first term is positive and can be estimated from below by const('u 1 '2 + 'u2 '2 ) due to the assumption. The second term is subordinated to the first one √ √ β β(0) ρu1 ρu1 −√ √ 0 ρ ρ(0) u , u 2 2 β β(0) −√ m − m(0) √ x x ρ ρ(0) 1 u2 2 xu 1 + 2 dx const x 0 1 u22 u22 2 2 ρu1 + 2 dx + const u1 + 2 dx. const x x 0 The relative bound const can be chosen less than 1 and the second term is bounded for any > 0. Case C. Suppose that m(0) −
β(0)2 = 0 and ρ(0)
d β2 m− |x=0 < 0. dx ρ
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
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The quadratic form can be decomposed as √ √ρu √ρu 1 1 1 −β/ ρ u2 , u2 + Lmin U, U = √ −β/ ρ β 2 /ρ x x √ρu √ρu 1 1 0 0 u2 , u2 . + 2 0 m − β /ρ x x Consider the vector function x β(t) h (t) dt, xh (x) , V = 0 ρ(t) where the scalar h has been introduced investigating Case A. Calculating the the quadratic form
β2 h ,h Lmin V , V = m − ρ and estimating the norm 1 2 2 2 x h + 'V ' = 0
0 1
x
β h dt ρ
2 dx
[x 2 h + const h 2 ] dx. 2
0
Since (m − (β 2 /ρ)) |x=0 < 0, the quotient LU, U 'U '2 tends to −∞ as → 0. The operator is not semibounded in this case. Case D. Suppose that β2 |x=0 = 0 and m− ρ
β2 m− |x=0 0. ρ
The operator Lmin is semibounded in this case due to the following estimate √ρu √ρu 1 1 0 0 u2 , u2 0 m − β 2 /ρ x x
1
= 0
m− x2
β2 ρ
|u2 |2 dx const'u2 '2 ,
(57)
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which is valid since the function (m − (β 2 /ρ))/x 2 from below. The Cases A–D cover all the possibilities. The Theorem is proven.
✷
Appendix A. On the Essential Spectrum of the Triple Sum of Operators in Banach Space The following simple lemma will be used to calculate the essential spectrum of the separable sum of pseudodifferential operators. It allows one to pass to the limit in formula (58) below when the point λ reaches the discrete spectrum. LEMMA A.1. Let T, Y, P be bounded operators acting in a Banach space X. Suppose that a certain dotted neighborhood of λ = 0 does not belong to the spectrum of the operator T and the point λ = 0 is not in the essential spectrum of the operator T. Suppose in addition that Y(T − λ)−1 P ∈ S∞
(58)
is a compact operator in the dotted neighborhood. Let RT be the parametrix of the operator T ([13]) ˙ TRT = ˙ I. RT T =
(59)
Then the operator YRT P is compact YRT P ∈ S∞ .
(60)
Proof. The following calculations prove the lemma YRT P := Y(T − λ)−1 (T − λ)RT P = ˙ Y(T − λ)−1 (I − λRT )P = ˙ −λY(T − λ)−1 RT P = ˙ −λYRT (I − λRT )−1 RT P = ˙ 0,
(61)
where the second equality from the end is valid for all λ, 0 < |λ| < 1/'RT ' not from the spectrum of the operator T ˙ R(I − λR)−1 . (T − λ)R = ˙ I − λR ⇒ (T − λ)−1 = The lemma is proven.
✷
These conditions imply that the point λ = 0 is a finite type eigenvalue of T ([13]) or does not
belong to the spectrum of T at all. In the last case the proof of the lemma is trivial.
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THEOREM A.1. Let M be the operator sum of three bounded operators Q, Y and P acting in a certain Banach space M = Q + Y + P,
(62)
such that the complement in C of the essential spectrum of the operator Q is connected. Suppose that the following two operators are compact for any λ from the regular set of Q P
1 Y ∈ S∞ ; Q−λ
Y
1 P ∈ S∞ . Q−λ
(63)
Then the essential spectrum of the operator M can be calculated as follows σess (M) \ σess (Q) = [σess (Q + Y) ∪ σess(Q + P)] \ σess(Q).
(64)
Proof. It has been proven in [13] (Corollary 8.5, page 204) that if the complement in C of the essential spectrum of a certain bounded operator is connected, then any number λ from the spectrum of the operator, but not from the essential spectrum is a finite type eigenvalue ([13]), i.e. the pole of the resolvent with finite rank Laurent coefficients with negative indices. Lemma A.1 implies that the operators YRQ (λ)P,
PRQ (λ)Y
(65)
are compact operators, where RQ (λ) is one of the parametrix of the operator Q at point λ ˙ RQ (λ)(Q − λ) = ˙ I. (Q − λ)RQ (λ) =
(66)
Then the following equalities can be proven ˙ Q + Y + P − λ; (Q + Y − λ)RQ (λ)(Q + P − λ) = ˙ Q + Y + P − λ. (Q + P − λ)RQ (λ)(Q + Y − λ) =
(67)
Let us prove the first equality only, since the prove of the second equality is similar. Formulas (65) imply that (Q + Y − λ)RQ (λ)(Q + P − λ) = ˙ (I + YRQ (λ))(Q + P − λ) = ˙ Q − λ + YRQ (λ)(Q − λ) + P + YRQ (λ)P = ˙ Q + Y + P − λ. We are going to prove now formula (64) for the essential spectra of the operators M, Q, Y, and P following the idea of [17], where a similar fact has been proven to the sum of two operators. Let us prove the following inclusion first σess (M) \ σess (Q) ⊂ [σess (Q + Y) ∪ σess(Q + P)] \ σess(Q).
(68)
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Suppose that λ does not belong to the essential spectra of the operators Q, Q + Y, and Q + P, then the operators Q + Y − λ,
RQ (λ),
Q+P−λ
are Fredholm operators as a product of three Fredholm operators. Then formulas (67) imply that the operator Q+Y+P−λ is a Fredholm operator. Hence the point λ does not belong to the essential spectrum of the operator Q + Y + P. In the second step let us prove the inclusion σess (M) \ σess (Q) ⊃ [σess (Q + Y) ∪ σess(Q + P)] \ σess(Q).
(69)
Suppose that λ does not belong to the essential spectra of the operators M and Q, i.e. that the operators M − λ and Q − λ are Fredholm operators. We are going to use Proposition 8.2 from [17] (see also [13]) stating that if the operators A and B are two bounded operators acting in a certain Banach space and the operators AB and BA are Fredholm operators, then the operators A and B are also Fredholm operators. Formulas (67) imply that the operators RQ (λ)(Q + Y − λ)RQ (λ)(Q + P − λ) and RQ (λ)(Q + P − λ)RQ (λ)(Q + Y − λ) are Fredholm operators. Then the proposition implies that the operators RQ (λ)(Q+ Y − λ) and RQ (λ)(Q + P − λ) are Fredholm operators. It follows from (66) that the operators Q + Y − λ= ˙ (Q − λ)RQ (λ)(Q + Y − λ) and Q + P − λ= ˙ (Q − λ)RQ (λ)(Q + P − λ) are Fredholm operators. It follows that λ does not belong to the essential spectra of the operators Q + Y and Q + P. Inclusion (69) is proven. Formulas (68) and (69) imply (64). The Theorem is proven. ✷ Remark. It is possible to get read of the condition that the complement in C of the essential spectrum the operator Q is connected. Then it is necessary to suppose that the operators YRQ (λ)P, PRQ (λ)Y are compact for any λ outside the essential spectrum of Q. It is possible to construct three operators Q, P, Y satisfying all conditions of the theorem except the connectivity of C \ σess (Q) but not satisfying formula (64). This counterexample can be prepared using bilateral shift in the Hilbert space X = ?2Z (?2N , H ⊕ H ), where H is a certain infinite dimensional axillary Hilbert space.
ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I
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PROPOSITION A.1. Let M be any n × n matrix separable pseudodifferential operator generated in the Hilbert space L2 (R, Cn ) by the symbol M(y, p) = Q + Y (y) + P (p),
p=
1 d , i dy
(70)
where Q is a constant diagonalizable matrix with simple spectrum, and the matrix functions Y (y) and P (p) are essentially bounded and satisfy the following two asymptotic conditions lim Y (y) = 0
x→∞
lim P (p) = 0.
p→∞
(71)
Then the essential spectrum of the operator M is given by σess (M) = σess (Q + P) ∪ σess (Q + Y).
(72)
Proof. The essential spectra of both operators Q + Y and Q + P contain the essential spectrum of Q σess (Q) = σ (Q), where σ (Q) is the spectrum of the matrix Q. To prove this fact one can use perturbation theory and the fact that the matrices Y (y), y → ∞ and P (p), p → ∞ are asymptotically small ([24]). Theorem A.1 implies that σess (Q + Y + P) \ {0} ⊃ σess(PN (Q + Y)PN ) \ {0} ⊃ σ (Q) \ {0}, n
(using A = Q + Y, B = P ). It follows that σess (M) \ {0} = (σess (Q + P) ∪ σess (Q + Y)) \ {0}.
(73)
We are going to remove the set {0} from the last formula. Applying the same analysis for the operator M − I we obtain that σess (M − I) \ {0} = (σess (Q − I + P) ∪ σess (Q − I + Y)) \ {0}.
(74)
This implies that σess (M(y, p)) \ { } = (σess (Q + P) ∪ σess (Q + Y)) \ { }, for arbitrary real and hence σess (M) = (σess (Q + P) ∪ σess (Q + Y)). The proposition is proven.
(75) ✷
In the special case case n = 1 and when the symbols Y (y) and P (p) are piecewise continuous the last proposition can be derived from Theorem 3 in [35] (see also [36]). The advantage of our approach is its transparency compared with the technique of C ∗ algebras used in [35].
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PAVEL KURASOV AND SERGUEI NABOKO
Remark. The condition concerning the simplicity of the spectrum of matrix Q can be removed in the special case where all matrices Q, Y (y), and P (p) are Hermitian. Proof. Consider the family of small Hermitian perturbations Q , > 0 of the matrix Q such that ' Q − Q ' and the spectrum of Q is simple. Such matrix Q satisfy the conditions of the theorem and hence σess (Q + Y + P) = σess (Q + P) ∪ σess (Q + Y).
(76)
Let us denote by F δ the δ-neighborhood of any set F ⊂ R F δ := {x ∈ R : dist(x, F ) δ}. Let A and B be two bounded self-adjoint operators acting in a certain Hilbert space. Then the essential spectra of the operators A and A+B are related by the following formula ([3]) σess (A + B) ⊂ (σess(A))'B' . From (76) we immediately obtain that σess (Q + Y + P(p)) ⊂ [σess (Q + Y) ∪ σess (Q + P)]2 ; σess (Q + Y) ∪ σess(Q + P) ⊂ [σess (Q + Y + P)]2 . Since the essential spectra are closed sets and is arbitrary small, we conclude that σess (Q + Y + P) = σess (Q + Y) ∪ σess (Q + P). This completes the proof.
(77) ✷
Appendix B. Elementary Lemmas on Calkin Calculus The following lemmas are necessary for the transformation of the resolvent. LEMMA B.1. Let the real valued function f (y) be positive bounded and separated from zero 0 < c f (y) C
(78)
for some c, C ∈ R+ . Let the function g(y) be bounded and the operator L ≡ pf (y)p + g(y)
(79)
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be self-adjoint and invertible in L2 (R). Suppose that the operator pL−1 p be bounded. Then for any bounded function h(y) such that limy→∞ h(y) = 0 the following equality holds in Calkin algebra ˙ pL−1 ph =
h . f
(80)
Comment 1. The rˆole of the function h is to regularize the equality which does not hold in Calkin algebra pL−1 = ˙ 1/f . Therefore the regularizing function h cannot be cancelled in (80). To construct a counter example let us first consider similar problem on the whole axis for which all calculations are trivial. Let the functions f and g be constant functions f = 1, g = 1. Then the operator pL−1 p − 1 = p(p 2 + 1)−1 p − 1 = −
1 p2 + 1
obviously is not compact, since it is a multiplication operator in the Fourier representation. Comment 2. In [4] similar result has been obtained in the regular case. Here an abstract proof of a generalization of the result is presented. We hope that the algebraic character of the proof will enable us to generalize these results to a wider classes of PDO and CDO. The advantage of our approach is that no information concerning the Green’s function is used. Proof. Consider first the case where the function h(y) is a C ∞ (R) function. We need this condition in order to avoid to consider the closure of bounded operators considered below. All these operators are well defined by their differential expressions on W21 (R). The following identity holds (at least in W21 (R)) 1 1 L = I. ((p + i)L−1 (p − i)) p−i p+i Multiplying the latter equality by the operator of multiplication by decreasing function h one can get the operator equality valid on W21 (R) 1 1 (pfp) h+ p−i p+i 1 1 g h = h. + ((p + i)L−1 (p − i)) p−i p+i
((p + i)L−1 (p − i))
The second term in left-hand side is a compact operator as the multiplication of the bounded operator (p + i)L−1 (p − i)(1/(p − i))g and the compact operator The latter condition could follow from the previous conditions for sufficiently smooth
function f .
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PAVEL KURASOV AND SERGUEI NABOKO
(1/(p + i))h (since the functions 1/(p + i) and h are decreasing function of p and y respectively). Hence the following equality holds in Calkin algebra ((p + i)L−1 (p − i))
1 1 (pfp) h= ˙ h. p−i p+i
Similarly taking into account that i 1 h= 1− h= ˙h p p+i p+i we get the following equality ((p + i)L−1 (p − i))
1 pf h = ˙ h. p−i
Multiplying by f −1 the latter equality, one gets (p + i)L−1 (p − i)
h 1 ph = ˙ , p−i f
using the fact the function f is boundedly invertible. Multiplying the latter equality by factor i p =1− p+i p+i from the left one gets in Calkin algebra p p p h = ˙ (p + i)L−1 (p − i) h = pL−1 ph. f p+i p−i p−i Let us consider the case of decreasing bounded but otherwise arbitrary function h. Every such function can be estimated from above by a certain positive ˜ |h(y)| h. ˜ We have already proven the decreasing to zero C ∞ (R) function h, Lemma for the function h˜ p p p h˜ ˜ = ˙ (p + i)L−1 (p − i) h = pL−1 p h. f p+i p−i p−i Of cause the multiplication by the contraction operator of multiplication by the bounded function h/h˜ preserves the equality in Calkin algebra. Finally one gets (80) for arbitrary h satisfying the conditions of the Lemma. ✷ The following lemma is well-known. (It is a special case of problems treated systematically in [19].) LEMMA B.2. Let the following conditions be satisfied
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(i) f c > 0, (ii) f, g ∈ C 1 (R), (iii) (pfp + g)|W22 (R) is invertible, then the operator (p + i)(pfp + g)−1 (p − i) defined originally on the dense set W21 (R) is bounded in L2 (R). COROLLARY. Under conditions of the lemma the operator p(pfp + g)−1 p|W21 (R) is also bounded, since the operator p/(p ± i) is a contraction. Remark. Condition (iv) can be substituted by a stronger condition (iv ) the real valued function g is positive definite g(x) c˜0 > 0. Really conditions (i), (ii), (iii) (iv ) imply (iv), since the estimate (pfp + g)u, u = fpu, pu + gu, u c0 'pu'2 + c˜0 'u'2 implies that the operator (pfp + g)|W 2 (R) has bounded inverse. 2
LEMMA B.3. Suppose that conditions (i)–(iii) of Lemma B.2 be satisfied. Let in addition the limits lim g(y),
lim f (y),
y→∞
y→∞
be finite. Then the difference between the inverse Hain–Lüst and asymptotic Hain– Lüst operators is a compact operator, moreover (p + i)[T −1 (µ) − Tas−1 (µ)] ∈ S∞ ; [T −1 (µ) − Tas−1 (µ)](p − i) ∈ S∞ . Proof. Consider the following chain of equalities (p + i)[T −1 (µ) − Tas−1 (µ)] ∈ S∞ = (p + i)T −1 (µ){Tas (µ) − T (µ)]T −1 (µ) ˜ −1 (µ) = (p + i)T −1 (µ)(p f˜p + g)T ˜ −1 (µ), = (p + i)T −1 (µ)(p − i)(p − i)−1 (p f˜p + g)T where f˜ = f − lim f (y) ∈ C 1 (R), y→∞
g˜ = g − lim g(y) ∈ L∞ (R) ∩ C 1 (R), y→∞
˜ = 0. lim f˜(y) = lim g(y)
y→∞
y→∞
(81)
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PAVEL KURASOV AND SERGUEI NABOKO
The operator (p + i)T −1 (µ)(p − i) is bounded and the operator (p − i)−1 (p f˜p + g)T ˜ −1 (µ) is compact operators, since the operators (p − i)−1 g˜ and f˜pT −1 (µ) are compact. (Here we use the fact that any pseudodifferential operator determined by the symbol ϕ(x)ψ(p) is compact if ϕ, ψ ∈ C(R) and limx→∞ ϕ(x) = 0, limp→∞ ϕ(p) = 0.) The lemma is proven. ✷ LEMMA B.4. Let conditions (i)-(iii) of Lemma B.2 be satisfied. Suppose in addition that the continuous functions α, γ ∈ C(R) are continuous and have finite limits at infinity. Then the following equality holds in Calkin algebra ˙ (α(∞)p + γ (∞))Tas−1 (µ). (αp + γ )T −1 (µ) =
(82)
Proof. Lemma B.3 implies that (αp + γ )T −1 (µ) = ˙ (αp + γ )Tas−1 (µ) = (α(∞)p + γ (∞))Tas−1 (µ) + ((α − α(∞))p + γ − γ (∞))Tas−1 (µ) = (α(∞)p + γ (∞))Tas−1 (µ) + (α − α(∞))pTas−1 (µ) + (γ − γ (∞))Tas−1 (µ) = ˙ (α(∞)p + γ (∞))Tas−1 (µ). The lemma is proven.
✷
Remark. All lemmas proven in this appendix for the operators acting in L2 (R) are in fact valid for the corresponding operators restricted to L2 (R+ ). To make the operators self-adjoint in L2 (R+ ) one needs to introduce some additional symmetric boundary condition at the origin, for example the Dirichlet boundary condition discussed in the paper. Let us mention here the necessary modifications of Lemma B.1 only. The other lemmas can be treated in the same way. Let L be a self-adjoint operator in L2 (R+ ) determined by (79) and certain boundary condition at the origin. Consider the extension of L to the operator acting in L2 (R) determined by the same expression, where the functions f (y) and g(y) are continued for negative values of y as even functions. Then equality (80) holds in Calkin algebra for the extended operator. Taking into account that the resolvent of the extended operator differs from the orthogonal sum of two copies of the resolvents of the initial operator taken on the positive and negative semiaxes separately by a finite rank operator. (Note that the functions from the domain of both operators satisfy proper separating boundary conditions at the origin.) We have used here the Glazman splitting method ([3]). As a result, we obtain the necessary equality for the operators in L2 (R+ ). Acknowledgements We would like to thank the referee for valuable remarks which allowed us to improve the original manuscript.
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The work of the authors was supported by the Royal Swedish Academy of Sciences.
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
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Adam, J. A.: Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics, Phys. Rep. 142 (1986), 263–356. Adamyan, V., Langer, H., Mennicken, R. and Saurer, J.: Spectral components of self-adjoint block operator matrices with unbounded entries, Math. Nachr. 178 (1996), 43–80. Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Pitman, Boston, 1981. Atkinson, F. V., Langer, H., Mennicken, R. and Shkalikov, A.: The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5–20. Descloux, J. and Geymonat, G.: Sur le spectre essentiel d’un operateur relatif á la stabilité d’un plasma en géometrie toroidal, C.R. Acad. Sci. Paris 290 (1980), 795–797. Everitt, W. N. and Bennewitz, C.: Some remarks of the Titchmarsch–Weyl m-coefficient, In: Tribute to Åke Pleijel, Univ. of Uppsala, 1980. Everitt, W. N. and Markus, L.: Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators, Amer. Math. Soc., Providence, RI, 1999. Faierman, M., Lifschitz, A., Mennicken, R. and Möller, M.: On the essential spectrum of a differentially rotating star, ZAMM Z. Angew. Math. Mech. 79 (1999), 739–755. Faierman, M., Mennicken, R. and Möller, M.: The essential spectrum of a system of singular ordinary differential operators of mixed order. I. The general problem and an almost regular case, Math. Nachr. 208 (1999), 101–115. Faierman, M., Mennicken, R. and Möller, M.: The essential spectrum of a system of singular ordinary differential operators of mixed order. II. The generalization of Kako’s problem, Math. Nachr. 209 (2000), 55–81. Faierman, M. and Möller, M.: On the essential spectrum of a differentially rotating star in the axisymmetric case, Proc. Royal Soc. Edinburgh A, 130 (2000), 1–23. Goedbloed, J. P.: Lecture notes on ideal magnetohydrodynamics, In: Rijnhiuzen Report, FormInstituut voor Plasmafysica, Nieuwegein, 1983, 83–145. Gohberg, I., Goldberg, S. and Kaashoek, M.: Classes of Linear Operators, vol. 1, Birkhäuser, Basel, 1990. Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems, 2nd edn, Birkhäuser, Basel, 1986. Grubb, G. and Geymonat, G.: The essential spectrum of elliptic systems of mixed order, Math. Anal. 227 (1977), 247–276. Hain, K. and Lüst, R.: Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mit Volumenströmmen, Z. Naturforsch. A 13 (1958), 936–940. Hardt, V., Mennicken, R. and Naboko, S.: System of singular differential operators of mixed order and applications to 1-dimensional MHD problems, Math. Nachr. 205 (1999), 19–68. Hartman, P.: Ordinary Differential Equations, John Wiley, New York, 1964. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols I–IV, 2nd edn, Springer-Verlag, Berlin, 1994. Kako, T.: On the essential spectrum of MHD plasma in toroidal region, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), 53–56. Kako, T.: On the absolutely continuous spectrum of MHD plasma confined in the flat torus, Math. Methods Appl. Sci. 7 (1985), 432–442. Kako, T.: Essential spectrum of linearized operator for MHD plasma in cylindrical region, Z. Angew. Math. Phys. 38 (1987), 433–449.
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Kako, T. and Descloux, J.: Spectral approximation for the linearized MHD operator in cylindrical region, Japan J. Indust. Appl. Math. 8 (1991), 221–244. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn, Springer, Berlin, 1976. Konstantinov, A. and Mennicken, R.: On the Friedrichs extension of some block operator matrices, Integral Equations Operator Theory 42 (2002), 472–481. Krasnosel’skii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevskii, P. E.: Integral Operators in Spaces of Summable Functions, Noordhoff Publishing, Leiden, 1976. Kurasov, P.: On the Coulomb potential in one dimension, J. Phys. A 29 (1996), 1767–1771; Response to ‘Comment on “On the Coulomb potential in one dimension”,’ J. Phys. A 30 (1997), 5579–5581. Langer, H. and Möller, M.: The essential spectrum of a non-elliptic boundary value problem, Math. Nachr. 178 (1996), 233–248. Lifchitz, A. E.: Magnetohydrodynamics and Spectral Theory, Kluwer Acad. Publ., Dordrecht, 1989. Mennicken, R., Naboko, S. and Tretter, Ch.: Essential spectrum of a system of singular differential operators, to appear in Proc. Amer. Math. Soc. Möller, M.: On the essential spectrum of a class of operators in Hilbert space, Math. Nachr. 194 (1998), 185–196. Naboko, S.: Nontangential boundary values of operator-valued R-functions in a half-plane, St. Petersburg (Leningrad) Math. J. 1 (1990), 1255–1278. Naimark, M. A.: Linear Differential Operators, Ungar, New York, 1968. Olver, F. W.: Introduction to Asymptotic and Special Functions, Academic Press, New York, 1974. Power, S. C.: Essential spectra of piecewise continuous Fourier integral operators, Proc. Roy. Irish. Acad. 81 (1981), 1–7. Poer, S. C.: Fredholm theory of piecewise continuous Fourier integral operators on Hilbert space, J. Operator Theory 7 (1982), 51–60. Raikov, G. D.: The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry, Arch. Rational Mech. Anal. 116 (1991), 161–198. Rofe-Beketov, F.: Self-adjoint extensions of differentila operators in a space of vectorfunctions, Soviet. Math. Doklady 10 (1969), 188–192. Rofe-Beketov, F. and Kholkin, A.: Spectral Analysis of Differential Operators. Relation of the Spectral and Oscillatory Properties, Mariupol, 2001. Schechter, M.: On the essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13 (1966), 205–215. Schechter, M.: Spectra of Partial Differential Operators, 2nd edn, North-Holland, Amsterdam, 1986. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, 2nd edn, vols I–IV. Academic Press, New York, 1984. de Snoo, H.: Regular Sturm–Liouville problems whose coefficients depend rationally on the eigenvalue parameter, Math. Nachr. 182 (1996), 99–126. Zelenko, L. B. and Rofe-Beketov, F. S.: The limit spectrum of systems of first order differential equations with slowly changing coefficients (Russian), Differencial’nye Uravnenija 7 (1971), 1982–1991 (in Russian).
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A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion BERNHARD G. BODMANN Department of Physics, Princeton University, 337 Jadwin Hall, Princeton, NJ 08544, U.S.A. e-mail: [email protected] (Received: 12 December 2001) Abstract. First we discuss the construction of self-adjoint Berezin–Toeplitz operators on weighted Bergman spaces via semibounded quadratic forms. To ensure semiboundedness, regularity conditions on the real-valued functions serving as symbols of these Berezin–Toeplitz operators are imposed. Then a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups is derived. All results are the consequence of a relation of Berezin–Toeplitz operators to Schrödinger operators defined via certain quadratic forms. The probabilistic expression is derived in conjunction with the Feynman–Kac–Itô formula. Mathematics Subject Classifications (2000): 81S10, 47D08. Key words: weighted Bergman spaces, Berezin–Toeplitz operators, Schrödinger operators, semigroups, Feynman–Kac–Itô formula.
1. Introduction The results presented here are inspired by a concept of Daubechies and Klauder [14, 15] which provides a probabilistic expression for the unitary group generated by Hamiltonians arising from the so-called anti-Wick quantization prescription. Based on geometric considerations, several works [1, 2, 26, 27] advocate natural generalizations of this expression which can be related [25, 29] to Hamiltonians from the more universal Berezin–Toeplitz quantization scheme [6]. The generalization presented in this paper evolved from a pattern behind the construction in [14, 15], according to which certain Berezin–Toeplitz operators can be realized as limits of monotone families of Schrödinger operators. Hereby a Berezin–Toeplitz operator Tf is understood as the compression of a suitable multiplication operator Mf : ψ → f ψ from a Hilbert space of square-integrable functions to a closed subspace of analytic functions. In this context the real-valued function f is called a symbol and can be interpreted as a classical observable corresponding to the operator Tf . In quantum mechanics Berezin–Toeplitz oper-
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ators model a variety of systems with canonical or other degrees of freedom [5, 6]. A Schrödinger operator H , on the other hand, can be thought of as a second-order differential operator of d-dimensional Euclidean space Rd , formally d on a subset 2 written H = k=1 (i∂k + ak ) + υ with the vector potential a: → Rd and the scalar potential υ: → R. Fairly general conditions have been worked out under which this formal expression characterizes H uniquely as a self-adjoint operator [11, 13, 20, 21, 28]. The first task in this paper is to find conditions on f which guarantee that Tf can be defined as a self-adjoint operator. Secondly we derive the analogue of the probabilistic expression in [14] for Berezin–Toeplitz semigroups in this general setting. Unlike the results [14, 16] derived in analogy of the anti-Wick situation and the strategies advocated for their generalization [1, 2, 26, 27], we stay with a probabilistic representation which is based on standard Brownian motion [36]. This way one may benefit from the associated repertory of probabilistic techniques. A minor difference with the original [14] is that we are concerned with Berezin–Toeplitz semigroups, not with the unitary groups associated with quantum mechanical time evolution. The structure and contents of this paper are as follows: After fixing the notation, we start in Section 3 with a review of Hilbert spaces of analytic functions which are known as weighted Bergman spaces. In Sections 4 and 5 we construct Berezin–Toeplitz operators on these spaces and state sufficient conditions on their symbols which guarantee self-adjointness and semiboundedness. Hereby an essential tool is that certain Berezin–Toeplitz operators can be extended to self-adjoint Schrödinger operators which are defined via quadratic forms. In Section 6 we continue the spirit of [14, 15] and derive a probabilistic expression for Berezin–Toeplitz semigroups which uses a realization of Berezin– Toeplitz operators as limits of monotone families of Schrödinger operators and the probabilistic representation of Schrödinger semigroups with the help of the Feynman–Kac–Itô formula [11, 38, 40]. This paper is not entirely self-contained. For the relevant background information, the reader is referred to [37] and [11] which neatly comprise the essential building blocks for the results presented here.
2. Basic Definitions Let R denote the real numbers. By convention, D is always an open, simply connected set in the plane of complex numbers C := {z = z1 + iz2 : z1,2 ∈ R}. The boundary of a set A ⊂ C is written as ∂A, its complement in C as C \ A or Ac . We will denote the real and imaginary parts of a complex number z as z1 and z2 , respectively, and the complex conjugate as z¯ = z1 − iz2 . All functions appearing in this text are tacitly understood to be measurable, each one in its appropriate sense. The positive and negative parts f + and f − of a real-
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valued function f are defined by f ± := max{±f, 0}, such that f = f + − f − . The indicator function of a set A is denoted as χA . Several spaces of complex-valued functions are used in the text. The space of arbitrarily often differentiable functions with compact support inside D is referred to as Cc∞ (D). Concerning differentiability, D is hereby regarded as a subset of the real vector space C ∼ = R2 . The space of Lebesgue-essentially bounded functions on D is denoted as L∞ (D). DEFINITION 2.1. A complex-valued function φ: D → C defined on D is called analytic in D if it is differentiable and satisfies the Cauchy–Riemann differential equations, which are stated as (∂1 + i∂2 )φ(z1 + iz2 ) = 0,
(1)
with the partial derivatives ∂1,2 := ∂/∂z1,2. DEFINITION 2.2. A positive function g: D → ]0, ∞[ is said to be essentially bounded away from zero on compacts inside D if for any given compact set C ⊂ D there exists a δ > 0 such that ess inf g(z) := sup inf g(z) δ. z∈C
A z∈A
(2)
Hereby the supremum is taken over all Lebesgue-measurable subsets A ⊂ C such that the Lebesgue measure λ vanishes on their difference, λ(C \ A) = 0. DEFINITION 2.3. A real-valued function f : C → R belongs to the Kato class K in two dimensions [24, 40] if the following condition is satisfied: |f (y)|ln|z − y| dλ(y) = 0. (3) lim sup r0 z∈C
{|y−z|
Whenever this property only holds locally, that is, χC f ∈ K for all compact sets C in C, we write f ∈ Kloc . It is useful to know that the local Kato property implies local integrability with respect to the Lebesgue measure, Kloc ⊂ L1loc (C). If a function satisfies f + ∈ Kloc and f− ∈ K then it is called Kato decomposable, symbolized as f ∈ K± . In order to apply these notions to functions which are at first only defined on a subset D ⊂ C, they are by convention extended to be zero on Dc .
3. Hilbert Spaces of Analytic Functions DEFINITION 3.1. Given a positive function g: D → ]0, ∞[ we associate with it the so-called weighted Bergman space L2a (gλ) := {φ: D → C, analytic in D and (φ, φ) < ∞},
(4)
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a vector space which is endowed with the inner product (φ, ψ) := φ(z)ψ(z)g(z) dλ(z)
(5)
D
where λ stands for the Lebesgue measure on the complex plane. Remarks 3.2. As a special case, if D is a bounded domain and the weight function is constant, L2a (gλ) is the well-known Bergman space [7, 30]. The inner product suggests that L2a (gλ) can be identified with a vector-subspace of L2 (gλ). To be precise, we recall that the latter consists of equivalence classes of gλ-square-integrable functions on D which differ from each other on a set of Lebesgue measure zero. Consequently, we identify each function in L2a (gλ) with its equivalence class from L2 (gλ). Now we will state conditions which guarantee that L2a (gλ) forms a Hilbertsubspace of L2 (gλ). PROPOSITION 3.3. If g is essentially bounded away from zero on compacts inside D, then L2a (gλ) is complete with respect to the norm-topology induced by the inner product. Proof. Let (ψn )n∈N be a Cauchy sequence in L2a (gλ). First we show uniform convergence of ψn on compacts inside D. Consider a compact subset C ⊂ D and a safety radius r < infy∈∂D |z − y| for all z ∈ C. By assumption there is a lower bound δ > 0 such that (2) is satisfied. Using the mean value property for analytic functions, Jensen’s inequality in conjunction with the convex square-modulus function, and the lower bound for g we estimate sup |ψn (z ) − ψm (z )|2
z ∈C
1 = sup 2 πr z ∈C
B(r,z )
2 (ψn (z) − ψm (z)) dλ(z)
1 sup |ψn (z) − ψm (z)|2 dλ(z) π r 2 z ∈C B(r,z) 1 |ψn (z) − ψm (z)|2 g(z) dλ(z) π r 2 δ B(r,z)
1 ψn − ψm 2 . π r 2δ
(6)
(7)
(8) (9)
The right-hand side can be made arbitrarily small and thus the sequence (ψn ) converges uniformly on C. We can therefore conclude that the pointwise limit defines a function ψ : ψ(z) = limn→∞ ψn (z) which is also analytic in D.
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It remains to show that the convergence ψn → ψ is also in the sense of the norm. Due to pointwise convergence and Fatou’s lemma the inequality ψ − ψm lim inf ψn − ψm n→∞
(10) ✷
follows, therefore the Cauchy property entails norm convergence.
LEMMA 3.4. Under the same assumption on g as in the preceding proposition, the point-evaluation functionals Fz parameterized by z ∈ D which are defined according to L2a (gλ) −→ C, Fz :
ψ −→ ψ(z)
(11) (12)
are continuous linear mappings. Proof. Let C be a compact neighborhood of z and select a convergent sequence (ψn )n∈N in L2a (gλ). Since the sequence has the Cauchy property, we can use the chain of inequalities (6) to show that ψn (z) is also Cauchy, and therefore convergent. ✷ PROPOSITION 3.5. If g is essentially bounded away from zero on compacts inside D, the weighted Bergman space L2a (gλ) possesses a reproducing kernel. More explicitly, there is a kernel κ: D × D → C such that any function ψ in L2a (gλ) satisfies the integral equation κ(z , z)ψ(z) g(z) dλ(z). (13) ψ(z ) = D
Proof. To see this, we observe that due to the continuity of the functional Fz and the completeness of L2a (gλ) the Riesz representation theorem implies that there is a vector κz in L2a (gλ) such that (κz , ψ) = ψ(z)
for all ψ ∈ L2a (gλ).
(14)
Inserting the definition of the inner product (5) yields the desired integral Equa✷ tion (13) with the claimed kernel given by κ(z, z ) = (κz , κz ). The last proposition ensures that all bounded operators have integral kernels. COROLLARY 3.6. If g is essentially bounded away from zero on compacts inside D, then any bounded operator B on L2a (gλ) possesses an integral kernel given by B(z, z ) = (κz , Bκz ), which means that the image of ψ ∈ L2a (gλ) is expressed as B(z, z )ψ(z ) g(z ) dλ(z ). (15) Bψ(z) = D
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Proof. That B(z, z) is indeed an integral kernel results from the reproducing property (13) and Fubini’s theorem. The sesqui-analyticity follows from the pointevaluation property (14) and the analyticity of the functions in L2a (gλ). ✷ Remark 3.7. Since (15) makes sense even for ψ ∈ L2 (gλ), any bounded operator extends naturally via its integral kernel to L2 (gλ). From this point of view, κ is the integral kernel of an orthogonal projection operator, henceforth called K, which maps L2 (gλ) onto L2a (gλ). EXAMPLES 3.8. Various examples of Lie group representations are realized on specific weighted Bergman spaces. We list a few groups and their unitary irreducible action which is in all but the last example related to Moebius transformations on the associated representation spaces. For more details, see [32] or [31]. Unless otherwise stated we set D = C. (1) Heisenberg–Weyl group. Hereby the Hilbert space is specified by the weight 2 function g(z) = (1/π )e−|z| . The reproducing kernel is κ(z, z ) = ez¯z . This space is also known as Fock–Bargmann space [3]. The group representation D(α, β) with parameters α ∈ [0, 2π [, β ∈ C acts on a vector ψ by 2 ¯ D(α, β)ψ(z) = eiα e−|β| eβz ψ(z − β).
(16)
(2) SU(2) group. For each integer or half-integer j ∈ 12 N a (2j + 1)-dimensional space is defined by setting g(z) =
2j + 1 (1 + |z|2 )−2j −2 . π
The reproducing kernel is κ(z, z ) = (1 + z¯z )2j . The group parameterized by α, β ∈ C with |α|2 + |β|2 = 1 acts as αz − β¯ 2j . (17) D(α, β)ψ(z) = (βz + α) ¯ ψ βz + α¯ (3) SU(1, 1) group. There are two well-known ways to represent this group on weighted Bergman spaces. (a) The first one is described in [32]. Unlike the previous examples, here D is not the whole complex plane, but the unit disc D = {z: |z| < 1}, and 2k − 1 (1 − z¯z)2k−2 g(z) = π with a fixed number k ∈ {1, 3/2, 2, . . .}. The reproducing kernel is κ(z, z ) = (1 − z¯z )−2k . The group action is given with the parameters α, β ∈ C, |α|2 − |β|2 = 1 as αz + β¯ −2k . (18) ψ D(α, β)ψ(z) = (βz + α) ¯ βz + α¯
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(b) An alternative to [32] comes from the so-called Barut–Girardello representation [4] of the SU(1, 1) group. We set D = C again, select k as above and choose 2|z|2k−1 K2k−1 (2|z|) g(z) = π +(2k) with the gamma function + and the modified Bessel function ∞ 1 σ t −σ −1 exp(−t − r 2 /4t) dt Kσ (r) = (r/2) 2 0 for r > 0, σ ∈ R [19, 8.432(6)]. The kernel is given as the confluent hypergeometric limit function
κ(z, z ) = 0 F1 (2k, z¯z ) =
∞ n=0
(z¯z )n . (2k)n +(n + 1)
Hereby (2k)n denotes the Pochhammer symbol, defined by (2k)0 := 1 and the recursion (2k)n = (2k)n−1 (2k + n − 1). The group action, however, does not seem to be related to Moebius transformations and will be omitted here. Remarks 3.9. In all these examples the kernel can be constructed from the weight function with the help of the Gram–Schmidt orthogonalization procedure. Due to the rotational symmetry g(z) = g(|z|), monomials pn : z → zn with differing degree n ∈ {0, 1, 2, . . .} are orthogonal, so the kernel is diagonalized in terms of the basis functions as κ(z, z ) = n cn pn (z)pn(z ) with suitable normalization constants cn . Here the summation runs for all the examples over the nonnegative integers, except for the SU(2) case, because there, only monomials with maximal degree 2j are square integrable. In Section 6 we will present a probabilistic approach to construct the reproducing kernel, which does not rely on special symmetries of g. 4. Self-adjoint Berezin–Toeplitz Operators Defined by Quadratic Forms In the remaining text we assume that L2a (gλ) is complete. DEFINITION 4.1. Given the Hilbert space L2a (gλ) and a real-valued function f : D → R, we consider the sesquilinear form Q(tf ) × Q(tf ) −→ C, tf :
(19)
(ψ, φ) −→
f ψφg dλ, D
(20)
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with form domain 2 2 |f ψ |g dλ < ∞ . Q(tf ) := ψ ∈ La (gλ) :
(21)
D
When it is interpreted as quadratic form, tf is written as tf (ψ) := tf (ψ, ψ). The next concern is a condition to guarantee that tf is closed and semibounded, which in turn ensures that there is a self-adjoint operator associated with tf via the Friedrichs representation theorem. LEMMA 4.2. The sesquilinear form belonging to a nonnegative function f 0 is closed. Proof. We need to show that Q(tf ), equipped with the form-norm •tf defined by ψtf := (tf (ψ) + ψ2 )1/2
for ψ ∈ Q(tf ),
(22)
is complete. Suppose (ψn )n∈N is a Cauchy sequence with respect to the form-norm. Due to the estimate ψ ψtf the sequence is convergent in L2a (gλ), ψn → ψ. Using pointwise convergence and Fatou’s lemma, we obtain ψ − ψn tf lim infm→∞ ψm − ψn tf and therefore the sequence (ψn ) converges with respect to the form-norm. ✷ THEOREM 4.3. If the form tf + belonging to the positive part f + 0 of a function f = f + − f − is densely defined and the negative part can be incorporated in tf as a form-bounded perturbation tf − (ψ) c1 tf + (ψ) + c2 ψ2
(23)
with relative form bound c1 < 1 and a constant c2 0, then tf is closed on Q(tf ) = Q(tf + ) and has a (greatest) lower bound c ∈ R, such that tf (ψ) cψ2 . Proof. This is the so-called KLMN theorem. For the proof, see [33, Theorem X.17]. ✷ THEOREM 4.4. If the form tf is closed and has the greatest lower bound c as in the preceding theorem, then it belongs to a unique self-adjoint operator Tf which is characterized in terms of the square-root Tf − c by the domain D( Tf − c) = Q(tf ) and the equality ( Tf − c φ, Tf − c ψ) + c(φ, ψ) = tf (φ, ψ)
for all φ, ψ ∈ Q(tf ).
(24)
Proof. Again, we refer to the literature [34, Theorem VIII.15] or [41, Theorem 5.36], where this result is known as the Friedrichs representation theorem. ✷
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Remarks 4.5. In the context of weighted Bergman spaces we call Tf a selfadjoint Berezin–Toeplitz operator and the function f its symbol. For ψ ∈ Dmin := {ψ ∈ L2a (gλ), f ψ ∈ L2 (gλ)} the identity Tf ψ = K(f ψ) relates Tf to the traditional way of defining a Berezin–Toeplitz operator as a composition of a multiplication operator with the orthogonal projection K. In that scheme Tf would arise as so-called Friedrichs extension from the quadratic form of the traditionally defined semibounded operator. A disadvantage of defining Tf by a semibounded form is that in general nothing is known about its domain. The situation is different, however, if Dmin is a domain of essential self-adjointness for Tf . Such situations have been investigated in detail [12, 23] for the case of the Fock–Bargmann space (see Example 1 in Section 3).
5. Relation to Schrödinger Operators This section shows how a Berezin–Toeplitz operator can be extended to a family of Schrödinger operators. A major benefit is that the knowledge about Schrödinger operators can be used to find sufficient conditions for the semiboundedness of tf in terms of g and f . These ensure self-adjointness of the corresponding Berezin– Toeplitz operator. 5.1. EMBEDDING THE WEIGHTED BERGMAN SPACE Consider the unitary mapping L2 (gλ) −→ L2 (D), U:
ψ −→
√
gψ,
(25) (26)
which simply amounts to a redistribution of the weight function. This mapping identifies L2a (gλ) with a closed subspace of L2 (D).
= U KU † of K is an orthogonal Remark. Note that the unitary equivalent K projection operator √ which maps L2 (D) onto U (L2a (gλ)) and has the integral kernel κ(z, ˜ z ) = κ(z, z ) g(z)g(z ). However, unlike the case for L2a (gλ), this kernel is continuous√if and only if g is continuous. Following the previous notation we define κ˜ z := g(z ) U κz . 5.2. A QUADRATIC FORM AND ITS NULL SPACE The purpose of this subsection is to construct a Schrödinger operator which is nonnegative and has U (L2a (gλ)) as its null eigenspace.
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DEFINITION 5.1. We define the quadratic form Cc∞ (D) −→ R,
(27)
r: ψ −→ (i∂1 − ∂2 + a)ψ2 ,
(28)
with the function
a(z) := (−i∂1 + ∂2 ) ln g(z),
(29)
where g is such that the local integrability condition |a|2 ∈ L1loc (D) holds. This form is closeable and the resulting form-closure domain Q(r) is contained in the set L2 (D). We say that the corresponding self-adjoint operator R obeys Dirichlet boundary conditions. Remarks 5.2. Despite the construction of R via (28), in general the null-space N (R) := R −1 ({0}) is only a closed subspace of U (L2a (gλ)). On the other hand, if instead of Cc∞ (D) we start with the maximal form domain, which means the set of all ψ ∈ L2 (D) for which the expression on the right-hand side of (28) makes sense and is finite, then the resulting self-adjoint operator is said to obey Neumann boundary conditions and possesses the null-space U (L2a (gλ)). For this reason it seems like an artificial complication to introduce Dirichlet boundary conditions here. Nevertheless, the preceding definition is needed as a preparation for the probabilistic expression in Section 6. If a is absolutely continuous we can formally write R as a Schrödinger-type operator R = (i∂1 + a1 )2 + (i∂2 + a2 )2 + υ
(30)
with υ := ∂2 a1 − ∂1 a2 . However, in order to consider R as the usual form sum [11, Remark 2.7] we need additional regularity assumptions on a and υ; see also the following subsection. Next we establish conditions which guarantee the inclusion of U (L2a (gλ)) in the domain of R. PROPOSITION 5.3. If either D = C or the operator Tf specified by f (z) = infy∈∂D |z − y|−2 is bounded on L2a (gλ), then the image U (L2a (gλ)) is contained in the form-closure domain Q(r) and the identity N (R) = U (L2a (gλ)) follows. Proof. In case D = C the argument follows a standard procedure, compare with [39] or [13, Theorem 1.13]. The more general case treated here demands an adaptation. The proof proceeds in two steps: First we will show that for any function ψ ∈ U (L2a (gλ)), the space of compactly supported, in D essentially bounded functions L∞ c (D), provides a sequence which converges to ψ in the sense of the form norm.
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297
The second step is a standard mollifier argument, which shows that Cc∞ (D) is dense in L∞ c (D) ∩ Q(r). Step 1: Consider a function ψ ∈ U (L2a (gλ)). We perform two alterations: a mollified cutoff near ∂D (if Dc is nonempty) and a smooth cutoff towards infinity. The details are as follows: Let Dn := {z ∈ D : |z − y| 1/n for all y ∈ ∂D}. Obviously the sequence of characteristic functions χDn converges pointwise to one on D as n → ∞. In addition, consider a real-valued function η ∈ C0∞ (C) which is nonnegative and satisfies η(0) = maxz∈C η(z) = 1 as well as the gradient bound |∇η| 1. We define a sequence ηn (z) = η(z/n) which also tends to one from below. Furthermore, let the nonnegative function δ1 ∈ Cc∞ (C) be an approximate δ-function, which means C δ1 (z) dλ = 1, and δ1 = 0 for all |z| > 1. We define a sequence of approximations δn (z) := n2 δ1 (nz) to smear out χDn by convolution, χDn (y)δn (x − y) dλ(y). (χDn ∗ δn )(z) := D
Now consider φn := ψηn (χDn ∗δn ). Clearly φn → ψ in L2 (D). To show that the convergence is also with respect to the form-norm, we use the triangle inequality, (i∂1 − ∂2 + a)φn = ψ(i∂1 − ∂2 )ηn (χDn ∗ δn ) ψ(χDn ∗ δn )(i∂1 − ∂2 )ηn + ψηn (i∂1 − ∂2 )(χDn ∗ δn ).
(31)
The first term on the right-hand side vanishes in the limit n → ∞ by dominated convergence; the second term needs a closer look. Note that the neigborhood of the boundary contains the support supp(i∂1 − ∂2 )(δn ∗ χDn ) ⊂ {z ∈ D : |z − y| > 2/n ∀y ∈ ∂D}. By Hölder’s inequality we can estimate |∇δn ∗ χDn | nλ(supp δ1 ) max |∇δ1 |. In√consequence there is some constant c 0 such that |(i∂1 − ∂2 )(χDn ∗ δn )| c f with the function f given in the statement of the theorem. Hence by the assumption on the boundedness of Tf we can apply dominated convergence and the second term in (31) also vanishes as n → ∞. Step 2: Let ψ ∈ L∞ c (D) ∩ Q(r) be given. We borrow an argument from [11, Lemma B.3, Assertion 3)]. Consider the previously defined δn for only suitably large n such that ψn := ψ ∗ δn ∈ Cc∞ (D). We have limn→∞ φ ∗ δn − φ = 0 for all φ ∈ L2 (D). Therefore it remains to show that after the estimate (i∂1 − ∂2 + a)(ψn − ψ) (i∂1 − ∂2 )(ψn − ψ) + a(ψn − ψ) each term on the right-hand side converges to zero.
(32)
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The first term is taken care of by the identity (i∂1 − ∂2 )(ψn − ψ) = ((i∂1 − ∂2 )ψ) ∗ δn − (i∂1 − ∂2 )ψ
(33)
because the assumption |a|2 ∈ Kloc implies a ∈ L2loc (D) and (i∂1 − ∂2 )ψ (i∂1 − ∂2 + a)ψ + aψ < ∞.
(34)
To ensure that the second term in (32) vanishes we pass to a subsequence of ψn which is almost everywhere convergent and select a compact set C which contains the support of all but finitely many ψn . The bound |ψn (x)| χC ψ∞ for x ∈ D together with a ∈ L2loc (D) allows dominated convergence which completes the proof. ✷ Remark 5.4. When D is a proper subset of the complex numbers, the condition on the functions in U (L2a (gλ)) amounts to controlling their decay towards the boundary ∂D, which relates to the intuitive understanding of Dirichlet boundary conditions.
5.3. EXTENDING BEREZIN – TOEPLITZ OPERATORS TO SCHRÖDINGER OPERATORS
Henceforth, we say f and g are admissible if they are such that |a|2 ∈ Kloc and ∂2 a1 − ∂1 a2 , f ∈ K± . For such f and g we define a family {s(ν) f }ν>0 of quadratic forms Q(s(ν) f ) −→ R, s(ν) f :
(35)
ψ −→ νr(ψ) +
D
f |ψ|2 dλ.
(36)
The domain is independent of ν given by + Q(s(ν) f ) := Q(r) ∩ {ψ : f ψ < ∞}, because the negative part of the second term in (36) is, due to the assumption, infinitesimally form-bounded with respect to r. With the above regularity assumptions on a, υ = ∂2 a1 − ∂1 a2 and f the corresponding self-adjoint, semibounded operator Sf(ν) is a Schrödinger operator defined in the usual form sense [11], which is implicitly understood in the expression Sf(ν) = ν[(i∂1 + a1 )2 + (i∂2 + a2 )2 + υ] + f.
(37)
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THEOREM 5.5. If f and g obey the conditions |a|2 ∈ Kloc
and
∂2 a1 − ∂1 a2 ,
f ∈ K± ,
and the conclusion of Proposition 5.3 that N (R) = U (L2a (gλ)) holds, then the form tf is semibounded and closed on Q(tf + ). Therefore, f defines a self-adjoint semibounded operator Tf on the closure Q(tf ) ⊂ L2a (gλ). Proof. First we note s(ν) f (U ψ) = tf (ψ) for any ν > 0 and ψ ∈ Q(tf ). Thus, we only need to show that the restriction of s(ν) f to the closed subspace N (r) = U (L2a (gλ)) is again a closed and semibounded form. To show closedness, assume a sequence (ψn )n∈N in N (r) which is Cauchy with respect to the form-norm. Then by the closedness of s(ν) f the sequence has a limit (ν) ψ ∈ Q(sf ). However, this limit must also be contained in N (r), because N (r) is a closed subspace and the sequence (ψn )n∈N converges with respect to the usual norm on L2 (D). Semiboundedness follows from the inequality inf{s(ν) f (ψ) : ψ = 1} inf{s(ν) f (ψ) : ψ ∈ N (r)
and
ψ = 1}.
✷
(38)
Remarks 5.6. As stated, the above theorem does not imply that tf is densely defined. Therefore, Tf might be self-adjoint only on a Hilbert-subspace of L2a (gλ). In analogy with Theorem 4.3, it is sufficient for the closedness and semiboundedness of tf when for some ν > 0 the negative part f − can be incorporated as a form-bounded perturbation of s(ν) f + with relative form bound strictly less than one. However, this condition is not as easy to characterize in terms of f as the stronger assumption in the preceding theorem.
6. Probabilistic Representation of Berezin–Toeplitz Semigroups In this section we derive a probabilistic expression for the sesqui-analytic integral kernel of Berezin–Toeplitz semigroups. The major steps are the reconstruction of Tf via the monotone convergence of s(ν) f for ν → ∞ and the application of the Feynman–Kac–Itô formula. DEFINITION 6.1. For a given z, z ∈ C and t, ν > 0 we define the integral with respect to the pinned Wiener measure 1 −|z−z |2 /4t ν e E[•] (39) (•) dµ(ν) z,0;z ,t := 4π tν via the expectation E[•] with respect to the two-dimensional Brownian bridge measure with diffusion constant ν. Both measures are concentrated on the set of
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continuous paths {b: [0, t] → C, b(0) = z and b(t) = z } which are pinned at the start and endpoint. As Gaussian stochastic process the Brownian bridge is uniquely determined by its mean s E[b(s)] = z + (z − z) , t and covariance
s ∈ [0, t]
(40)
rs E[b(r)b(s)] − E[b(r)]E[b(s)] = 4ν min{r, s} − , t
(41)
E[b(r)b(s)] − E[b(r)]E[b(s)] = 0 r, s ∈ [0, t].
(42)
DEFINITION 6.2. Given D, the random variable TD := inf{s > 0 : b(s) ∈ Dc } is called the first exit time of the process. By convention, we define TD to be infinite on the set for which b never leaves D. PROPOSITION 6.3. If f and g are admissible and N (R) = U (L2a (gλ)) as in the conclusion of Proposition 5.3, then for ν → ∞, the semigroup generated by Sf(ν) converges strongly, (ν)
(∞)
= U e−t Tf E U † ψ, lim e−t Sf ψ = e−t Sf Eψ
ν→∞
(43)
:= U EU † . where ψ ∈ L2 (D), E = E † E projects onto the closure Q(tf ), and E (ν) Proof. The limit ν → ∞ of sf yields a nondensely defined form (ν) s(∞) f : ψ → lim sf (ψ) ν→∞
(44)
with the domain Q(s(∞) f ) = U (Q(tf )). This last equality follows from Proposition 5.3 and the definition of the embedding. The monotone convergence implies that sf(∞) is closed [37] and semiboundedness follows from that of s(ν) f for any ν > 0. All these properties hold then for tf (∞) as well, via the identity sf (U ψ) = tf (ψ) valid for all ψ ∈ Q(tf ), and thus give rise to a semibounded self-adjoint operator Tf = U † Sf(∞) U on the closure Q(tf ) ⊂ L2a (gλ). By the monotone convergence of forms the self-adjoint operators associated with s(ν) f converge in the strong resolvent sense [37], which in turn implies strong convergence of the semigroups they generate [34, Theorem S.14]. ✷ THEOREM 6.4. Provided f and g are such that |a|2 ∈ Kloc and υ, f ∈ K± , and the conclusion of Proposition 5.3 holds, then for t > 0 the continuous integral kernel of the semigroup generated by Sf(ν) converges in the limit ν → ∞ pointwise
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2 (D)), to the kernel of the semigroup associated with the generator Sf(∞) on E(L (ν)
(∞)
z ). lim e−t Sf (z, z ) = (e−t Sf E)(z,
(45)
ν→∞
Proof. The proof heavily borrows from the strategy of [9] which is accommodated here to the case of unbounded f . The key to the present generalization is the use of monotone form convergence. Fundamental to the proof is the well-known Feynman–Kac–Itô formula [11] which expresses the continuous integral kernel of (ν)
the semigroup e−t Sf as the path integral t (ν) 1 −t Sf (z, z ) = exp [a(b(s)) db(s) − a(b(s)) db(s)]− e 2 0 {TD >t } t ds(νυ + f )(b(s)) dµ(ν) − z,0;z ,t ,
(46)
0
where in this case it does not matter whether the stochastic integral in the exponent is interpreted in the Itô sense or according to Fisk and Stratonovich [35], because ∂1 a1 + ∂2 a2 = 0. For notational convenience we fix a reference diffusion constant ν0 > 0 and abbreviate for α 0, w ∈ C (ν0 )
ηw(α) (z) := e−t Sαf (z, w).
(47)
As preparation for the main part of the proof we state three properties of ηw(α) : (1) Each ηw(α) is a bounded and continuous function [11, Theorem 4.1] and lies in L2 (D), which follows from the inequality (ν0 )
ηw(α) sup ηw(α) = e−t Sαf 2,∞ ,
(48)
w∈D
(ν)
where the last term is the finite operator norm of e−t Sαf considered as mapping from L2 (D) to L∞ (D) [11, Estimate (2.39)]. (2) The mapping w → ηw(α) is a strongly continuous mapping, because of the (ν0 ) )(w, w ) and due to the continuity of the identity (ηw(α) , ηw(α) ) = exp(−2tSαf kernel [11, Theorem 6.1]. (3) In addition, the mapping α → ηw(α) is also strongly continuous. To see this, we employ to represent the difference of the two integral kernels in (α) (46) (α |ηw (z) − ηw ) (z)|2 dλ(z) as one path integral. Now we can bound the ab t solute value of the path integrand by 2 exp( 0 (−ν0 u + α0 f − ) ds) with some large α0 , which shows that dominated convergence applies in the limit α → α. As the main part of the proof we show that (ν)
(∞)
κ˜ z ) lim e−t Sf (z, z ) = (κ˜ z , e−t Sf E
ν→∞
(49)
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which by an analogue of Corollary 3.6 in connection with Subsection 5.1 consti on L2 (D). tutes the continuous integral kernel for exp(−tSf(∞) )E To see (49), we use the semigroup property and rewrite the integral kernel as scalar product (ν)
(ν−2ν0 ) )ηz 0 e−t Sf (z, z ) = (ηz(ν0 /ν) , exp(−tS(ν−2ν 0 )f/ν
(ν /ν)
)
(50)
which converges in the limit ν → ∞ to (ν)
(∞)
η(0) lim e−t Sf (z, z ) = (ηz(0) , e−t Sf E z ).
ν→∞
(51)
This can be deduced from the strong continuity of ηz(α) in α and the strong convergence stated in Theorem 6.3 together with the uniform boundedness of the (ν−2ν0 ) ) according to the Banach–Steinhaus theorem. operators exp(−tS(ν−2ν 0 )f/ν To finish the proof, we observe that the right-hand side of (51) is an integral
on L2 (D) which is, in addition, continuous in z and z kernel for exp(−tSf(∞) )E and therefore coincides with the right-hand side of (49). The continuity of (51) is
it can be checked that it indeed constitutes an
exp(−tS0(ν0 ) ) = E clear, and with E integral kernel. ✷ COROLLARY 6.5. Combined with the Feynman–Kac–Itô formula (46) and the identity (∞)
)(z, z), e−t Tf (z, z ) = (U † e−t Sf EU
(52)
the result (45) provides a probabilistic expression for the continuous integral kernel of the semigroup generated by Tf on Q(tf ) ⊂ L2a (gλ). This is the generalized Daubechies–Klauder formula, which states that for admissible f and g and for t > 0, e−t Tf (z, z ) = lim
ν→∞
(z ) g(z)
t 1 exp [a(b(s)) db(s) − a(b(s)) db(s)]− 2 0 {TD >t } t ds(νυ + f )(b(s)) dµ(ν) − z,0;z ,t .
(53)
0
In particular, the choice f = 0 yields the reproducing kernel of the weighted Bergman space which g characterizes. Remarks 6.6. According to [8, 9], due to the significance of ν the expression for the integral kernel of the Berezin–Toeplitz semigroup in (53) is called an ultradiffusive limit. By rescaling the Brownian bridge as in [9, Equation (11)], one
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Table I. No.
D
a(z)
υ(z)
iz
−2
C
1 π e 2j +1 2 −2j −2 π (1 + |z| )
2(j +1)iz 1+|z|2
− 4(j +1) 2 2
{|z|2 < 1}
2k−1 (1 − |z|2 )2k−2 π
− 4(k−1) 2 2
C
2|z|2k−1 π+(2k) K2k−1 (2|z|)
2(k−1)iz 1−|z|2 Z2k−2 iz Z2k−1
C
1 2 3(a) 3(b)
g(z) −|z|2
(1+|z| ) (1−|z| )
−
2 (Z 2k−2 Z2k−1 + 2 Z2k−1 2 |z|2 (Z2k−2 −
Z2k−1 Z2k−3 ))
may alternatively restate (53) as a long-time limit. Another version of the result pointed out there implies that the integrand in (53) can be re-expressed in terms of a complex Itô stochastic integral. All these versions will be omitted here. Reading from the right to the left, (53) can be interpreted as a quantization formula, which constructs from the functions f and g the semigroup generated by Tf and thereby specifically selects the relevant Hilbert space Q(tf ) on which Tf is properly defined as a self-adjoint operator. In the context of quantization, ln g is interpreted as a Kähler potential on D. To our knowledge, without additional symmetry requirements, the setting considered here still lacks the proof of a correspondence principle [10, 17], for which probabilistic techniques might provide helpful tools. EXAMPLES 6.7. In Table I we revisit the examples from Section 3 again and list the functions a and υ belonging to each weight function g considered there. In the last row we used the abbreviation Zσ := |z|σ Kσ (2|z|). The following remarks examine the validity of Equation (53) for each case. Remarks 6.8. The primary condition for the validity of (53) in a specific situation is that the weight function leads to admissible a and υ. If this is satisfied, any Kato-decomposable symbol f allows a probabilistic expression for the semigroup generated by Tf . By inspection we decide whether each example can be used in the formula (53). For the examples 1 and 2 the function a is continuous and υ is even bounded, hence they admit the probabilistic representation according to (53). Unfortunately, for the Example 3(a) either k = 1 and Proposition 5.3 is not satisfied or the scalar potential υ is not Kato decomposable because of a strong negative singularity towards the boundary of the disc. Therefore the probabilistic representation is not valid in either case. However, the desired monotone form
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convergence can be recovered with the modification of the domain suggested in Remark 5.2 which is associated with Neumann boundary conditions. On the other hand, the vector and scalar potentials a and υ emerging from the Barut–Girardello representation 3(b) are admissible for k > 1, which can be read off from the asymptotics: 1 ∞ −σ −1 −1/t |σ | lim r Kσ (2r) = t e dt r→0 2 0 for σ = 0, and limr→0 K0 (2r)/ ln 2r = −1. Within the setting of Example 1 the identity (53) is in essence a result by [14]. The identity corresponding to Example 2 has already been worked out in [9], where it is also compared to a similar formula derived in [14]. These last two alternatives are among the few known ways to obtain a mathematically well-founded path integral for spin. The weighted Bergman space of Example 3(b) was not previously known to admit a formula of type (53), which can now serve as an alternative to path-integral formulas that do not contain genuine path measures [18, 22].
7. Conclusion In this paper we have tried to indicate the key principles behind the construction of Daubechies and Klauder and have thus derived a natural generalization. As a major spin-off we developed criteria for self-adjointness of Berezin–Toeplitz operators. Once the relation to Schrödinger operators is established, this is an immediate consequence. As to further ramifications, we point out that with a suitable analyticity argument one could obtain from (53) the probabilistic expression for the unitary group e−it Tf which was a primary motivation for [14, 15]. It might also be interesting to investigate random Berezin–Toeplitz operators, for which the probabilistic representation seems to offer an appropriate analytic framework. Finally, it deserves mentioning that the concept of path transformations is also applicable in the context of (53) in order to relate the resolvents of certain Berezin–Toeplitz operators (in preparation). Acknowledgements It is a pleasure to thank Hajo Leschke for his scientific guidance through the initial stage and Simone Warzel for her valuable criticism and participation in the struggle for the clear picture. Thanks are extended to John R. Klauder for encouragement, inspiration and lots of resourceful advice. I am also indebted to Kazuyuki Fujii for drawing my attention to the Barut–Girardello representation. The Studienstiftung des deutschen Volkes is acknowledged for financial support.
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Mathematical Physics, Analysis and Geometry 5: 307–318, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Geometrical Lagrangian for a Supersymmetric Yang–Mills Theory on the Group Manifold M. F. BORGES Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag 7701 Rondebosch, Cape Town, South Africa and UNESP, State University of São Paulo, Department of Computing, 15054-000, São José do Rio Preto, Brazil. e-mail: [email protected] (Received: 12 April 2001; in final form: 14 November 2001) Abstract. Perhaps one of the main features of Einstein’s General Theory of Relativity is that spacetime is not flat itself but curved. Nowadays, however, many of the unifying theories like superstrings on even alternative gravity theories such as teleparalell geometric theories assume flat spacetime for their calculations. This article, an extended account of an earlier author’s contribution, it is assumed a curved group manifold as a geometrical background from which a Lagrangian for a supersymmetric N = 2, d = 5 Yang–Mills – SYM, N = 2, d = 5 – is built up. The spacetime is a hypersurface embedded in this geometrical scenario, and the geometrical action here obtained can be readily coupled to the five-dimensional supergravity action. The essential idea that underlies this work has its roots in the Einstein–Cartan formulation of gravity and in the ‘group manifold approach to gravity and supergravity theories’. The group SYM, N = 2, d = 5, turns out to be the direct product of supergravity and a general gauge group G: G = G ⊗ SU(2, 2/1). Mathematics Subject Classifications (2000): 83E50, 81T13. Key words: group manifold, supergravity, supersymmetry, super Yang–Mills theory.
1. Introduction The understanding of low energy phenomena may indicate the need for field theories with gauge symmetries as a convenient scheme to describe the fundamental interactions between elementary particles. In many ways, physics is dominated by the striking successes of quantum electrodynamics and the trends in the description of fundamental interactions (chromodynamics and Glashow–Weinberg–Salam theory). As a result of this, the theory of gravitation is sometimes required to conform to the principles and fashions prevalent in elementary particle physics. Therefore, it seems logical that all gauge theories could be interpreted in only one fundamental principle. In other words, the different symmetries related to the fundamental interactions at a low level of energy could be the fragments of a higher symmetry, related to interactions at a more elevated level of energy. Supergravity arises naturally in such a picture. Supergravity theories are an embedding of Einstein’s theory of gravity in a broader framework, with two main aims: renormalization (or better, finiteness of quantum gravity) and (super) unification with other interactions.
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In such theories, the internal symmetries (gauge groups of Yang–Mills) and the spacetime symmetries (gauge groups of Poincaré) are unified into one algebraic structure. In this way also the internal symmetries show a geometrical feature that extend the geometrical interpretation of pure gravity. A gauge structure for the theory of gravitation was carried out by Cartan [1] in the twenties and followed by Kibble and Sciama in the sixties in an attempt of treat Einstein’s gravitation as a Yang–Mills theory. The so-called Cartan–Kibble–Sciama gauge formulation of gravity was extended by Regge and Ne’eman [3] and soon after by Regge and collaborators [4], in Turin, in an approach called ‘group manifold approach to supersymmetric theories’. In this article, we will give an extended account of that approach and of previous results obtained by the author in [5] and by Borges and Masalskiene in [6], in the attempt to construct an extended gravity theory of the Yang–Mills type. With this aim, a geometrical Lagrangian for the N = 2, d = 5 supersymmetric Yang–Mills theory, where d stands for the dimension of the spacetime manifold and N is the number of generations of supersymmetry transformations, is built up on the groupmanifold G. The procedure to obtain the Lagrangian in this paper is, in many ways, an application of the general scheme provided by Castellani, D’Auria and Fré [7], and by the fact that the theory is directly coupled to the N = 2, d = 5 supergravity on the group manifold [8], that means G = G ⊗ SU(2, 2/1), where G is a general gauge group. In general, the construction of a Lagrangian for supersymmetric theories is a complicated task. Although a geometrical first-order N = 2 supergravity action in five dimensions was established some time ago [8, 9], a minimal off-shell secondorder supergravity version has only recently been developed [10]. There are well known formulations for d = 4, N = 2 [11, 12] as well as for d = 6, N = 2 [13], and even a geometrical and generic first-order N = 2 Yang–Mills case obtained by Fré et al. [14]. However, a specific N = 2, d = 5 first-order Yang–Mills theory readily coupled to supergravity needs to be worked out. We intend to bridge that gap by constructing such a geometric coupling Yang–Mills action.
2. Geometrical Action for a Supersymmetric Five-Dimensional Yang–Mills Theory The geometrical supersymmetric Yang–Mills theory is said to be ‘impure’, contaminated by the presence of 0-form matter fields. In a ‘pure’ theory, instead, the only fields present are the 1-forms µA , which constitute the so-called pseudoconnection. Geometrical Yang–Mills theories have another feature: they can be naturally coupled to supergravity, if the latter is a pure theory. For instance, the globally supersymmetric N = 2, d = 5 Yang–Mills becomes locally supersymmetric in the geometrical formalism by adjoining the action of the N = 2, d = 5 supergravity which is a ‘pure’ theory of supergravity. Two motivations evidenc-
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ing the importance of supersymmetric Yang–Mills theories within the context of unification, may be cited as follows: (i) their possible connection with other cases (for example, with pure supergravity theories formulated in higher-dimensional spacetimes) through the technique of dimensional reduction, thereby improving the knowledge of the various supersymmetric actions; (ii) the second motivation leads us to quantization: it is well known that when the couplings – matter coupled to supergravity – are just those which arise naturally when a higher extended supergravity theory is considered as a theory with less symmetry, then part of what was the gravitational multiplet appears in a particular way which reflects the larger symmetry of the theory. A general hypothesis for the curvature of the N = 2, d = 5 Yang–Mills theory has then proposed in [5] as the following: dA = F abV a ∧ V b + iλA ∧ m ψA ∧ V m + if σ ∧ A ∧ A ; DλA = mA V m − iF ab ∧ ab A − φa ∧ a A , i Dσ = φa ∧ V a + λA ∧ A ; 2 m ∧ V + iA [a ∧ b] A , DF ab = Gab m
(1)
where V is the Vierbein associated to the graviton, is the graviton field, λ is the Dirac spinorial field, and σ the scalar field. F stands for the curvature associated with the gauge group G. Determination through the analysis of the Bianchi identities of a compatible system of equations for the parameters of the curvature, indicated that the hypothesis made for them is acceptable. The group manifold G of this theory is then determined by the fact that the theory is directly coupled to supergravity, which means G = G ⊗ SU(2, 2/1).
(2)
The theory has also presented a bundle structure where H = G ⊗ SO(1, 4) ⊗ U(1) would be the fibre and the quotient space, G/H , the base space of the principal fibre, identified with the superspace. Besides its symmetry group properties, this theory has presented another important feature. The Bianchi identities are satisfied as both the Dirac equation and the homogeneous Maxwell equations hold [5]: m mA = 0;
G[ab/m] = 0.
(3)
This result is related to the closure of the algebra of supersymmetry transformations ‘on shell’. Finally, through the explicit determination of the curvature by means of the Bianchi identities, we are able to build up from now onwards the geometrical action of this N = 2, d = 5 supersymmetric Yang–Mills theory, which will become readily coupled to the N − 2, d = 5 supergravity. Before getting on with that, however, some basic requirements must be taken into account as preliminary ‘precautions’:
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(1) the action is an integral of 5-forms developed on an arbitrary hypersurface M5 (5 = number of dimensions of spacetime), immersed in the whole manifold G, (2) the action must be stationary relating to the variations of fields, and to the variations of the hypersurface M5 . It implies that the equations of motion interpreted in differential forms and resulting from the variations of the Lagrangian related to the fields of the theory will have validity on the whole manifold, (3) the 5-form Lagrangian is gauge invariant under the group G ⊗ SO(1, 4) ⊗ U(1)
(4)
with a general form given by LYM = A + νA θ A + νAB θ A ∧ θ B ,
(5)
where A , νA , νAB are polynomials with constant coefficients and θ A , θ B are the ‘curvatures’ of the manifold for the ‘purely’ Yang–Mills case. The Lagrangian, LYM will be built up with all the fields present in the theory in first-order form: A, λA , σ, Fab , φ a , Va , A , A . The Hodge operator will not be used. We will assume, for simplicity, throughout the calculations that the group G is Abelian. Such a requirement, however, doesn’t constitute any over-simplification for the theory. In fact, we have that DF = DdA + D[A, A] = DdA,
as D[A, A] = D(( 12 Cθν Aθ ∧ Aν )) = 0.
Consequently, the terms in [A, A] do not make any contribution to the parametrization of the curvatures. (4) LYM is not trivial. That means that if curvatures of supergravity are zero, then the equations of motion must be satisfactorily worked out, having as content both the curvature of the gauge group G and of the covariant derivatives DλA , Dσ , DF ab , whose parametrization is already completed (1). (5) Finally, we would require that the projection of the equations of motion on spacetime will lead us to the equations of Maxwell, Dirac and Klein–Gordon (usually present in a simple version of a Yang–Mills theory in four dimensions). The N = 2, d = 5 Yang–Mills action, AYM , can then be written as LMaxwell + LDirac + LKlein–Gordon + AYM = M5
+ others terms to be determined,
(6)
where LMaxwell , LDirac , LKlein–Gordon are the Lagrangians of Maxwell, Dirac and Klein–Gordon written in first-order formalism and in five dimensions. The Maxwell action is, in second-order formalism, written as √ (7) AMaxwell = Fuv F uv −gd5 X.
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The last, as presented, is useless for our purposes. The fields Fab and A should be treated independently as, for example, in the first-order formalism. In the second order one, Fab is related to A by F ab =
1 (∂a Ab − ∂b Aa ). 2
(8)
Consequently, the problem is reduced to finding out through the Maxwell action, in five dimensions and in first-order formalism, the relation between A and Fab , but making no use of the Hodge operator. That will lead us to a new term to be added to (8) in its first-order version. The so-called first-order version of the Maxwell action (8) is written as (9) AMaxwell = Fab F ab V i ∧ V j ∧ V K ∧ V l ∧ V m εijklm, where V i = V i µ dx µ ; g µν = Vaµ Vbν ηab ; √ det V = −g; ε µνρσ d5 x = dx µ ∧ dx ν ∧ dx ρ ∧ dx σ ∧ dx ; √ d5 x −g = V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm
(10) (11) (12) (13) (14)
and ∧ is the edge product. The new, hypothetical Maxwell action is then taken to be (with the new term mentioned above, added in): dAF ab V c ∧ V d ∧ V e εabcde + AMaxwell = + KF ab F ab V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm.
(15)
The variation of (15) to F ab produces dA ∧ V c ∧ V d ∧ V e εabcde + +2KF ab V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm = 0; dA = Xlm V l ∧ V m + (terms with A ).
(16) (17)
In projecting (16) on the five-dimensional spacetime (= terms in V ∧V ∧V ∧V ∧V ), we obtain that 1 (18) ; Xlm = Flm . K = − 20 There, in spacetime M5 , we find out as desired the correct relation between Fab and A, given by dA = Fab V a ∧ V b = Fµν dx µ dx ν = Dµ Aν dx µ dx ν .
(19)
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Consequently, our Yang–Mills action (partial) assumes the following form AYM = dAF ab V c ∧ V d ∧ V e εabcde + 1 Fab F ab V i ∧ V j ∧ V k ∧ V m εijklm + + − 20 + other terms to be determined.
(20)
The Dirac term in the above Lagrangian, in first-order formalism, may be written as iλA a DλA ∧ V b ∧ V c ∧ V d ∧ V e εabcde .
(21)
In fact the projection of it on the spacetime gives us the usual Dirac Lagrangian: iλA m Dm λA ,
(22)
where DλA = Dm λA V m + (terms with A ).
(23)
The following terms to be found are those related to the Klein–Gordon Lagrangian. The procedure taken is analogous to that for the Maxwell case. In the first-order formalism, the fields σ and φ a are considered as independent fields. The connection between them should arise out of the equation of motion. In the action (20), the following terms corresponding to the Klein–Gordon action will be introduced; 1 a φ φa εijklmV i ∧ V j ∧ V k ∧ V l ∧ V m + AKlein–Gordon = 2 + W φ a dσ V b ∧ V c ∧ V d ∧ V e εabcde .
(24)
The variation of (24) related to φ a in considering Dσ = Zm V m + (terms with A ),
(25)
is such that the projection of this result on the spacetime will give us W = −5
(26)
Za = φa .
(27)
and
Consequently, dσ = φa V a = Da σ V a .
(28)
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The action (24) is then implying that φa is the covariant derivative of σ . With the last steps taken into account, the Lagrangian of Yang–Mills (20) may then be written as AYM = dA F ab ∧ V e ∧ V d ∧ V e εabcde − 1 Fab F ab V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm + − 20 1 + 2 aφa φ a V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm − − 5aφ a dσ V b ∧ V c ∧ V d ∧ V e εabcde + + ibλA a DλA ∧ V b ∧ V c ∧ V d ∧ V e εabcde + + other terms to be added.
(29)
a and b are parameters to be determined. To find out all the other remaining terms of the Yang–Mills action, the startingpoint will be the partial action (29), with the terms of Maxwell, Dirac, and Klein– Gordon already identified. Let us consider (29). The variation of it relating to Fab , leads, after substitution of dA as indicated in (1), to the following equation: Flm V l ∧ V m ∧ V c ∧ V d ∧ V e εabcde + +iλA m A ∧ V m ∧ V c ∧ V d ∧ V e εabcde+ +iσ A ∧ A ∧ V c ∧ V d ∧ V e εabcde + 1 Fab V i ∧ V j ∧ V k ∧ V l εijklm = 0. − 10
(30)
Consequently, to avoid the possibility that the only solution for λA and σ is a trivial one, we must add new terms to the action. These will be constructed only from the fields of the theory and consistent, after their variations related to Fab , φa and λA , in such a way as to make (30) identically zero, but without having zero as a solution. Hypothetically, we will introduce in (29) terms of the following type: −iF ab λA m A ∧ Vm ∧ Vc ∧ Vd ∧ Ve ε abcde ; −iF ab σ A ∧ A ∧ V c ∧ V d ∧ V e εabcde ; 5 iaφ a λA A ∧ V b ∧ V c ∧ V d ∧ V e εabcde ; 2
(31) (32) (33)
cdAλA abA V a ∧ V b ; edσ λA ab A Ve ∧ Vd ∧ Ve ε abcde ; if λA ab λA λA ∧ B ∧ V c ∧ V d ∧ V e εabcde ; igλA λB δAC c ab B ∧ V c ∧ V d ∧ V e εabcde ; ihλA ab λA B c B ∧ Va ∧ Vb ∧ Vc ; ilλA a λB δAC X bcBC ∧ Va ∧ Vb ∧ Vc ,
(34) (35) (36) (37) (38) (39)
where X bcBC = C bc B .
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The final expression resulting from the variation of (29) related to λA with all the terms mentioned above, will be the following: iF ab m A V m ∧ V e ∧ V d ∧ V e εabcde + + 52 iaφ a A V b ∧ V c ∧ V d ∧ V e εabcde + +2ib a mA V m ∧ V b ∧ V c ∧ V d ∧ V e εabcde + +24bF abcdA V a ∧ V b ∧ V c ∧ V d εabcde + +2ibF am m A V b ∧ V c ∧ V d ∧ V e εabcde − −2ibφ a A ∧ V b ∧ V c ∧ V d ∧ V e εabcde − −4bφm am A V b ∧ V c ∧ V d ∧ V e εabcde− −2b a λA B b B ∧ V c ∧ V d ∧ V e εabcde + +cF abcdA V a ∧ V b ∧ V c ∧ V d + +icσ abA B ∧ B ∧ V a ∧ V b − − 18 ciλB δAC C cd B ∧ V m ∧ V r ∧ V s εcdmrs + + 14 cimλB δAC C ab B ∧ V m ∧ Va ∧ Vb − − 12 c bs λB δAC C cd B ∧ V m ∧ V ∧ Vb εmcdrs − − 14 c rs λB δAC C cd B ∧ Vc ∧ V a ∧ V b εdabrs+ + 18 cibc λA B a B ∧ V c ∧ V a ∧ V b + 1 r c λA B m B ∧ V s ∧ V a ∧ V b εmabrs− + 16 1 rs − 16 c λA B ∧ B ∧ V m ∧ V a ∧ V b εmabrs− −eφl ab A V l V e ∧ V d ∧ V e εabcde + + 18 eiλB δAC C ab B ∧ V c ∧ V d ∧ V e εabcde + + 34 ie K λB δAC C ab B ∧ Va ∧ Vb ∧ Vk + + 12 e bm δAC C am B V c ∧ V d ∧ V e εabcde + + 38 eilm λA B c BVc ∧ V l ∧ V m − 1 eb λA B c B Va ∧ Vd ∧ Ve ε abcde + − 16 +2if ab λA B B V e ∧ V d ∧ V e εabcde+ +2igλB δAC C ab B V c ∧ V d ∧ V e εabcde + +2ih ab λA B c B Va ∧ Vb ∧ Vc + +2il c λB B δAC C ab B Vc ∧ Va ∧ Vb + 1 eiab λA B B Vc ∧ Vd ∧ Ve ε abcde = 0. + 16
(40)
From a final analysis of Equation (40), we find the presence of only one term with three , −icσ ab A B ∧ B ∧ V a ∧ V b .
(41)
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That term will imply further inclusion in the action (40) of another term, again to avoid a trivial solution. This term will be icσ λA ab A B ∧ B ∧ V a ∧ V b .
(42)
Finally, by projecting terms of the same type as in (44), for convenient tangent vectors, one obtains the following set of equations for the parameters: 5 a − 2b = 0, c + 24b = 0, 1 − 4b = 0, 3e − c = 0, 1 2 1 1 c + 38 e + 2h = 0, e + 8b = 0, −2b − 16 c − 16 e = 0, 8 1 1 − 16 c + 16 e + 2f = 0, − 18 c + 18 e + 2g = 0, 3 1 − c + + 2l = 0. (43) 4 4 From these equations unique values for the parameters could be obtained: b = 14 ;
e = −2; g = − 14 ;
a = 15 ;
h = 34 ;
c = −6;
f = − 18 ;
l = 32 .
(44)
Further variations of (35) related to σ and A, with all the new terms previously discussed, would lead us to the introduction of other terms in the action, to avoid triviality. Such terms are imdA ∧ σ ∧ A m A ∧ Vm ,
(45)
pσ 2 A m A B ∧ B ∧ Vm ,
(46)
and
from the variation related to σ , and inAdA ∧ A ∧ A ,
(47)
from the variation related to A. The parameters m, p and n are determined without any incompatibility. Their values are m = −6,
n=
3 2
and
p = −3.
(48)
The Yang–Mills action, taking into account all the terms reported here and the calculated parameters, will be (dAF ab V c ∧ V d ∧ V e εabcde − AYM = M5 1 Fab F ab V i ∧ V j ∧ V k ∧ V l ∧ V m εijklm+ − 20
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1 a + 10 φ φa V i V j ∧ V k ∧ V l ∧ V m εijklm − −φ a dσ V b ∧ V c ∧ V d ∧ V e εabcde + + 14 λA a DλA V b ∧ V c ∧ V d ∧ V e εabcde − − iF ab λA m A V m ∧ V c ∧ V d ∧ V e εabcde − − iF ab σ A A V c ∧ V d ∧ V e εabcde + + 12 iφ a λA A V b ∧ V c ∧ V d ∧ V e εabcde − − 6dAλA abA V a ∧ V b − − 2(dσ )λA ab A V c ∧ V d ∧ V e εabcde − − 6iσ λA abA B B V a ∧ V b − − 18 iλA ab λA B ∧ B ∧ V c ∧ V d ∧ V e εabcde − − 14 iλA λB δAC C ab B ∧ V c ∧ V d ∧ V e εabcde + + 34 iλA ab λA B c B ∧ Va ∧ Vb ∧ Vc + + 32 iλA c λB δAC c ab B ∧ Vc ∧ Va ∧ Vb − − 6idAσ (A m A ) ∧ Vm − 3σ 2 A m A ∧ B ∧ B ∧ Vm + + 32 iA ∧ dA ∧ A ∧ A ).
(49)
3. Conclusions In this paper, the author has assumed a curved group-manifold as the geometrical scenario from which a classical alternative gravity theory of the Yang–Mills type is considered in a fuller account, completing some results obtained in an earlier paper [5]. The spacetime is presented as a hypersurface embedded in this geometrical background. This work is deeply routed in a previous article by Regge and Ne’eman [3]. Nevertheless, here dynamics is controlled by geometry in the sense that first the curvatures and Bianchi identities were established [5, 6], and then the Lagrangian and motion field equations worked out. Using the basic tools of differential geometry, exterior forms and exterior derivative, gravitation and its extension as a Yang–Mills theory is then described over a group manifold G, G = G ⊗ SU(2, 2/1). The group G has the same relationship to the Poincaré group as curved spacetime does to Minkovski spacetime, except that the existence of a metric is now replaced by the use of the vierbein. So far, one of the main motivations standing behind the present work is the close relationship that today connects quantum gravity and Yang–Mills theories in the nonperturbative strings theory scenario, and 11-dimensional supergravity framework, as exemplified by many others such as Lukas et al. [15], Green [16], and in the Gunaydin and Zagermann [17] formulation of d = 5, N = 2 matter coupled supergravity, where the physical theory on spacetime can also be recovered by truncating the supergravity sector. The author expects to present a quantum
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version of the main results here reported, which is still under investigation, in the near future. Acknowledgements It is a great pleasure to thank George Ellis, Brian Hahn and Di Loureiro for their hospitality in Cape Town, South Africa. I should also thank George Ellis for a brief comment on Sciama’s [2] earlier contribution towards describing all forces of nature by means of non-Riemannian geometries, and the possible physical content of those in terms of a Vierbien formalism. References 1.
2.
3. 4. 5. 6.
7. 8. 9.
10. 11. 12. 13. 14.
15.
Cartan, E.: Sur les varietes a connexions affines et la theorie de la relativité generalisée, Ann. Ecole Norm. Sup. 40 (1923), 325; 41 (1924), 1; 42 (1925), 3; Reprinted in Oeuvres completes, vol. 3, Gauthier-Villars, Paris, 1955. English version: On Manifolds with an Affine Connection and the Theory of General Relativity, Bibliopolis, Bologna, Italy, 1986. Kibble, T. W.: Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961), 212; Sciama, D. W.; Les bases physiques de la théorie du champ unifié, Ann. Inst. H. Poincaré 17 (1961), 1. Ne’eman, Y. and Regge, T.: Gauge theory of gravity and supergravity on a group manifold, Riv. Nuovo Cimento 1 (1978), 5. D’Adda, A., D’Auria, R., Fré, P. and Regge, T.: Geometrical formulation of supergravity theories on orthosympletie supergroup manifold, Riv. Nuovo Cimento 3 (1980), 6. Borges, M. F.: Bianchi identities for an N = 2, d = 5 supersymmetric Yang–Mills theory on the group manifold, J. Geom. Phys. 20 (1996), 142. Borges, M. F. and Masalskiene, S. R. M.: Geometric extended gravity theory of Yang–Mills type, In: The Eighth Marcel Grossmann Meeting, Proc. meeting held at the Hebrew Univ. of Jerusalem, 22–27 June 1997, World Scientific, Singapore, 1999. Castellani, L., D’Auria, R. and Fré, P.: Supergravity and Superstrings. A Geometric Perspective, vol. 1, Mathematical Foundations, World Scientific, Singapore, 1991. D’Auria, R., Fré, P., Maina, E. and Regge, T.: Geometrical first order supergravity in 5 spacetime dimensions, Ann. Phys. 135 (1981), 237. D’Auria, R., Fré, P., Maina, E. and Regge, T.: A new group theoretical technique for the analysis of Bianchi identities and its application to the auxiliary field problem of d = 5 supergravity, Ann. Phys. 139 (1982), 93. Zucker, M.: Minimal off-shell supergravity in five dimensions, Nuclear Phys. B 570 (2000), 267. Breitenlohner, P. and Sohnius, M. F.: Superfields, auxiliary fields and tensor calculus for N = 2 extended supergravity, Nuclear Phys. B 165 (1980), 483. De Wit, B., Van Holten, J. and Van Proeyen, S. A.: Transformation rules of N = 2 supergravity multiplets, Nuclear Phys. B 167 (1980), 186. Bergshoeff, E., Sezgin, E. and Salan, A.: Supersymmetric R 2 actions, conformal invariance and the Lorents Chern–Simons term in 6 and 10 dimensions, Nuclear Phys. B 279 (1987), 659. Andrianopoli, L., Bertolini, M., Ceresole, A., D’Auria, R., Ferrara, S., Fré, P. and Magri, T.: N = 2 supergravity and N = 2 super-Yang–Mills theory on general scolar manifolds: simpletic covariance, gaugings and the mommentum map, J. Geom. Phys. 23(2) (1997), 111. Lukas, A., Ovrut, B. A., Stelle, K. S. and Waldran, D.: Heterotic M-theory in five dimensions, hep-th/9806051.
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Green, M. B.: Superstrings, M theory and quantum gravity, Classical Quantum Gravity, Millenium Issue, 16 (1999), A77. Gunaydin, M. and Zagermann, M.: The gauging of five-dimensional N = 2 Maxwell–Einstein supergravity theories coupled to tensor multiplets, Nuclear Phys. B 572 (2000), 1–2, 131.
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Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua A. H. VARTANIAN Department of Mathematics, Winthrop University, Rock Hill, SC 29733, U.S.A. e-mail: [email protected] (Received: 8 January 2002; in final form: 7 August 2002) Abstract. For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leadingorder asymptotics as |t| → ∞ (x/t ∼ O(1)) of solutions (u = u(x, t)) to the Cauchy problem for the defocusing nonlinear Schrödinger equation (Df NLSE), i∂t u + ∂x2 u − 2(|u|2 − 1)u = 0, with finite-density initial data i(1 ∓ 1)θ (1 + o(1)), θ ∈ [0, 2π). u(x, 0) = exp x→±∞ 2 The Df NLSE dark soliton position shifts in the presence of the continuum are also obtained. Mathematics Subject Classifications (2000): Primary: 35Q15, 37K40, 35Q55, 37K15; secondary: 30E20, 30E25, 81U40. Key words: asymptotics, direct and inverse scattering, reflection coefficient, Riemann–Hilbert problems, singular integral equations.
1. Introduction In direct detection systems making use of polarisation-preserving single-mode (PPSM) optical fibres, return-to-zero bright soliton (strictly speaking, soliton-like) pulses, which propagate in the anomalous group velocity dispersion (GVD) regime (wavelengths >1.3 µm in standard telecommunications fibres), have been shown to be effective toward the partial resolution of the deleterious problem of performance degradation caused by, for example, dispersive pulse spreading [1]. For coherent communications systems, nonreturn-to-zero dark soliton pulses, which propagate in the normal GVD regime (wavelengths <1.3 µm) and consist of a rapid dip in the intensity of a broad pulse of a continuous wave background, offer an analogous benefit [2–4].
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A model for dark soliton pulse propagation in PPSM optical fibres in the picosecond time scale, which describes the slowly varying amplitude of the complex field envelope, u = u(x, t), in normalised and dimensionless form, is the Cauchy problem for the defocusing nonlinear Schrödinger equation (Df NLSE) with finite-density, or nonvanishing, initial data [1–4], i∂t u + ∂x2 u − 2(|u|2 − 1)u = 0,
(x, t) ∈ R × R,
i(1 ∓ 1)θ (1 + o(1)), exp 2
u(x, 0) := u0 (x)
=
x→±∞
(1)
where u0 (x) ∈ C∞ (R), θ ∈ [0, 2π ) (see Equation (3)), and o(1) is to be understood in the sense that, l i(1 ∓ 1)θ d k = 0. u0 (x) − exp ∀(k, l) ∈ Z0 × Z0 , |x| x→±∞ dx 2 It is shown in [5] that, for initial data satisfying l i(1 ∓ 1)θ d k |x| = 0, u0 (x) − exp x→±∞ dx 2
(k, l) ∈ Z0 × Z0 ,
the closure of the set of soliton, or reflectionless, potentials of the Df NLSE in the topology of uniform convergence of functions on compact sets of R remains an invariant set of the model ∀t ∈ R and not just for t = 0 (see, also, [6]). When (temporal) dark solitons are launched sufficiently close together in optical fibres, they interact not only through soliton-soliton interactions, but also through soliton-radiation-tail interactions. Such interactions manifest as a jitter in the arrival times of dark solitons, potentially resulting in their shift outside of some predetermined timing window and giving rise to errors in the detected information [4]. Physically, the optical pulse adjusts its width as it propagates along the optical fibre to evolve into a (multi-) dark soliton pulse/mode, and a part, however small, of the pulse energy is shed in the form of an asymptotically decaying dispersive wavetrain, manifesting as a low-level broadband background radiation (a continuum of linear-like radiative waves/modes). Modulo an O(1) position shift due to cummulative interactions with other dark solitons and the (dispersive) continuum, the dark soliton pulse/mode maintains its robust/stable properties. From the physical and theoretical point of view, therefore, it is important to understand how the dark solitons and continuum interact, and to be able to derive an explicit functional form for this process, namely, to study the asymptotics as |t| → ∞ (x/t ∼ O(1)) of solutions to the Cauchy problem for the Df NLSE with finitedensity initial data having a (not the only one possible) decomposition of the form u0 (x) := usol (x) + urad (x), where u0 (x) satisfies the conditions stated heretofore, usol (x) ‘generates’ the multi- or N-dark soliton component of the solution, and urad (x) is the ‘small’ nondark-soliton part giving rise to the dispersive component of the solution. In this paper, the leading- (O(1)) and next-to-leading-order
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
321
(O(|t|−1/2 )) terms of the asymptotic expansion as |t| → ∞ (x/t ∼ O(1)) of the solution to the Cauchy problem for the Df NLSE with finite-density initial data are derived: they represent, respectively, the N-dark soliton component, and the dispersive continuum and nontrivial interaction/overlap of the N-dark solitons with the continuum. Within the framework of the inverse scattering method (ISM) [7–9] (see, also, [10]), it is well known that the Df NLSE is a completely integrable nonlinear evolution equation (NLEE) having a representation as an infinite-dimensional Hamiltonian system [11, 12]. Even though the analysis of completely integrable NLEEs with rapidly decaying, e.g., Schwartz class, initial data on R have received the vast majority of the attention within the ISM framework, there have been a handful of works devoted exclusively to the direct and inverse scattering analysis of completely integrable NLEEs belonging to the ZS-AKNS class with nonvanishing (as |x| → ∞) values of the initial data [13–15] (see, also, [16, 17]). Other, very interesting classes of finite-density-type initial data for completely integrable NLEEs have also been considered [18–32]. Within the ISM framework, the asymptotic analysis of the solution to the Cauchy problem for the Df NLSE with finite-density initial data is divided into two steps: (1) the analysis of the solitonless (pure radiative, or continuous) component of the solution; and (2) the inclusion of the N-dark soliton component via the application of a ‘dressing’ procedure to the solitonless background [33–37]. The complete details of the asymptotic analysis that constitutes stage (1) of the two-step asymptotic paradigm above, which is quite technical and whose results are essential in order to obtain those of the present paper, can be found in [38]: this paper addresses stage (2) of the above programme via the matrix Riemann–Hilbert problem (RHP) approach [7, 12, 39–47]. It is important to note that, to the best of the author’s knowledge as at the time of the presents, the first to obtain the asymptotics of solutions to the Df NLSE for finite-density initial data in the solitonless sector were Its and Ustinov [48, 49]. This paper is organized as follows. In Section 2, the necessary facts from the direct and inverse scattering analysis for the Df NLSE with finite-density initial data are given, the (matrix) RHP analysed asymptotically as |t| → ∞ (x/t ∼ O(1)) is stated, and the results of this paper are summarised in Theorems 2.2.1–2.2.4 (and Corollaries 2.2.1 and 2.2.2). In Section 3, an augmented RHP, which is equivalent to the original one stated in Section 2, is formulated, and it is shown that, as t → +∞, modulo exponentially small terms, the solution of the augmented RHP converges to the solution of an explicitly solvable, model RHP. In Section 4, the model RHPis solved asymptotically as t → +∞, from which the asymptotics of x u(x, t) and ±∞ (|u(x , t)|2 − 1) dx are derived, and, in Appendix A, the – analogous – asymptotic analysis is succinctly reworked for the case when t → −∞. In Appendices B and C, respectively, formulae which are necessary in order to obtain the remaining asymptotic results of this paper are presented, and a panoramic view of the matrix RH theory in the L2 -Sobolev space is given [44–46, 50].
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2. The Riemann–Hilbert Problem and Summary of Results In this section, a synopsis of the direct/inverse spectral analysis for Equation (1) is given, the matrix RHP studied asymptotically as |t| → ∞ (x/t ∼ O(1)) is stated, and the results of this paper are summarised in Theorems 2.2.1–2.2.4. Before doing so, however, it will be convenient to introduce the notation used throughout this work. NOTATIONAL CONVENTIONS
(1) I =
1
is the 2 × 2 identity matrix, 0 0 0= , 0 0 0 1 0 −i , σ2 = , and σ1 = 1 0 i 0 are the Pauli matrices, 0 1 0 0 and σ− = σ+ = 0 0 1 0 0
0 1
σ3 =
1 0 0 −1
are, respectively, the raising and lowering matrices, sgn(x) := 0 if x = 0 and x|x|−1 if x = 0, and R± := {x; ±x > 0}; (2) for a scalar ω and a 2 × 2 matrix ϒ, ωad(σ3 ) ϒ := ωσ3 ϒω−σ3 ; (3) for each segment of an oriented contour D, according to the given orientation, the ‘+’ side is to the left and the ‘−’ side is to the right as one traverses the contour in the direction of orientation, that is, for a matrix Aij (·), i, j ∈ {1, 2}, (Aij (·))± denote the nontangential limits (Aij (z))± :=
lim
z →z z ∈± side of D
Aij (z );
(4) for a matrix Aij (·), i, j ∈ {1, 2}, to have boundary values in the L2 sense on an oriented contour D, it is meant that |A(z ) − (A(z))± |2 |dz| = 0, lim z →z z ∈± side of D
D
where |A(·)| denotes the Hilbert–Schmidt norm, |A(·)| :=
2
1/2 Aij (·) Aij (·)
,
i,j =1
with (•) denoting complex conjugation of (•);
323
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
(5) for 1 p < ∞ and D some point set, p
LM2 (C) (D)
:= f : D → M2 (C); f (·)LpM
2 (C)
(D)
1/p
:=
|f (z)| |dz| p
<∞ ,
D
and, for p = ∞, L∞ M2 (C) (D)
:= max sup |g (z)| < ∞ ; := g: D → M2 (C); g(·)L∞ (D) ij M (C) i,j ∈{1,2} z∈D
2
(6) for D an unbounded domain of R, SC (D) (respectively, SM2 (C) (D)) denotes the Schwartz space on D, namely, the space of all infinitely continuously differentiable (smooth) C-valued (respectively, M2 (C)-valued) functions which together with all their derivatives tend to zero faster than any positive power of | • |−1 as | • | → ∞, that is, l
k d ∞ f (x) SC (D) := C (D) ∩ f : D → C; f (·)k,l := sup x dx x∈R < ∞, (k, l) ∈ Z0 × Z0 and SM2 (C) (D) := F : D → M2 (C); Fij (·) ∈ C∞ (D), i, j ∈ {1, 2}
∩ G: D → M2 (C); Gij (·)k,l l k d := max sup x Gij (x) < ∞, (k, l) ∈ i,j ∈{1,2} x∈R dx ∞ (∗) := Ck0 (∗); Z0 × Z0 , and C∞ 0 k=0
(7) for D an unbounded domain of R,
SC1 (D) := SC (D) ∩ {h(z); h(·)L∞ (D) := sup |h(z)| < 1}; (8) F (·)∩p∈J LpM
2 (C)
(∗)
:=
p∈J
z∈D
F (·)LpM
2 (C)
(∗) , with i ν
card(J ) < ∞;
(9) for (µ, ν ) ∈ R × R, the function (• − µ) : C \ (−∞, µ) → C: • → eiν ln(•−µ) , with the branch cut taken along (−∞, µ) and the principal branch of the logarithm chosen, ln(•−µ) := ln |• − µ|+ i arg(• − µ), arg(• − µ) ∈ (−π, π ); (10) a contour, D, say, which is the finite union of piecewise-smooth, simple, closed curves, is said to be orientable if its complement, C\D, can always be divided into two, possibly disconnected, disjoint open sets ✵+ and ✵− , either of which has finitely many components, such that D admits an orientation
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A. H. VARTANIAN
so that it can either be viewed as a positively oriented boundary D + for ✵+ or as a negatively oriented boundary D − for ✵− [45], i.e., the (possibly disconnected) components of C \ D can be coloured by + or − in such a way that the + regions do not share boundary with the − regions, except, possibly, at finitely many points [46]; (11) for γ a nullhomologous path in a region D ⊂ C,
1 dζ = 0 . int(γ ) := ζ ∈ D \ γ ; indγ (ζ ) := 2π i γ ζ − ζ 2.1. THE RHP FOR THE Df NLSE In this subsection, the main results from the direct/inverse scattering analysis of the Cauchy problem for the Df NLSE are succinctly recapitulated: since the proofs of these results are given in [38], only final results are stated. PROPOSITION 2.1.1. The necessary and sufficient condition for the compatibility of the following linear system (Lax pair), for arbitrary ζ ∈ C, ∂x $(x, t; ζ ) = U(x, t; ζ )$(x, t; ζ ),
(2)
∂t $(x, t; ζ ) = V(x, t; ζ )$(x, t; ζ ),
where U(x, t; ζ ) = −iλ(ζ )σ3 +
0 u u 0
V(x, t; ζ ) = −2i(λ(ζ )) σ3 + 2λ(ζ ) 2
,
0 u u 0
−i
uu − 1 ∂x u uu − 1 ∂x u
σ3 ,
and λ(ζ ) = 12 (ζ + ζ −1 ), with ∂∗ ζ = 0, ∗ ∈ {x, t}, is that u = u(x, t) satisfies the Df NLSE. One proves Proposition 2.1.1 via the isospectral deformation condition (∂∗ ζ = 0, ∗ ∈ {x, t}), and invoking the Frobenius compatibility condition, ∂t ∂x $(x, t; ζ ) = ∂x ∂t $(x, t; ζ ) ⇒ ∂t U(x, t; ζ ) − ∂x V(x, t; ζ )+ + [U(x, t; ζ ), V(x, t; ζ )] = 0, ζ ∈ C, where [A, B] := AB − BA is the matrix commutator (note that tr(U(x, t; ζ )) = tr(V(x, t; ζ )) = 0). Remark 2.1.1. Note that, if u(x, t) is a solution of the Df NLSE with $(x, t; ζ ) the corresponding solution of system (2), $(x, t; ζ )Q(ζ ), with Q(ζ ) ∈ M2 (C), also solves system (2). The ISM analysis for the Df NLSE is based on the direct scattering problem for the (self-adjoint) operator (cf. Proposition 2.1.1) 1 (ζ + ζ −1 ) iu0 (x) D 2 , O := iσ3 ∂x − 1 iu0 (x) (ζ + ζ −1 ) 2
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
325
where u(x, 0) := u0 (x) satisfies u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), with i(1 ∓ 1)θ u0 (±∞) := exp , θ ∈ [0, 2π ) 2 (see Equation (3)), u0 (x) ∈ C∞ (R),
u0 (x) − u0 (±∞) ∈ SC (R± ).
and
DEFINITION 2.1.1. Let u(x, t) be the solution of the Df NLSE with u(x, 0) := u0 (x) where
=
x→±∞
u0 (±∞)(1 + o(1)),
i(1 ∓ 1)θ u0 (±∞) := exp , 2
θ ∈ [0, 2π )
(see Equation (3)), u0 (x) ∈ C∞ (R),
u0 (x) − u0 (±∞) ∈ SC (R± ).
and
Define $ ± (x, 0; ζ ) as the (Jost) solutions of the first equation of system (2), O D $ ± (x, 0; ζ ) = 0, with the following asymptotics: i(1∓1)θ 1 −iζ −1 ± σ3 4 + o(1) e−ik(ζ )xσ3 , e $ (x, 0; ζ ) = x→±∞ iζ −1 1 where k(ζ ) = 12 (ζ − ζ −1 ). COROLLARY 2.1.1. Let u(x, t) be the solution of the Cauchy problem for the Df NLSE and $(x, t; ζ ) the corresponding solution of system (2) with the asymptotics stated in Definition 2.1.1. Then $(x, t; ζ ) satisfies the symmetry reductions σ1 $(x, t; ζ ) σ1 = $(x, t; ζ ) and $(x, t; ζ −1 ) = ζ $(x, t; ζ )σ2 . PROPOSITION 2.1.2. Set ± $11 (ζ ) $ ± (x, 0; ζ ) := ± (ζ ) $21 Then +
$11 (ζ ) + $21 (ζ )
+ $12 (ζ ) + $22 (ζ )
and
and
− $12 (ζ ) − $22 (ζ )
− $11 (ζ ) − $21 (ζ )
± $12 (ζ ) ± $22 (ζ )
.
have analytic continuation to C+ (respectively,
have analytic continuation to C− ), the monodromy (scat-
tering) matrix, T(ζ ), is defined by $ − (x, 0; ζ ) := $ + (x, 0; ζ )T(ζ ),
Im(ζ ) = 0,
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A. H. VARTANIAN
where
T(ζ ) = with
a(ζ ) b(ζ )
b(ζ ) a(ζ )
,
+ − + − (ζ )$11 (ζ ) − $12 (ζ )$21 (ζ ) , a(ζ ) = (1 − ζ −2 )−1 $22 + − + − (ζ ) $21 (ζ ) − $12 (ζ )$11 (ζ ) , b(ζ ) = (1 − ζ −2 )−1 $22 a(ζ −1 ) = a(ζ ),
|a(ζ )|2 − |b(ζ )|2 = 1,
b(ζ −1 ) = −b(ζ ),
and det($ ± (x, 0; ζ ))|ζ =±1 = 0. COROLLARY 2.1.2. Let the reflection coefficient associated with the direct scattering problem for the operator O D be defined by r(ζ ) := b(ζ )/a(ζ ). Then r(ζ −1 ) = −r(ζ ). Remark 2.1.2. Note that, even though a(ζ ) (respectively, a(ζ )) has an analytic continuation off Im(ζ ) = 0 to C+ (respectively, C− ) and is continuous on C+ (respectively, C− ), b(ζ ) does not, in general, have an analytic continuation to C\R. Furthermore, for the finite-density initial data considered here, it is shown in [14] that, using Volterra-type integral representations for the elements of $ ± (x, 0; ζ ) and a successive approximations argument, r(ζ ) ∈ SC (R) (see, also, Part 1 of [12]). LEMMA 2.1.1. Let u(x, t) be the solution of the Cauchy problem for the Df NLSE and $ ± (x, 0; ζ ) the corresponding (Jost) solutions of O D $ ± (x, 0; ζ ) = 0 given in Definition 2.1.1. Then $ ± (x, 0; ζ ) have the following asymptotics: $ − (x, 0; ζ ) = e
ζ →∞
iθ 2 σ3
x i −∞ (|u0 (x )|2 − 1) dx iu0 (x) eiθ + O(ζ −2 ) e−ik(ζ )xσ3 ,
1 I+ ζ
$ + (x, 0; ζ ) x 1 i +∞ (|u0 (x )|2 − 1) dx I+ = ζ →∞ iu0 (x) ζ + O(ζ −2 ) e−ik(ζ )xσ3 , iθ
−iu0 (x) x −i +∞ (|u0 (x )|2 − 1) dx
$ − (x, 0; ζ ) = (ζ −1 σ2 e− 2 σ3 + O(1))e−ik(ζ )xσ3 , ζ →0
$ + (x, 0; ζ ) = (ζ −1 σ2 + O(1))e−ik(ζ )xσ3 . ζ →0
−iu0 (x)e−iθ −i −∞ (|u0 (x )|2 − 1) dx x
+
+
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ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
COROLLARY 2.1.3. The following asymptotics are valid: +∞ iθ 2 −1 −2 2 (|u0 (x )| − 1) dx ζ + O(ζ ) , a(ζ ) = e 1 + i ζ →∞
−∞
− iθ2
a(ζ ) = e ζ →0
(1 + O(ζ )),
r(ζ ) = O(ζ −1 ), ζ →∞
r(ζ ) = O(ζ ); ζ →0
in particular, r(0) = 0. In [38] it is shown that, for u(x, 0) := u0 (x) satisfying u0 (x) =x→±∞ u0 (±∞)× ), θ ∈ [0, 2π ) (see Equation (3)), u0 (x) ∈ (1 + o(1)), with u0 (±∞) := exp( i(1∓1)θ 2 ∞ C (R), and u0 (x) − u0 (±∞) ∈ SC (R± ), σO D := spec(O D ) = σd ∪ σc (σd ∩ σc = ∅), where σd is the finitely denumerable ‘discrete’ spectrum given by σd = +a ∪ +a , where +a := {ςn ; a(ζ )|ζ =ςn = 0, ςn = eiφn , φn ∈ (0, π ), n ∈ {1, 2, . . . , N}}, with +∞ N (ζ − ςn ) ln(1 − |r(µ)|2 ) dµ exp − , a(ζ ) = e (ζ − ςn ) (µ − ζ ) 2π i −∞ n=1 +∞ N ln(1 − |r(µ)|2 ) dµ < 2π, φn − 0 θ = −2 µ 2π −∞ n=1 iθ 2
ζ ∈ C+ , (3)
and +a ∩ +a = ∅ (card(σd ) = 2N), and σc is the ‘continuous’ spectrum given by σc = {ζ ; Im(ζ ) = 0}, with orientation from −∞ to +∞ (card(σc ) = ∞). Furthermore, it is shown in [38] that, for r(ζ ) ∈ SC1 (R) and |r(±1)| = 1, a(s + iε) =
(−s)N exp i θ2 + N n=1 φn + P.V. R\{s}
ε↓0
× (1 + o(1)),
ln(1−|r(µ)|2 ) dµ (µ−s) 2π
(1 − |r(s)|2 )κsgn(s) s ∈ {±1},
×
where P.V. denotes the principal value integral, with κ± real, possibly zero, constants, and (trace identity)
+∞ −∞
(|u(x , t)|2 − 1) dx = −2
N n=1
sin(φn ) −
+∞ −∞
ln(1 − |r(µ)|2 )
dµ . (4) 2π
The ‘inverse part’ of the ISM analysis is invoked by re-introducing the t-dependence, namely, studying the ∂t $(x, t; ζ ) = V(x, t; ζ )$(x, t; ζ ) component of system (2). The scattering map (S) u0 (x) → r(ζ ) = R(u0 (·)), which is a bijection for u0 (x) satisfying the finite-density initial conditions and r(ζ ) ∈ SC1 (R), linearises the Df NLSE flow in the sense that, since a(ζ, t) = a(ζ ) is the ‘generator’ of the integrals of motion and b(ζ, t) = b(ζ ) exp(4ik(ζ )λ(ζ )t) [12],
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A. H. VARTANIAN
r(ζ, t) := b(ζ, t)/a(ζ, t) evolves in the scattering data phase space according to the rule r(ζ, t) = r(ζ ) exp(4ik(ζ )λ(ζ )t). Set [38] − $11 (x,t ;ζ ) + $ (x, t; ζ ) 12 −a(ζ ) , ζ ∈ C+ , $21 (x,t ;ζ ) + $ (x, t; ζ ) 22 a(ζ ) (x, t; ζ ) := 2 − $12 (x,t ;ζ ) + $ (x, t; ζ ) a(ζ ) , ζ ∈ C− , 11 − $22 (x,t ;ζ ) + $21 (x, t; ζ ) a(ζ )
(x, t; ζ ) has the asymptotics [38] with $ ± (x, t; ζ ) the solutions of system (2): 2 (x, t; ζ ) 2 x 1 i +∞ (|u(x , t)|2 − 1) dx I+ = ζ →∞ iu(x, t) ζ + O(ζ −2 ) e−ik(ζ )(x+2λ(ζ )t )σ3 ,
−iu(x, t) −i +∞ (|u(x , t)|2 − 1) dx x
+
(x, t; ζ ) = (ζ −1 σ2 + O(1))e−ik(ζ )(x+2λ(ζ )t )σ3 . 2 ζ →0
LEMMA 2.1.2 ([38]). Let u(x, t) be the solution of the Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) := u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), where i(1 ∓ 1)θ , u0 (±∞) := exp 2 +∞ N ln(1 − |r(µ)|2 ) dµ < 2π, sin(φn ) − 0 θ = −2 µ 2π −∞ n=1 u0 (x) ∈ C∞ (R), and u0 (x) − u0 (±∞) ∈ SC (R± ). Set (x, t; ζ ) exp(ik(ζ )(x + 2λ(ζ )t)σ3 ). m(x, t; ζ ) := 2 Then: (1) the bounded discrete set σd is finite; (2) the poles of m(x, t; ζ ) are simple; (3) the first (respectively, second) column of m(x, t; ζ ) has poles in C+ (respecN tively, C− ) at {ςn }N n=1 (respectively, {ςn }n=1 ); and (4) m(x, t; ζ ): C \ (σd ∪ σc ) → M2 (C) solves the following RHP: (i) m(x, t; ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; (ii) m± (x, t; ζ ) := lim ζ →ζ m(x, t; ζ ) satisfy the jump condition ±Im(ζ )>0
m+ (x, t; ζ ) = m− (x, t; ζ )G(x, t; ζ ),
ζ ∈ R,
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ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
where G(x, t; ζ ) = exp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3 ))
1+r(ζ )r(ζ −1) r(ζ )
r(ζ −1 ) 1
,
and r(ζ ), the reflection coefficient associated with the direct scattering problem for the operator O D , satisfies r(ζ ) =ζ →0 O(ζ ), r(ζ ) =ζ →∞ O(ζ −1 ), r(ζ −1 ) = −r(ζ ), and r(ζ ) ∈ SC1 (R); N (iii) for the simple poles of m(x, t; ζ ) at {ςn}N n=1 and {ςn }n=1 , there exist nilpotent matrices, with degree of nilpotency 2, such that m(x, t; ζ ) satisfies the polar conditions Res(m(x, t; ζ ); ςn ) = lim m(x, t; ζ )gn (x, t)σ− , ζ →ςn
Res(m(x, t; ζ ); ςn ) = σ1 Res(m(x, t; ζ ); ςn ) σ1 ,
n ∈ {1, 2, . . . , N}, n ∈ {1, 2, . . . , N},
where gn (x, t) = gn exp(2ik(ςn )(x + 2λ(ςn )t)), with +∞ iθ ln(1 − |r(µ)|2 ) dµ iθγn × gn := |γn |e (ςn − ςn ) exp − + 2 (µ − ςn ) 2π i −∞ N π ςn − ςk , θγn = ± ; × ςn − ςk 2 k=1 k=n
(iv) det(m(x, t; ζ ))|ζ =±1 = 0; (v) m(x, t; ζ ) =ζ →0 ζ −1 σ2 + O(1); (vi) m(x, t; ζ ) = ζ →∞ I + O(ζ −1 ); ζ ∈C\(σd ∪σc )
(vii) m(x, t; ζ ) possesses the symmetry reductions m(x, t; ζ ) = σ1 m(x, t; ζ ) σ1 and m(x, t; ζ −1 ) = ζ m(x, t; ζ )σ2 . For r(ζ ) ∈ SC1 (R): (i) the RHP for m(x, t; ζ ) formulated above is uniquely as(x, t; ζ ) = m(x, t; ζ ) exp(−ik(ζ )(x + 2λ(ζ )t)σ3 ) ymptotically solvable; and (ii) 2 solves system (2) with u(x, t) := i
lim
ζ →∞ ζ ∈C\(σd ∪σc )
(ζ(m(x, t; ζ ) − I))12
(5)
the solution of the Cauchy problem for the Df NLSE, and
x +∞
(|u(x , t)|2 − 1) dx := −i
lim
ζ →∞ ζ ∈C \(σd ∪σc )
(ζ(m(x, t; ζ ) − I))11 .
(6)
Remark 2.1.3. In this paper, for r(ζ ) ∈ SC1 (R), the solvability of the RHP for m(x, t; ζ ) formulated in Lemma 2.1.2 is proved, via explicit construction, for all sufficiently large |t| (x/t ∼ O(1)): the solvability of the RHP in the solitonless sector, σd ≡ ∅, for r(ζ ) ∈ SC1 (R), as |t| → ∞ and |x| → ∞ such that z0 := x/t ∼ O(1) and ∈ R \ {−2, 0, 2}, was proved in [38].
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2.2. SUMMARY OF RESULTS In this subsection, the results of this work are summarised in Theorems 2.2.1– 2.2.4: before doing so, however, the following preamble is necessary. Recall from Subsection 2.1 that ςn := eiφn , φn ∈ (0, π ), n ∈ {1, 2, . . . , N}. Set ςn := ξn + iηn , where ξn = Re(ςn ) = cos(φn ) ∈ (−1, 1), and ηn = Im(ςn ) = sin(φn ) ∈ (0, 1). Throughout this paper, it is assumed that: (1) ξi = ξj ∀i = j ∈ {1, 2, . . . , N}; and (2) the following ordering (enumeration) for the elements of the discrete spectrum (solitons), σd , is taken, ξ1 > ξ2 > · · · > ξN . Remark 2.2.1. Throughout this paper, the ‘symbols’ cS (♦), c(6, 7, 8), c(z1 , z2 , z3 , z4 ), c(•), and c, appearing in the various error estimates, are to be understood as follows: (1) for ±♦ > 0, cS (♦) ∈ SC (R± ); (2) for ±6 > 0, c(6, 7, 8) ∈ L∞ C (R± × C∗ × C∗ ), where C∗ := C \ {0}; (3) for (z1 , z2 ) ∈ R± × R± , c(z1 , z2 , z3 , z4 ) ∈ ∞ 2 ∗ ∗ L∞ C (R± × C × C ); (4) for ±• > 0, c(•) ∈ LC (D± ), where D+ := (0, 2) ∗ and D− := (−2, 0); and (5) c ∈ C . Even though the symbols cS (♦), c(6, 7, 8), c(z1 , z2 , z3 , z4 ), c(•), and c are not, in general, equal, and should properly be denoted as c1 (·), c2 (·), etc., the simplified notations cS (♦), c(6, 7, 8), c(z1 , z2 , z3 , z4 ), c(•), and c are retained throughout in order to eschew a flood of superfluous notation as well as to maintain consistency with the main theme of this work, namely, to derive explicitly the leading-order asymptotics and the classes to which the errors belong without regard to their precise z0 -dependence. Remark 2.2.2. In Theorems 2.2.1–2.2.4 below, one should keep, everywhere, the upper (respectively, lower) signs as t → +∞ (respectively, t → −∞). THEOREM 2.2.1. For r(ζ ) ∈ SC1 (R), let m(x, t; ζ ) be the solution of the Riemann–Hilbert problem formulated in Lemma 2.1.2. Let u(x, t), the solution of the Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) := u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), where i(1 ∓ 1)θ , u0 (±∞) := exp 2 N +∞ ln(1 − |r(µ)|2 ) dµ < 2π, sin(φn ) − 0 θ = −2 µ 2π −∞ n=1 u0 (x) ∈ C∞ (R), and u0 (x) − u0 (±∞) ∈ SC (R± ), be defined by Equation (5). Then, for θγm = εb π/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 := x/t < −2 and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (0, π )}, √ u(x, t) ν(λ1 ) −i(θ ± (1)+s ± ) ( uC (x, t) + uSC (x, t))+ uS (x, t) + √ =e |t|(λ1 − λ2 ) (z02 + 32)1/4 S ln|t| c (λ1 )c(λ2 , λ3 , λ4 ) cS (λ2 )c(λ1 , λ3 , λ4 ) + , (7) +O (λ1 − λ2 )t λ1 (z02 + 32) λ2 (z02 + 32)
331
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
where
ln(1 − |r(µ)|2 ) dµ , j ∈ {0, 1}, µj 2π −∞ λ2 λ2 +∞ ln(1 − |r(µ)|2 ) dµ − + θ (l) = , l ∈ {0, 1}, µl 2π 0 λ1 +
0
θ (j ) =
+
λ1
λ1 = − 12 (a1 − (a12 − 4)1/2 ),
λ2 = λ−1 1 ,
λ3 = − 12 (a2 − i(4 − a22 )1/2 ),
λ4 = λ3 ,
a1 = 14 (z0 − (z02 + 32)1/2 ),
(8) (9)
(10)
a2 = 14 (z0 + (z02 + 32)1/2 ),
0 < λ2 < λ1 , |λ3 |2 = 1, a1 a2 = −2, s+ = 2
N
φk ,
s− = 2
k=m+1
φk ,
ν(z) = −
k=1
1 ln(1 − |r(z)|2 ), (11) 2π
−2iφm +9± (x,t )
1 + εb εP e , (1 + εb εP e9± (x,t )) N −1 m−1 sin( 1 (φm + φk )) sin( 1 (φm + φk )) 2 2
uS (x, t) = εP = sgn
m−1
k=1 N−m
sin( 12 (φm − φk ))
k=m+1
sin( 12 (φm − φk ))
, = (−1) xm± ), 9 (x, t) = −2 sin(φm )(x + 2t cos(φm ) −
(13) (14)
±
xm±
(12)
N sin( 12 (φm + φk )) ln(|γm |) sgn(m − k) ± ± ln = 2 sin(φm ) k=1 2 sin(φm ) sin( 12 (φm − φk )) λ2 +∞ 0 λ1 dµ ln(1 − |r(µ)|2 ) 1 , (15) + − − ± 2 2 0 (µ − 2µ cos(φm ) + 1) 2π λ1 −∞ λ2 ±
uC (x, t) = ieis (λ1 e∓i(:
± (z ,t )±(2∓1) π ) 0 4
:± (z0 , t) = ± arg r(λ1 ) ± 4
+ λ2 e±i(:
± (z ,t )±(2∓1) π ) 0 4
),
(16)
arg(λ1 − eiφk )−
k∈J ±
− arg ;(iν(λ1 )) ± t (λ1 − λ2 )(z0 + λ1 + λ2 )+ + ν(λ1 ) ln|t| + 3ν(λ1 ) ln(λ1 − λ2 )+ + 12 ν(λ1 ) ln(z02 + 32) ∓ <± (λ1 ) ± 12 <± (0),
(17)
332
A. H. VARTANIAN
<+ (z) = <− (z) =
k∈J +
:=
1 π 1 π
N
−∞ λ2
+ +
λ1
7
λ1
ln|µ − z| d ln(1 − |r(µ)|2 ),
λ2 +∞
0
k=m+1 ,
uSC (x, t) =
0
k∈J −
:=
ln|µ − z|d ln(1 − |r(µ)|2 ),
m−1 k=1
u(k) SC (x, t),
(20)
π ± = −2iεb εP csc(φm ) sin(s ) cos : (z0 , t) ± (2 ∓ 1) × 4 × sinh(9± (x, t)), ± εP (cos(φm )eis + u(2) SC (x, t) = 2iεb π 9± (x,t ) ± ± e , + 2 sin(φm ) sin(s )) cos : (z0 , t) ± (2 ∓ 1) 4 ±
u(1) SC (x, t)
±
u(3) SC (x, t)
4iεb εP λ21 sin(φm ) sin(s ± )e9 (x,t ) = × (λ21 − 2λ1 cos(φm ) + 1)2 π ± ± × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) ± (2 ∓ 1) 4 π , ± (λ1 − λ2 ) sin(φm ) sin :± (z0 , t) ± (2 ∓ 1) 4 ±
2iεb εP λ1 cos(φm )e9 (x,t ) × 2 (λ1 − 2λ1 cos(φm ) + 1) π ± ± − × 2 cos(s − φm ) cos : (z0 , t) ± (2 ∓ 1) 4 π ± ± ∓ − (λ1 + λ2 ) cos(s ) cos : (z0 , t) ± (2 ∓ 1) 4 π , ∓ (λ1 − λ2 ) sin(s ± ) sin :± (z0 , t) ± (2 ∓ 1) 4 ± 4εb π εP sin(φm )e9 (x,t ) ± cos : × (x, t) = − (z , t) ± (2 ∓ 1) u(5) 0 SC (1 − e29± (x,t )) 4
u(4) SC (x, t) =
±
(19)
, ;(·) is the gamma function [51], and
k=1
with
(18)
±
× (e−is + cos(s ± − φm )e−iφm +29 (x,t )), ± 4 εP λ1 sin(φm )e9 (x,t ) (6) × uSC (x, t) = (1 − e29± (x,t ))(λ21 − 2λ1 cos(φm ) + 1) ± ± εP cos(φm )e−iφm +9 (x,t )))× × (e9 (x,t )(1 + εb
333 π + × −2 εP cos(s ± − φm ) cos :± (z0 , t) ± (2 ∓ 1) 4 π ± ± ± + εP (λ1 + λ2 ) cos(s ) cos : (z0 , t) ± (2 ∓ 1) 4 π ± ± + ± εP (λ1 − λ2 ) sin(s ) sin : (z0 , t) ± (2 ∓ 1) 4
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
±
εP e9 (x,t ))× + (1 − εb π × 2iεb sin(s ± − φm ) cos :± (z0 , t) ± (2 ∓ 1) − 4 π ± ± ± − iεb (λ1 + λ2 ) sin(s ) cos : (z0 , t) ± (2 ∓ 1) 4 π ± ± , ± iεb (λ1 − λ2 ) cos(s ) sin : (z0 , t) ± (2 ∓ 1) 4 ±
u(7) SC (x, t)
8λ21 sin2 (φm )e−iφm +29 (x,t ) =− × (1 − e29± (x,t ))(λ21 − 2λ1 cos(φm ) + 1)2 π ± × ((λ1 + λ2 ) cos(φm ) − 2) cos :± (z0 , t) ± (2 ∓ 1) 4 π ± × ± (λ1 − λ2 ) sin(φm ) sin : (z0 , t) ± (2 ∓ 1) 4 ± 1 − εb εP e9 (x,t ) 9± (x,t ) ± ± εP e ) sin(s ) + i cos(s ) . × (1 + εb 1 + εb εP e9± (x,t )
For the conditions stated in the formulation of the theorem, as t → ±∞ and x → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (−π, 0)}, √ ν(ℵ4 ) −i(ψ ± (1)+s ± ) (uC (x, t)+ uS (x, t) + √ u(x, t) = −e |t|(ℵ3 − ℵ4 ) (z02 + 32)1/4 S c (ℵ3 )c(ℵ4 , ℵ1 , ℵ2 ) cS (ℵ4 )c(ℵ3 , ℵ1 , ℵ2 ) + × + uSC (x, t)) + O |ℵ3 |(z02 + 32) |ℵ4 |(z02 + 32) ln|t| , (21) × (ℵ3 − ℵ4 )t where
ℵ4
ln(1 − |r(µ)|2 ) dµ , j ∈ {0, 1}, µj 2π ℵ3 −∞ ℵ3 +∞ ln(1 − |r(µ)|2 ) dµ − , l ∈ {0, 1}, + ψ (l) = µl 2π ℵ4 0 +
ψ (j ) =
+
0
(22) (23)
334
A. H. VARTANIAN
ℵ1 = − 12 (a1 − i(4 − a12 )1/2 ),
ℵ2 = ℵ1 ,
ℵ3 = − 12 (a2 − (a22 − 4)1/2 ),
ℵ4 = ℵ−1 3 ,
(24)
ℵ4 < ℵ3 < 0, |ℵ1 |2 = 1, ±
1 + εb εP e−2iφm +✵ (x,t ) , (25) (1 + εb εP e✵± (x,t )) N −1 m−1 (− sin( 1 (φm + φk ))) (− sin( 1 (φm + φk ))) 2 2
uS (x, t) = εP = sgn
k=1 m−1
sin( 12 (φm − φk ))
k=m+1
sin( 12 (φm − φk ))
= (−1) , ✵ (x, t) = −2 sin(φm )(x + 2t cos(φm ) − xm± ), ±
xm±
N sin( 12 (φm + φk )) ln(|γm |) sgn(m − k) ± ± ln = 2 sin(φm ) k=1 2 sin(φm ) sin( 12 (φm − φk )) ℵ4 0 ℵ3 +∞ dµ ln(1 − |r(µ)|2 ) 1 , + − − ± 2 2 −∞ (µ + 2µ cos(φm ) + 1) 2π ℵ3 0 ℵ4 ±
± (z ,t )±(2∓1) π 0 4
uC (x, t) = ieis (ℵ3 e±i(2
2± (z0 , t) = ± arg r(ℵ4 ) ± 4
) + ℵ4 e∓i(2± (z0 ,t )±(2∓1) π4 ) ),
(26) (27)
(28)
(29)
arg(ℵ4 + eiφk ) − arg ;(iν(ℵ4 ))±
k∈J ±
± t (ℵ4 − ℵ3 )(z0 + ℵ3 + ℵ4 ) + ν(ℵ4 ) ln|t|+ + 3ν(ℵ4 ) ln(ℵ3 − ℵ4 ) + 12 ν(ℵ4 ) ln(z02 + 32) ∓ >± (ℵ4 )± ± 12 >± (0), 1 > (z) = π +
1 > (z) = π −
ℵ4
−∞ ℵ3 ℵ4
+ +
0
(30)
ℵ3 +∞
ln|µ − z| d ln(1 − |r(µ)|2 ),
(31)
ln|µ − z| d ln(1 − |r(µ)|2 ),
(32)
0
and uSC (x, t) =
7 k=1
u(k) SC (x, t),
(33)
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
with
π ± ± × (x, t) = 2iε ε csc(φ ) sin(s ) cos 2 (z , t) ± (2 ∓ 1) u(1) b P m 0 SC 4 × sinh(✵± (x, t)), ±
is u(2) SC (x, t) = −2iεb εP (cos(φm )e +
π ✵± (x,t ) ± , e + 2 sin(φm ) sin(s )) cos 2 (z0 , t) ± (2 ∓ 1) 4 ±
±
u(3) SC (x, t)
4iεb εP ℵ24 sin(φm ) sin(s ± )e✵ (x,t ) = × (ℵ24 + 2ℵ4 cos(φm ) + 1)2 π ± × ((ℵ4 + ℵ3 ) cos(φm ) + 2) cos 2± (z0 , t) ± (2 ∓ 1) 4 π ± (ℵ4 − ℵ3 ) sin(φm ) sin 2± (z0 , t) ± (2 ∓ 1) , 4 ±
2iεb εP ℵ4 cos(φm )e✵ (x,t ) × = (ℵ24 + 2ℵ4 cos(φm ) + 1) π ± ± + × 2 cos(s − φm ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π ± ± ± + (ℵ4 + ℵ3 ) cos(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π ± ± , ± (ℵ4 − ℵ3 ) sin(s ) sin 2 (z0 , t) ± (2 ∓ 1) 4 ± 4εb εP sin(φm )e✵ (x,t ) π (5) ± cos 2 (z0 , t) ± (2 ∓ 1) × uSC (x, t) = 4 (1 − e2✵± (x,t ))
u(4) SC (x, t)
±
±
× (e−is + cos(s ± − φm )e−iφm +2✵ (x,t )), ± 4εP ℵ4 sin(φm )e✵ (x,t ) (6) × uSC (x, t) = − (1 − e2✵± (x,t ))(ℵ24 + 2ℵ4 cos(φm ) + 1) ± ± × (e✵ (x,t )(1 + εb εP cos(φm )e−iφm +✵ (x,t )))× π + × 2εP cos(s ± − φm ) cos 2± (z0 , t) ± (2 ∓ 1) 4 π ± ± ± + εP (ℵ4 + ℵ3 ) cos(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π − ± εP (ℵ4 − ℵ3 ) sin(s ± ) sin 2± (z0 , t) ± (2 ∓ 1) 4 ± (x,t )
− (1 − εb εP e✵
)×
335
336
A. H. VARTANIAN
π ± ± × 2iεb sin(s − φm ) cos 2 (z0 , t) ± (2 ∓ 1) + 4 π ± ± ∓ + iεb (ℵ4 + ℵ3 ) sin(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π ± ± , ∓ iεb (ℵ4 − ℵ3 ) cos(s ) sin 2 (z0 , t) ± (2 ∓ 1) 4 ±
u(7) SC (x, t)
8ℵ24 sin2 (φm )e−iφm +2✵ (x,t ) =− × (1 − e2✵± (x,t ))(ℵ24 + 2ℵ4 cos(φm ) + 1)2 π ± ± × ((ℵ4 + ℵ3 ) cos(φm ) + 2) cos 2 (z0 , t) ± (2 ∓ 1) 4 π × ± (ℵ4 − ℵ3 ) sin(φm ) sin 2± (z0 , t) ± (2 ∓ 1) 4 ± 1 − εb εP e✵ (x,t ) ✵± (x,t ) ± ± ) sin(s ) + i cos(s ) . × (1 + εb εP e 1 + εb εP e✵± (x,t )
THEOREM 2.2.2. For r(ζ ) ∈ SC1 (R), let m(x, t; ζ ) be the solution of the Riemann–Hilbert problem formulated in Lemma 2.1.2. Let u(x, t), the solution of the Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) := u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), where i(1 ∓ 1)θ , u0 (±∞) := exp 2 +∞ N ln(1 − |r(µ)|2 ) dµ < 2π, sin(φn ) − 0 θ = −2 µ 2π −∞ n=1 u0 (x) ∈ C∞ (R), x and u0 (x)− u0 (±∞) ∈ SC (R± ), be defined by Equation (5), and +∞ (|u(x , t)|2 − 1) dx be defined by Equation (6). Let ? ∈ {±1}. Then, for θγm = εb π/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 < −2 and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (0, π )}, x (|u(x , t)|2 − 1) dx sgn(?)∞ = S?±
?± + E S (x, t)+ +H √ ν(λ1 ) C (x, t) + E SC (x, t))+ (E +√ |t|(λ1 − λ2 ) (z02 + 32)1/4 S ln|t| c (λ1 )c(λ2 , λ3 , λ4 ) cS (λ2 )c(λ1 , λ3 , λ4 ) + , (34) +O (λ1 − λ2 )t λ1 (z02 + 32) λ2 (z02 + 32)
337
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
where
! N 2 k=m+1 sin(φk ), ? = +1, = ? = −1, −2 m k=1 sin(φk ), ! m−1 2 k=1 sin(φk ), ? = +1, S?− = N −2 k=m sin(φk ), ? = −1,
+
− θ (0), θ (0), ? = +1, + − H? = H? = − −θ (0), ? = −1, −θ + (0), S?+
(35)
? = +1, ? = −1,
(36)
±
εP sin(φm )e9 (x,t ) S (x, t) = 2εb , E (1 + εb εP e9± (x,t )) π ± ± , EC (x, t) = −2 cos(s ) cos : (z0 , t) ± (2 ∓ 1) 4
(37) (38)
and SC (x, t) = E
7
(k) SC (x, t), E
(39)
k=1
with (1) SC (x, t) = E
(2) SC (x, t) = E
±
8λ21 sin2 (φm ) cos(s ± )e29 (x,t ) × (1 + εb εP e9± (x,t ))2 (λ21 − 2λ1 cos(φm ) + 1)2 π ± ± × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) ± (2 ∓ 1) 4 π ± , ± (λ1 − λ2 ) sin(φm ) sin : (z0 , t) ± (2 ∓ 1) 4 ±
4λ1 εP sin(φm )e9 (x,t ) × ± (1 − e29 (x,t ))(λ21 − 2λ1 cos(φm ) + 1) ± × 2( εP cos(φm ) sin(s ± − φm )e9 (x,t ) − εb sin(s ± ))× π ± − × cos : (z0 , t) ± (2 ∓ 1) 4 ± (x,t )
εP cos(φm ) sin(s ± )e9 − (λ1 + λ2 )( π ± ± × cos : (z0 , t) ± (2 ∓ 1) 4
± (x,t )
εP cos(φm ) cos(s ± )e9 ± (λ1 − λ2 )( π , × sin :± (z0 , t) ± (2 ∓ 1) 4
− εb sin(s ± + φm ))×
− εb cos(s ± + φm ))×
338
A. H. VARTANIAN
(3) SC (x, t) = E
±
εP λ1 cos(φm )e9 (x,t ) 2εb × (λ21 − 2λ1 cos(φm ) + 1) π ± ± − × 2 cos(s − φm ) cos : (z0 , t) ± (2 ∓ 1) 4 π ± ± ∓ − (λ1 + λ2 ) cos(s ) cos : (z0 , t) ± (2 ∓ 1) 4 π , ∓ (λ1 − λ2 ) sin(s ± ) sin :± (z0 , t) ± (2 ∓ 1) 4 ±
4εb εP λ21 sin(φm ) sin(s ± )e9 (x,t ) (4) SC (x, t) = × E (λ21 − 2λ1 cos(φm ) + 1)2 π ± ± × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) ± (2 ∓ 1) 4 π , ± (λ1 − λ2 ) sin(φm ) sin :± (z0 , t) ± (2 ∓ 1) 4 π (5) ± ± × εP csc(φm ) sin(s ) cos : (z0 , t) ± (2 ∓ 1) ESC (x, t) = −2εb 4 × sinh(9± (x, t)), 4 sin(φm ) sin(s ± − φm ) π 29± (x,t ) (6) ± cos : (z0 , t) ± (2 ∓ 1) e , ESC (x, t) = 4 (1 − e29± (x,t )) π 9± (x,t ) (7) ± ± e εP cos(s − φm ) cos : (z0 , t) ± (2 ∓ 1) , ESC (x, t) = 2εb 4
εP , 9± (x, t), and :± (z0 , t) given in Theorem 2.2.1, and θ ± (·), {λn }4n=1 , s ± and ν(·), Equations (8)–(9), (10), (11), (13), (14)–(15), and (17)–(19), respectively. For the conditions stated in the formulation of the theorem, as t → ±∞ and x → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (−π, 0)},
x
(|u(x , t)|2 − 1) dx
sgn(?)∞ = S?±
+ H?± + ES (x, t)+ √ ν(ℵ4 ) (EC (x, t) + ESC (x, t))+ +√ |t|(ℵ3 − ℵ4 ) (z02 + 32)1/4 S ln|t| c (ℵ3 )c(ℵ4 , ℵ1 , ℵ2 ) cS (ℵ4 )c(ℵ3 , ℵ1 , ℵ2 ) + , +O (ℵ − ℵ )t 2 2 3 4 |ℵ3 |(z0 + 32) |ℵ4 |(z0 + 32) (40)
339
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
where
? = +1, −2 m k=1 sin(φk ), = N 2 k=m+1 sin(φk ), ? = −1, ! −2 N k=m sin(φk ), ? = +1, − S? = m−1 ? = −1, 2 k=1 sin(φk ),
−
+ ψ (0), ψ (0), ? = +1, + − H? = H? = + −ψ (0), ? = −1, −ψ − (0), !
S?+
(41)
? = +1, ? = −1,
(42)
±
2εb εP sin(φm )e✵ (x,t ) , (1 + εb εP e✵± (x,t )) π ± ± , EC (x, t) = 2 cos(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4
ES (x, t) =
(43) (44)
and ESC (x, t) =
7
(k) (x, t), ESC
(45)
k=1
with ±
(1) (x, t) = ESC
8ℵ24 sin2 (φm ) cos(s ± )e2✵ (x,t ) × (1 + εb εP e✵± (x,t ))2 (ℵ24 + 2ℵ4 cos(φm ) + 1)2 π ± ± × ((ℵ4 + ℵ3 ) cos(φm ) + 2) cos 2 (z0 , t) ± (2 ∓ 1) 4 π ± , ± (ℵ4 − ℵ3 ) sin(φm ) sin 2 (z0 , t) ± (2 ∓ 1) 4 ±
(2) (x, t) ESC
4ℵ4 εP sin(φm )e✵ (x,t ) × = (1 − e2✵± (x,t ))(ℵ24 + 2ℵ4 cos(φm ) + 1) ± × 2(εP cos(φm ) sin(s ± − φm )e✵ (x,t ) − εb sin(s ± ))× π ± + × cos 2 (z0 , t) ± (2 ∓ 1) 4 ± (x,t )
+ (ℵ4 + ℵ3 )(εP cos(φm ) sin(s ± )e✵ π ± ∓ × cos 2 (z0 , t) ± (2 ∓ 1) 4
± (x,t )
∓ (ℵ4 − ℵ3 )(εP cos(φm ) cos(s ± )e✵ π , × sin 2± (z0 , t) ± (2 ∓ 1) 4
− εb sin(s ± + φm ))×
− εb cos(s ± + φm ))×
340
A. H. VARTANIAN ±
2εb εP ℵ4 cos(φm )e✵ (x,t ) × (ℵ24 + 2ℵ4 cos(φm ) + 1) π ± ± + × 2 cos(s − φm ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π ± ± ± + (ℵ4 + ℵ3 ) cos(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4 π , ± (ℵ4 − ℵ3 ) sin(s ± ) sin 2± (z0 , t) ± (2 ∓ 1) 4
(3) (x, t) = − ESC
±
4εb εP ℵ24 sin(φm ) sin(s ± )e✵ (x,t ) × (ℵ24 + 2ℵ4 cos(φm ) + 1)2 π ± ± × ((ℵ4 + ℵ3 ) cos(φm ) + 2) cos 2 (z0 , t) ± (2 ∓ 1) 4 π , ± (ℵ4 − ℵ3 ) sin(φm ) sin 2± (z0 , t) ± (2 ∓ 1) 4 π (5) ± ± × ESC (x, t) = −2εb εP csc(φm ) sin(s ) cos 2 (z0 , t) ± (2 ∓ 1) 4 × sinh(✵± (x, t)), 4 sin(φm ) sin(s ± − φm ) π 2✵± (x,t ) (6) ± cos 2 (z0 , t) ± (2 ∓ 1) e , ESC (x, t) = − 4 (1 − e2✵± (x,t )) π ✵± (x,t ) (7) ± ± e , ESC (x, t) = 2εb εP cos(s − φm ) cos 2 (z0 , t) ± (2 ∓ 1) 4 (4) (x, t) = − ESC
and ψ ± (·), {ℵn }4n=1 , εP , ✵± (x, t), and 2± (z0 , t) given in Theorem 2.2.1, Equations (22)–(23), (24), (26), (27)–(28), and (30)–(32), respectively. One important application of the asymptotic results obtained in this paper is related to the so-called N-dark soliton scattering, namely, the explicit calculation of the nth dark soliton position shift in the presence of the (nontrivial) continuous spectrum. Note that, unlike bright solitons of the focusing NLSE (with rapidly decaying, in the sense of Schwartz, initial data), which undergo both position and phase shifts [7, 12, 52], dark solitons of the Df NLSE (for the finite-density initial data considered here) only undergo a position shift [13]. This leads to the following (see, also, Corollary 2.2.2) COROLLARY 2.2.1. Set xn+ − xn− + xn :=
and
+xn := xn+ − xn− ,
n ∈ {1, 2, . . . , N}.
As t → ±∞ and x → ∓∞ such that z0 := x/t < −2 and (x, t) ∈ {(x, t); x + 2t cos(φn ) = O(1), φn ∈ (0, π )},
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
341
N sin( 12 (φn + φk )) sgn(n − k) + ln + xn = 1 sin(φ ) sin( (φ − φ )) n n k 2 k=1 λ2 +∞ 0 λ1 dµ ln(1 − |r(µ)|2 ) , + − − + 2 (µ − 2µ cos(φn ) + 1) 2π λ1 −∞ 0 λ2 and, as t → ±∞ and x → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x + 2t cos(φn ) = O(1), φn ∈ (−π, 0)}, N sin( 12 (φn + φk )) sgn(n − k) + ln +xn = 1 sin(φ ) sin( (φ − φ )) n n k 2 k=1 ℵ4 0 ℵ3 +∞ dµ ln(1 − |r(µ)|2 ) . + − − + 2 (µ + 2µ cos(φn ) + 1) 2π ℵ3 0 −∞ ℵ4 Proof. Follows from the definition of + xn and +xn , and Theorem 2.2.1, Equations (15) and (28). ✷ THEOREM 2.2.3. For r(ζ ) ∈ SC1 (R), let m(x, t; ζ ) be the solution of the Riemann–Hilbert problem formulated in Lemma 2.1.2. Let u(x, t), the solution of the Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) := u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), where i(1 ∓ 1)θ , u0 (±∞) := exp 2 +∞ N ln(1 − |r(µ)|2 ) dµ < 2π, sin(φn ) − 0 θ = −2 µ 2π −∞ n=1 u0 (x) ∈ C∞ (R), and u0 (x) − u0 (±∞) ∈ SC (R± ), be defined by Equation (5). Then, for θγm = εb π/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 := x/t ∈ (−2, 0) and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (0, π )}, ± εP e−2iφm +98 (x,t )) −i(@ ± (1)+s ± ) (1 + εb + u(x, t) = e ± (1 + εb εP e98 (x,t )) min {sin(φk )| cos(φk )−cos(φm )|} −4|t | k=m∈{1,2,...,N} , (46) +O e εP , respectively, are given in Theorem 2.2.1, Equations (11) and (13), where s ± and 0 ln(1 − |r(µ)|2 ) dµ + , @ (j ) = µj 2π −∞ (47) +∞ ln(1 − |r(µ)|2 ) dµ − , j ∈ {0, 1}, @ (j ) = µj 2π 0 ± xm,8 ), (48) 9± 8 (x, t) = −2 sin(φm )(x + 2t cos(φm ) −
342
A. H. VARTANIAN
and ± xm,8
sin( 12 (φm + φk )) ± ln =± 1 2 sin(φ ) sin( (φ − φ )) m m k 2 k=1 +∞ 0 1 dµ ln(|γm|) ln(1 − |r(µ)|2 ) − ± + , (49) 2 − 2µ cos(φ ) + 1) 2π 2 0 (µ 2 sin(φm ) m −∞ N sgn(m − k)
and, as t → ±∞ and x → ±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (−π, 0)}, −i(@ ± (1)+s ± )
u(x, t) = −e
±
(1 + εb εP e−2iφm +97 (x,t )) ±
(1 + εb εP e97 (x,t ))
+
−4|t | min {| sin(φ ) cos(φ )−cos(φ )|} m k k k=m∈{1,2,...,N} , +O e
(50)
where εP is given in Theorem 2.2.1, Equation (26), ± 9± 7 (x, t) = −2 sin(φm )(x + 2t cos(φm ) − xm,7 ),
(51)
and ± xm,7
sin( 12 (φm + φk )) ± ln =± 1 2 sin(φ ) sin( (φ − φ )) m m k 2 k=1 0 +∞ dµ ln(|γm |) ln(1 − |r(µ)|2 ) 1 + . (52) − ± 2 2 −∞ (µ + 2µ cos(φm ) + 1) 2π 2 sin(φm ) 0 N sgn(m − k)
THEOREM 2.2.4. For r(ζ ) ∈ SC1 (R), let m(x, t; ζ ) be the solution of the Riemann–Hilbert problem formulated in Lemma 2.1.2. Let u(x, t), the solution of the Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) := u0 (x) =x→±∞ u0 (±∞)(1 + o(1)), where i(1 ∓ 1)θ , u0 (±∞) := exp 2 +∞ N ln(1 − |r(µ)|2 ) dµ < 2π, sin(φn ) − 0 θ = −2 µ 2π −∞ n=1 u0 (x) ∈ C∞ (R), x and u0 (x)− u0 (±∞) ∈ SC (R± ), be defined by Equation (5), and +∞ (|u(x , t)|2 − 1) dx be defined by Equation (6). Let ? ∈ {±1}. Then, for θγm = εb π/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 ∈ (−2, 0) and
343
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
(x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (0, π )}, x (|u(x , t)|2 − 1) dx sgn(?)∞ ±
S?±
=
+
± H8,?
+
2εb εP sin(φm )e98 (x,t ) ±
(1 + εb εP e98 (x,t ))
+
−4|t | min {sin(φk )| cos(φk )−cos(φm )|} k=m∈{1,2,...,N} , +O e
(53)
?± are given in Theorem 2.2.2, where εP is given in Theorem 2.2.1, Equation (13), S ± Equation (35), 98 (x, t) are given in Theorem 2.2.3, Equations (48)–(49),
+
− ? = +1, ? = +1, @ (0), @ (0), + − H8,? = H8,? = (54) − + −@ (0), ? = −1, −@ (0), ? = −1, and @ ± (·) are given in Theorem 2.2.3, Equation (47), and, as t → ±∞ and x → ±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (−π, 0)}, x (|u(x , t)|2 − 1) dx sgn(?)∞ ±
=
S?±
+
± H7,?
+
2εb εP sin(φm )e97 (x,t ) ±
(1 + εb εP e97 (x,t ))
+
min {| sin(φk ) cos(φk )−cos(φm )|} −4|t | k=m∈{1,2,...,N} , +O e
(55)
where εP is given in Theorem 2.2.1, Equation (26), S?± are given in Theorem 2.2.2, Equation (41), 9± 7 (x, t) are given in Theorem 2.2.3, Equations (51)–(52), and
−
+ @ (0), @ (0), ? = +1, ? = +1, + − H7,? = (56) H7,? = + − −@ (0), ? = −1, −@ (0), ? = −1. COROLLARY 2.2.2. Set + − xn,8 − xn,8 +xn8 :=
and
+ − +xn7 := xn,7 − xn,7 ,
n ∈ {1, 2, . . . , N}.
As t → ±∞ and x → ∓∞ such that z0 := x/t ∈ (−2, 0) and (x, t) ∈ {(x, t); x+ 2t cos(φn ) = O(1), φn ∈ (0, π )}, +xn8
sin( 12 (φn + φk )) + ln = 1 sin(φ ) sin( (φ − φ )) n n k 2 k=1 +∞ 0 dµ ln(1 − |r(µ)|2 ) , − + 2 0 −∞ (µ − 2µ cos(φn ) + 1) 2π N sgn(n − k)
344
A. H. VARTANIAN
and, as t → ±∞ and x → ±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x + 2t cos(φn ) = O(1), φn ∈ (−π, 0)}, +xn7
sin( 12 (φn + φk )) + ln = 1 sin(φ ) sin( (φ − φ )) n n k 2 k=1 0 +∞ ln(1 − |r(µ)|2 ) dµ − + . 2 + 2µ cos(φ ) + 1) 2π (µ n −∞ 0 N sgn(n − k)
Proof. Follows from the definition of +xn8 and +xn7 , and Theorem 2.2.3, Equations (49) and (52). ✷ Remark 2.2.3. In this paper, the complete details of the asymptotic analysis are presented for the case t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ {(x, t); x + 2t cos(φn ) = O(1), φn ∈ (0, π )}, and the final results for the analogous asymptotic analysis as t → −∞ and x → +∞ such that z0 < −2 and (x, t) ∈ {(x, t); x + 2t cos(φn ) = O(1), φn ∈ (0, π )} are given in Appendix A. The remaining cases are treated similarly, and one uses the results of Appendix B to obtain the corresponding leading-order asymptotic expansions.
3. The Model RHP In this section, the RHP studied asymptotically (as t → +∞) in Section 4, the so-called model RHP, is derived: it is obtained from the (normalised at ∞) RHP for m(x, t; ζ ) formulated in Lemma 2.1.2 via an ingenius method due to Deift et al. [34] (see below). Set m := {(x, t); x + 2t cos(φm ) = O(1), φm ∈ (0, π )}, m ∈ {1, 2, . . . , N}: note that the mth dark soliton ‘trajectory’ in the (x, t)plane, R2 , belongs to m . From Lemma 2.1.2(iii), and the dark soliton ordering adopted in Subsection 2.2, one notes that, as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m : (1) for n = m, gn (x, t) m = O(1); (2) for n < m, gn (x, t) m = O(exp(−4t sin(φn )| cos(φn ) − cos(φm )|)) → 0; and (3) for n > m, gn (x, t) m = O(exp(4t sin(φn )| cos(φn ) − cos(φm )|)) → ∞. Thus (cf. Remark 2.1.3), since the RHP for m(x, t; ζ ) formulated in Lemma 2.1.2 is asymptotically solvable for the (x, t)-sector stated above, one deduces that, along the trajectory of the (arbitrarily fixed) mth dark soliton: (1) for n = m, 0 and Res(m(x, t; ζ ); ςn ) = 00 O(1) ; (2) for n < m, Res(m(x, t; ζ ); ςn ) = O(1) O(1) 0 O(1) 0 → 0 and Res(m(x, t; ζ ); ςn ) = 00 O() → 0, Res(m(x, t; ζ ); ςn ) = O() O() 0 O() where := exp(−4t sin(φn )| cos(φn ) − cos(φm )|); and (3) for n > m, Res(m(x, t; 0 O(−1 ) ∞0 0∞ −1 ) 0 ς ) = → and Res(m(x, t; ζ ); → . ζ ); ςn ) = O( −1 −1 n O( ) 0 ∞0 0 O( ) 0∞ Hence, along the trajectory of the (arbitrarily fixed) mth dark soliton, there are
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
345
exponentially growing polar (residue) conditions for solitons n with n ∈ {m + 1, m + 2, . . . , N}. In a paper dealing with the Toda Rarefaction Problem [34], Deift et al. showed how this problem could be dealt with. Proceeding from the construction of Zhou [44–46] related to the singular RHP (see the synopsis below Theorem C.1.4 in Appendix C), one uses the method of Deift et al. to ‘replace’ the poles which give rise to the exponentially growing residue conditions by jump matrices on mutually disjoint, and disjoint with respect to σc , ‘small’ circles (see [46], Section 2, Remark 2.18, for a discussion about the radii of these circles) in such a way that the jump matrices on these small circles behave like I + exponentially decreasing terms "N (as t → +∞), thus constructing the augmented contour σaugmented := σc ∪ ( n=m+1 ∂(small circles)). Thus, instead of the original RHP, one obtains an augmented (and normalised at ∞) RHP with 2(N − m) fewer poles and 2(N − m) additional circles with jump conditions stated on them. Finally, by ‘removing’ the 2(N − m) small circles from the augmented RHP, one arrives at an asymptotically solvable, equivalent, ‘model’ RHP, in the sense that a solution of the equivalent RHP gives a solution of the augmented RHP and vice versa; in particular, if there are two RHPs, (X1 (λ), υ1 (λ), ;1 ) and (X2 (λ), υ2 (λ), ;2 ), say, with ;2 ⊂ ;1 and υ1 (λ) ;1 \;2 =t →+∞ I + o(1), then, modulo o(1) estimates, their solutions, X1 (λ) and X2 (λ), respectively, are asymptotically equal. Actually, as will be shown below (see Lemma 3.5), the solution of the model RHP approximates, up to terms that are exponentially small (as t → +∞), the solution of the augmented RHP (hence the original RHP). The reason for introducing the factor δ(ζ ) in Lemma 3.1 below is given in Section 4 of [38]. Remark 3.1. For notational convenience, all explicit x, t dependencies are hereafter suppressed, except where absolutely necessary and/or where confusion may arise. LEMMA 3.1. For r(ζ ) ∈ SC1 (R), let m(ζ ): C\(σd ∪σc ) → M2 (C) be the solution of the RHP formulated in Lemma 2.1.2. Set m(ζ ) := m(ζ )(δ(ζ ))−σ3 , where δ(ζ ) = exp
0 −∞
+
λ1
λ2
ln(1 − |r(µ)|2 ) dµ , (µ − ζ ) 2π i
with λ1 and λ2 given in Theorem 2.2.1, Equation (10), δ(ζ )δ(ζ ) = 1, δ(ζ )δ(ζ −1 ) = δ(0), and (δ(·))±1 L∞ (C) := supζ ∈C |(δ(ζ ))±1 | < ∞. Then m(ζ ): C\(σd ∪σc ) → M2 (C) solves the following RHP: (i) m(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ;
346
A. H. VARTANIAN
(ii) m± (ζ ) := lim
ζ →ζ ±Im(ζ )>0
m(ζ ) satisfy the jump condition
m+ (ζ ) = m− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 ))G(ζ ), where
ζ ∈ R,
(1 − r(ζ )r(ζ ))δ− (ζ )(δ+ (ζ ))−1 −r(ζ ) δ− (ζ )δ+ (ζ ) r(ζ )(δ− (ζ )δ+ (ζ ))−1 (δ− (ζ ))−1 δ+ (ζ ) " (iii) m(ζ ) has simple poles in σd = N n=1 ({ςn } ∪ {ςn }) with
G(ζ ) =
Res(m(ζ ); ςn ) = lim m(ζ )gn (δ(ςn ))−2 σ− , ζ →ςn
Res(m(ζ ); ςn ) = σ1 Res(m(ζ ); ςn ) σ1 ,
;
n ∈ {1, 2, . . . , N},
n ∈ {1, 2, . . . , N},
where gn := |gn |eiθgn exp(2ik(ςn )(x + 2λ(ςn )t)), with +∞ sin(φn ) ln(1 − |r(µ)|2 ) dµ × |gn | = 2|γn | sin(φn ) exp 2 −∞ (µ − 2µ cos(φn ) + 1) 2π N sin( 12 (φn + φk )) , × sin( 12 (φn − φk )) k=1 k=n
θg n
+∞ θ π (µ − cos φn ) ln(1 − |r(µ)|2 ) dµ − = θγn + − − 2 2 (µ2 − 2µ cos(φn ) + 1) 2π −∞ N π φk , θγn = ± ; − 2 k=1 k=n
(iv) det(m(ζ ))|ζ =±1 = 0; (v) m(ζ ) =ζ →0 ζ −1 (δ(0))σ3 σ2 + O(1); (vi) m(ζ ) = ζ →∞ I + O(ζ −1 ); ζ ∈C\(σd ∪σc )
(vii) m(ζ ) = σ1 m(ζ ) σ1 and m(ζ −1 ) = ζ m(ζ )(δ(0))σ3 σ2 . Let u(x, t) := i and
x +∞
lim
ζ →∞ ζ ∈C\(σd ∪σc )
(ζ(m(ζ )(δ(ζ ))σ3 − I))12 ,
(|u(x , t)|2 − 1) dx := −i
lim
ζ →∞ ζ ∈C\(σd ∪σc )
(ζ(m(ζ )(δ(ζ ))σ3 − I))11 .
(57)
(58)
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. Proof. The RHP for m(ζ ) (respectively, Equations (57) and (58)) follows from the RHP for m(ζ ) formulated in Lemma 2.1.2 (respectively, Equations (5) and (6)) ✷ upon using m(ζ ) := m(ζ )(δ(ζ ))−σ3 , with δ(ζ ) given in the lemma.
347
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
DEFINITION 3.1. For m ∈ {1, 2, . . . , N} and {ςn }N n=m+1 ⊂ C+ (respectively, N {ςn }n=m+1 ⊂ C− ), define the clockwise (respectively, counter-clockwise) oriented circles Kn := {ζ ; |ζ − ςn | = εnK } (respectively, Ln := {ζ ; |ζ − ςn | = εnL }), with εnK (respectively, εnL ) chosen sufficiently small such that Kn ∩ Kn = Ln ∩ Ln = Kn ∩ Ln = Kn ∩ σc = Ln ∩ σc = ∅ ∀n = n ∈ {m + 1, m + 2, . . . , N}. Remark 3.2. Note that the orientation for Kn (⊂ C+ ) and Ln (⊂ C− ) is consistent with Equation (C.1) (see Appendix C). LEMMA 3.2. For r(ζ ) ∈ SC1 (R), let m(ζ ): C \(σd ∪ σc ) → M2 (C) be the solution of the RHP formulated in Lemma 3.1. Set m(ζ ), " ζ ∈ C \ (σc ∪ ( N n=m+1 (Kn ∪ int(Kn ) ∪ Ln ∪ int(Ln )))), −2 g (δ(ς )) m(ζ ) I − n n σ , − (ζ −ςn ) m 6 (ζ ) := ζ ∈ int(Kn ), n ∈ {m + 1, m + 2, . . . , N}, (δ(ςn ))−2 σ+ , m(ζ ) I + gn (ζ −ς ) n ζ ∈ int(Ln ), n ∈ {m + 1, m + 2, . . . , N}. " "N 6 Then m (ζ ): C \ ((σd \ N n=m+1 ({ςn } ∪ {ςn })) ∪ (σc ∪ ( n=m+1 (Kn ∪ Ln )))) → M2 (C) solves the following RHP: " (i) m 6 (ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C\(σc ∪( N n=m+1 (Kn ∪ Ln ))); m 6 (ζ ) satisfy the jump condition (ii) m6± (ζ ) := lim ζ →ζ ζ ∈± side of σc ∪(∪N (K ∪Ln )) n=m+1 n
6
6
m+ (ζ ) = m− (ζ )υ 6 (ζ ),
N #
ζ ∈ σc ∪
(K n ∪ L n ) ,
n=m+1
where
exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 ))G(ζ ), ζ ∈ R, gn (δ(ςn ))−2 υ 6 (ζ ) = I + (ζ −ςn ) σ− , ζ ∈ Kn , n ∈ {m + 1, m + 2, . . . , N}, I + gn (δ(ςn ))−2 σ , ζ ∈ L , n ∈ {m + 1, m + 2, . . . , N}, (ζ −ςn )
+
n
with G(ζ ) given in Lemma 3.1(ii); " (iii) m 6 (ζ ) has simple poles in σd \ N n=m+1 ({ςn } ∪ {ςn }) with Res(m 6 (ζ ); ςn ) = lim m 6 (ζ )gn (δ(ςn ))−2 σ− , ζ →ςn
Res(m 6 (ζ ); ςn ) = σ1 Res(m 6 (ζ ); ςn ) σ1 , (iv) det(m 6 (ζ ))|ζ =±1 = 0;
n ∈ {1, 2, . . . , m},
n ∈ {1, 2, . . . , m};
348
A. H. VARTANIAN
(v) m 6 (ζ ) =ζ →0 ζ −1 (δ(0))σ3 σ2 + O(1); " "N (vi) as ζ → ∞, ζ ∈ C\((σd \ N n=m+1 ({ςn }∪{ςn }))∪(σc ∪( n=m+1 (Kn ∪ Ln )))), m6 (ζ ) = I + O(ζ −1 ); (vii) m 6 (ζ ) = σ1 m 6 (ζ ) σ1 and m6 (ζ −1 ) = ζ m 6 (ζ )(δ(0))σ3 σ2 . " "N For ζ ∈ C \ ((σd \ N n=m+1 ({ςn } ∪ {ςn })) ∪ (σc ∪ ( n=m+1 (Kn ∪ Ln )))), let u(x, t) := i lim (ζ(m 6 (ζ )(δ(ζ ))σ3 − I))12 , ζ →∞
and
x +∞
(|u(x , t)|2 − 1) dx := −i lim (ζ(m(ζ )(δ(ζ ))σ3 − I))11 . ζ →∞
(59)
(60)
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. Proof. The RHP for m 6 (ζ ) (respectively, Equations (59) and (60)) follows from the RHP for m(ζ ) formulated in Lemma 3.1 (respectively, Equations (57) and (58)) ✷ upon using the definition of m 6 (ζ ) in terms of m(ζ ) given in the lemma. " Remark 3.3. Even though the set (of first-order poles) N n=m+1 ({ςn } ∪ {ςn }), giving rise to the exponentially growing residue conditions, has " been removed from the specification of the RHP and replaced by jump matrices on N n=m+1 (Kn ∪ Ln ), it should be noted that these jump matrices are also exponentially growing (as t → +∞). These lower/upper diagonal, exponentially growing jump matrices are now replaced, via a finite sequence of transformations, by upper/lower diagonal jump matrices which converge to I as t → +∞. " LEMMA 3.3. For m ∈ {1, 2, . . . , N}, let σd := σd \ nn=m+1 ({ςn } ∪ {ςn }), σc := " σc ∪ ( N n=m+1 (Kn ∪ Ln )), where Kn and Ln are given in Definition 3.1, and σO D := σd ∪ σc (σd ∩ σc = ∅). Set 6 $ + −σ3 m (ζ ) N , k=m+1 (dk (ζ )) " ζ ∈ C \ (σc ∪ ( N n=m+1 (int(Kn ) ∪ int(Ln )))), $ m 6 (ζ )(J (ζ ))−1 N − −σ3 , Kn k=m+1 (dk (ζ )) m8 (ζ ) := ζ ∈ int(Kn ), n ∈ {m + 1, m + 2, . . . , N}, $ − −σ3 , m 6 (ζ )(JLn (ζ ))−1 N k=m+1 (dk (ζ )) ζ ∈ int(Ln ), n ∈ {m + 1, m + 2, . . . , N}, where N # ζ − ςn + , ζ ∈ C \ σc ∪ (int(Kn ) ∪ int(Ln )) , dn (ζ ) = ζ − ςn n=m+1
ζ − ς , ζ ∈ int( K ), n n dn− (ζ ) = (ζ − ςn )−1 , ζ ∈ int(Ln ),
349
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
J (ζ ) (∈ SL(2, C)) and JLn (ζ ) (∈ SL(2, C)), respectively, are holomorphic in "N "KNn k=m+1 int(Kk ) and l=m+1 int(Ll ), with JKn (ζ ) $N
dk+ (ζ ) CnK gn (δ(ςn ))−2 k=m+1 d − (ζ ) − (ζ −ςn )2 k=n k
=
$N k=m+1 k=n
(ζ −ςn )
−gn (δ(ςn ))−2 JLn (ζ )
(ζ − ςn )
=
CL
− (ζ −ςn n )2
$N
$
$
(ζ − ςn )
dk− (ζ ) N k=m+1 (dk+ (ζ ))−1 k=n
$
,
$N
,
(ζ −ςn )
−1
−2i
N j=m+1 j=n
= CnL = −4 sin (φn )(gn ) (δ(ςn )) e N sin( 12 (φn + φk )) 2 . × sin( 12 (φn − φk )) k=m+1 2
(dk+ (ζ ))−1 k=m+1 dk− (ζ ) k=n dk− (ζ ) N k=m+1 dk+ (ζ ) k=n
dk+ (ζ ) k=m+1 dk− (ζ ) k=n $N dk− (ζ ) CnL gn (δ(ςn ))−2 $N dk− (ζ ) k=m+1 d + (ζ ) − k=m+1 (d + (ζ ))−1 (ζ −ςn )2 k=n k=n k k
gn (δ(ςn ))−2
and CnK
$N
CnK (ζ −ςn )2
dk− (ζ ) N k=m+1 dk+ (ζ ) k=n
dk+ (ζ ) dk− (ζ )
k=m+1 k=n
(dk+ (ζ ))−1 dk− (ζ )
2
φj
×
k=n
Then m (ζ ): C \ σO D → M2 (C) solves the following (augmented) RHP: 8
(i) m8 (ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; m8 (ζ ) satisfy the following jump conditions, (ii) m8± (ζ ) := lim ζ →ζ ζ ∈±side of σ D O
m8+ (ζ ) = m8− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 ))G8 (ζ ), where G8 (ζ ) =
(1 − r(ζ )r(ζ ))δ− (ζ )(δ+ (ζ ))−1 $ + −2 r(ζ )(δ− (ζ )δ+ (ζ ))−1 N k=m+1 (dk (ζ ))
and
−r(ζ ) δ− (ζ )δ+ (ζ )
ζ ∈ R,
$N
+ 2 k=m+1 (dk (ζ ))
(δ− (ζ ))−1 δ+ (ζ )
CnK 8 m− (ζ ) I + (ζ −ς σ+ , n) ζ ∈ K , n ∈ {m + 1, m + 2, . . . , N}, n 8 m+ (ζ ) = CL 8 m− (ζ ) I + (ζ −ςn n ) σ− , ζ ∈ Ln , n ∈ {m + 1, m + 2, . . . , N};
(iii) m8 (ζ ) has simple poles in σd with Res(m (ζ ); ςn ) = lim m (ζ )gn (δ(ςn )) 8
8
−2
ζ →ςn
n ∈ {1, 2, . . . , m}, Res(m8 (ζ ); ςn ) = σ1 Res(m8 (ζ ); ςn ) σ1 ,
N
(dk+ (ςn ))−2
k=m+1
n ∈ {1, 2, . . . , m};
σ− ,
,
350
A. H. VARTANIAN
(iv) det(m8 (ζ ))|ζ =±1 = 0; $ + σ3 (v) m8 (ζ ) =ζ →0 ζ −1 (δ(0))σ3 ( N k=m+1 (dk (0)) )σ2 + O(1); 8 −1 (vi) m (ζ ) = ζ →∞ I + O(ζ ); ζ ∈C \σ D O
(vii) m (ζ ) = σ1 m8 (ζ ) σ1 and m8 (ζ −1 ) = ζ m8 (ζ )(δ(0))σ3 ( 8
Let
u(x, t) := i lim
ζ m8 (ζ )(δ(ζ ))σ3
ζ →∞ ζ ∈C\σ D O
and
N
$N
+ σ3 k=m+1 (dk (0)) )σ2 .
(dk+ (ζ ))σ3 − I
(61)
,
k=m+1
12
x
(|u(x , t)|2 − 1) dx +∞ := −i lim
ζ →∞ ζ ∈C\σ OD
ζ m8 (ζ )(δ(ζ ))σ3
N
(dk+ (ζ ))σ3 − I
k=m+1
(62)
. 11
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. lemma, one shows that, for Proof. From the definition of m8 (ζ ) given in the " 8 8 8 8 m ∈ {1, 2, . . . , N}, m+ (ζ ) = m− (ζ )υK (ζ ), ζ ∈ N n=m+1 Kn , and m+ (ζ ) = n " 8 8 m− (ζ )υL (ζ ), ζ ∈ N n=m+1 Ln , where n
N gn (δ(ςn ))−2 σ− = (dk+ (ζ ))−σ3 , JKn (ζ ) I + (ζ − ς ) n k=m+1 k=m+1 N N gn (δ(ςn ))−2 8 + σ3 −1 σ+ (JLn (ζ )) (dk (ζ )) (dk− (ζ ))−σ3 . I+ υL (ζ ) = n (ζ − ς ) n k=m+1 k=m+1
8 υK (ζ ) n
N
(dk− (ζ ))σ3
8
8
Now, as in [34], demanding that υK (ζ ) (respectively, υL (ζ )) have the following n
n
8 (ζ ) = I + CnK (ζ − ςn )−1 σ+ upper (respectively, lower) diagonal structure, υK n
(respectively, υL8 (ζ ) = I + CnL (ζ − ςn )−1 σ− ), one arrives at n
JKn (ζ ) = $
dk+ (ζ ) N k=m+1 d − (ζ ) k
−
−2
(δ(ςn )) − gn(ζ −ςn )
JLn (ζ ) = $N dk+ (ζ ) k=m+1 d − (ζ ) k − L C
n − (ζ −ς n)
$N
$N
(dk+ (ζ ))−1 k=m+1 d − (ζ ) k $N dk− (ζ ) k=m+1 d + (ζ ) k
CnK gn (δ(ςn ))−2 (ζ −ςn )2
dk (ζ ) k=m+1 (d + (ζ ))−1 k
$N
(dk+ (ζ ))−1 k=m+1 d − (ζ ) k $N dk− (ζ ) k=m+1 d + (ζ ) k
CnK (ζ −ςn )
dk+ (ζ ) gn (δ(ςn ))−2 $N k=m+1 (ζ −ςn ) dk− (ζ ) − L −2 $N $ dk (ζ ) dk− (ζ ) Cn gn (δ(ςn )) N + 2 −1 k=m+1 d + (ζ ) − k=m+1 (ζ −ς ) (d n k k (ζ ))
, ,
351
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
with det(JKn (ζ )) = det(JLn (ζ )) = 1. Choosing dn± (ζ ) as in the lemma, one shows that Res(JKn (ζ ); ςn ) $ N
=
k=m+1 k=n
dk+ (ζ ) dk− (ζ )
−
CnK gn (δ(ςn ))−2 (ζ −ςn )2
$N k=m+1 k=n
(dk+ (ζ ))−1 dk− (ζ )
ζ =ςn
0
Res(JLn (ζ ); ςn ) 0 $N = 0
dk− (ζ ) k=m+1 dk+ (ζ ) k=n
0
−
CnL gn (δ(ςn ))−2 (ζ −ςn )2
0
$
dk− (ζ ) N k=m+1 (dk+ (ζ ))−1 k=n
,
0 :
ζ =ςn
choosing CnK and CnL as in the lemma, one gets that Res(JKn (ζ ); ςn ) = Res(JLn (ζ ); ςn ) = 0; thus, JKn (ζ ) (respectively, JLn (ζ )) is holomorphic in "N "N n=m+1 int(Kn ) (respectively, n=m+1 int(Ln )). The remainder of the proof follows from Lemma 3.2 and the definition of m8 (ζ ) given in the lemma via straightforward algebraic calculations. ✷ Remark 3.4. One notes from the proof of Lemma 3.3 that, for m ∈ {1, 2, . . . , N}, with ηn := sin(φn ) ∈ (0, 1) and ξn := cos(φn ) ∈ (−1, 1), as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , −4t ηn |ξn −ξm | CnK e 8 σ+ = I + O σ+ , υK (ζ ) = I + n (ζ − ςn ) (ζ − ςn ) ζ ∈ Kn , n ∈ {m + 1, m + 2, . . . , N}, −4t ηn |ξn −ξm | CnL e 8 σ− = I + O σ− , υL (ζ ) = I + n (ζ − ςn ) (ζ − ςn ) ζ ∈ Ln , n ∈ {m + 1, m + 2, . . . , N}; hence, as t → +∞, υD8n (ζ ) → I (uniformly), where D ∈ {K, L}. One also notes " from Lemmae 3.1–3.3 that, for ζ ∈ N n=m+1 int(Kn ), ζ −ς $N CnK $N + + −1 −1 n − k=m+1 (dk (ζ )) k=m+1 (dk (ζ )) ζ −ςn (ζ −ς ) k=n , n $k=n m8 (ζ ) = m(ζ ) N ζ −ςn + 0 d (ζ ) k=m+1 k ζ −ςn k=n
and, for ζ ∈ ∪N n=m+1 int(Ln ), $ m8 (ζ ) = m(ζ )
ζ −ςn ζ −ςn
CnL (ζ −ςn )
N k=m+1 k=n
$N
(dk+ (ζ ))−1
k=m+1 k=n
dk+ (ζ )
ζ −ςn ζ −ςn
$
0
N k=m+1 k=n
dk+ (ζ )
;
hence, modulo singular terms like (ζ −ςn )−1 and (ζ −ςn )−1 , and recalling that (see above), as t → +∞, CnK and CnL are O(exp(−4tηn |ξn − ξm |)), one deduces that,
352
A. H. VARTANIAN
since the RHP for m(ζ ) formulated in Lemma 3.1 is asymptotically "N solvable [38], 8 there are no exponentially growing factors for m (ζ ) when ζ ∈ n=m+1 (int(Kn ) ∪ int(Ln )). By estimating the error along the trajectory of the mth dark soliton (m ∈ {1, 2, . . . , N}) when the jump matrices on {Kn , Ln }N n=m+1 are removed from the specification of the RHP for m8 (ζ ), one arrives at an asymptotically solvable, model RHP (see Lemma 3.5 below); however, since the proof of Lemma 3.5 relies substantially on the Beals–Coifman (BC) construction [41] for the solution of a matrix (and appropriately normalised) RHP on an oriented and unbounded contour, it is convenient to present, with some requisite preamble, a succinct and self-contained synopsis of it at this juncture. But first, the following result is necessary. PROPOSITION 3.1 ([38]). The solution of the RHP for m8 (ζ ): C \ σO D → M2 (C) formulated in Lemma 3.3 has the (integral equation) representation m8− (µ)(υ 8 (µ) − I) dµ 8 8 −1 8 8 , m (ζ ) = (I + ζ +0 )P (ζ ) md (ζ ) + (µ − ζ ) 2π i σc ζ ∈ C \ σO D ,
where m8d (ζ )
=I+
m Res(m8 (ζ ); ςn ) n=1
(ζ − ςn )
σ1 Res(m8 (ζ ); ςn ) σ1 + , (ζ − ςn )
v 8 (·) is a generic notation for the jump matrices of m8 (ζ ) on σc (Lemma 3.3(ii)), and +80 and P 8 (ζ ) are specified below. The solution of the above (integral) equation can be written as the ordered factorisation m8 (ζ ) = (I + ζ −1 +80 )P 8 (ζ )m8d (ζ )mc (ζ ),
ζ ∈ C \ σO D ,
where m8d (ζ ) = σ1 m8d (ζ ) σ1 (∈ SL(2, C)) has the representation given above, P 8 (ζ ) = σ1 P 8 (ζ ) σ1 is chosen so that +80 is idempotent, I + ζ −1 +80 (∈ M2 (C)) is holomorphic in a punctured neighbourhood of the origin, with +80 = σ1 +80 σ1 8 (∈ GL(2, C)) such that det(I + ζ −1 +0 )|ζ =±1 = 0, and having the finite, order 2, matrix involutive structure 8 +8 ei(k+1/2)π (1 + (+8 )2 )1/2 e−iϑ 8 8 , k ∈ Z, +0 = (1 + (+8 )2 )1/2 eiϑ +8 e−i(k+1/2)π where +8 and ϑ 8 are obtained from the relation +80 = P 8 (0)m8d (0)mc (0)(δ(0))σ3 × $ 8 8 8 8 + σ3 ( N k=m+1 (dk (0)) )σ2 , and satisfying tr(+0 ) = 0, det(+0 ) = −1, and +0 +0 = I, c c and m (ζ ): C \ σc → SL(2, C) solves the following RHP: (1) m (ζ ) is piecewise (sectionally) holomorphic ∀ζ ∈ C \ σc ; (2) mc± (ζ ) := lim ζ →ζ mc (ζ ) ζ ∈± side of σc
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
353
satisfy, for ζ ∈ σc , the jump condition mc+ (ζ ) = mc− (ζ )υ c (ζ ), where υ c (ζ ) = exp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3 ))G8 (ζ ), ζ ∈ R, with G8 (ζ ) given in Lemma 3.3(ii), υ c (ζ ) = I + CnK (ζ − ςn )−1 σ+ , ζ ∈ Kn , and υ c (ζ ) = I + CnL (ζ − ςn )−1 σ− , ζ ∈ Ln , n ∈ {m + 1, m + 2, . . . , N}, with CnK and CnL given in Lemma 3.3; (3) mc (ζ ) = ζ →∞ I + O(ζ −1 ); and (4) mc (ζ ) = σ1 mc (ζ ) σ1 . ζ ∈C\σc
The BC formulation [41] now follows. One agrees to call a contour ; 8 oriented if: (1) C\; 8 has finitely many open connected components; (2) C\; 8 is the disjoint union of two, possibly disconnected, open regions, denoted by ✵+ and ✵− ; and (3) ; 8 may be viewed as either the positively oriented boundary for ✵+ or the negatively oriented boundary for ✵− (C \ ; 8 is coloured by two colours, ±). Let ; 8 , as a closed set, be the union of finitely many oriented simple piecewise-smooth arcs. Denote the set of all self-intersections of ; 8 by ; 8 (with card(; 8 ) < ∞ assumed throughout). Set ; 8 := ; 8 \ ; 8 . The BC construction for the solution of a (matrix) RHP, in the absence of a discrete spectrum and spectral singularities [45, 53], on an oriented contour ; 8 consists of finding an M2 (C)-valued function X(λ) such that: ;8, (1) X(λ) is piecewise holomorphic ∀λ ∈ C\; 8 ; (2) X+ (λ) = X− (λ)υ(λ), λ ∈ for some ‘jump’ matrix υ(λ): ; 8 → GL(2, C); and (3) uniformly as λ → ∞, 8 −1 ;8, λ ∈ C \ ; , X(λ) = I + O(λ ). Let υ(λ) := (I − w− (λ))−1 (I + w+ (λ)), λ ∈ be a factorisation for υ(λ), where w± (λ) are some upper/lower, or lower/upper, 8 matrices, with degree of triangular (depending on the 'orientationp of ; )8 nilpotent 8 nilpotency 2, and w± (λ) ∈ p∈{2,∞} LM2 (C)(; ) (if ; is unbounded, one requires that w± (λ) = λ→∞8 0). Define w(λ) := w+ (λ) + w− (λ), and introduce the Cauchy λ∈ ;
operators on L2M2 (C) (; 8 ), (C± f )(λ) :=
lim
λ →λ λ ∈± side of ; 8
;8
f (z) dz , (z − λ ) 2π i
where f (·) ∈ L2M2 (C) (; 8 ), with C± : L2M2 (C) (; 8 ) → L2M2 (C) (; 8 ) bounded in operator normD , and (C± f )(·)L2 (∗) const.||f (·)L2 (∗) . Introduce the BC M2 (C) M2 (C) operator: Cw f := C+ (f w− ) + C− (f w+ ),
f (·) ∈ L2M2 (C)(∗);
moreover, since C \; 8 can be coloured by two colours (±), C± are complementary projections [45], namely, C+2 = C+ , C−2 = −C− , C+ C− = C− C+ = 0 (the null operator), and C+ − C− = id (the identity operator): in the case that C+ and −C− are complementary, the contour ; 8 can always be oriented in such a way that the ± regions lie on the ± sides of the contour, respectively. Specialising the BC construction to the solution of the RHP for mc (ζ ) on σc formulated in Proposition 3.1, and writing υ c (ζ ) as the following (bounded) algebraic factorisation D C ± N (; 8 ) < ∞, where N (∗) denotes the space of all bounded linear operators acting from 2 LM (C) (∗) into L2M (C) (∗). 2 2
354
A. H. VARTANIAN
c c υ c (ζ ) := (I − w− (ζ ))−1 (I + w+ (ζ )), ζ ∈ σc , the integral representation for mc (ζ ) is given by the following
LEMMA 3.4 (Beals and Coifman [41]). Let c c (ζ ))−1 = mc− (ζ )(I − w− (ζ ))−1 , µc (ζ ) = mc+ (ζ )(I + w+
ζ ∈ σc .
If µc (ζ ) ∈ I + L2M2 (C) (σc ) := {I + h(·); h(·) ∈ L2M2 (C)(σc )}D solves the linear singular integral equation c c ) + C− (w+ ), (id − Cwc )(µc (ζ ) − I) = Cwc I = C+ (w−
ζ ∈ σc ,
where id is the identity operator on L2M2 (C) (σc ), then the solution of the RHP for mc (ζ ) is µc (z)w c (z) dz , ζ ∈ C \ σc , mc (ζ ) = I + (z − ζ ) 2π i σc c c (ζ ) + w− (ζ ). where µc (ζ ) = ((id − Cwc )−1 I)(ζ ), and w c (ζ ) := w+
Finally, one arrives at, and is in a position to prove, the following " LEMMA 3.5. For m ∈ {1, 2, . . . , N}, set σd := m n=1 ({ςn } ∪ {ςn }), and let σc = {ζ ; Im(ζ ) = 0} with orientation from −∞ to +∞. Let χ(ζ ): C \ (σd ∪ σc ) → M2 (C) solve the following RHP: (i) χ(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; (ii) χ± (ζ ) := lim ζ →ζ χ (ζ ) satisfy the jump condition ζ ∈± side of σc
χ+ (ζ ) = χ− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 ))G8 (ζ ), (iii) χ(ζ ) has simple poles in σd with
N
Res(χ (ζ ); ςn ) = lim χ(ζ )gn (δ(ςn ))−2 ζ →ςn
ζ ∈ R;
(dk+ (ςn ))−2 σ− ,
k=m+1
n ∈ {1, 2, . . . , m}, Res(χ (ζ ); ςn ) = σ1 Res(χ (ζ ); ςn ) σ1 ,
n ∈ {1, 2, . . . , m};
(iv) det(χ(ζ ))|ζ =±1 = 0; $ + σ3 (v) χ(ζ ) =ζ →0 ζ −1 (δ(0))σ3 ( N k=m+1 (dk (0)) )σ2 + O(1); −1 (vi) χ(ζ ) = ζ →∞ I + O(ζ ); ζ ∈C\(σd ∪σc ) $ + σ3 (vii) χ(ζ ) = σ1 χ (ζ ) σ1 and χ (ζ −1 ) = ζ χ(ζ )(δ(0))σ3 ( N k=m+1 (dk (0)) )σ2 . D For f (ζ )
f (∞)2 2
LM
2 (C)
(∗)
∈
I + L2M (C) (∗), f (·)I+L2 2 M
)1/2 [44].
2 (C)
(∗)
:=
(f (∞)2L∞
M2 (C) (∗)
+ f (·) −
355
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
Then, as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m8 (ζ ): C \ σO D → M2 (C) has the following asymptotics: m8 (ζ ) = (I + O(F (ζ ) exp(− t)))χ(ζ ), where := 4 min
m∈{1,2,...,N} n∈{m+1,m+2,...,N}
{sin(φn )| cos(φn ) − cos(φm )|} (> 0), and, for i, j ∈
{1, 2}, (F (ζ ))ij =ζ →∞ O(|ζ |−1 ) and (F (ζ ))ij =ζ →0 O(1). Furthermore, let N u(x, t) := i lim (dk+ (ζ ))σ3 − I + ζ χ(ζ )(δ(ζ ))σ3 ζ →∞ ζ ∈C\(σd ∪σc )
k=m+1
12
+ O(exp(− t)), and
(63)
x
(|u(x , t)|2 − 1) dx +∞ := −i
lim
ζ →∞ ζ ∈C\(σd ∪σc )
ζ χ(ζ )(δ(ζ ))σ3
N
(dk+ (ζ ))σ3 − I
k=m+1
+ 11
+ O(exp(− t)).
(64)
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. Remark 3.5. The solution of the (normalised at ∞) RHP for χ(ζ ): C \ (σd ∪ σc ) → M2 (C) formulated in Lemma 3.5 has a factorised representation analogous to that of m8 (ζ ) given in Proposition 3.1 (with appropriate change(s) of notation). Proof. Define E(ζ ) := m8 (ζ )(χ(ζ ))−1 . From this definition, the RHPs for m (ζ ) and χ(ζ ) formulated in Lemmae 3.3 and 3.5, respectively, Proposition 3.1, and Remark 3.5, one shows that, for m ∈ {1, 2, . . . , N} and n ∈ {m + 1, m + 2, . . . , N}, E(ζ ) solves the following RHP:"(1) E(ζ ) is piecewise (sectionally) n n holomorphic ∀ζ ∈ C \ IE , where IE = N n=m+1 IE , with IE := Kn ∪ Ln (with orientations preserved); (2) E± (ζ ) := lim ζ →ζ E(ζ ) satisfy the jump 8
ζ ∈± side of IE
condition E+ (ζ ) = E− (ζ )υE (ζ ), ζ ∈ IE , where ! " EKn (ζ ), ζ ∈ N I+W Kn (⊂ IE ), n ∈ {m + 1, m + 2, . . . , N}, υE (ζ ) = "n=m+1 Ln N I + WE (ζ ), ζ ∈ n=m+1 Ln (⊂ IE ), n ∈ {m + 1, m + 2, . . . , N}, ELn (ζ ) = CnL (ζ − ςn )−1 X7 (ζ ), EKn (ζ ) = CnK (ζ − ςn )−1 X6 (ζ ), W with W −χ11 (ζ )χ21 (ζ ) (χ11 (ζ ))2 , X6 (ζ ) = −(χ21 (ζ ))2 χ11 (ζ )χ21 (ζ ) χ12 (ζ )χ22 (ζ ) −(χ12 (ζ ))2 , X7 (ζ ) = (χ22 (ζ ))2 −χ12 (ζ )χ22 (ζ )
356
A. H. VARTANIAN
and CnK and CnL given in Lemma 3.3; (3) det(E(ζ ))|ζ =±1 = 1; (4) E(ζ ) =ζ →0 O(1) and E = ζ →∞ I + O(ζ −1 ); and (5) E(ζ ) = σ1 E(ζ ) σ1 and E(ζ −1 ) = E(ζ ). Note, ζ ∈C\IE
in particular, that E(ζ ) has no jump discontinuity on R, and no poles. Recall, now, the BC construction (see the paragraph preceding Lemma 3.4). Write the following E E (ζ ))−1 (I + w+ (ζ )), (bounded) algebraic factorisation for υE (ζ ), υE (ζ ) = (I − w− " Kn N E E ζ ∈ IE , and choose [46] w− (ζ ) = 0; hence, w+ (ζ ) = WE (ζ ), ζ ∈ n=m+1 Kn , " E E ELn (ζ ), ζ ∈ N and w+ (ζ ) = W n=m+1 Ln . Let µ (ζ ) be the solution of the BC E linear singular integral equation (idE − CwE )µ (ζ ) = I, ζ ∈ IE , where idE is the identity operator on L2M2 (C) (IE ), and, for f (·) ∈ L2M2 (C) (IE ), set CwE f := E E E ) + C− (f w+ ) = C− (f w+ ), with C+ (f w− f (z) dz (C± f )(ζ ) := . lim ζ →ζ IE (z − ζ ) 2π i ζ ∈± side of IE
It was shown in [38] that (idE − CwE )−1 N (IE ) < ∞, where N (∗) denotes the space of bounded linear operators from L2M2 (C) (∗) to L2M2 (C) (∗). According to the BC construction, the solution of the (normalised at ∞) RHP for E(ζ ) has the integral representation µE (z)w E (z) dz , ζ ∈ C \ IE , E(ζ ) = I + (z − ζ ) 2π i IE E (ζ ). Since where µE (ζ ) = ((idE − CwE )−1 I)(ζ ), and w E (ζ ) = l∈{±} wlE (ζ ) = w+ (cf. Definition 3.1), for i = j ∈ {m + 1, m + 2, . . . , N}, Ki ∩ Li = Ki ∩ Kj = Li ∩ Lj = ∅, it follows that N EKn (z) dz ELn (z) dz µE (z)W µE (z)W + , E(ζ ) = I + (z − ζ ) 2π i (z − ζ ) 2π i Kn Ln n=m+1 ζ ∈ C \ IE . ELn (ζ ), EKn (ζ ) and W From the second resolvent identity and the expressions for W one shows that N dz dz CnK X6 (z) CnL X7 (z) + + E(ζ ) − I = Kn (z − ςn )(z − ζ ) 2π i Ln (z − ςn )(z − ζ ) 2π i n=m+1 CnK ((idE − CwE )−1 CwE I)(z)X6 (z) dz + + (z − ςn )(z − ζ ) 2π i Kn L Cn ((idE − CwE )−1 CwE I)(z)X7 (z) dz , ζ ∈ C \ IE . + (z − ςn )(z − ζ ) 2π i Ln Using the Cauchy–Schwarz inequality for integrals, one arrives at ( ( K N ( ( |Cn | I 6 ( ( X (·)L2 (Kn ) ( |E(ζ ) − I| M2 (C) 2π (· − ς )(· − ζ ) ( n=m+1
n
+ L2M
2
(C) (Kn )
357
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
( ( ( ( I |CnL | 7 ( ( X (·)L2 (Ln ) ( + M (C) 2 2π (· − ςn )(· − ζ ) (L2
+
M2 (C) (Ln )
|CnK | (idE − CwE )−1 N (Kn ) (CwE I)(·)L2 (Kn ) × M2 (C) 2π ( ( ( ( I ( + × X6 (·)L2 (Kn ) ( ( (· − ς )(· − ζ ) ( 2 M2 (C)
+
n
LM
2 (C)
( Kn )
|CnL |
(idE − CwE )−1 N (Ln ) (CwE I)(·)L2 (Ln ) × M2 (C) ( ( ( ( I 7 ( ( × X (·)L2 (Ln ) ( , ζ ∈ C \ IE . M2 (C) (· − ςn )(· − ζ ) (L2 (Ln ) +
2π
M2 (C)
One shows that, for ζ ∈ C \ IE , ) ) ( ( 1/2 2π ( ( 2 dω 2 I ( ( =: FKn (ζ ), ( (· − ς )(· − ζ ) ( 2 K K e−iω |2 K ε |ζ − ε ε n 0 n n n LM (C) (Kn ) 2 ) ) ( ( 1/2 2π ( ( I 2 dω 2 ( ( =: F (ζ ), ( (· − ς )(· − ζ ) ( 2 L L iω 2 εn |ζ − εn e | εnL Ln n 0 L (Ln ) M2 (C)
with FDn (ζ ) =ζ →∞ O(|ζ |−1 ) and FDn (ζ ) =ζ →0 O(1), D ∈ {K, L}. Again, via the Cauchy–Schwarz inequality for integrals, (CwE I)(·)L2
M2 (C) (Kn )
E E (C− (Iw+ ))(·)L2 (Kn ) C− N (Kn ) w+ (·)L2 (Kn ) M2 (C) M2 (C) ( ( ( ( CnK 6 ( C− N (Kn ) ( ( (· − ς ) X (·)( 2 n LM (C) (Kn ) 2 ( ( ( ( I K 6 ( ( C− N (Kn ) |Cn |X (·)L2 (Kn ) ( M2 (C) (· − ςn ) (L2 (Kn ) M2 (C) * π 2 K |CnK |C− N (Kn ) X6 (·)L2 (Kn ) , M2 (C) εn
with an analogous estimate for (CwE I)(·)L2
M2 (C) (Ln )
(CwE I)(·)L2
M2 (C) (Ln )
Hence, for ζ ∈ C \ IE , |E(ζ ) − I|
:
* π 2 L |CnL |C− N (Ln ) X7 (·)L2 (Ln ) . M2 (C) εn
358
A. H. VARTANIAN
K N |C L |FL (ζ ) |Cn |FKn (ζ ) + X6 (·)L2 (Kn ) + n + n X7 (·)L2 (Ln ) + M2 (C) M2 (C) K L π 2 ε π 2 ε n n n=m+1 √ 2 |CnK |2 FKn (ζ ) √ K (idE − CwE )−1 N (Kn ) C− N (Kn ) X6 (·)2L2 (Kn ) + + M2 (C) π εn √ L 2 2 |Cn | FLn (ζ ) (idE − CwE )−1 N (Ln ) C− N (Ln ) X7 (·)2L2 (Ln ) . + √ L M2 (C) π εn It is shown, a posteriori, in Section 4 that the RHP for χ (ζ ) formulated in the Lemma is asymptotically solvable, whence X6 (·)2L2 (K ) const. = c and M2 (C)
n
const. = c. Furthermore [38], (idE − CwE )−1 N (Dn )
X7 (·)2L2
M2 (C) (Ln )
const. (idE − CwE )−1 N (IE ) c (see above), D ∈ {K, L}. Recalling the expressions for CnK and CnL given in Lemma 3.3, that as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , (gn )−1 = O(exp(−4t sin(φn )|cos(φn ) − cos(φm )|)), and the definition E(·) − IL2
M2 (C) (C \IE )
:= max
sup |(E(ζ ) − I)ij |,
i,j ∈{1,2} ζ ∈C \IE
assembling the above, one arrives at E(·) − IL2 (C \IE ) M2 (C) O FE (ζ ) exp −4t
min
m∈{1,2,...,N} n∈{m+1,m+2,...,N}
{sin(φn )|cos(φn ) − cos(φm )|}
,
where FE (ζ ) =ζ →∞ O(|ζ |−1 ) and FE (ζ ) =ζ →0 O(1); hence, the asymptotic estimate for m8 (ζ ) stated in the lemma. Finally, from the asymptotics for E(ζ ) − I derived above, the ordered factorisation for m8 (ζ ) given in Proposition 3.1, and Equations (61) and (62), the large-ζ asymptotics lead one to Equations (63) and (64). ✷
4. Asymptotic Solution of the Model RHP In this section, the model (normalised at ∞) RHP for χ (ζ ) formulated in Lemma 3.5 is solved asymptotically as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, and the corresponding (asymptotic) for u(x, t), the solution of the Cauchy problem for the Df NLSE, and x results 2 (|u(x , t)| − 1) dx stated in Theorem 2.2.1 (for the upper sign) are derived. ±∞ LEMMA 4.1. The solution of the RHP for χ(ζ ): C \ (σd ∪ σc ) → M2 (C) formulated in Lemma 3.5 is given by the following ordered factorisation, χ (ζ ) = (I + ζ −1 +0 )P (ζ )md (ζ )χ c (ζ ),
ζ ∈ C \ (σd ∪ σc ),
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
359
where md (ζ ) = σ1 md (ζ ) σ1 (∈ SL(2, C)) has the representation m Res(χ(ζ ); ςn ) σ1 Res(χ(ζ ); ςn ) σ1 + , md (ζ ) = I + (ζ − ς ) (ζ − ς ) n n n=1 P (ζ ) = σ1 P (ζ ) σ1 is chosen (see Lemma 4.3 below) so that +0 is idempotent, I + ζ −1 +0 is holomorphic in a punctured neighbourhood of the origin, with +0 = σ1 +0 σ1 (∈ GL(2, C)) and det(I + ζ −1 +0 )|ζ =±1 = 0, and determined by the relation N + c σ3 σ3 (dk (0)) σ2 , +0 = P (0)md (0)χ (0)(δ(0)) k=m+1
and satisfying tr(+0 ) = 0, det(+0 ) = −1, and +0 +0 = I, and χ c (ζ ): C \ σc → SL(2, C) solves the following RHP: (1) χ c (ζ ) is piecewise (sectionally) holomorphic ∀ζ ∈ C \ σc ; (2) χ±c (ζ ) := lim ζ →ζ χ c (ζ ) satisfy, for ζ ∈ R, the jump ±Im(ζ )>0
condition χ+c (ζ )
= χ−c (ζ )e−ik(ζ )(x+2λ(ζ )t ) ad(σ3 ) × (1 − r(ζ )r(ζ ))δ− (ζ )/δ+ (ζ ) × $N r(ζ ) + −2 k=m+1 (dk (ζ )) δ− (ζ )δ+ (ζ )
(3) χ c (ζ ) =
ζ →∞ ζ ∈C\σc
$N r(ζ ) + 2 − (δ− (ζ )δ (d (ζ )) −1 k k=m+1 + (ζ )) ; δ+ (ζ )/δ− (ζ )
I + O(ζ −1 ); and (4) χ c (ζ ) = σ1 χ c (ζ ) σ1 .
Proof. One verifies that, modulo the explicit determination of +0 , P (ζ ), md (ζ ), and χ c (ζ ), the ordered factorisation for χ (ζ ) stated in the lemma, with the conditions on +0 , P (ζ ), md (ζ ), and χ c (ζ ) stated therein, solves the RHP for χ (ζ ) stated in Lemma 3.5. ✷ The determination of the asymptotics for the solution of the RHP for χ c (ζ ): C\ σc → SL(2, C) stated in Lemma 4.1 was the (principal) subject of study in [38], and is given by the following lemma: LEMMA 4.2. Let ε be an arbitrarily fixed, sufficiently small positive real number, and, for z ∈ {λ1 , λ2 }, with λ1 and λ2 given in Theorem 2.2.1, Equation (10), set U(z; ε) := {ζ ; |ζ − z| " < ε}. Then, as t → +∞ and x → −∞ such that z0 := x/t < −2, for ζ ∈ C \ z∈{λ1 ,λ2 } U(z; ε), χ c (ζ ) has the following asymptotics: c (ζ ) χ11
=1+O
cS (λ2 )c(λ1 , λ3 , λ3 ) ln t cS (λ1 )c(λ2 , λ3 , λ3 ) + , λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t
360
A. H. VARTANIAN
c χ12 (ζ )
√
−i(:+ (z0 ,t )+ π ) π + 4 λ2 ei(: (z0 ,t )+ 4 ) ν(λ1 ) λ12iν(λ1) λ1 e + + =e √ (ζ − λ1 ) (ζ − λ2 ) t (λ1 − λ2 ) (z02 + 32)1/4 S c (λ1 )c(λ2 , λ3 , λ3 ) cS (λ2 )c(λ1 , λ3 , λ3 ) ln t , +O + λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t i<+ (0) 2
c χ21 (ζ )
c (ζ ) χ22
√
i(:+ (z0 ,t )+ π ) π + 1) 4 λ1 e λ2 e−i(: (z0 ,t )+ 4 ) ν(λ1 ) λ−2iν(λ 1 + + √ =e (ζ − λ1 ) (ζ − λ2 ) t (λ1 − λ2 ) (z02 + 32)1/4 S cS (λ2 )c(λ1 , λ3 , λ3 ) ln t c (λ1 )c(λ2 , λ3 , λ3 ) + , +O λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t +
− i< 2 (0)
=1+O
cS (λ2 )c(λ1 , λ3 , λ3 ) ln t cS (λ1 )c(λ2 , λ3 , λ3 ) + , λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t
where λ3 , ν(·), :+ (z0 , t), and <+ (·), respectively, are given in Theorem 2.2.1, Equations (10), (11), (17), and (18), (· − λk )−1 L∞ (C \∪z∈{λ1 ,λ2 } U(z;ε)) < ∞, k ∈ {1, 2}, χ c (ζ ) = σ1 χ c (ζ ) σ1 , and (χ c (0)σ2 )2 = I (+ O(t −1 ln t)). Sketch of proof. Proceeding as in the proof of Lemma 6.1 in [38] and particularising it to the case of the RHP for χ c (ζ ) stated in Lemma 4.1, one arrives at πν
c (ζ ) χ11
iπ
r(λ1 )(δB0 )−2 e 2 e 4
√ × I 0 2π i(ζ − λ1 )β21B XB t +∞ i iπ iπ z2 e− 4 ∂z D−iν (z) − e 4 zD−iν (z) z−iν e− 4 dz+ × 2 0
=1−
r(λ1 )(1 − |r(λ1 )|2 )−1 (δB0 )−2 e− 4 × + √ I 0 3π ν 2π i(ζ − λ1 )β21B e 2 XB t +∞ i − 3π i 3π i z2 e 4 ∂z D−iν (z) − e 4 zD−iν (z) z−iν e− 4 dz− × 2 0 3π i
r(λ1 )(δA0 )−2 e− 2 (−1)iν e 4 − √ × I 0 2π i(ζ − λ2 )β21A XA t +∞ i 3π i z2 − 3π4 i ∂z Diν (z) + e 4 zDiν (z) ziν e− 4 dz+ e × 2 0 πν
3π i
r(λ1 )(1 − |r(λ1 )|2 )−1 (δA0 )−2 (−1)iν e− 4 × √ I 0 πν 2π i(ζ − λ2 )β21A e 2 XA t iπ
+
361
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
+∞ i − iπ iπ z2 4 4 e ∂z Diν (z) + e zDiν (z) ziν e− 4 dz+ × 2 0 S c (λ1 )c(λ2 , λ3 , λ3 )(δB0 )−2 +O √ + (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) XB ln t cS (λ2 )c(λ1 , λ3 , λ3 )(δA0 )−2 √ , + (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) XA t πν iπ 3π i r(λ1 )(δB0 )2 e 2 e− 4 r(λ1 )(1 − |r(λ1 )|2 )−1 (δB0 )2 e 4 c × χ12 (ζ ) = √ − √ 3π ν 2π i(ζ − λ1 )XB t 2π i(ζ − λ1 )e 2 XB t +∞ z2 Diν (z)ziν e− 4 dz+ ×
0
+
r(λ1 )(1 − |r(λ1 )|2 )−1 (δA0 )2 e 4 r(λ1 )(δA0 )2 e− 2 e− 4 − √ √ πν 2π i(ζ − λ2 )(−1)iν XA t 2π i(ζ − λ2 )e 2 (−1)iν XA t 3π i
πν
+∞
×
iπ
×
z2
D−iν (z)z−iν e− 4 dz+
0
+O
c (ζ ) χ21
cS (λ1 )c(λ2 , λ3 , λ3 )(δB0 )2 √ + (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) XB ln t cS (λ2 )c(λ1 , λ3 , λ3 )(δA0 )2 √ , + (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) XA t
r(λ1 )(1 − |r(λ1 )|2 )−1 (δB0 )−2 e− r(λ1 )(δB0 )−2 e 2 e 4 =− √ − √ 3π ν 2π i(ζ − λ1 )XB t 2π i(ζ − λ1 )e 2 XB t +∞ z2 D−iν (z)z−iν e− 4 dz− × iπ
πν
3π i 4
×
0
−
r(λ1 )(δA0 )−2 e− 2 e 4 r(λ1 )(1 − |r(λ1 )|2 )−1 (δA0 )−2 e− 4 − √ √ πν 2π i(ζ − λ2 )(−1)−iν XA t 2π i(ζ − λ2 )e 2 (−1)−iν XA t 3π i
πν
+∞
×
z2
Diν (z)ziν e− 4 dz+
0
+O
cS (λ1 )c(λ2 , λ3 , λ3 )(δB0 )−2 √ + (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) XB ln t cS (λ2 )c(λ1 , λ3 , λ3 )(δA0 )−2 √ , + (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) XA t
r(λ1 )(δB0 )2 e 2 e− 4 πν
c (ζ ) χ22
iπ
√ × I 0 2π i(ζ − λ1 )β12B XB t +∞ i − iπ iπ z2 4 4 e ∂z Diν (z) + e zDiν (z) ziν e− 4 dz− × 2 0
=1+
iπ
×
362
A. H. VARTANIAN 3π i
r(λ1 )(1 − |r(λ1 )|2 )−1 (δB0 )2 e 4 √ × I 0 3π ν 2π i(ζ − λ1 )β12B e 2 XB t +∞ i 3π i z2 − 3π4 i ∂z Diν (z) + e 4 zDiν (z) ziν e− 4 dz+ e × 2 0
−
πν
r(λ1 )(δA0 )2 e− 2 e−
3π i 4
√ × I 0 2π i(ζ − λ2 )β12A (−1)iν XA t +∞ i − 3π i 3π i z2 e 4 ∂z D−iν (z) − e 4 zD−iν (z) z−iν e− 4 dz− × 2 0
+
iπ
r(λ1 )(1 − |r(λ1 )|2 )−1 (δA0 )2 e 4 √ × I 0 πν 2π i(ζ − λ2 )β12A e 2 (−1)iν XA t +∞ i iπ z2 − iπ 4 4 e ∂z D−iν (z) − e zD−iν (z) z−iν e− 4 dz+ × 2 0 S c (λ1 )c(λ2 , λ3 , λ3 )(δB0 )2 √ + +O (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) XB ln t cS (λ2 )c(λ1 , λ3 , λ3 )(δA0 )2 √ , + (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) XA t $ + −2 where r(ζ ) = r(ζ ) N (|r(λ1 )| = |r(λ1 )|), ν = ν(λ1 ), k=m+1 (dk (ζ )) −
− 2 Z(λ1 ) e × δB0 = |λ1 − λ3 |−iν (2t (λ1 − λ2 )3 λ−3 1 ) it × exp − (λ1 − λ2 )(z0 + λ1 + λ2 ) , 2 it 0 iν 3 −3 iν2 Z(λ2 ) (λ1 − λ2 )(z0 + λ1 + λ2 ) , exp δA = |λ2 − λ3 | (2t (λ1 − λ2 ) λ2 ) e 2 0 i ln|µ − λ1 | d ln(1 − |r(µ)|2 )+ Z(λ1 ) = 2π −∞ λ1 i ln|µ − λ1 | d ln(1 − |r(µ)|2 ), + 2π λ2 0 i ln|µ| d ln(1 − |r(µ)|2 )+ Z(λ2 ) = −Z(λ1 ) + 2π −∞ λ1 i ln|µ| d ln(1 − |r(µ)|2 ), + 2π λ2 ) ) |λ1 − λ3 | 2(λ1 − λ2 ) |λ2 − λ3 | 2(λ1 − λ2 ) , XA = , XB = λ1 λ1 λ2 λ2 √ √ π ν iπ πν iπ 2π e− 2 e 4 2π e− 2 e− 4 IB 0 IB 0 IA0 IA0 , β12 = β21 = , β12 = β21 = r(λ1 ) ;(iν) r(λ1 ) ;(iν) iν
363
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
;(·) is the gamma function [51], and D∗ (·) is the parabolic cylinder function [51]. −1/2 2 Using Equation (10), one shows that |λk − λ3 |λ−1 (z0 + 32)1/4 , k ∈ k = (2λk ) 1 {1, 2}. Using the identities [51] ∂z Dz1 (z) = 2 (z1 Dz1 −1 (z) − Dz1 +1 (z)), zDz1 (z) = Dz1 +1 (z) + z1 Dz1 −1 (z), and |;(iν)|2 = π/(ν sinh(π ν)), and the integral [51] +∞ 2 D−z1 (z)zz2 −1 e−z /4 dz 0 √ π exp(− 12 (z1 + z2 ) ln 2);(z2 ) = , Re(z2 ) > 0, ;( 12 (z1 + z2 ) + 12 ) from the above expressions for χijc (ζ ), i, j ∈ {1, 2}, and repeated application of πν the relation |r(λ1 );(iν)|νe 2 = (2π ν)1/2 , one obtains the result stated in the lemma. Furthermore, one shows that the symmetry reduction χ c (ζ ) = σ1 χ c (ζ ) σ1 is satisfied, and verifies that (χ c (0)σ2 )2 = I + O(t −1 ln t). ✷ PROPOSITION 4.1. For m ∈ {1, 2, . . . , N}, set Res(χ(ζ ); ςn ) :=
an
bn , cn dn c c −cn χ12 (ςn )/χ22 (ςn ),
n∈
c c (ςn )/χ22 (ςn ), dn = and {1, 2, . . . , m}. Then bn = −an χ12 m {an , cn }n=1 satisfy the following (nonsingular) system of 2m linear inhomogeneous algebraic equations, ∗ c g1 χ12 (ς1 ) a1 c (ς2 ) a2 g2∗ χ12 A . B . . .. . a gm∗ χ c (ςm) 12 m , c = g ∗ χ c (ς ) 1 1 22 1 ∗ c c2 g2 χ22 (ς2 ) B A .. .. . . cm gm∗ χ c (ςm) 22
where Aij :=
det(χ c (ς ))+g ∗W(χ c (ς ),χ c (ς )) i i 12 i 22 i , χ c (ς ) −
Bij := −
22 i c (ς )χ c (ς )−χ c (ς )χ c (ς )) gi∗ (χ12 i 22 j 22 i 12 j c (ς ) (ςi −ςj )χ22 j
i = j ∈ {1, 2, . . . , m}, , i = j ∈ {1, 2, . . . , m},
c c c c gi∗ (χ22 (ςi )χ22 (ςj ) − χ12 (ςi )χ12 (ςj )) c (ςi − ςj )χ22 (ςj )
,
gj∗ = |gj |eiθgj exp(2ik(ςj )(x + 2λ(ςj )t))(δ(ςj ))−2
i, j ∈ {1, 2, . . . , m}, N k=m+1
j ∈ {1, 2, . . . , m},
(dk+ (ςj ))−2 ,
364
A. H. VARTANIAN
with |gj | and θgj given in Lemma 3.1(iii), and c c χ (z) χ (z) c c 22 . W(χ12 (z), χ22 (z)) = 12c c ∂z χ12 (z) ∂z χ22 (z) Proof. Recall from Lemma 4.1 that χ(ζ ): C \ (σd ∪ σc ) → M2 (C) has the factorised representation χ (ζ ) = (I + ζ −1 +0 )P (ζ )× m Res(χ (ζ ); ςn ) σ1 Res(χ(ζ ); ςn ) σ1 + χ c (ζ ), × I+ (ζ − ς ) (ζ − ς ) n n n=1 where χ c (ζ ) is given in Lemma 4.2. For m ∈ {1, 2, . . . , N}, set Res(χ (ζ ); ςn ) := an bn , whence σ1 Res(χ(ζ ); ςn ) σ1 = dbn acn ; thus, c d n
n
n
1 χ(ζ ) = I + +0 P (ζ )× ζ m an × ×
ak dk 1 + ζ −ς + k=1 ζ −ς + m k=1 ζ −ςk n k k=n m m ck bk cn k=1 ζ −ς + k=1 ζ −ς ζ −ςn + k
k=n
c (ζ ) χ11 c (ζ ) χ21
c χ12 (ζ ) c χ22 (ζ )
k
n
m m bk ck bn k=1 ζ −ςk + k=1 ζ −ςk ζ −ςn + k=n
dk ak dn 1 + ζ −ς + mk=1 ζ −ς + m k=1 ζ −ςk n k
×
k=n
(65)
.
As in the BC construction [41], one now Taylor expands χ c (ζ ) about {ςn }m n=1 : χijc (ζ ) = χijc (ςn ) + (∂ζ χijc (ςn ))(ζ − ςn ) + O((ζ − ςn )2 ), i, j ∈ {1, 2}, where ∂ζ χijc (ςn ) = ∂ζ χijc (ζ )|ζ =ςn . Recalling from Lemma 3.5(iii) that χ (ζ ) satisfies the polar (residue) conditions Res(χ(ζ ); ςn ) = limζ →ςn χ (ζ )gn∗ σ− and Res(χ (ζ ); ςn ) = σ1 Res(χ(ζ ); ςn ) σ1 , n ∈ {1, 2, . . . , m}, with gn∗ given in the proposition, assembling the above, one shows that the only nontrivial conditions are c c (ςn ) + bn χ22 (ςn ) = 0, an χ12 c c cn χ12 (ςn ) + dn χ22 (ςn ) = 0, c c (ςn ) + bn χ21 (ςn ) an χ11
=
c (ςn ) an gn∗ ∂ζ χ12
+ 1+
+
c (ςn ) bn gn∗ ∂ζ χ22
+
m k=1 k=n
m k=1 k=n
m ak dk + g ∗ χ c (ςn )+ ςn − ςk k=1 ςn − ςk n 12
m bk ck + g ∗ χ c (ςn )+ ςn − ςk k=1 ςn − ςk n 22
gn∗ c c (ςn ) + bn χ22 (ςn ) ) , + lim ( an χ12 ζ →ςn 2 34 5 ζ − ςn 0 c (ςn ) cn χ11
+
c dn χ21 (ςn )
365
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
=
c (ςn ) cn gn∗ ∂ζ χ12
+
m k=1 k=n
c dn gn∗ ∂ζ χ22 (ςn )
+
bk ck + g ∗ χ c (ςn )+ ςn − ςk k=1 ςn − ςk n 12 m
+ 1+
m k=1 k=n
m dk ak + g ∗ χ c (ςn )+ ςn − ςk k=1 ςn − ςk n 22
gn∗ c c (ςn ) + dn χ22 (ςn ) ) . + lim ( cn χ12 ζ →ςn 2 34 5 ζ − ςn 0 c (ςn )/ From the first two equations of the above system, one gets that bn = −an χ12 c c c χ22 (ςn ) and dn = −cn χ12 (ςn )/χ22 (ςn ) (whence, det acnn bdnn = 0): using the
latter (two) relations, it follows from the last two equations of the above system that, for n ∈ {1, 2, . . . , m}, an An =
m ak g ∗ Bnk n
ςn − ςk
k=1 k=n
cn An =
ςn − ςk
k=1 k=n
m ck g ∗ Dnk n
k=1
m ck g ∗ Bnk n
+
+
ςn − ςk
m ak g ∗ Dnk n
k=1
ςn − ςk
c + gn∗ χ12 (ςn ),
c + gn∗ χ22 (ςn ),
where An = Bnk = Dnk =
c c det(χ c (ςn )) + gn∗ W(χ12 (ςn ), χ22 (ςn )) , c χ22 (ςn ) c c c c χ12 (ςn )χ22 (ςk ) − χ12 (ςk )χ22 (ςn ) , c χ22 (ςk ) c c c c χ22 (ςn )χ22 (ςk ) − χ12 (ςn )χ12 (ςk ) c χ22 (ςk )
;
thus, the (rank 2m) linear inhomogeneous algebraic system for {an , cn }m n=1 stated in the proposition. The nondegeneracy of the (2m × 2m) coefficient matrix is a consequence of the asymptotic solvability of the original RHP formulated in Lemma 2.1.2 [38] (see, also, Equation (66) below). ✷ PROPOSITION 4.2. As t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, for n ∈ {1, 2, . . . , m − 1}, +
bn = O(t −1/2 (z02 + 32)−1/4 e− גt ),
+
dn = O(t −1/2 (z02 + 32)−1/4 e− גt ),
an = O(e− גt ), cn = O(e− גt ), where ג+ := 4 min
+
m∈{1,2,...,N} n∈{1,2,...,m−1}
+
{sin(φn )|cos(φn ) − cos(φm )|} (> 0), and
366
A. H. VARTANIAN
gm∗ gm∗ (ςm − ςm )−1 1 cS (z0 ) ln t + =: am = am0 + √ am1 + O (z02 + 32)1/2 t (1 + gm∗ gm∗ (ςm − ςm )−2 ) t c c 1 gm∗ gm∗ (ςm − ςm )−1 (gm∗ ∂ζ χ 12 (ςm ) + gm∗ ∂ζ χ 12 (ςm )) +√ + ∗ ∗ −2 2 (1 + gm gm (ςm − ςm ) ) t c 12 (ςm ) cS (z0 ) ln t gm∗ χ +O , + (1 + gm∗ gm∗ (ςm − ςm )−2 ) (z02 + 32)1/2 t c 1 gm∗ gm∗ (ςm − ςm )−1 χ 1 1 cS (z0 ) ln t 12 (ςm ) + =: − √ bm = √ bm + O 2 1/2 ∗ ∗ t (z0 + 32) t t (1 + gm gm (ςm − ςm )−2 ) cS (z0 ) ln t , +O (z02 + 32)1/2 t gm∗ 1 1 cS (z0 ) ln t 0 cm = cm + √ cm + O + =: (z02 + 32)1/2 t (1 + gm∗ gm∗ (ςm − ςm )−2 ) t c c 12 (ςm ) − gm∗ gm∗ ∂ζ χ 12 (ςm ) 1 gm∗ gm∗ (ςm − ςm )−1 χ + +√ ∗ ∗ −2 (1 + gm gm (ςm − ςm ) ) t c c 12 (ςm ) + gm∗ ∂ζ χ 12 (ςm )) cS (z0 ) ln t gm∗ (gm∗ ∂ζ χ , +O + (z02 + 32)1/2 t (1 + gm∗ gm∗ (ςm − ςm )−2 )2 c gm∗ χ 1 1 1 cS (z0 ) ln t 12 (ςm ) + =: − dm = √ dm + O √ 2 1/2 ∗ ∗ t (z0 + 32) t t (1 + gm gm (ςm − ςm )−2 ) cS (z0 ) ln t , +O (z02 + 32)1/2 t where c (ζ ) χ 12
√ −i(:+ (z0 ,t )+ π ) i<+ (0) π + 1) 4 λ2 ei(: (z0 ,t )+ 4 ) ν(λ1 ) e 2 λ2iν(λ λ1 e 1 + , =√ (ζ − λ1 ) (ζ − λ2 ) (λ1 − λ2 ) (z02 + 32)1/4
with ν(·), λ1 , λ2 , λ3 , <+ (·), and :+ (z0 , t) specified in Lemma 4.2, and cS (z0 ) =
cS (λ1 )c(λ2 , λ3 , λ3 ) cS (λ2 )c(λ1 , λ3 , λ3 ) + √ . √ λ1 (λ1 − λ2 ) λ2 (λ1 − λ2 )
Proof. Noting that, as t → +∞ and x → −∞ such that z0 < −2 and (x, t) ∈ m , gn∗ m = O(1), n = m, and gn∗ m = O(exp(−4t sin(φn )|cos(φn )−cos(φm )|)),
367
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
n ∈ {1, 2, . . . , m − 1}, one deduces from Proposition 4.1 that {an , cn }m n=1 solve
A1
o(1)
o(1) .. .
A2 .. .
g∗ B
− ςmm −ςm11 o(1) .. . .. . g∗ D − ςmm −ςm11
... .. . .. .
o(1) .. .
o(1) ∗B gm mm−1 − ··· ςm −ςm−1 Am
o(1) .. . .. . g∗ D
... ... .. .. . . .. .. . .
o(1) .. . .. .
g∗ D
− ςmm −ςm11 · · · · · · − ςmm −ςmm m
... ... .. .. . . .. .. . .
o(1) .. . .. .
A1 .. . .. .
··· ···
g∗ D − ςmm −ςmm m
g∗ B − ςmm −ςm11
A2 .. .
... .. . .. .
o(1)
···
g∗ B − ςmm −ςmm−1 m−1
Am
o(1)
o(1) .. .
a1 a2 .. . .. . am c1 c2 .. . . .. cm
c c = [o(1), . . . , o(1), gm∗ χ12 (ςm ), o(1), . . . , o(1), gm∗ χ22 (ςm ) ]T , 34 5 2 2 34 5 m
m
where T denotes transposition, An , Bnk , and Dnk are given in the proof of Proposition 4.1, o(1) := O(exp(−4t min m∈{1,2,...,N} {sin(φn )|cos(φn ) − cos(φm )|})), and n∈{1,2,...,m−1} c c (·) and χ22 (·) are given in Lemma 4.2. Solving the above system for {an , cn }m χ12 n=1 via the Cauchy–Binet formula, or Cramer’s rule, recalling the expressions for χijc (ζ ), c (ζ ) and cS (z0 ) as in the proposii, j ∈ {1, 2}, given in Lemma 4.2, setting χ 12 c c (ςn )/χ22 (ςn ) and dn = tion, and recalling from Proposition 4.1 that bn = −an χ12 m−1 c c −cn χ12 (ςn )/χ22 (ςn ), one gets the estimates for {an , bn , cn , dn }n=1 and the explicit – asymptotic expansion – formulae for {am , bm , cm , dm } stated in the proposition. Furthermore, setting A B , Y := B A with A and B defined in Proposition 4.1, from the asymptotic estimates above for {an , bn , cn , dn }m n=1 , and recalling that, as a consequence of the asymptotic solvability of the original RHP formulated in Lemma 2.1.2, det(Y) ≡ 0, an application $ 2m 2 of Hadamard’s inequality (|det(Y)|2 2m j =1 i=1 |Yij | , where Yij denotes the (i j )-element of Y) shows that m sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm ) 2φ(x,t ) 2 2 e + 1+ 0 < |det(Y)| sin2 ( 12 (φm + φj )) j =1 cS (z0 ) ln t , (66) +O (z02 + 32)1/2 t where φ(x, t) := −2 sin(φm )(x + 2t cos φm ),
(67)
368
A. H. VARTANIAN
P (φm , φk ) :=
N −1 m−1 sin( 1 (φm + φk )) sin( 1 (φm + φk )) 2 2
sin( 12 (φm − φk )) sin( 12 (φm − φk )) k=m+1 λ2 +∞ 0 λ1 + − − × Q(φm ) := exp
,
(68)
k=1
0
−∞
λ
λ
1 2 sin(φm ) ln(1 − |r(µ)|2 ) dµ , × 2 (µ − 2µ cos(φm ) + 1) 2π
(69) ✷
and cS (z0 ) is given in the proposition.
The following lemma is proved via the higher-order generalisation [54] of the Deift–Zhou (DZ) nonlinear steepest descent method [55] (see, also, [56]), but its proof is far beyond the scope of the present work (it shall be presented elsewhere). Remark 4.1. Even though in Lemma 4.3 below, in the sensus strictu of asymptotic analysis, the exponentially small terms should be neglected, and thus not written out explicitly, in lieu of the t −p/2 (ln t)q corrections, p 1, q ∈ {0, 1, . . . , p − 1}, they are written there, and there only (see, also, Appendix A, Lemma A.1.7), in order to bring to the reader’s attention the fact that there are additional, albeit exponentially small, terms that are due to the remaining solitons: thereafter, exponentially small terms are neglected. LEMMA 4.3. As t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, ζ +a + a+ 1
P (ζ ) =
3
ζ +a2+
ζ +a4+
ζ +a4+
ζ +a2+
a3+
ζ +a1+
,
where a1+
=
a2+
=1+
p−1 1 ∞ apq (z0 )(ln t)q p=1 q=0
t p/2
+
−4t min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{1,2,...,m−1} , +O e a3+
=
p−1 3 ∞ apq (z0 )(ln t)q p=1 q=0
a4+
= 1+
t p/2
−4t min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{1,2,...,m−1} +O e ,
p−1 4 ∞ apq (z0 )(ln t)q p=1 q=0
t p/2
−4t min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{1,2,...,m−1} +O e ,
k (z0 ) ∈ cS (z0 ), k ∈ {1, 3, 4}, and P (ζ ) = σ1 P (ζ ) σ1 . apq
369
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
Remark 4.2. Even though Lemma 4.3 is not proven in this paper, it will be shown that (see the proof of Proposition 4.6 below), up to the leading-order terms retained in this work, namely, terms that are cS (z0 ) ln t 3 , a10 (z0 ) = 0; O (z02 + 32)1/2 t thus, actually, a3+ =
p−1 3 ∞ apq (z0 )(ln t)q
t p/2
p=2 q=0
+ O exp −4t =O
+
min
m∈{1,2,...,N} n∈{1,2,...,m−1}
{sin(φn )|cos(φn ) − cos(φm )|}
c (z0 ) ln t . (z02 + 32)1/2 t S
Furthermore, to cS (z0 ) ln t , O (z02 + 32)1/2 t the asymptotic expansion for a4+ plays, in fact, no role in the final formulae of this paper. As a possible prelude to a motivation of why ai+ , i ∈ {1, 2, 3, 4}, have, modulo exponentially small terms, the asymptotic expansions stated in Lemma 4.3, one can apply the higher-order generalisation of the DZ method [54] to the proof of Lemma 6.1 in [38] to show that χijc (ζ ), i, j ∈ {1, 2}, have the asymptotic expansion χijc (ζ )
= δij +
p−1 ∞ (χijc (z0 ))pq (fij (ζ ))pq (ln t)q
t p/2
p=1 q=0
,
where δij is the Kronecker delta, c c (·))10 = (χ22 (·))10 = 0, (χ11
and
(fij (·))pq L∞ (C\"z∈{λ
1 ,λ2 }
U(z;ε))
< ∞.
However, as stated heretofore, these details are omitted in this paper (it is the author’s conjecture that a3+ = a4+ = 0, namely, P (ζ ) is diagonal). 3 1 2 (z0 ) =: a1 , a10 (z0 ) =: a2 , a10 (z0 ) =: a3 , and PROPOSITION 4.3. Set a10 4 a10 (z0 ) =: a4 . Then as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, 0 N cm iδ −1 (0)e2i k=m+1 φk + ςm N 1 bm c0 c1 iδ −1 (0)e2i k=m+1 φk −(a1 − a2 ) m − √ + m + + ςm ςm ςm t
(+0 )11 = −
370
A. H. VARTANIAN
√
(+0 )12
(+0 )21
a0 a 0 2δ(0) ν(λ1 ) cos(:+ (z0 , t) + π4 ) + a3 1 − m − 1 − m + √ ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln t , +O (z02 + 32)1/2 t N am0 =− 1− iδ(0)e−2i k=m+1 φk + ςm N 1 am0 dm1 am iδ(0)e−2i k=m+1 φk −(a1 − a2 ) 1 − + + + + √ ςm ςm ςm t √ 0 0 2δ −1 (0) ν(λ ) cos(:+ (z , t) + π ) cm cm 1 0 4 − √ + + a3 ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln t , +O (z02 + 32)1/2 t N am0 = 1− iδ −1 (0)e2i k=m+1 φk + ςm N 1 dm1 am0 am iδ −1 (0)e2i k=m+1 φk (a1 − a2 ) 1 − + − − + √ ςm ςm ςm t √ 0 2δ(0) ν(λ ) cos(:+ (z , t) + π ) 0 cm cm 1 0 4 + √ + − a3 2 ςm ςm (λ1 − λ2 ) (z0 + 32)1/4 cS (z0 ) ln t , +O (z02 + 32)1/2 t
(+0 )22 =
0 N cm iδ(0)e−2i k=m+1 φk + ςm N 1 0 1 cm cm bm am0 iδ(0)e−2i k=m+1 φk (a1 − a2 ) √ + + − a3 1 − + + ςm ςm ςm ςm t √ am0 2δ −1 (0) ν(λ1 ) cos(:+ (z0 , t) + π4 ) √ + + 1− ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln t . +O (z02 + 32)1/2 t
Proof. Recall from Lemma 4.1 that +0 = P (0)md (0)χ (0)(δ(0)) c
σ3
N k=m+1
(dk+ (0))σ3
σ2 .
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
371
Collect, now, the following facts: (1) from Lemma 4.3, + + a1 /a2 a3+ /a4+ ; P (0) = a3+ /a4+ a1+ /a2+ (2) from the expression for md (ζ ) given in Lemma 4.1, the definition (cf. Propo sition 4.1) Res(χ (ζ ); ςn ) = acnn bdnn , n ∈ {1, 2, . . . , m}, and the asymptotics for
{an , bn , cn , dn }m n=1 given in Proposition 4.2, one shows that md (0) =
1−
am ςm
−
− ςbmm −
dm ςm
cm ςm
− ςbmm − 1−
am ςm
cm ςm − ςdmm
+
−4t min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{1,2,...,m−1} , + O2×2 e where O2×2 () denotes a 2 × 2 matrix each of whose entries are O(); (3) from c c Lemma 4.2 and the formula for χ 12 (ζ ) (= χ 21 (ζ )) given in Proposition 4.2, one shows that c √1 χ 1 (0) cS (z0 ) ln t c t 12 ; + O2×2 χ (0) = √1 χ c (0) 1 (zo2 + 32)1/2 t t 21 $ $ + −1 (dk+ (0))σ3 )σ2 = iδ −1 (0)( N and (4) (δ(0))σ3 ( N k=m+1 k=m+1 (dk (0)) )σ− $N + − iδ(0)( k=m+1 dk (0))σ+ . Using the results of (1)–(4), and recalling the expression for +0 given above, one arrives at (+0 )11
(+0 )12
(+0 )21
c N am dm a1+ −1 χ 12 (0) bm cm 1− = + iδ (0) √ − + (dk+ (0))−1 + − ςm ςm ςm ςm a2 t k=m+1 N + c a am dm χ (0) bm cm − √ + (d + (0))−1 , − 12 + 3+ iδ −1 (0) 1 − ςm ςm ςm ςm k=m+1 k a4 t N c a1+ am dm (0) bm cm χ 21 = + iδ(0) − 1 − − + dk+ (0)+ + √ ςm ςm ςm ςm a2 t k=m+1 N c a3+ χ 21 dm cm (0) am bm 1− − + dk+ (0), + + + iδ(0) − √ ςm ςm ςm ςm a4 t k=m+1 N + a am dm χ c (0) bm cm = 1 iδ −1 (0) 1 − − + (dk+ (0))−1 + − 12 √ + ς ς ς ς t m m m m a2 k=m+1 N + c a am dm χ (0) bm cm 1− √ − + (dk+ (0))−1 , − + 3 iδ −1 (0) 12 + ς ς ς ς t m m m m a4 k=m+1
372
A. H. VARTANIAN
(+0 )22
N c χ 21 dm cm (0) am bm 1− = iδ(0) − √ − + dk+ (0)+ + ςm ςm ςm ςm t a2+ k=m+1 N + c am a3 dm cm χ 21 (0) bm iδ(0) − 1 − − + dk+ (0). + √ + ςm ςm ςm ςm t a+ k=m+1
a1+
4
Using the asymptotic expansions for {am , bm , cm , dm } (respectively, {ai+ }4i=1 ) given in Proposition 4.2 (respectively, Lemma 4.3), one arrives at the leading-order results stated in the proposition. ✷ Remark 4.3. In Propositions 4.4 and 4.6 below, one should keep, everywhere, the upper (respectively, lower) signs for θγm = +π/2 (respectively, θγm = −π/2). PROPOSITION 4.4. Let φ(x, t), P (φm , φk ), and Q(φm ) be defined by Equations (67), (68), and (69), respectively. Then, for θγm = ±π/2, as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, 2i sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ) , (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) + 2 sin(φm )|γm |δ −1 (0)ei(φm +s )+φ(x,t )P (φm , φk )Q(φm ) 0 , cm = ∓ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))
am0 = −
am1
=
√ 16iλ21 sin2 (φm )|γm |3 ν(λ1 ) P 3 (φm , φk )Q3 (φm ) cos(s + )e3φ(x,t) ∓ × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π + ∓ + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4
∓
√ 2λ1 sin(φm )|γm | ν(λ1 ) P (φm , φk )Q(φm )eφ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) cos(φm + s + )× × 2 cos(s ) cos : (z0 , t) + 4 π π + + + − (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) + + × cos : (z0 , t) + 4 4 π − i(λ1 + λ2 ) sin(φm + s + )× + 2i sin(s + ) cos :+ (z0 , t) + 4 π π + + + + i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) + , × cos : (z0 , t) + 4 4
1 = bm
+
√ + 2iλ1 sin(φm )|γm |2 ν(λ1 ) δ(0)e−i(φm +s )+2φ(x,t) P 2 (φm , φk )Q2 (φm ) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π − (λ1 + λ2 ) cos(φm + s + )× × 2 cos(s + ) cos :+ (z0 , t) + 4
373 π π − (λ1 − λ2 ) sin(φm + s + ) sin :+ (z0 , t) + + × cos :+ (z0 , t) + 4 4 π + + − i(λ1 + λ2 ) sin(φm + s + )× + 2i sin(s ) cos : (z0 , t) + 4 π π + i(λ1 − λ2 ) cos(φm + s + ) sin :+ (z0 , t) + , × cos :+ (z0 , t) + 4 4
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
1 =− cm
√ + 16λ21 sin2 (φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P 2 (φm , φk )Q2 (φm ) cos(s + ) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
π + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + + 4 π + − + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4 −
√ + 2iλ1 sin(φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P 2 (φm , φk )Q2 (φm ) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) cos(φm + s + )× × 2 cos(s ) cos : (z0 , t) + 4 π π + + + − (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) + − × cos : (z0 , t) + 4 4 π + i(λ1 + λ2 ) sin(φm + s + )× − 2i sin(s + ) cos :+ (z0 , t) + 4 π π + + + − i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) + + × cos : (z0 , t) + 4 4
+
+
√ + 8λ21 sin2 (φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P 2 (φm , φk )Q2 (φm ) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
π + + (λ1 − λ2 ) sin(φm )× × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π cos(s + ) − i ((λ1 + λ2 ) cos(φm ) − 2)× × sin :+ (z0 , t) + 4 π π + + + + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + sin(s ) , × cos : (z0 , t) + 4 4 dm1
=
√ 2λ1 sin(φm )|γm | ν(λ1 ) P (φm , φk )Q(φm )eφ(x,t) ± × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) cos(φm + s + )× × 2 cos(s ) cos : (z0 , t) + 4 π π + + + − (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) + + × cos : (z0 , t) + 4 4
+
374
A. H. VARTANIAN
π + 2i sin(s + ) cos :+ (z0 , t) + − i(λ1 + λ2 ) sin(φm + s + )× 4 π π + + + + i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) + , × cos : (z0 , t) + 4 4 where s + is given in Theorem 2.2.1, Equation (11). 1 0 1 , cm , cm , dm1 } given in ProposiProof. Recalling the definitions of {am0 , am1 , bm c tion 4.2, substituting into them the expressions for gm∗ and χ 12 (ζ ) given in Propositions 4.1 and 4.2, respectively, using standard trigonometric identities, and defining φ(x, t), P (φm , φk ), and Q(φm ) as in Equations (67), (68), and (69), respectively, one obtains, after tedious, but otherwise straightforward calculations, the result stated in the proposition. PROPOSITION 4.5. As t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, √ i<+ (0) ν(λ1 ) e 2 λ12iν(λ1) + × u(x, t) = i (+0 )12 + a3 + bm + cm + √ t (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln t π π + + , (70) × (λ1 e−i(: (z0 ,t )+ 4 ) + λ2 ei(: (z0 ,t )+ 4 ) ) + O (z02 + 32)1/2 t x
(|u(x , t)|2 − 1) dx +∞ = −i (+0 )11 + +i
x −∞
a1+
−
0 −∞
+
a2+ λ1
+ am + dm + 2i
N
dµ ln(1 − |r(µ)| ) 2π
2
λ2
sin(φk )+
k=m+1
cS (z0 ) ln t +O , (71) (z02 + 32)1/2 t
(|u(x , t)|2 − 1) dx
x
N
+∞
dµ . (72) ln(1 − |r(µ)|2 ) 2π +∞ −∞ n=1 x Proof. Recall Equations (63), (64), and (65) for u(x, t), +∞ (|u(x , t)|2 −1) dx , and χ(ζ ), respectively. Using the result for P (ζ ) (respectively, χijc (ζ ), i, j ∈ {1, 2}) stated in Lemma 4.3 (respectively, Lemma 4.2), noting that 0 λ1 ±1 2 dµ ζ −1 + O(ζ −2 ) = 1±i + ln(1 − |r(µ)| ) (δ(ζ )) ζ →∞ 2π λ2 −∞ =
(|u(x , t)| − 1) dx − 2 2
and N k=m+1
(dk+ (ζ ))±1
= 1±
ζ →∞
sin(φn ) −
N k=m+1
(ςk − ςk ) ζ −1 + O(ζ −2 ),
375
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
and using the asymptotic estimates for {an , bn , cn , dn }m−1 n=1 given in Proposition 4.2, one forms the large-ζ asymptotics for χ (ζ ) given in Equation (65) to show that (ζ(χ (ζ )(δ(ζ ))σ3
N
(dk+ (ζ ))σ3 − I))11
k=m+1
=
ζ →∞ ζ ∈C\(σd ∪σc )
(+o )11 + +i
a1+
0 −∞
−
+
a2+ λ1
+ am + dm +
ln(1 − |r(µ)|2 )
λ2 S
(ζ(χ (ζ )(δ(ζ ))
N
(ςk − ςk )+
k=m+1
c (z0 ) ln t +O 2 (z0 + 32)1/2 t σ3
N
dµ + 2π
+ O(e−Qt ),
(dk+ (ζ ))σ3 − I))12
k=m+1
√
i<+ (0)
1) ν(λ1 ) e 2 λ2iν(λ + 1 = (+0 )12 + a3 + bm + cm + √ × ζ →∞ t (λ1 − λ2 ) (z02 + 32)1/4 ζ ∈C\(σ ∪σc ) d
+
+
× (λ1 e−i(: (z0 ,t )+ 4 ) + λ2 ei(: (z0 ,t )+ 4 ) )+ cS (z0 ) ln t + O(e−Qt ), +O (z02 + 32)1/2 t π
π
where Q := 4 min m∈{1,2,...,N} {sin(φn )|cos(φn ) − cos(φm )|} (> 0). Neglecting exn=m∈{1,2,...,N} ponentially small terms (cf. Remark 4.1), from the expressions for u(x, t) and x 2 +∞ (|u(x , t)| − 1) dx given, respectively, in Equations (63) and (64), and the trace identity (cf. Equation (4)) x +∞ +∞ (|u(x , t)|2 − 1) dx = + (|u(x , t)|2 − 1) dx −∞
= −2
−∞ N
x
sin(φn ) −
n=1
one obtains the results stated in the proposition.
+∞ −∞
ln(1 − |r(µ)|2 )
dµ , 2π ✷
PROPOSITION 4.6. As t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, for θγm = ±π/2, 2 sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) + (+0 )11 = i ± (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) √ π ν(λ1 ) + cos(s + )+ −2 cos : (z0 , t) + +√ 4 t (λ1 − λ2 ) (z02 + 32)1/4
376
A. H. VARTANIAN
+
4 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) sin(s + − φm ) cos(:+ (z0 , t) + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )
π 2φ(x,t) 4 )e
+
8λ21 sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm )(1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )e2φ(x,t)
× π + × ((λ1 + λ2 ) cos(φm ) − 2) cos :+ (z0 , t) + 4 π + cos(s + )+ + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4 4λ1 sin(φm ) cos(φm )|γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ) × + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))(λ21 − 2λ1 cos(φm ) + 1) π − (λ1 + λ2 ) sin(s + )× × 2 sin(s + − φm ) cos :+ (z0 , t) + 4 π π + + + + (λ1 − λ2 ) cos(s ) sin : (z0 , t) + + × cos : (z0 , t) + 4 4 cS (z0 ) ln t , +O (z02 + 32)1/2 t +
(1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2
+
+
(+0 )12 = −ie−i(θ (1)+s ) + + + 2 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm )e−i(θ (1)+φm +s )+2φ(x,t ) + + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) 1 2i Im(a1 − a2 ) sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm )e−i(θ + (1)+φm +s+ )+2φ(x,t) ± +√ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) t ± ∓
√ + + 4i sin(φm )|γm |P (φm , φk )Q(φm ) ν(λ1 ) e−i(θ (1)+2s )+φ(x,t) cos(:+ (z0 , t) + √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) (λ1 − λ2 ) (z02 + 32)1/4
π 4)
∓
√ + + 16iλ21 sin2 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) e−i(θ (1)+s )+3φ(x,t) cos(s + ) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
π + + × ((λ1 + λ2 ) cos(φm ) − 2) sin(φm ) cos : (z0 , t) + 4 π 2 + + + (λ1 − λ2 ) sin (φm ) sin : (z0 , t) + 4 π + + i ((λ1 + λ2 ) cos(φm ) − 2) cos(φm ) cos :+ (z0 , t) + 4 π + + + (λ1 − λ2 ) sin(φm ) cos(φm ) sin : (z0 , t) + 4
+ Im(a1 − a2 )e−i(θ − ×
+ (1)+s + )
−
√ + + 4λ1 sin(φm )|γm |P (φm , φk )Q(φm ) ν(λ1 ) e−i(θ (1)+s )+φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π ± (λ1 + λ2 ) sin(s + )× ∓ 2 sin(s + − φm ) cos :+ (z0 , t) + 4
377 π π ∓ (λ1 − λ2 ) cos(s + ) sin :+ (z0 , t) + + × cos :+ (z0 , t) + 4 4 cS (z0 ) ln t , +O (z02 + 32)1/2 t
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
Im(a1 − a2 ) =
√ ν(λ1 ) sin(s + ) cos(:+ (z0 , t) + π4 )(1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) ± ± √ (λ1 − λ2 ) (z02 + 32)1/4 sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t)
√ 4λ21 ν(λ1 ) sin(φm )|γm |P (φm , φk )Q(φm ) sin(s + )eφ(x,t ) × ± √ (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π + ± + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4 √ 2λ1 ν(λ1 ) cos(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) √ × ± 2 (λ1 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4 π − × 2 cos(s + − φm ) cos :+ (z0 , t) + 4 π + + − − (λ1 + λ2 ) cos(s ) cos : (z0 , t) + 4 π ± − (λ1 − λ2 ) sin(s + ) sin :+ (z0 , t) + 4 ±
√ 2 ν(λ1 )|γm |P (φm , φk )Q(φm ) cos(s + − φm ) cos(:+ (z0 , t) + √ (λ1 − λ2 ) (z02 + 32)1/4
+O
π φ(x,t) 4 )e
+
cS (z0 ) ln t , (z02 + 32)1/2 t
Re(a1 − a2 ) = Re(a3 ) = Im(a3 ) = 0, where θ + (·) is given in Theorem 2.2.1, Equation (8). Recall from Lemma 4.1 that: (1) +0 = P (0)md (0)χ c (0)(δ(0))σ3 × $Proof. N ( k=m+1 (dk+ (0))σ3 )σ2 ; (2) tr(+0 ) = 0; (3) det(+0 ) = −1; and (4) +0 +0 = I. Taking the determinant of both sides of the above expression for +0 and using the S (z ) ln t 0 fact that det(+0 ) = −1, it follows that, modulo terms that are O( (z2c+32) 1/2 t ), and 0
always ignoring exponentially small terms, det(P (0)) = (det(md (0)))−1 . Before proceeding further, this will be verified; in particular, since md (ζ ) ∈ SL(2, C), it must be the case that, modulo terms that are cS (z0 ) ln t , det(md (0)) = 1. O (z02 + 32)1/2 t
378
A. H. VARTANIAN
From Lemma 4.3, keeping only leading-order terms, one shows that 2Re(a1 − a2 ) cS (z0 ) ln t det(P (0)) = 1 + +O , √ (z02 + 32)1/2 t t and, from the proof of Proposition 4.1, the estimates of Proposition 4.2, and noting that am0 c0 c0 am0 1− − m m =1 1− ςm ςm ςm ςm and
+∞ −∞
(1 − µ2 ) ln(1 − |r(µ)|2 ) dµ =0 (µ2 − 2µ cos(φm ) + 1) µ
(which is proven using the symmetry reduction r(ζ −1 ) = −r(ζ )), one shows that (det(md (0)))−1
1 0 1 1 2 am0 dm1 cm am cm bm = 1 + √ Re 1 − + + + + ςm ςm ςm ςm ςm ςm t cS (z0 ) ln t ; +O (z02 + 32)1/2 t
thus, from the – yet to be verified – identity det(P (0)) = (det(md (0)))−1 , and the above, it follows that 1 0 1 1 am0 dm1 cm am cm bm + + + . Re(a1 − a2 ) = Re 1 − ςm ςm ςm ςm ςm ςm If the formulae presented thus far are correct, then one must be able to show from them that the right-hand side of the latter relation equals zero. From Proposition 4.2 and repeated application of standard trigonometric identities, one shows, after a very lengthy and tedious algebraic calculation, that (θγm = ±π/2) 0 1 c1 b c Re m m + m ςm ςm ςm =±
√ 16λ21 sin3 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) (1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )e3φ(x,t) × √ (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )3
π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π cos(s + )± + (λ1 − λ2 ) sin(φm ) sin :+ (z0 , t) + 4 ±
√ 8λ1 sin2 (φm ) cos(φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) e3φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
379 π − (λ1 + λ2 ) sin(s + )× × 2 sin(s + − φm ) cos :+ (z0 , t) + 4 π π + + + + (λ1 − λ2 ) cos(s ) sin : (z0 , t) + , × cos : (z0 , t) + 4 4
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
and 1 am a0 d1 Re 1 − m + m ςm ςm ςm =∓
√ 16λ21 sin3 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) (1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )e3φ(x,t) × √ (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )3
π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π + (λ1 − λ2 ) sin(φm ) sin :+ (z0 , t) + cos(s + )∓ 4 ∓
√ 8λ1 sin2 (φm ) cos(φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) e3φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) sin(s + )× × 2 sin(s − φm ) cos : (z0 , t) + 4 π π + (λ1 − λ2 ) cos(s + ) sin :+ (z0 , t) + ; × cos :+ (z0 , t) + 4 4
+
thus, adding, 1 0 1 1 am0 dm1 cm am cm bm + + + = 0, Re 1 − ςm ςm ςm ςm ςm ςm whence Re(a1 − a2 ) = 0. Recalling the expression for (+0 )11 given in Proposition 4.3, the estimates and expansions of Proposition 4.2, and the fact – just established – that Re(a1 − a2 ) = 0, one shows that 1 1 cS (z0 ) ln t β γ α , (+0 )11 = √ (+0 )11 + i (+0 )11 + √ (+0 )11 + O (z02 + 32)1/2 t t t where (+0 )α11 2 Im(a1 − a2 ) sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) √ 2 ν(λ1 ) cos(:+ (z0 , t) + π4 ) sin(s + ) − + √ (λ1 − λ2 ) (z02 + 32)1/4 − Re(a3 ) sin(θ + (1) + s + ) − Im(a3 ) cos(θ + (1) + s + )+
:= ∓
380
A. H. VARTANIAN
+
√ 4 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) cos(:+ (z0 , t) + π4 ) cos(s + − φm )e2φ(x,t) + √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) (λ1 − λ2 ) (z02 + 32)1/4
+
2 Re(a3 ) sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) cos(s + + φm + θ + (1))e2φ(x,t) − (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )
− +
2 Im(a3 ) sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) sin(s + + φm + θ + (1))e2φ(x,t) + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) √ 8λ21 sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) sin(s + )e2φ(x,t) × √ 2 2 (1 − |γm | P (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
π + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + + 4 π + + (λ1 − λ2 ) sin(φm ) sin :+ (z0 , t) + 4 +
√ 4λ1 sin(φm ) cos(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) e2φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) cos(s + )× × 2 cos(s − φm ) cos : (z0 , t) + 4 π π + + + − (λ1 − λ2 ) sin(s ) sin : (z0 , t) + , × cos : (z0 , t) + 4 4
+
β
(+0 )11 := ±
2 sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) , (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))
γ
(+0 )11
√ 2 ν(λ1 ) cos(:+ (z0 , t) + π4 ) cos(s + ) := − √ + (λ1 − λ2 ) (z02 + 32)1/4 + Re(a3 ) cos(θ + (1) + s + ) − Im(a3 ) sin(θ + (1) + s + )+ +
√ 4 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) cos(:+ (z0 , t) + π4 ) sin(s + − φm )e2φ(x,t) + √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) (λ1 − λ2 ) (z02 + 32)1/4
+
2 Re(a3 ) sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) sin(s + + φm + θ + (1))e2φ(x,t) + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )
+
2 Im(a3 ) sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) cos(s + + φm + θ + (1))e2φ(x,t) + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )
√ 8λ21 sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) cos(s + ) √ × + (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 (1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))e2φ(x,t ) × × (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))2 π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π + + (λ1 − λ2 ) sin(φm ) sin :+ (z0 , t) + 4
381
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
+
√ 4λ1 sin(φm ) cos(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) ν(λ1 ) e2φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − (λ1 + λ2 ) sin(s + )× × 2 sin(s − φm ) cos : (z0 , t) + 4 π π + + + + (λ1 − λ2 ) cos(s ) sin : (z0 , t) + , × cos : (z0 , t) + 4 4
+
and θ + (·) is specified in the proposition. Recalling that tr(+0 ) = 0, it follows that Re((+0 )11 ) = 0; thus, (+0 )α11 = 0, which gives a relation for Im(a1 − a2 ), but, since Re(a3 ) and Im(a3 ) are as yet undetermined, this is not enough. Towards this end, one uses the condition det(+0 ) = (+0 )11 (+0 )11 − (+0 )12 (+0 )12 = −1 (Note: if the conditions tr(+0 ) = 0 and det(+0 ) = −1 are satisfied, then it follows that +0 +0 = I is also satisfied, so it is enough to use the condition det(+0 ) = −1). From the formula for (+0 )11 given above, and the expression for (+0 )12 given in Proposition 4.3, one shows that a0 a0 a0 2 1 − m + √ Re (a1 − a2 ) 1 − m × (+0 )12 (+0 )12 = 1 − m ςm ςm ςm t a0 a0 d1 am1 + m − Re 1 − m + × 1− m ςm ςm ςm ςm 0√ ν(λ1 ) δ(0) cos(:+ (z0 , t) + π4 ) am0 2cm √ − + Re 1 − ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 0 am0 cm a3 cS (z0 ) ln t . +O − Re 1 − ςm ςm (z02 + 32)1/2 t Using the estimates given in Proposition 4.2, and recalling that Re(a1 − a2 ) = 0, one gets that (+0 )12 (+0 )12 4 sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ) + =1+ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) 4 sin2 (φm )|γm |4 P 4 (φm , φk )Q4 (φm )e4φ(x,t ) + + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))2 √ 2 4 sin(φm )|γm |P (φm , φk )Q(φm ) ν(λ1 ) cos(s + ) cos(:+ (z0 , t) + π4 )eφ(x,t) ± +√ ∓ √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) (λ1 − λ2 ) (z02 + 32)1/4 t ±
√ 8 sin2 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) sin(s + − φm ) cos(:+ (z0 , t) + √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ1 − λ2 ) (z02 + 32)1/4
±
2 sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) (Re(a3 ) cos(s + + θ + (1))− (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))
π 3φ(x,t) 4 )e
±
382
A. H. VARTANIAN
4 sin2 (φm )|γm |3 P 3 (φm , φk )Q3 (φm )e3φ(x,t ) × (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))2 × (Re(a3 ) sin(s + + θ + (1) + φm ) + Im(a3 ) cos(s + + θ + (1) + φm ))± √ 16λ21 sin3 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) cos(s + ) ± √ × (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 − Im(a3 ) sin(s + + θ + (1))) ±
(1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))e3φ(x,t ) × (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))3 π + + × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) + 4 π + ± + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4
×
±
√ 8λ1 sin2 (φm ) cos(φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) e3φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π + − × 2 sin(s − φm ) cos : (z0 , t) + 4 π + + + − (λ1 + λ2 ) sin(s ) cos : (z0 , t) + 4 cS (z0 ) ln t π + + +O . + (λ1 − λ2 ) cos(s ) sin : (z0 , t) + 4 (z02 + 32)1/2 t
+
From the expression for (+0 )11 given above, and using the fact that (+0 )α11 = 0, one shows that (+0 )11 (+0 )11 4 sin2 (φm )|γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ) = + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))2 √ 2 4 sin(φm )|γm |P (φm , φk )Q(φm ) ν(λ1 ) cos(s + ) cos(:+ (z0 , t) + π4 )eφ(x,t) ± +√ ∓ √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) ) (λ1 − λ2 ) (z02 + 32)1/4 t ±
√ 8 sin2 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) sin(s + − φm ) cos(:+ (z0 , t) + √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ1 − λ2 ) (z02 + 32)1/4
π 3φ(x,t) 4 )e
2 Re(a3 ) sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) cos(s + + θ + (1))+ ± (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) 2 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) sin(s + + φm + θ + (1))e2φ(x,t ) ± + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) 2 Im(a3 ) sin(φm )|γm |P (φm , φk )Q(φm )eφ(x,t ) − sin(s + + θ + (1))+ ± (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))
±
383
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
2 sin(φm )|γm |2 P 2 (φm , φk )Q2 (φm ) cos(s + + φm + θ + (1))e2φ(x,t ) ± + (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t )) √ 16λ21 sin3 (φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) cos(s + ) √ × ± (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 (1 + |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))e3φ(x,t ) × × (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t ))3 π + × ((λ1 + λ2 ) cos(φm ) − 2) cos :+ (z0 , t) + 4 π + ± + (λ1 − λ2 ) sin(φm ) sin : (z0 , t) + 4 ±
√ 8λ1 sin2 (φm ) cos(φm )|γm |3 P 3 (φm , φk )Q3 (φm ) ν(λ1 ) e3φ(x,t) × √ (1 − |γm |2 P 2 (φm , φk )Q2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
π × 2 sin(s + − φm ) cos :+ (z0 , t) + − 4 π + + + − (λ1 + λ2 ) sin(s ) cos : (z0 , t) + 4 cS (z0 ) ln t π + + +O . + (λ1 − λ2 ) cos(s ) sin : (z0 , t) + 4 (z02 + 32)1/2 t Now, taking note of the relation a0 a0 β β 1 − m = −1, (+0 )11 (+0 )11 − 1 − m ςm ςm one substitutes the above-derived formulae for |(+0 )11 |2 and |(+0 )12 |2 into S (z ) ln t 0 |(+0 )11 |2 − |(+0 )12 |2 = −1, and, modulo terms that are O( (z2c+32) 1/2 t ), gets exact 0
cancellation at O(1) and O(t −1/2 ); thus, one concludes that Re(a3 ) = Im(a3 ) = 0. Recalling that (+0 )α11 = 0, and using the fact that Re(a3 ) = Im(a3 ) = 0, from the expression for (+0 )11 given above, one obtains, after some straightforward algebra, the expressions for Im(a1 − a2 ) and (+0 )11 stated in the Proposition. From Proposition 4.2, and the fact that Re(a3 ) = Im(a3 ) = Re(a1 − a2 ) = 0, one obtains, upon recalling the expression for (+0 )12 given in Proposition 4.3, the formula for ✷ (+0 )12 given in the proposition.
LEMMA 4.4. As t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, x2, . . . , N}, u(x, t), the solution of the Cauchy problem for the Df NLSE, and ±∞ (|u(x , t)|2 − 1) dx have the leading-order asymptotic expansions (for the upper sign) stated in Theorem 2.2.1, x Equations (7)–(20). Proof. The asymptotic expansions for u(x, t) and ±∞ (|u(x , t)|2 −1) dx follow from Proposition 4.2, Proposition 4.4, Equations (70)–(72), and Proposition 4.6 after tedious, but otherwise straightforward algebraic calculations. ✷
384
A. H. VARTANIAN
Appendix A. Asymptotic Analysis as t → −∞ In a silhouette of the asymptotic analysis for u(x, t) and x this appendix, 2 (|u(x , t)| − 1) dx as t → −∞ and x → +∞ such that z0 := x/t < −2 and ±∞ (x, t) ∈ m , m ∈ {1, 2, . . . , N}, is presented. Since the calculations are analogous to those of Sections 3 and 4, only final results/statements, with in one instance a sketch of a proof, are given: one mimics the scheme of the calculation in Sections 3 and 4 to arrive at the corresponding asymptotic results. The analogue of Lemma 3.1 is LEMMA A.1.1. For r(ζ ) ∈ SC1 (R), let m(ζ ): C \ (σd ∪ σc ) → M2 (C) be the solution of the RHP formulated in Lemma 2.1.2. Set m (ζ ) := m(ζ )( δ (ζ ))−σ3 , where λ2 +∞ ln(1 − |r(µ)|2 ) dµ , + δ (ζ ) = exp (µ − ζ ) 2π i λ1 0 with λ1 and λ2 given in Theorem 2.2.1, Equation (10), δ(ζ ) δ (ζ ) δ (ζ −1 ) = δ (ζ ) = 1, ±1 ±1 (ζ ): C\(σd ∪σc ) → δ (0), and (δ (·)) L∞ (C) := supζ ∈C |(δ (ζ )) | < ∞. Then m M2 (C) solves the following RHP: (i) m (ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; (ζ ) satisfy the jump condition (ii) m ± (ζ ) := lim ζ →ζ m ±Im(ζ )>0
− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 )) G(ζ ), m + (ζ ) = m
ζ ∈ R,
where G(ζ ) =
δ− (ζ )( δ+ (ζ ))−1 (1 − r(ζ )r(ζ )) r(ζ )(δ− (ζ )δ+ (ζ ))−1
−r(ζ ) δ− (ζ ) δ+ (ζ ) (δ− (ζ ))−1 δ+ (ζ )
;
" (iii) m (ζ ) has simple poles in σd = N n=1 ({ςn } ∪ {ςn }) with (ζ )gn ( δ (ςn ))−2 σ− , n ∈ {1, 2, . . . , N}, Res( m(ζ ); ςn ) = lim m ζ →ςn
m(ζ ); ςn ) σ1 , Res( m(ζ ); ςn ) = σ1 Res( where gn is defined in Lemma 3.1(iii); (iv) det( m(ζ ))|ζ =±1 = 0; δ (0))σ3 σ2 + O(1); (v) m (ζ ) =ζ →0 ζ −1 ( (vi) m (ζ ) = ζ →∞ I + O(ζ −1 );
n ∈ {1, 2, . . . , N},
ζ ∈C\(σd ∪σc )
(ζ ) σ1 and m (ζ −1 ) = ζ m (ζ )( δ (0))σ3 σ2 . (vii) m (ζ ) = σ1 m Let u(x, t) := i
lim
ζ →∞ ζ ∈C\(σd ∪σc )
(ζ( m(ζ )( δ (ζ ))σ3 − I))12 ,
(73)
385
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
and
x +∞
(|u(x , t)|2 − 1) dx := −i
lim
ζ →∞ ζ ∈C\(σd ∪σc )
(ζ( m(ζ )( δ (ζ ))σ3 − I))11 .
(74)
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. The analogue of Definition 3.1 is DEFINITION A.1.1. For m ∈ {1, 2, . . . , N} and {ςn }m−1 n=1 ⊂ C+ (respectively, m−1 {ςn }n=1 ⊂ C− ), define the clockwise (respectively, counter-clockwise) oriented n := {ζ ; |ζ − ςn | = n := {ζ ; |ζ − ςn | = εnK } (respectively, L εnL }), with circles K K L n = εn (respectively, εn ) chosen sufficiently small such that Kn ∩ Kn = Ln ∩ L n = K n ∩ σc = L n ∩ σc = ∅ ∀n = n ∈ {1, 2, . . . , m − 1}. n ∩ L K The analogue of Lemma 3.2 is (ζ ): C \ (σd ∪ σc ) → M2 (C) be the LEMMA A.1.2. For r(ζ ) ∈ SC1 (R), let m solution of the RHP formulated in Lemma A.1.1. Set " m (ζ ), ζ ∈ C \ (σc ∪ ( m−1 n=1 (Kn ∪ int(Kn ) ∪ Ln ∪ int(Ln )))), ( δ(ςn ))−2 6 n ), n ∈ {1, 2, . . . , m − 1}, σ− , ζ ∈ int(K (ζ ) I − gn(ζ m (ζ ) := m −ςn ) ( δ(ςn ))−2 n ), n ∈ {1, 2, . . . , m − 1}. σ+ , ζ ∈ int(L m (ζ ) I + gn(ζ −ςn ) " "m−1 Then m 6 (ζ ): C \ ((σd \ m−1 n=1 ({ςn } ∪ {ςn })) ∪ (σc ∪ ( n=1 (Kn ∪ Ln )))) → M2 (C) solves the following RHP: " (i) m 6 (ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ (σc ∪ ( m−1 n=1 (Kn ∪ n ))); L m 6 (ζ ) satisfy the jump condition (ii) m 6± (ζ ) := lim ζ →ζ " n ∪L n )) ζ ∈± side of σc ∪( m−1 (K n=1
m 6+ (ζ )
=
m 6− (ζ ) υ 6 (ζ ),
ζ ∈ σc ∪
m−1 #
n ∪ L n ) , (K
n=1
where
exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 )) G(ζ ), ζ ∈ R, gn ( δ(ςn ))−2 n , n ∈ {1, 2, . . . , m − 1}, υ 6 (ζ ) = I + (ζ −ςn ) σ− , ζ ∈ K ( δ(ςn ))−2 n , n ∈ {1, 2, . . . , m − 1}, σ+ , ζ ∈ L I + gn(ζ −ςn )
with G(ζ ) given in Lemma A.1.1(ii);
386
A. H. VARTANIAN
(iii) m 6 (ζ ) has simple poles in σd \
"m−1 n=1
({ςn } ∪ {ςn }) with
Res( m6 (ζ ); ςn ) = lim m 6 (ζ )gn ( δ (ςn ))−2 σ− , ζ →ςn
n ∈ {m, m + 1, . . . , N}, m6 (ζ ); ςn ) σ1 , Res( m6 (ζ ); ςn ) = σ1 Res(
n ∈ {m, m + 1, . . . , N};
(iv) det( m6 (ζ ))|ζ =±1 = 0; δ (0))σ3 σ2 + O(1); (v) m 6 (ζ ) =ζ →0 ζ −1 ( " "m−1 (vi) as ζ → ∞, ζ ∈ C \ ((σd \ m−1 n=1 ({ςn } ∪ {ςn })) ∪ (σc ∪ ( n=1 (Kn ∪ Ln )))), m 6 (ζ ) = I + O(ζ −1 ); 6 (ζ ) σ1 and m 6 (ζ −1 ) = ζ m 6 (ζ )( δ (0))σ3 σ2 . (vii) m 6 (ζ ) = σ1 m For ζ ∈ C \ ((σd \
"m−1
n=1 ({ςn } ∪ 6
{ςn })) ∪ (σc ∪ (
"m−1 n=1
n ∪ L n )))), let (K
m (ζ )(δ (ζ ))σ3 − I))12 , u(x, t) := i lim (ζ( ζ →∞
and
x +∞
(75)
(|u(x , t)|2 − 1) dx := −i lim (ζ( m(ζ )( δ (ζ ))σ3 − I))11 .
(76)
ζ →∞
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. The analogue of Lemma 3.3 is " LEMMA A.1.3. For m ∈ {1, 2, . . . , N}, let σd := σd \ m−1 n=1 ({ςn } ∪ {ςn }), σc := "m−1 n ∪ L n )), where K n and L n are given in Definition A.1.1, and σc ∪ ( n=1 (K σO D := σd ∪ σc (σd ∩ σc = ∅). Set $ + −σ3 m 6 (ζ ) m−1 , k=1 (dk (ζ )) " m−1 n ) ∪ int(L n )))), ζ ∈ C \ (σc ∪ ( n=1 (int(K $ − −σ3 m , 6 (ζ )(JKn (ζ ))−1 m−1 k=1 (dk (ζ )) 8 m (ζ ) := n ), n ∈ {1, 2, . . . , m − 1}, ζ ∈ int(K $ − 6 −σ3 L (ζ ))−1 m−1 (ζ )( J , m k=1 (dk (ζ )) n n ), n ∈ {1, 2, . . . , m − 1}, ζ ∈ int(L where dn± (ζ ) are given in Lemma 3.3, JKn (ζ ) (∈ SL(2, C)) and JLn (ζ ) (∈ SL(2, "m−1 " C)), respectively, are holomorphic in m−1 k=1 int(Kk ) and l=1 int(Ll ), with $m−1 dk+ (ζ ) CnK gn (δ(ςn ))−2 $m−1 (dk+ (ζ ))−1 − (ζ ) k=1 d − (ζ ) − k=1 + −1 K $ (ζ −ςn )2 d (d (ζ )) C m−1 k=n k n k k=n k k=1 (ζ −ςn ) (ζ −ςn )2 dk− (ζ ) , JKn (ζ ) = k=n − − $ $ dk (ζ ) dk (ζ ) −gn ( δ (ςn ))−2 m−1 (ζ − ςn ) m−1 k=1 k=1 d + (ζ ) d + (ζ ) k=n
k
k=n
k
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
(ζ − ςn )
JLn (ζ ) =
L C
− (ζ −ςn n )2
$m−1 k=1 k=n
$m−1 k=1 k=n
dk+ (ζ ) dk− (ζ )
dk+ (ζ ) k=1 − n n k=n dk (ζ ) −2 $m−1 dk− (ζ ) CnL gn ( $ dk− (ζ ) δ(ςn )) m−1 k=1 d + (ζ ) − k=1 (d + (ζ ))−1 (ζ −ςn )2 k=n k k=n k
g ( δ (ς ))
dk− (ζ ) (dk+ (ζ ))−1
$m−1 −2
387 ,
(ζ −ςn )
and nK C
−2i
m−1
nL = −4 sin (φn )(gn ) ( =C δ (ςn )) e m−1 sin( 1 (φn + φk )) 2 2 . × 1 sin( (φn − φk )) k=1 2 −1
2
2
j=1 j=n
φj
×
k=n
Then m 8 (ζ ): C \ σO D → M2 (C) solves the following (augmented) RHP: (i) m 8 (ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; m 8 (ζ ) satisfy the following jump conditions, (ii) m 8± (ζ ) := lim ζ →ζ ζ ∈± side of σ D O
8− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 )) G8 (ζ ), m 8+ (ζ ) = m
ζ ∈ R,
where G8 (ζ ) =
(1 − r(ζ )r(ζ )) δ− (ζ )( δ+ (ζ ))−1 $ r(ζ )( δ− (ζ ) δ+ (ζ ))−1 m−1 (d + (ζ ))−2 k=1
k
$ + 2 −r(ζ ) δ− (ζ ) δ+ (ζ ) m−1 k=1 (dk (ζ )) , ( δ− (ζ ))−1 δ+ (ζ )
and ! m 8+ (ζ )
=
m 8− (ζ ) I + 8 m − (ζ ) I +
nK C σ , (ζ −ςn ) + nL C σ , − (ζ −ςn )
n , n ∈ {1, 2, . . . , m − 1}, ζ ∈K n , n ∈ {1, 2, . . . , m − 1}; ζ ∈L
(iii) m 8 (ζ ) has simple poles in σd with (ζ )gn ( δ (ςn )) Res( m (ζ ); ςn ) = lim m 8
8
−2
ζ →ςn
m−1
(dk+ (ςn ))−2
σ− ,
k=1
n ∈ {m, m + 1, . . . , N}, m8 (ζ ); ςn ) σ1 , Res( m8 (ζ ); ςn ) = σ1 Res(
n ∈ {m, m + 1, . . . , N};
(iv) det( m8 (ζ ))|ζ =±1 = 0; $ + σ3 δ (0))σ3 ( m−1 (v) m 8 (ζ ) =ζ →0 ζ −1 ( k=1 (dk (0)) )σ2 + O(1); 8 −1 (vi) m (ζ ) = ζ →∞ I + O(ζ ); ζ ∈C\σ D O
8 (ζ ) σ1 and m 8 (ζ −1 ) = ζ m 8 (ζ )( δ (0))σ3 ( (vii) m 8 (ζ ) = σ1 m
$m−1 k=1
(dk+ (0))σ3 )σ2 .
388
A. H. VARTANIAN
Let u(x, t) := i lim
ζ →∞ ζ ∈C\σ D O
and
δ (ζ ))σ3 ζ m 8 (ζ )(
m−1
(dk+ (ζ ))σ3 − I
,
k=1
(77)
12
x
(|u(x , t)|2 − 1) dx +∞ δ (ζ ))σ3 ζ m 8 (ζ )(
:= −i lim
ζ →∞ ζ ∈C\σ OD
m−1
(dk+ (ζ ))σ3 − I
k=1
.
(78)
11
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. The analogue of Proposition 3.1 is PROPOSITION A.1.1 ([38]). The solution of the RHP for m 8 (ζ ): C \ σO D → M2 (C) formulated in Lemma A.1.3 has the (integral equation) representation m 8− (µ)( υ 8 (µ) − I) dµ 8 8 −1 8 8 , d (ζ ) + m (ζ ) = (I + ζ +0 )P (ζ ) m (µ − ζ ) 2π i σc ζ ∈ C \ σO D ,
where m 8d (ζ )
=I+
N Res( m8 (ζ ); ςn ) n=m
(ζ − ςn )
σ1 Res( m8 (ζ ); ςn ) σ1 + , (ζ − ςn )
8 (ζ ) on σc (Lemma A.1.3(ii)), v 8 (·) is a generic notation for the jump matrices of m 8 8 (ζ ) are specified below. The solution of the above (integral) equa 0 and P and + tion can be written as the ordered factorisation 8 (ζ ) 80 )P m8d (ζ ) mc (ζ ), m 8 (ζ ) = (I + ζ −1 + 8
8
ζ ∈ C \ σO D ,
d (ζ ) σ1 (∈ SL(2, C)) has the representation given above, where m d (ζ ) = σ1 m 8 8 80 is idempotent, I + ζ −1 + 80 (∈ M2 (C)) P (ζ ) = σ1 P (ζ ) σ1 is chosen so that + 80 σ1 80 = σ1 + is holomorphic in a punctured neighbourhood of the origin, with + 8 0 )|ζ =±1 = 0, and having the finite, order 2, (∈ GL(2, C)) such that det(I + ζ −1 + matrix involutive structure 8 ei(k+1/2)π 8 )2 )1/2 e−iϑ 8 + (1 + ( + 8 0 = , k ∈ Z, + 8 )2 )1/2 eiϑ 8 8 e−i(k+1/2)π (1 + (+ +
389
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
8 and ϑ 8 are obtained from the relation where + m−1 8 8 + 8 c σ σ (0) 0 = P + md (0) m (0)( δ (0)) 3 (dk (0)) 3 σ2 , k=1
80 ) det(+
80 + 80 = I, and m = 0, = −1, and + c (ζ ): C \ and satisfying c (ζ ) is piecewise (sectionally) σc → SL(2, C) solves the following RHP: (1) m holomorphic ∀ζ ∈ C \ σc ; 80 ) tr(+
(2)
m c± (ζ ) :=
lim
ζ →ζ ζ ∈± side of σc
m c (ζ )
c− (ζ ) υ c (ζ ), ζ ∈ σc , where υ c (ζ ) = satisfy the jump condition m c+ (ζ ) = m 8 8 G (ζ ), ζ ∈ R, with G (ζ ) given in Lemexp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3 )) c K −1 nL (ζ − ma A.1.3(ii), υ (ζ ) = I + Cn (ζ − ςn ) σ+ , ζ ∈ Kn , and υ c (ζ ) = I + C −1 K L ςn ) σ− , ζ ∈ Ln , n ∈ {1, 2, . . . , m − 1}, with Cn and Cn given in Lemma A.1.3; c (ζ ) = σ1 m c (ζ ) σ1 . (3) m c (ζ ) = ζ →∞ I + O(ζ −1 ); and (4) m ζ ∈C\σc
The analogue of Lemma 3.5 is
" LEMMA A.1.4. For m ∈ {1, 2, . . . , N}, set σd := N n=m ({ςn } ∪ {ςn }), and let σd ∪σc ) → σc = {ζ ; Im(ζ ) = 0} with orientation from −∞ to +∞. Let X(ζ ): C\( M2 (C) solve the following RHP: (i) X(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc ; (ii) X± (ζ ) := lim ζ →ζ X(ζ ) satisfy the jump condition ζ ∈± side of σc
G8 (ζ ), X+ (ζ ) = X− (ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3 )) (iii) X(ζ ) has simple poles in σd with δ (ςn )) Res(X(ζ ); ςn ) = lim X(ζ )gn (
−2
ζ →ςn
m−1
ζ ∈ R;
(dk+ (ςn ))−2
σ− ,
k=1
n ∈ {m, m + 1, . . . , N}, Res(X(ζ ); ςn ) = σ1 Res(X(ζ ); ςn ) σ1 ,
n ∈ {m, m + 1, . . . , N};
(iv) det(X(ζ ))|ζ =±1 = 0; $ + σ3 δ (0))σ3 ( m−1 (v) X(ζ ) =ζ →0 ζ −1 ( k=1 (dk (0)) )σ2 + O(1); (vi) X(ζ ) = ζ →∞ I + O(ζ −1 ); ζ ∈C\( σd ∪σc ) $ + σ3 δ (0))σ3 ( m−1 (vii) X(ζ ) = σ1 X(ζ ) σ1 and X(ζ −1 ) = ζ X(ζ )( k=1 (dk (0)) )σ2 . Then, as t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m 8 (ζ ): C \ σO D → M2 (C) has the following asymptotics: |t|)))X(ζ ), m 8 (ζ ) = (I + O(F(ζ ) exp(−
390
A. H. VARTANIAN
where := 4 min m∈{1,2,...,N} {sin(φn )|cos(φn ) − cos(φm )|} (> 0), and, for i, j ∈ n∈{1,2,...,m−1} {1, 2}, (F(ζ ))ij =ζ →∞ O(|ζ |−1 ) and (F(ζ ))ij =ζ →0 O(1). Furthermore, let m−1 u(x, t) := i lim (dk+ (ζ ))σ3 − I + ζ X(ζ )( δ (ζ ))σ3 ζ →∞ ζ ∈C\( σd ∪σc )
k=1
12
+ O(exp(− |t|)), and
(79)
x
(|u(x , t)|2 − 1) dx +∞ := −i
lim
ζ →∞ ζ ∈C\( σd ∪σc )
ζ X(ζ )( δ (ζ ))σ3
m−1
(dk+ (ζ ))σ3 − I
k=1
+ 11
+ O(exp(− |t|)).
(80)
Then u(x, t) is the solution of the Cauchy problem for the Df NLSE. The analogue of Lemma 4.1 is LEMMA A.1.5. The solution of the RHP for X(ζ ): C \ ( σd ∪ σc ) → M2 (C) formulated in Lemma A.1.4 is given by the following ordered factorisation, (ζ ) 0 )P md (ζ )M c (ζ ), X(ζ ) = (I + ζ −1 +
ζ ∈ C \ ( σd ∪ σc ),
d (ζ ) σ1 (∈ SL(2, C)) has the (series) representation where m d (ζ ) = σ1 m N Res(X(ζ ); ςn ) σ1 Res(X(ζ ); ςn )σ1 + , m d (ζ ) = I + ζ − ςn ζ − ςn n=m (ζ ) σ1 is chosen (see Lemma A.1.7 below) so that + (ζ ) = σ1 P 0 is idempoP 0 is holomorphic in a punctured neighbourhood of the origin, with tent, I + ζ −1 + 0 σ1 (∈ GL(2, C)) and det(I + ζ −1 + 0 )|ζ =±1 = 0, and determined by 0 = σ1 + + $m−1 + c σ3 0 ) = 0, md (0)M (0)(δ (0)) ( k=1 (dk (0))σ3 )σ2 , and satisfying tr(+ +0 = P (0) c det(+0 ) = −1, and +0 +0 = I, and M (ζ ): C \ σc → SL(2, C) solves the following RHP: (1) M c (ζ ) is piecewise (sectionally) holomorphic ∀ζ ∈ C \ σc ; c (ζ ) := lim ζ →ζ M c (ζ ) satisfy, for ζ ∈ R, the jump condition (2) M± ±Im(ζ )>0
c c (ζ ) = M− (ζ )e−ik(ζ )(x+2λ(ζ )t ) ad(σ3 ) × M+ $m−1 + r(ζ ) 2 )) δ (ζ )/ δ (ζ ) − (d (ζ )) (1 − r(ζ )r(ζ − + −1 k k=1 (δ− (ζ )δ+ (ζ )) $m−1 + ; × r(ζ ) −2 δ− (ζ ) δ+ (ζ )/ k=1 (dk (ζ )) δ (ζ ) δ (ζ ) −
(3) M c (ζ ) =
ζ →∞ ζ ∈C\σc
+
I + O(ζ −1 ); and (4) M c (ζ ) = σ1 M c (ζ ) σ1 .
391
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
The analogue of Lemma 4.2 is LEMMA A.1.6. Let ε be an arbitrarily fixed, sufficiently small positive real number, and, for z ∈ {λ1 , λ2 }, with λ1 and λ2 given in Theorem 2.2.1, Equation (10), set U(z; ε) := {ζ ; |ζ − z| < ε}. Then, as t → −∞ and x → +∞ such " that z0 := x/t < −2, for ζ ∈ C \ z∈{λ1 ,λ2 } U(z; ε), M c (ζ ) has the following asymptotics: c (ζ ) M11
=1+O c (ζ ) M12
cS (λ2 )c(λ1 , λ3 , λ3 ) ln|t| cS (λ1 )c(λ2 , λ3 , λ3 ) + , λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t
√ i(:− (z0 ,t )− 3π ) 3π − 4 λ2 e−i(: (z0 ,t )− 4 ) ν(λ1 ) λ1−2iν(λ1) λ1 e + + √ =e (ζ − λ1 ) (ζ − λ2 ) |t|(λ1 − λ2 ) (z02 + 32)1/4 S cS (λ2 )c(λ1 , λ3 , λ3 ) ln|t| c (λ1 )c(λ2 , λ3 , λ3 ) + , +O λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t i<− (0) 2
c (ζ ) M21
√
−i(:− (z0 ,t )− 3π ) 3π − 4 λ2 ei(: (z0 ,t )− 4 ) ν(λ1 )λ12iν(λ1) λ1 e + + =e √ (ζ − λ1 ) (ζ − λ2 ) |t|(λ1 − λ2 ) (z02 + 32)1/4 S cS (λ2 )c(λ1 , λ3 , λ3 ) ln|t| c (λ1 )c(λ2 , λ3 , λ3 ) + , +O λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t −
− i< 2 (0)
c (ζ ) M22
=1+O
cS (λ2 )c(λ1 , λ3 , λ3 ) ln|t| cS (λ1 )c(λ2 , λ3 , λ3 ) + , λ2 (z02 + 32) (ζ − λ1 ) λ1 (z02 + 32) (ζ − λ2 ) (λ1 − λ2 )t
where λ3 , ν(·), :− (z0 , t), and <− (·), respectively, are given in Theorem 2.2.1, Equations (10), (11), (17), and (19), (· − λk )−1 L∞ (C \"z∈{λ ,λ } U(z;ε)) < ∞, k ∈ 1 2
{1, 2}, M c (ζ ) = σ1 M c (ζ ) σ1 , and (M c (0)σ2 )2 = I (+ O(t −1 ln|t|)). Sketch of proof. Proceeding as in the proof of Lemma 6.1 in [38] and particularising it to the case of the RHP for M c (ζ ) stated in Lemma A.1.5, one arrives at 3π ν 3π i r(λ1 )( δB0 )−2 e− 2 e 4 c M11 (ζ ) = 1 + 0 √ × I B 21 2π i(ζ − λ1 )β XB |t| +∞ i 3π i z2 − 3π4 i 4 ∂z Diν (z) + e zDiν (z) ziν e− 4 dz− e × 2 0 2 −1 0 −2 − iπ r(λ1 )| ) (δB ) e 4 r(λ1 )(1 − | × − 0 √ I πν B B |t| 21 2π i(ζ − λ1 )β e− 2 X
392
A. H. VARTANIAN
i − iπ iπ z2 4 4 × e ∂z Diν (z) + e zDiν (z) ziν e− 4 dz+ 2 0 πν iπ 0 −2 − −iν r(λ1 )( δA ) e 2 (−1) e 4 + × 0 √ I A 21 2π i(ζ − λ2 )β XA |t| +∞ i iπ z2 − iπ e 4 ∂z D−iν (z) − e 4 zD−iν (z) z−iν e− 4 dz− × 2 0 2 −1 0 −2 − 3π4 i r(λ1 )(1 − | r(λ1 )| ) (δA ) e − 0 πν √ × I A iν 2 2π i(ζ − λ2 )β21 e (−1) XA |t| +∞ i 3π i 3π i z2 e 4 ∂z D−iν (z) − e− 4 zD−iν (z) z−iν e− 4 dz+ × 2 0 S δB0 )−2 c (λ1 )c(λ2 , λ3 , λ3 )( + +O √ B (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) X δA0 )−2 ln|t| cS (λ2 )c(λ1 , λ3 , λ3 )( , + √ A t (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) X iπ 3π ν 3π i r(λ1 )(1 − | r(λ1 )|2 )−1 ( δB0 )2 e 4 r(λ1 )( δB0 )2 e− 2 e− 4 c M12 (ζ ) = − × √ √ πν B |t| B |t| 2π i(ζ − λ1 )X 2π i(ζ − λ1 )e− 2 X +∞ z2 D−iν (z)z−iν e− 4 dz+ ×
+∞
0
3π i r(λ1 )|2 )−1 ( δA0 )2 e 4 r(λ1 )(1 − | + √ − πν A |t| 2π i(ζ − λ2 )e 2 (−1)−iν X +∞ πν iπ δA0 )2 e− 2 e− 4 r(λ1 )( z2 Diν (z)ziν e− 4 dz+ − √ −iν A |t| 0 2π i(ζ − λ2 )(−1) X S δB0 )2 c (λ1 )c(λ2 , λ3 , λ3 )( + +O √ B (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) X δA0 )2 ln |t| cS (λ2 )c(λ1 , λ3 , λ3 )( , + √ A t (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) X iπ 3π ν 3π i r(λ1 )( r(λ1 )(1 − | r(λ1 )|2 )−1 ( δB0 )−2 e− 4 δB0 )−2 e− 2 e 4 c − × M21 (ζ ) = − √ √ πν B |t| B |t| 2π i(ζ − λ1 )X 2π i(ζ − λ1 )e− 2 X +∞ z2 Diν (z)ziν e− 4 dz− ×
0
3π i r(λ1 )(1 − | r(λ1 )|2 )−1 ( δA0 )−2 e− 4 − − √ πν A |t| 2π i(ζ − λ2 )e 2 (−1)iν X +∞ π ν iπ 2 r(λ1 )( δA0 )−2 e− 2 e 4 −iν − z4 D (z)z e dz+ − √ −iν A |t| 0 2π i(ζ − λ2 )(−1)iν X
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
δB0 )−2 cS (λ1 )c(λ2 , λ3 , λ3 )( + √ B (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) X ln |t| δA0 )−2 cS (λ2 )c(λ1 , λ3 , λ3 )( , + √ A t (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) X 3π ν 3π i r(λ1 )( δB0 )2 e− 2 e− 4 c (ζ ) = 1 − M22 0 √ × I B 2π i(ζ − λ1 )β12 XB |t| +∞ i 3π i 3π i z2 e 4 ∂z D−iν (z) − e− 4 zD−iν (z) z−iν e− 4 dz+ × 2 0 iπ r(λ1 )(1 − | r(λ1 )|2 )−1 ( δB0 )2 e 4 + 0 √ × πν I B B |t| 12 2π i(ζ − λ1 )β e− 2 X +∞ i iπ z2 − iπ 4 4 e ∂z D−iν (z) − e zD−iν (z) z−iν e− 4 dz− × 2 0 0 2 − π2ν iν − iπ r(λ1 )(δA ) e (−1) e 4 − 0 √ × I A 12 2π i(ζ − λ2 )β XA |t| +∞ i − iπ iπ z2 4 4 e ∂z Diν (z) + e zDiν (z) ziν e− 4 dz+ × 2 0 3π i 2 −1 0 2 r(λ1 )| ) (δA ) (−1)iν e 4 r(λ1 )(1 − | × + 0 πν √ I A 12 A |t| 2π i(ζ − λ2 )β e2X +∞ i 3π i z2 − 3π4 i ∂z Diν (z) + e 4 zDiν (z) ziν e− 4 dz+ e × 2 0 S δB0 )2 c (λ1 )c(λ2 , λ3 , λ3 )( + +O √ B (ζ − λ1 )|λ1 − λ3 | (λ1 − λ2 ) X δA0 )2 ln |t| cS (λ2 )c(λ1 , λ3 , λ3 )( , + √ A t (ζ − λ2 )|λ2 − λ3 | (λ1 − λ2 ) X +O
where r(ζ ) = r(ζ )
$m−1 k=1
r(λ1 )| = |r(λ1 )|), ν = ν(λ1 ), (dk+ (ζ ))−2 (|
iν 2 Y(λ1 ) × δB0 = |λ1 − λ3 |iν (2|t|(λ1 − λ2 )3 λ−3 1 ) e it × exp − (λ1 − λ2 )(z0 + λ1 + λ2 ) , 2 − iν2 Y(λ2 ) e × δA0 = |λ2 − λ3 |−iν (2|t|(λ1 − λ2 )3 λ−3 2 ) it (λ1 − λ2 )(z0 + λ1 + λ2 ) , × exp 2 λ2 i ln|µ − λ1 | d ln(1 − |r(µ)|2 )+ Y(λ1 ) = 2π 0
393
394
A. H. VARTANIAN
+
i 2π
+∞
ln|µ − λ1 | d ln(1 − |r(µ)|2 ),
λ1
λ2 i Y(λ2 ) = −Y(λ1 ) + ln|µ| d ln(1 − |r(µ)|2 )+ 2π 0 +∞ i ln|µ| d ln(1 − |r(µ)|2 ), + 2π λ1 √ π ν 3π i 0 0 2π e− 2 e 4 I I B B , XA = XA , β12 = β21 = XB = XB , r(λ1 );(iν) √ πν 3π i 0 0 2π e− 2 e− 4 I I A A , β12 = β21 = r(λ1 ) ;(iν) ;(·) is the gamma function [51], and D∗ (·) is the parabolic cylinder function [51]. Proceeding, now, as at the end of the sketch of the proof of Lemma 4.2, one obtains the result stated in the lemma. Furthermore, one shows that the symmetry reduction M c (ζ ) = σ1 M c (ζ ) σ1 is satisfied, and verifies that (M c (0)σ2 )2 = I + O(t −1 ln|t|). ✷ The analogue of Proposition 4.1 is PROPOSITION A.1.2. For m ∈ {1, 2, . . . , N}, set Res(X(ζ ); ςn ) := c c −an M12 (ςn )/M22 (ςn ),
an
bn dn
,
cn c −cn M12 (ςn )/
dn = n ∈ {m, m + 1, . . . , N}. Then bn = c (ςn ), and {an , cn }N satisfy the following (nonsingular) system of 2(N −m+1) M22 n=m linear inhomogeneous algebraic equations, g D M c (ςm ) m 12 am c D M g am+1 m+1 12 (ςm+1 ) .. . A B . . . c D M (ς ) g aN N N 12 , = c D cm gm M22 (ςm ) cm+1 g D M c (ς ) m+1 B . m+1 22 A .. ... c D cN gN M22 (ςN ) where ij := A
det(Mc (ς ))+g DW(Mc (ς ),Mc (ς )) i i 12 i 22 i , c (ς ) M22 i
i = j ∈ {m, m + 1, . . . , N},
c (ς )M c (ς )−M c (ς )M c (ς )) i − giD (M12 22 j 22 i 12 j , i = j ∈ {m, m + 1, . . . , N}, (ςi −ςj )M c (ςj ) 22
ij := − B
c c c c giD (M22 (ςi )M22 (ςj ) − M12 (ςi )M12 (ςj )) c (ςi − ςj )M22 (ςj )
i, j ∈ {m, m + 1, . . . , N},
,
395
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
gjD
iθgj
= |gj |e
exp(2ik(ςj )(x + 2λ(ςj )t))( δ (ςj ))−2
m−1
(dk+ (ςj ))−2 ,
k=1
j ∈ {m, m + 1, . . . , N}, with |gj | and θgj given in Lemma 3.1(iii), and c c M12 (z) M (z) 22 c c . W(M12 (z), M22 (z)) = c c ∂z M12 (z) ∂z M22 (z) The analogue of Proposition 4.2 is PROPOSITION A.1.3. As t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, for n ∈ {m + 1, m + 2, . . . , N}, − |t |
an = O(e−ג
−ג− |t |
cn = O(e
where ג− := 4 min am =
a0m
),
),
− |t |
bn = O(t −1/2 (z02 + 32)−1/4 e−ג dn = O(t
m∈{1,2,...,N} n∈{m+1,m+2,...,N}
−1/2
(z02
+ 32)
−1/4 −ג− |t |
e
),
),
{sin(φn )|cos(φn ) − cos(φm )|} (> 0), and
1 1 cS (z0 ) ln|t| + √ am + O |t| (z02 + 32)1/2 t
gmD gmD (ςm − ςm )−1 + (1 + gmD gmD (ςm − ςm )−2 ) D D c c 12 12 gm gm (ςm − ςm )−1 (gmD ∂ζ M (ςm ) + gmD ∂ζ M (ςm )) 1 + +√ D D −2 2 |t| (1 + gm gm (ςm − ςm ) ) c 12 cS (z0 ) ln|t| (ςm ) gmD M +O , + (z02 + 32)1/2 t (1 + gmD gmD (ςm − ςm )−2 ) 1 1 cS (z0 ) ln|t| bm = √ bm + O |t| (z02 + 32)1/2 t c 12 cS (z0 ) ln|t| (ςm ) 1 gmD gmD (ςm − ςm )−1 M +O , =: − √ |t| (1 + gmD gmD (ςm − ςm )−2 ) (z02 + 32)1/2 t 1 1 cS (z0 ) ln|t| 0 cm = cm + √ cm + O |t| (z02 + 32)1/2 t gmD + =: (1 + gmD gmD (ςm − ςm )−2 ) D D c c 12 12 gm gm (ςm − ςm )−1 M (ςm ) − gmD gmD ∂ζ M (ςm ) 1 + +√ |t| (1 + gmD gmD (ςm − ςm )−2 ) c c 12 12 (ςm ) + gmD ∂ζ M (ςm )) cS (z0 ) ln|t| gmD (gmD ∂ζ M , +O + (z02 + 32)1/2 t (1 + gmD gmD (ςm − ςm )−2 )2 =:
396
A. H. VARTANIAN
1 cS (z0 ) ln|t| dm = √ d1m + O |t| (z02 + 32)1/2 t c 12 cS (z0 ) ln|t| gmD M (ςm ) 1 +O , =: − √ |t| (1 + gmD gmD (ςm − ςm )−2 ) (z02 + 32)1/2 t where c 12 (ζ ) M
√ i<− (0) 3π 3π − − ν(λ1 ) e 2 λ1−2iν(λ1) λ1 ei(: (z0 ,t )− 4 ) λ2 e−i(: (z0 ,t )− 4 ) + , =√ (ζ − λ1 ) (ζ − λ2 ) (λ1 − λ2 ) (z02 + 32)1/4
with ν(·), λ1 , λ2 , λ3 , <− (·), and :− (z0 , t) specified in Lemma A.1.6, and cS (z0 ) given in Proposition 4.2. Furthermore, setting A B := , Y B A
)|2 0 < | det(Y
N
sin2 (φm )|γm |2 P −2 (φm , φk )Q−2 (φm ) 2φ(x,t ) 2 e 1+ sin2 ( 12 (φm + φj )) j =m cS (z0 ) ln|t| , (81) +O (z02 + 32)1/2 t
where φ(x, t), P (φm , φk ), and Q(φm ) are defined in Equations (67), (68), and (69), respectively. The analogue of Lemma 4.3 is (see, also, Remark 4.1) LEMMA A.1.7. As t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, ζ +a − a− 1 −
3
a2 (ζ ) = ζ + P a−
ζ + a4−
ζ + a4−
ζ + a2−
3
ζ + a1−
,
where a2− = 1 + a1− =
p−1 1 ∞ apq (z0 )(ln|t|)q p=1 q=0
|t|p/2
+
−4|t | min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{m+1,m+2,...,N} , +O e a3−
=
p−1 3 ∞ apq (z0 )(ln|t|)q p=1 q=0
|t|p/2
+
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
397
−4|t | min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{m+1,m+2,...,N} , +O e a4−
=1+
p−1 4 ∞ apq (z0 )(ln|t|)q p=1 q=0
|t|p/2
+
−4|t | min m∈{1,2,...,N} {sin(φn )|cos(φn )−cos(φm )|} n∈{m+1,m+2,...,N} +O e , k (ζ ) = σ1 P (ζ ) σ1 . (z0 ) ∈ cS (z0 ), k ∈ {1, 3, 4}, and P apq
The analogue of Proposition 4.3 is 1 2 3 (z0 ) =: a1 , a10 (z0 ) =: a2 , a10 (z0 ) =: a3 , and PROPOSITION A.1.4. Set a10 4 a4 . Then as t → −∞ and x → +∞ such that z0 := x/t < −2 a10 (z0 ) =: and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, 0 )11 (+ 1 −1 2i m−1 m−1 k=1 φk i δ bm c0m −1 (0)e c0 c1 2i φ −( a1 − = − i δ (0)e k=1 k + √ a2 ) m − + m + ςm ςm ςm ςm |t| √ 3π − 0 δ (0) ν(λ1 ) cos(: (z0 , t) − 4 ) a0 a 2 √ + + a3 1 − m − 1 − m ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t 0 )12 (+ −2i m−1 m−1 k=1 φk a0m i δ (0)e a0 −2i φ k=1 k + −( a1 − =− 1− √ a2 ) 1 − m + i δ (0)e ςm ςm |t| √ 1 3π −1 − δ (0) ν(λ1 ) cos(: (z0 , t) − 4 ) d1 c0 c0 2 am + m + a3 m − m + + √ ςm ςm ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t 0 )21 (+ −1 2i m−1 m−1 k=1 φk i δ a0m (0)e a0 −1 2i φ ( = 1− a1 − a2 ) 1 − m − i δ (0)e k=1 k + √ ςm ςm |t| √ 1 3π − δ (0) ν(λ1 ) cos(: (z0 , t) − 4 ) d1 c0 2 am c0 + m − a3 m + m √ + − ςm ςm ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t 0 )22 (+ 1 −2i m−1 m−1 k=1 φk c0m i δ (0)e c0 c1 bm −2i φ k=1 k + ( = i δ (0)e √ a1 − a2 ) m + + m − ςm ςm ςm ςm |t|
398
A. H. VARTANIAN
√
−1 δ (0) ν(λ1 ) cos(:− (z0 , t) − a0 2 a0 − a3 1 − m + 1 − m √ ςm ςm (λ1 − λ2 ) (z02 + 32)1/4 cS (z0 ) ln|t| . +O (z02 + 32)1/2 t
3π ) 4
+
The analogue of Proposition 4.4 is PROPOSITION A.1.5. Let φ(x, t), P (φm , φk ), and Q(φm ) be defined by Equations (67), (68), and (69), respectively. Then, for θγm = ±π/2, as t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, a0m =−
2i sin(φm )|γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t ) , (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t ))
c0m 2 sin(φm )|γm | δ −1 (0)ei(φm +s )+φ(x,t )P −1 (φm , φk )Q−1 (φm ) , =∓ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t )) −
a1m =∓
√ 16iλ21 sin2 (φm )|γm |3 ν(λ1 ) P −3 (φm , φk )Q−3 (φm ) cos(s − )e3φ(x,t) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
3π − − × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) − 4 3π ∓ − (λ1 − λ2 ) sin(φm ) sin :− (z0 , t) − 4
∓
√ 2λ1 sin(φm )|γm | ν(λ1 ) P −1 (φm , φk )Q−1 (φm )eφ(x,t) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
3π − − (λ1 + λ2 ) cos(φm + s − )× × 2 cos(s ) cos : (z0 , t) − 4 3π 3π − − − + (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) − + × cos : (z0 , t) − 4 4 3π − − − i(λ1 + λ2 ) sin(φm + s − )× + 2i sin(s ) cos : (z0 , t) − 4 3π 3π − − − − i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) − , × cos : (z0 , t) − 4 4
b1m =
−
√ − 2iλ1 sin(φm )|γm |2 ν(λ1 ) δ (0)e−i(φm +s )+2φ(x,t) P −2 (φm , φk )Q−2 (φm ) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
3π − (λ1 + λ2 ) cos(φm + s − )× × 2 cos(s − ) cos :− (z0 , t) − 4
399 3π 3π − − − + (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) − + × cos : (z0 , t) − 4 4 3π − i(λ1 + λ2 ) sin(φm + s − )× + 2i sin(s − ) cos :− (z0 , t) − 4 3π 3π − − − − i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) − , × cos : (z0 , t) − 4 4
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
c1m =−
√ − 16λ21 sin2 (φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P −2 (φm , φk )Q−2 (φm ) cos(s − ) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
3π − × ((λ1 + λ2 ) cos(φm ) − 2) cos :− (z0 , t) − 4 3π − − − (λ1 − λ2 ) sin(φm ) sin : (z0 , t) − 4 −
√ − 2iλ1 sin(φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P −2 (φm , φk )Q−2 (φm ) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
3π − × 2 cos(s ) cos : (z0 , t) − − (λ1 + λ2 ) cos(φm + s − )× 4 3π 3π − + − + (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) − − × cos : (z0 , t) − 4 4 3π + i(λ1 + λ2 ) sin(φm + s − )× − 2i sin(s − ) cos :− (z0 , t) − 4 3π 3π − − − + i(λ1 − λ2 )cos(φm + s )sin : (z0 , t) − + × cos : (z0 , t) − 4 4 +
−
√ − 8λ21 sin2 (φm )|γm |2 ν(λ1 ) δ −1 (0)ei(φm +s )+2φ(x,t) P −2 (φm , φk )Q−2 (φm ) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
3π − − × ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) − 4 3π cos(s − )− − (λ1 − λ2 ) sin(φm ) sin :− (z0 , t) − 4 3π − − − i ((λ1 + λ2 ) cos(φm ) − 2) cos : (z0 , t) − 4 3π − − sin(s ) , − (λ1 − λ2 ) sin(φm ) sin : (z0 , t) − 4 d1m =±
√ 2λ1 sin(φm )|γm | ν(λ1 ) P −1 (φm , φk )Q−1 (φm )eφ(x,t) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
3π − (λ1 + λ2 ) cos(φm + s − )× × 2 cos(s − ) cos :− (z0 , t) − 4
400
A. H. VARTANIAN
3π 3π − − − × cos : (z0 , t) − + (λ1 − λ2 ) sin(φm + s ) sin : (z0 , t) − + 4 4 3π − i(λ1 + λ2 ) sin(φm + s − )× + 2i sin(s − ) cos :− (z0 , t) − 4 3π 3π − − − − i(λ1 − λ2 ) cos(φm + s ) sin : (z0 , t) − , × cos : (z0 , t) − 4 4 where s − is given in Theorem 2.2.1, Equation (11). The analogue of Proposition 4.5 is PROPOSITION A.1.6. As t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, 0 )12 + a3− + bm + cm + u(x, t) = i (+ √
i<− (0)
ν(λ1 ) e 2 λ1−2iν(λ1) 3π − (λ1 ei(: (z0 ,t )− 4 ) + +√ 2 1/4 |t|(λ1 − λ2 ) (z0 + 32) cS (z0 ) ln|t| −i(:− (z0 ,t )− 3π ) 4 , ) +O + λ2 e (z02 + 32)1/2 t x
(|u(x , t)|2 − 1) dx +∞ 0 )11 + = −i (+ a1− − a2− + am + dm + 2i
x −∞
m−1
sin(φk )+
dµ + + ln(1 − |r(µ)|2 ) 2π 0 λ1 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t x N 2 2 (|u(x , t)| − 1) dx = (|u(x , t)| − 1) dx − 2 sin(φn )− +i
(82)
λ2
+∞
k=1
+∞
−
+∞ −∞
(83)
n=1
dµ . ln(1 − |r(µ)|2 ) 2π
(84)
The analogue of Proposition 4.6 is PROPOSITION A.1.7. As t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, 2, . . . , N}, for θγm = ±π/2, 0 )11 (+ 2 sin(φm )|γm |P −1 (φm , φk )Q−1 (φm )eφ(x,t ) + =i ± (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t ))
401
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
√ 3π ν(λ1 ) − cos(s − )+ − 2 cos : (z0 , t) − +√ 4 |t|(λ1 − λ2 ) (z02 + 32)1/4 +
4 sin(φm )|γm |2 P −2 (φm , φk )Q−2 (φm ) sin(s − − φm ) cos(:− (z0 , t) − (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )
3π 2φ(x,t) 4 )e
+
8λ2 sin2 (φ )|γ |2 P −2 (φ , φ )Q−2 (φ )(1 + |γ |2 P −2 (φ , φ )Q−2 (φ )e2φ(x,t) ) cos(s − )e2φ(x,t)
m m m k m m m k m × + 1 (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 3π − × ((λ1 + λ2 ) cos(φm ) − 2) cos :− (z0 , t) − 4 3π + − (λ1 − λ2 ) sin(φm ) sin :− (z0 , t) − 4 4λ1 sin(φm ) cos(φm )|γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t ) × + (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t ))(λ21 − 2λ1 cos(φm ) + 1) 3π − sin(s − − φm ) − (λ1 + λ2 ) sin(s − )× × 2 cos : (z0 , t) − 4 3π 3π − − − − (λ1 − λ2 ) cos(s ) sin : (z0 , t) − + × cos : (z0 , t) − 4 4 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t
0 )12 (+
−
−
2 sin(φ )|γ |2 P −2 (φ , φ )Q−2 (φ )e−i(θ
− (1)+φ
m +s
− )+2φ(x,t)
m m m k m = −ie−i(θ (1)+s ) + + (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) ) − − 1 2i Im( a1 − a2 ) sin(φm )|γm |2 P −2 (φm , φk )Q−2 (φm )e−i(θ (1)+φm +s )+2φ(x,t) ± +√ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) ) t
± ∓
√ − − 4i sin(φm )|γm |P −1 (φm , φk )Q−1 (φm ) ν(λ1 ) e−i(θ (1)+2s )+φ(x,t) cos(:− (z0 , t) − 3π 4 ) ∓ √ 2 2 −2 −2 2φ(x,t) 1/4 (1 − |γm | P (φm , φk )Q (φm )e ) (λ1 − λ2 ) (z0 + 32) √ − − 16iλ21 sin2 (φm )|γm |3 P −3 (φm , φk )Q−3 (φm ) ν(λ1 ) e−i(θ (1)+s )+3φ(x,t) cos(s − ) × √ (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )2 (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4
3π − − × ((λ1 + λ2 ) cos(φm ) − 2) sin(φm ) cos : (z0 , t) − 4 3π + − (λ1 − λ2 ) sin2 (φm ) sin :− (z0 , t) − 4 3π − − + i ((λ1 + λ2 ) cos(φm ) − 2) cos(φm ) cos : (z0 , t) − 4 3π + − (λ1 − λ2 ) sin(φm ) cos(φm ) sin :− (z0 , t) − 4
a2 )e−i(θ + Im( a1 − −
− (1)+s − )
−
√ − − 4λ1 sin(φm )|γm |P −1 (φm , φk )Q−1 (φm ) ν(λ1 ) e−i(θ (1)+s )+φ(x,t) × √ 2 (1 − |γm |2 P −2 (φm , φk )Q−2 (φm )e2φ(x,t) )(λ1 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4
402
A. H. VARTANIAN
3π ∓ 2 sin(s − − φm ) cos :− (z0 , t) − ± (λ1 + λ2 ) sin(s − )× 4 3π 3π − − − × cos : (z0 , t) − ± (λ1 − λ2 ) cos(s ) sin : (z0 , t) − + 4 4 cS (z0 ) ln|t| , +O (z02 + 32)1/2 t a2 ) Im( a1 − ×
=±
√ 2 −2 (φ , φ )Q−2 (φ )e2φ(x,t) ) ν(λ1 ) sin(s − ) cos(:− (z0 , t) − 3π m k m 4 )(1 − |γm | P ± √ (λ1 − λ2 ) (z02 + 32)1/4 sin(φm )|γm |P −1 (φm , φk )Q−1 (φm )eφ(x,t)
√ 4λ21 ν(λ1 ) sin(φm )|γm |P −1 (φm , φk )Q−1 (φm ) sin(s − )eφ(x,t ) × ± √ (λ21 − 2λ1 cos(φm ) + 1)2 (λ1 − λ2 ) (z02 + 32)1/4 3π − × ((λ1 + λ2 ) cos(φm ) − 2) cos :− (z0 , t) − 4 3π − ± − (λ1 − λ2 ) sin(φm ) sin : (z0 , t) − 4 √ 2λ1 ν(λ1 ) cos(φm )|γm |P −1 (φm , φk )Q−1 (φm )eφ(x,t ) √ × ± (λ21 − 2λ1 cos(φm ) + 1) (λ1 − λ2 ) (z02 + 32)1/4 3π − − − × 2 cos(s − φm ) cos : (z0 , t) − 4 3π − − + − (λ1 + λ2 ) cos(s ) cos : (z0 , t) − 4 3π − − ± + (λ1 − λ2 ) sin(s ) sin : (z0 , t) − 4 ±
√ 2 ν(λ1 )|γm |P −1 (φm , φk )Q−1 (φm ) cos(s − − φm ) cos(:− (z0 , t) − √ (λ1 − λ2 ) (z02 + 32)1/4
3π φ(x,t) 4 )e
+
cS (z0 ) ln|t| , +O (z02 + 32)1/2 t
a2 ) = Re( a3 ) = Im( a3 ) = 0, Re( a1 − − where θ (·) is given in Theorem 2.2.1, Equation (9). The analogue of Lemma 4.4 is LEMMA A.1.8. As t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ m , m ∈ {1, x2, . . . , N}, u(x, t), the solution of the Cauchy problem for the Df NLSE, and ±∞ (|u(x , t)|2 − 1) dx have the leading-order asymptotic expansions (for the lower sign) stated in Theorem 2.2.1, Equations (7)–(20).
403
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
Appendix B In order to obtain the results of Theorems 2.2.3 and 2.2.4, the following Lemma, which is the analogue of Lemmae 4.2 and A.1.6, is requisite. LEMMA B.1.1. Let ε be an arbitrarily fixed, sufficiently small positive real number, and, for λ ∈ J := {(s1 )±1 , (s2 )±1 }, where 1 s1 = − (a1 − i(4 − a12 )1/2) = eiϕ1 , 2 π (4 − a12 )1/2 ∈ 0, , a1 < 0, |a1 | < 2, ϕ1 := arctan |a1 | 2 1 s2 = − (a2 − i(4 − a22 )1/2) = eiϕ2 , 2 (4 − a22 )1/2 π ∈ , π , a2 > 0, |a2 | < 2, ϕ2 := − arctan |a2 | 2 with a1 and a2 given in Theorem 2.2.1, Equation (10), set U(λ; ε) := {z; |z − λ| < ε}. Then, for r(s1 ) = exp(−iε1 π/2)|r(s1 )|, ε1 ∈ {±1}, " r(s2 ) = exp(iε2 π/2)|r(s2 )|, ε2 ∈ {±1}, 0 < r(s2 )r(s2 ) < 1, and ζ ∈ C \ λ∈J U(λ; ε), as t → +∞ and x → −∞ such that z0 := x/t ∈ (−2, 0), mc (ζ ) has the following asymptotics, −4αt c(z0 ) e c(z0 ) c + , m11 (ζ ) = 1 + O (ζ − s1 ) (ζ − s2 ) βt 0 (µ−cos ϕ ) ln(1−|r(µ)|2 ) dµ 0 dµ ln(1−|r(µ)|2 ) 1 − 2a0 t +sin(ϕ1 ) −∞ −i ϕ1 + −∞ π (µ−cos ϕ1 )2 +sin2 ϕ1 π (µ−cos ϕ1 )2 +sin2 ϕ1 ε e e 1 + mc12 (ζ ) = 2(|r(s1 )|)−1 (b0 t)1/2 (ζ − s1 ) 0 (µ−cos ϕ ) ln(1−|r(µ)|2 ) dµ 0 dµ ln(1−|r(µ)|2 ) − 2a0 t −sin(ϕ3 )
ε2 e
−∞ (µ−cos ϕ )2 +sin2 ϕ π 3 3
i ϕ3 −
e
−∞
3 (µ−cos ϕ3 )2 +sin2 ϕ3
π
+ 2(|r(s2 )|)−1 (1 − r(s2 )r(s2 ))(b0 t)1/2 (ζ − s2 ) −4αt c(z0 ) e c(z0 ) + , +O (ζ − s1 ) (ζ − s2 ) βt 0 (µ−cos ϕ ) ln(1−|r(µ)|2 ) dµ 0 dµ ln(1−|r(µ)|2 ) 1 − 2a0 t +sin(ϕ1 ) −∞ i ϕ1 + −∞ π (µ−cos ϕ1 )2 +sin2 ϕ1 π (µ−cos ϕ1 )2 +sin2 ϕ1 ε e e 1 + mc21 (ζ ) = 2(|r(s1 )|)−1 (b0 t)1/2 (ζ − s1 ) 0 (µ−cos ϕ ) ln(1−|r(µ)|2 ) dµ 0 dµ ln(1−|r(µ)|2 ) 3 − 2a0 t −sin(ϕ3 ) −∞ −i ϕ3 − −∞ π (µ−cos ϕ3 )2 +sin2 ϕ3 π (µ−cos ϕ3 )2 +sin2 ϕ3 e ε2 e + + 2(|r(s2 )|)−1 (1 − r(s2 )r(s2 ))(b0 t)1/2 (ζ − s2 ) −4αt c(z0 ) e c(z0 ) + , +O (ζ − s1 ) (ζ − s2 ) βt −4αt c(z0 ) e c(z0 ) + , mc22 (ζ ) = 1 + O (ζ − s1 ) (ζ − s2 ) βt +
404
A. H. VARTANIAN
where 1 a0 = (z0 − a1 )(4 − a12 )1/2 (> 0), 2 1 a0 = − (z0 − a2 )(4 − a22 )1/2 (> 0), 2 1 2 b0 = (z0 + 32)1/2 (4 − a12 )1/2 (> 0), 2 1 b0 = (z02 + 32)1/2 (4 − a22 )1/2 (> 0), 2 β := min{b0 , b0 }, α := min{a0 , a0 }, and, for r(s1 ) = exp(iε1 π/2)|r(s1 )|, ε1 ∈ {±1}, " r(s2 ) = exp(−iε2 π/2)|r(s2 )|, ε2 ∈ {±1}, 0 < r(s1 )r(s1 ) < 1, and ζ ∈ C \ λ∈J U(λ; ε), as t → −∞ and x → +∞ such that z0 ∈ (−2, 0), −4α|t | c(z0 ) e c(z0 ) c + , m11 (ζ ) = 1 + O (ζ − s1 ) (ζ − s2 ) βt
− 2a0 |t|−sin(ϕ1 )
mc12 (ζ ) = −
ε2 e
−
i ϕ1 −
+∞
e
+∞ ln(1−|r(µ)|2 ) dµ 0 (µ−cos ϕ3 )2 +sin2 ϕ3 π
0
(µ−cos ϕ1 ) ln(1−|r(µ)|2 ) dµ π (µ−cos ϕ1 )2 +sin2 ϕ1
0
2(|r(s1
− 2a0 |t|+sin(ϕ3 )
−i ϕ3 +
+∞ (µ−cos ϕ3 ) ln(1−|r(µ)|2 ) dµ 0 π (µ−cos ϕ3 )2 +sin2 ϕ3
)|)−1 (1 −
+∞ 0
dµ ln(1−|r(µ)|2 ) (µ−cos ϕ1 )2 +sin2 ϕ1 π
−i ϕ1 −
+∞
e
0
(µ−cos ϕ1 ) ln(1−|r(µ)|2 ) dµ π (µ−cos ϕ1 )2 +sin2 ϕ1
r(s1 )r(s1 ))(b0 |t|)1/2 (ζ − s1 )
ln(1−|r(µ)|2 ) dµ (µ−cos ϕ3 )2 +sin2 ϕ3 π
i ϕ3 +
e
+∞ (µ−cos ϕ3 ) ln(1−|r(µ)|2 ) dµ 0 π (µ−cos ϕ3 )2 +sin2 ϕ3
2(|r(s2 )|)−1 (b0 |t|)1/2 (ζ − s2 )
c(z0 ) e−4α|t | c(z0 ) + , +O (ζ − s1 ) (ζ − s2 ) βt −4α|t | c(z0 ) e c(z0 ) c + , m22 (ζ ) = 1 + O (ζ − s1 ) (ζ − s2 ) βt
−
e 2(|r(s2 )|)−1 (b0 |t|)1/2 (ζ − s2 )
+∞
ε1 e
ε2 e
−4α|t | c(z0 ) e c(z0 ) + , (ζ − s1 ) (ζ − s2 ) βt
− 2a0 |t|−sin(ϕ1 )
ln(1−|r(µ)|2 ) dµ (µ−cos ϕ1 )2 +sin2 ϕ1 π
2(|r(s1 )|)−1 (1 − r(s1 )r(s1 ))(b0 |t|)1/2 (ζ − s1 )
− 2a0 |t|+sin(ϕ3 )
+O mc21 (ζ ) = −
0
ε1 e
−
+∞
+
−
+
where supζ ∈C \"λ∈J U(λ;ε) |(ζ − (sn )±1 )−1 | < ∞, and mc (ζ ) = σ1 mc (ζ ) σ1 .
Appendix C. Matrix Riemann–Hilbert Theory in the L2 Sobolev Space In this Appendix, the theoretical foundation for this paper is presented. Beginning from the Lax-pair isospectral deformation formulation for a completely integrable
405
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
NLEE, in the sense of the ISM, a succinct review of several basic and key facts from the 2 × 2 matrix RH factorisation theory on unbounded self-intersecting contours is presented: for complete details and proofs, see [40, 44–47, 50]. For simplicity, one begins with the solitonless sector, σd ≡ ∅, leading to the so-called ‘regular’ RHP: inclusion of the (nonempty and finitely denumerable) discrete spectrum, σd , is known as the ‘singular’ RHP, and is discussed below Theorem C.1.4. For a completely integrable system of NLEEs, in the sense of the ISM, write the spatial part of the associated Lax pair (see, for example, Proposition 2.1.1) as t; λ))$ (x, t; λ) = (J(λ) + R(x, (x, t; λ), where (x, t) ∈ R × [−T , T ], λ ∈ C, ∂x $ t; λ) is offJ(λ) := diag(z1 (λ), z2 (λ)) is rational with distinct entries, and R(x, diagonal. The orders of the poles of J (λ) and R(x, t; λ) must satisfy the following requirements (denote by PJ the set of poles of J(λ), and let k(λ ) denote the order t; λ) is a pole of J(λ); (2) if ∞ is of the pole of λ ∈ PJ): (1) every pole of R(x, t; λ) of order not greater a pole of J(λ) of order k(∞), then it is a pole of R(x, than k(∞) − 1; and (3) if λ is a finite pole of J (λ) of order k(λ ), then it is a t; λ) of order not greater than k(λ ). Hence, one has the following pole of R(x, t; λ): (1) representations for J(λ) and R(x, J(λ) =
k(λ )
Jλ ,j (λ − λ )−j +
λ ∈ PJ\{∞} j =1
k(∞)
J∞,l λl ,
l=0
where Jλ ,j and J∞,l are M2 (C)-valued, diagonal matrices with distinct elements; and (2) t; λ) = R(x,
k(λ )
λ ∈ PJ\{∞}
j =1
−j
rλ ,j (x, t)(λ − λ )
+
k(∞)−1
r∞,l (x, t)λl .
l=0
Remark C.1.1. Hereafter, for economy of notation, all explicit x, t dependencies are suppressed. the closure of {λ ∈ C; Re(z1 (λ) − z2 (λ)) = 0}. Decompose Denote by > " l into a finite union of piecewise smooth, simple, closed curves, > := l∈L > > , V := {λ; > l ∩ (card(L) < ∞). Denote by V the set of all self-intersections of > m = ∅, l = m ∈ {1, 2, . . . , card(L)}} (it is assumed throughout that card(V ) < > into two disjoint open subsets of C, + and − , ∞). Divide the complement of > " ± each of which have finitely many components, ± := l ± ∈L± ± l ± (card(L ) < admits an orientation so that it can be viewed either as a positively ∞), such that > + , for + , or as a negatively (clockwise) (counter-clockwise) oriented boundary, > ± − − oriented boundary, > , for ; moreover, for each component ± l ± , ∂l ± has no self-intersections. DEFINITION C.1.1. For an M2 (C)-valued function, f (λ), say, denote by f± (λ), respectively, the nontangential limits, if they exist, of f (λ) taken from ± . For
406
A. H. VARTANIAN
→ M2 (C), define f (0) (λ) := f (λ), and, for k ∈ Z1 , f (j ) (λ) := ∂λj f (λ), f (λ): > " l , = l∈L > j ∈ {1, 2, . . . , k}. For the piecewise smooth simple closed curve > 2 k and k ∈ Z1 , define the LM2 (C) (>) Sobolev space H (>, M2 (C)) as the set satisfying: (1) for l ∈ {1, 2, . . . , card(L)}, of all M2 (C)-valued functions on > 2 (j ) l ); and (2) for f > l , j ∈ {0, 1, . . . , k − 1}, exist pointwise and ∈ LM2 (C) (> (k) > l ∈ {1, 2, . . . , card(L)}, f (k−1)> l is locally absolutely continuous and f l ∈ 2 0 2 l ). For k = 0, denote H (> , M2 (C)) by LM (C) (> ). Define LM2 (C)(> 2 k ± k H (> , M2 (C)) := {f : > → M2 (C); f ∂±± ∈ H (∂± l ± , M2 (C)), l ± ± k ± l ∈ {1, 2, . . . , card(L )}, k ∈ Z1 }: the norm on H (> , M2 (C)), k ∈ Z1 , is k (j ) 1/2 defined as f (·)H k (> (·)2L2 (> . ,M2 (C)) := f (·)2,k := ( l∈L j =0 f )) M2 (C)
l
± , M2 (C)) is a Hilbert space: for k = 0, f (·)2,0 = With H (> this norm, 2 1/2 . ( l∈L f (·)L2 (> )) k
M2 (C)
l
) are defined as The Cauchy integral operators on L2M2 (C) (> f (z) dz : (C± f )(λ) := lim λ →λ (z − λ ) 2π i > ± λ ∈
). Since > = note that C+ −C− = id, where id is the identity operator on L2M2 (C) (> " l∈L >l , where >l , l ∈ {1, 2, . . . , card(L)}, are piecewise smooth and simple, the ) into L2M (C)(> ); moreover, Cauchy integral operators are bounded from L2M2 (C) (> 2 ± , that is, > , provides the Cauchy integral the aforementioned orientation for > 2 operators on LM2 (C) (>) with the crucial property that ±C± are complementary projections, that is, C+2 = C+ , C−2 = −C− , C+ C− = C− C+ = 0, where 0 is the ). Even though C± are not bounded in operator norm null operator on L2M2 (C) (> 6 k α , M2 (C)); moreover, injecon H (>, M2 (C)), C± are bounded on α∈{±} H k (> ± , M2 (C)) → H k (> ± , M2 (C)), and C± : H k (> ∓ , M2 (C)) → tively, C± : H k (> ' k k α (> , M2 (C)) := α∈{±} H (> , M2 (C)). Since, in the ISM, > is (usually) unH k ± bounded, the function f > ± = I ∈ H (> , M2 (C)), k ∈ Z0 ; hence, for D ∈ , > ± }, embed H k (D, M2 (C)), k ∈ Z0 , into a larger Hilbert space {> " ∪ ( α∈{±} α ) HIk (D, M2 (C)) consisting of M2 (C)-valued functions f (λ) on > with the limit f (∞) at ∞ such that f (λ)−f (∞) ∈ H k (D, M2 (C)), with the norm defined by f (·)||HIk (D,M2 (C)) := f (·)I,2,k := (|f (∞)|2 + f (·) − f (∞)22,k )1/2 . HIk (D, M2 (C)), k ∈ Z0 , is isomorphic to the Hilbert space direct sum of M2 (C) and H k (D, M2 (C)) (HIk (D, M2 (C)) ≈ M2 (C) ⊕ H k (D, M2 (C))). ± , M2 (C)) := {f (λ) ∈ HIk (> ± , M2 (C)); det(f (λ)) ≡ 0}; Define: (1) GHIk (> k ± k ± and (2) SHI (> , M2 (C)) := {f (λ) ∈ HI (> , M2 (C)); det(f (λ)) = 1}. If ± , M2 (C))), where χ±c (∞) := lim λ→∞ χ c (λ), χ±c (λ) − χ±c (∞) ∈ ran C± (⊂ H k (> λ∈± " denote by χ c (λ) the sectionally holomorphic function on α∈{±} α with bound, M2 (C)) := {χ c (λ); χ±c (λ) − χ±c (∞) ∈ ary values χ±c (λ). Define: (1) H k (C \ >
ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA
407
, M2 (C)) := {χ c (λ) ∈ H k (C \ > , M2 (C)); det(χ c (λ)) ≡ ran C± }; (2) GH k (C \ > k , M2 (C)) := {χ c (λ) ∈ H k (C\ > , M2 (C)); det(χ c (λ)) = 1}. 0}; and (3) SH (C\ > − , M2 (C)) ∗ GHIk (> + , M2 (C)) (A ∗ THEOREM C.1.1. Every v(λ) ∈ GHIk (> , admits an RH factorisation, v(λ) = B := {xy; x ∈ A, y ∈ B}), λ ∈ > (χ−c (λ))−1 (λ)χ+c (λ), where (λ) := diag
λ − λ+ λ − λ−
k1 λ − λ+ k2 , , λ − λ−
λ± ∈ ± ,
, M2 (C)) (ki , i ∈ {1, 2}, are called the partial inand χ c (λ) ∈ GH k (C \ > dices (uniquely determined by v(·) up to a permutation) of v(λ)); moreover, if , M2 (C)), and 2j =1 kj = det(v(λ)) = 1, χ c (λ) can be chosen to be in SH k (C \ > Ij (> , M2 (C)), for some j ∈ {0, 1, . . . , k}, k ∈ Z1 , is 0. The matrix χ c (λ) ∈ H said to be a solution of the RH factorisation problem of v(λ) if χ±c (λ) − χ±c (∞) ∈ ± , M2 (C)). When v(∞) = I and (λ) = I, χ±c (λ) can be uniquely ran C± ⊂ H k (> determined by letting χ±c (∞) = I (canonical normalisation), in which case, χ±c (λ), or χ c (λ) (χ c (∞) = I), is called the fundamental solution of the RHP of v(λ). For the ISM, v(∞) = I. Conversely, if v(λ) admits a factorisation v(λ) = − , M2 (C)) ∗ GHIk (> + , M2 (C)). (χ−c (λ))−1 (λ)χ+c (λ), then v(λ) ∈ GHIk (> PROPOSITION C.1.1. tr(R(λ)) = 0 ⇒ det(χ c (λ)) = const. DEFINITION C.1.2. A linear operator L on HIk (D, M2 (C)) is Fredholm if: (1) the complement of ran L is open in HIk (D; M2 (C)); and (2) dim ker(L) and dim coker(L) are finite. For L linear and Fredholm, i(L) := dim ker(L) − dim coker(L) is called the (Fredholm) index of L. THEOREM C.1.2. Let k ∈ Z1 . If v(λ) in Theorem C.1.1 can be represented as the following (algebraic) block triangular factorisation, v(λ) := (I−w − (λ))−1 (I+ , where w ± (λ) ∈ H k (> ± , M2 (C)), I±w ± (λ) ∈ GHIk (> ± , M2 (C)), w + (λ)), λ ∈ > ± of nilpotency 2, and if, as a linear operator on and w (λ) are nilpotent, ' with degree Ik (> Ik (> Ik (> , M2 (C)) := α∈{±} HIk (> α , M2 (C)), Cw : H , M2 (C)) → H , M2 (C)) H k − + is defined as (f ∈ HI (>, M2 (C))) f → C+ (f w ) + C− (f w ), then id − Cw , Ik (> , M2 (C)), is Fredholm, that is, i(id − where id is the identity operator on H Cw ) = dim ker(id−Cw )−dim coker(id−Cw ) = 0, dim ker(id−Cw ) = 2 kj >0 kj , and dim coker(id−Cw ) = −2 kj <0 kj , where ki , i ∈ {1, 2}, are the partial indices d(arg det(v(·))) = 0, of v(λ); moreover, i(id − Cw ) = 2 ind det(v(λ)) = π1 > 2 c where ind det(v(λ)), the index of det(v(λ)), equals j =1 kj . Define χ0 (λ) := ((id − Cw )−1 I)(λ): then the boundary values χ±c (λ) := χ0c (λ)(I ± w ± (λ)) ∈ ± , M2 (C)) ⊂ (I + H k (> ± , M2 (C))) ∩ GHIk (> ± , M2 (C)) (I + ran C± ) ∩ GHIk (> give the fundamental solution of the RH factorisation problem for v(λ).
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THEOREM C.1.3. If all the partial indices of v(λ) are zero (ki = 0, i ∈ {1, 2}), Ik (> , M2 (C)), namely, then the Fredholm operator id − Cw is invertible on H ker(id − Cw ) = ∅ (dim ker(id − Cw ) = 0). LEMMA C.1.1. The RHP of v(λ) := (I − w − (λ))−1 (I + w + (λ)) = , where w ± (λ) ∈ H k (> ± , M2 (C)), has a fundamental (χ−c (λ))−1 χ+c (λ), λ ∈ > 1 c c solution (χ (∞) = I, χ (λ) ≡ 0) only if 2π > d(arg det(v(·))) = 0. Conversely, k c c if χ (λ) ∈ HI (>,M2 (C)), k ∈ Z1 , χ (∞) = I is a solution of the RHP of , and 1 d(arg det(v(·))) = 0, then χ c (λ) is a fundamental solution; v(λ) on > 2π > furthermore, det(v(λ)) = 1 ⇒ det(χ c (λ)) = 1. PROPOSITION C.1.2. If the RHP of v(λ) := (I − w − (λ))−1 (I + w + (λ)) = , where w ± (λ) ∈ H k (> ± , M2 (C)), admits a fundamen(χ−c (λ))−1 χ+c (λ), λ ∈ > j I (> , M2 (C)) for some j ∈ Z1 , then it is unique in tal solution χ c (λ) ∈ H 0 2 LI (>, M2 (C)) := HI (>, M2 (C)). PROPOSITION C.1.3. If the RHP of v(λ) := (I − w − (λ))−1 (I + w + (λ)) = , where w ± (λ) ∈ H k (> ± , M2 (C)), admits a fundamental (χ−c (λ))−1 χ+c (λ), λ ∈ > j I (> , M2 (C)) for some j ∈ Z0 , then id − Cw is invertible on solution χ c (λ) ∈ H j I (> , M2 (C)) ∀j ∈ {0, 1, . . . , k}, k ∈ Z1 . H ± , M2 (C)). If id − Cw is PROPOSITION C.1.4. Suppose that w ± (λ) ∈ H k (> j I (> , M2 (C)) for any j k, k ∈ Z1 , then it is invertible ∀j k. invertible on H Denote the Schwarz reflection of an M2 (C)-valued function by f S (λ) := (f (λ))† , where † denotes Hermitian conjugation, and, for a subset of C, as the reflection about R. is a Schwarz reflection invariant contour about R, v(λ) ∈ THEOREM C.1.4. If > − , M2 (C)) ∗ SHIk (> + , M2 (C)), v(∞) = I, v(·) is positive definite on R, SHIk (> −1 S v (λ) > Re(v(λ)) R > 0, and v(λ) > \R = σ \R σ , where σ is a constant, invertible, finite-order matrix involution which changes the sign(s) of some (or all) of the elements of the matrix on which it (and its inverse) is multiplied, then all the partial indices of v(λ) are zero, ki = 0, i ∈ {1, 2}. In this case, the RHP for v(λ) is solvable. The singular RHP, that is, the RH factorisation problem with isolated singularities (in this work, first-order poles), is now introduced. Let ζ ∈ C. For the remainder of this Appendix, the same symbol is used to denote an M2 (C)-valued function analytic in a punctured neighbourhood of ζ and the germ (the set of equivalence classes of analytic continuations) at ζ it represents, with the algebra of all such germs denoted by Aζ , and SAζ := {ϕζ (λ) ∈ Aζ ; det(ϕζ (λ)) = 1}. Let ± ∪ D, M2 (C)) := D ⊂ C, with card(D) < ∞. Set D ± := D ∩ ∓ . Define H k (>
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409
6
± , M2 (C)) ⊕ ( ζ ∈D Aζ ). An element in α∈{±} H k (> α ∪ D, M2 (C)) is H k (> represented either as ϕ(λ) := (ϕc (λ), ϕζ (λ))ζ ∈D , or
, ϕc (λ), λ ∈ > ϕ(λ) := ϕζ (λ), λ ≈ ζ, ζ ∈ D, 6 α , M2 (C)), and ϕζ (λ) ∈ Aζ (in the above, the subwhere ϕc (λ) ∈ α∈{±} H k (> script c is used to connote ‘continuous’, while the subscript ζ (for ζ ∈ D) is used ‘discrete’). The Cauchy integral operators, C± , are defined on 6 to connote k α α∈{±} H (> ∪ D, M2 (C)) in the following sense: construct the augmented con ∪(∪ζ ∈D > ζ are sufficiently small, mutually disjoint, and ζ ), where > aug := > tour > disjoint with respect to >, disks oriented counter-clockwise (respectively, clockD − ). Since, with the above-given conditions wise) ∀ζ ∈ D + (respectively, ∀ζ ∈ 6 ζ , ζ ∈ D, and, for each ϕ(λ) ∈ α∈{±} H k (> α ∪ D, M2 (C)), ϕ(λ) λ∈> on > aug ∃, 6 k α it represents an element in α∈{±} H (>aug , M2 (C)); hence, (C± ϕ)(λ) are defined, ± and (C± ϕ)(λ) ∈ H k (> aug , M2 (C)). Hereafter, (C± ϕ)(λ) are to be understood as k ± elements in H (> ∪D, M2 (C)). For ζ ∈ D + , (C+ ϕ)(λ) extends analytically into ζ , and (C− ϕ)(λ) := (C+ ϕ)(λ) − ϕ(λ) extends analytically the disk bounded by > into the punctured disk; therefore, they represent germs in Aζ , denoted by fζ± , respectively. Similarly, for ζ ∈ D − , (C− ϕ)(λ) extends analytically into the disk ζ , and (C+ ϕ)(λ) := (C− ϕ)(λ) + ϕ(λ) extends analytically into the bounded by > punctured disk; therefore, they represent germs in Aζ , denoted by fζ∓ , respec± ± ∪ := (fc± , fζ± )ζ ∈D ∈ H k (> tively. Write fc± := (C± ϕ)(λ) λ∈> , and define f 6 k α D, M2 (C)). From the construction above, C± : α∈{±} H (> ∪ D, M2 (C)) → ± ∪D, M2 (C)), and (C± ϕ)(λ) = f ± . In this sense, C± are called the Cauchy H k (> ∪ D. The following notion of piecewiseintegral operators with singular support > holomorphic matrix-valued function has been used throughout this paper. For an M2 (C)-valued function, $(λ), say, the ‘symbol’ $(λ) := ($c (λ), $ζ (λ))ζ ∈D is said to be a piecewise-holomorphic matrix-valued function with respect to the ∪ D if $c (λ) is a piecewise-holomorphic matrix-valued function on contour > \ D and $ζ (λ) ∈ Aζ is analytic at each ζ ∈ D. The boundary values $± (λ), if they exist, of the (generalised) holomorphic matrix-valued function $(λ) := ($c (λ), $ζ (λ))ζ ∈D are defined by , ($c (λ))+ , λ ∈ > $+ (λ) := $c (λ), λ ≈ ζ, ζ ∈ D − , λ ≈ ζ, ζ ∈ D + , $ζ (λ), , ($c (λ))− , λ ∈ > (C.1) $− (λ) := $c (λ), λ ≈ ζ, ζ ∈ D + , λ ≈ ζ, ζ ∈ D − , $ζ (λ), ∪ D, M2 (C)) := {$(λ); where ($c (λ))± := lim λ →λ $c (λ ). Define H k (C \ > λ ∈±
∪ D, M2 (C)) := {$(λ) ∈ H k (C \ $± (λ) − $± (∞) ∈ ran C± }, and SH k (C \ > ∪ D, M2 (C)); det($(λ)) = 1}. >
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− ∪ D, M2 (C)) ∗ SHIk (> + ∪ D, M2 (C)) THEOREM C.1.5. Every v(λ) ∈ SHIk (> admits an RH factorisation v(λ) := (χ− (λ))−1 (λ)χ+ (λ), where χ(λ) ∈ SH k (C\ ∪ D, M2 (C)), (λ) is defined in Theorem C.1.1, and λ± ∈ D ± ∪ (± \ D ∓ ). > ∪ D is Schwarz reflection invariant with respect to R, THEOREM C.1.6. If > − ∪ D, M2 (C))∗SHIk (> + ∪ D, M2 (C)), v(∞) = I, Re(v(λ))λ∈R> v(λ) ∈ SHIk (> −1 S v (λ) λ∈(> 0, and v(λ) λ∈(> ∪D)\R = σ ∪D)\R σ , where σ is a constant, invertible, finite-order matrix involution which changes the sign(s) of some (or all) of the elements of the matrix on which it (and its inverse) is multiplied, then all the partial indices of v(λ) are zero, ki = 0, i ∈ {1, 2}. In this case, the RHP for v(λ) is solvable. Note that, for D ≡ ∅, Theorem C.1.6 reduces to Theorem C.1.4. The asymptotic analysis of the latter part of the above-given paradigm, related to the singular RHP (when D ≡ ∅ and card(D) < ∞), is the subject of the present asymptotic study. Using the results of this subsection, the very important Lemma 2.4 of [50], and the Deift–Zhou nonlinear steepest descent method [55], the (rigorous) asymptotic analysis, as |t| → ∞ and |x| → ∞ such that z0 := x/t ∼ O(1) and ∈ R \ {−2, 0, 2}, of the RHP for m(ζ ) formulated in Lemma 2.1.2, for σd ≡ ∅, was completed in [38]. Acknowledgements The author is very grateful to X. Zhou for the invitation to Duke University and for the opportunity to complete this work. The author is also grateful to the referees for helpful suggestions. References 1. 2. 3.
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Mathematical Physics, Analysis and Geometry 5: 415–416, 2002.
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Contents of Volume 5 (2002) Volume 5 No. 1
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N. N. KHURI / Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis
1–63
M. BEN CHROUDA and H. OUERDIANE / Algebras of Operators on Holomorphic Functions and Applications
65–76
MICHEL TALAGRAND / On the Gaussian Perceptron at High Temperature
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Volume 5 No. 2
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DANIEL BUMP, PERSI DIACONIS and JOSEPH B. KELLER / Unitary Correlations and the Fejér Kernel 101–123 ROSSELLA BARTOLO and ANNA GERMINARIO / Trajectories Joining Two Submanifolds under the Action of Gravitational and Electromagnetic Fields on Static Spacetimes 125–143 LECH ZIELINSKI / Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients 145–182 in Rd F. ALBERTO GRÜNBAUM and PLAMEN ILIEV / Heat Kernel Expansions on the Integers 183–200 Volume 5 No. 3
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G. RUDOLPH, M. SCHMIDT and I. P. VOLOBUEV / Classification of Gauge Orbit Types for SU(n)-Gauge Theories 201–241 PAVEL KURASOV and SERGUEI NABOKO / On the Essential Spectrum of a Class of Singular Matrix Differential Operators. I: Quasiregularity Conditions and Essential Self-adjointness 243–286 BERNHARD G. BODMANN / A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion 287–306
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M. F. BORGES / Geometrical Lagrangian for a Supersymmetric Yang– Mills Theory on the Group Manifold 307–318 A. H. VARTANIAN / Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua 319–413
