Mathematical Physics, Analysis and Geometry 7: 1–8, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Relating Thomas–Whitehead Projective Connections by a Gauge Transformation CRAIG ROBERTS Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO 63701-4799, U.S.A. e-mail:
[email protected] (Received: 13 August 2002) Abstract. Thomas–Whitehead projective connections, or TW-connections, are torsionfree linear connections, satisfying certain properties, on a naturally defined principal R-bundle over a manifold. The name credits T. Y. Thomas and J. H. C. Whitehead, who originally studied these connections in the 1920’s and 1930’s. Three equivalence classes of TW-connections will be considered. This leads to a necessary and sufficient condition for TW-connections to be related by a gauge transformation; namely, they induce the same projective structure on the base manifold, have identical Ricci tensor, and induce the identity element in the one-dimensional de Rham cohomology vector space of the base manifold. Mathematics Subject Classifications (2000): 53C05, 53C22, 53C80. Key words: bundle of volume elements, gauge transformation, projective structure, Ricci equivalence, structural equivalence, Thomas–Whitehead projective connection.
1. Introduction Thomas–Whitehead projective connections, or TW-connections, have their origin in the work of T. Y. Thomas (1925, 1926) and J. H. C. Whitehead (1931). Each represented a projective connection on a manifold by means of a torsionfree linear connection defined on another manifold of one more dimension. With this in mind, a TW-connection is a torsionfree linear connection, satisfying certain properties, on a naturally defined principal R-bundle over a manifold. It is shown in (Roberts, 1992) and (Roberts, 1995) that a TW-connection induces a projective structure on the base manifold and that an equivalence relation on the set of TW-connections may be defined by calling TW-connections equivalent whenever they induce the same projective structure on the base manifold. In this paper two refinements of this equivalence relation are studied. The first refinement defines TW-connections to be Ricci equivalent if they induce the same projective structure on the base manifold and have the same Ricci tensor. The second refinement defines TW-connections to be gauge equivalent if they are related by a gauge transformation of the principal R-bundle. It is shown that such a gauge transformation exists if and only if the TW-connections are Ricci equivalent and induce the identity
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element in the one-dimensional de Rham cohomology vector space of the base manifold. 2. Structurally Equivalent TW-connections The relevant definitions and results for TW-connections will be reviewed in this section. Further details may be obtained by referring to (Roberts, 1992) and (Roberts, 1995). The construction of the principal R-bundle begins with a real n-dimensional vector space V . If v is a nonzero element in the nth exterior product of V , then the set ε = {±v} will be called a volume element. Defining a volume element only up to sign allows for the consideration of both orientable and nonorientable manifolds. The set of all volume elements will be denoted by E(V ). If M is a smooth n-dimensional manifold and p an element of M, replacing V with Tp M, the tangent space of M at p, allows the bundle of volume elements over M, denoted by E(M), to be defined as E(Tp M). E(M) = p∈M
E(M) is the total space of a principal R-bundle over M with the projection map π : E(M) → M defined by π(ε) = p. The structure group is the reals R under addition, and the right action on E(M) is given by ε · a = ea ε. The fundamental vector field on E(M) corresponding to the element d/dt in the Lie algebra of R will be called the canonical fundamental vector field on E(M) and denoted by ξ . DEFINITION 1. A Thomas–Whitehead projective connection, or TW-connection, is a torsionfree linear connection ∇ on E(M) that is invariant with respect to the right action of R on E(M) and for which ∇ξ = −
1 (id), n+1
where id is the identity (1, 1)-tensor and ξ the canonical fundamental vector field on E(M). A TW-connection ∇ induces a projective structure on the base manifold M in and Y are the the following way. If ω is a connection 1-form on E(M) and X respective ω-horizontal lifts of smooth vector fields X and Y on M, then ) ∇Xω Y = π∗ (∇XY defines a torsionfree linear connection ∇ ω on M. For the same TW-connection ∇, but a different connection 1-form ω, the torsionfree linear connection ∇ ω belongs to the same projective structure as ∇ ω . Furthermore, the mapping ω → ∇ ω is a one-to-one correspondence between the connection 1-forms on E(M) and the
GAUGE-RELATED TW-CONNECTIONS
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torsionfree linear connections belonging to the same projective structure as ∇ ω . Consequently, the TW-connection ∇ induces a projective structure on the base manifold M. DEFINITION 2. TW-connections that induce the same projective structure on the base manifold M are structurally equivalent. Definition 2 is a change in terminology from (Roberts, 1992) and (Roberts, 1995) designed to distinguish this equivalence class of TW-connections from those that follow. If the base manifold M has dimension one, then all TW-connections are structurally equivalent in this special case. This follows since the induced connections on M all have the same geodesics up to a reparameterization, which implies the induced connections belong to the same projective structure on M. See (Spivak, 1979) and (Kobayashi, 1972). Structurally equivalent TW-connections are more generally characterized by the existence of a symmetric (0, 2)-tensor on E(M). THEOREM 1. The TW-connections ∇ and ∇ are structurally equivalent if and only if there is a unique symmetric (0, 2)-tensor β on E(M) such that the Lie derivative Lξ β = 0, and ∇ = ∇ + (ιξ β) ⊗ id + id ⊗ (ιξ β) − β ⊗ ξ, where (ιξ β) denotes a 1-form on E(M) defined by (ιξ β)(X) = β(ξ, X), for any smooth vector field X on E(M), and satisfies (ιξ β)(ξ ) = 0.
3. Ricci Equivalent TW-connections DEFINITION 3. The Ricci tensor of a TW-connection ∇ is a (0, 2)-tensor on E(M) defined by Ricci(X, Y ) = trace{V → R(V , X)Y }, where R is the curvature tensor of ∇ and X, Y , and V are smooth vector fields on E(M). The Ricci tensor of a TW-connection satisfies two special properties. First, since a TW-connection is invariant with respect to the right action of R on E(M), its Ricci tensor will be invariant with respect to this right action. Also, the Ricci tensor is 0 if one of the arguments is the canonical fundamental vector field ξ . These two properties imply that Ricci = π ∗ α, for a (0, 2)-tensor α on M. For structurally equivalent TW-connections, a straightforward, but rather lengthy, calculation shows the difference of their Ricci tensors may be expressed in terms of the symmetric (0, 2)-tensor β from Theorem 1.
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THEOREM 2. If ∇ and ∇ are structurally equivalent TW-connections and their respective Ricci tensors are denoted by Ricci and Ricci, then Ricci − Ricci = −(n + 1) d(ιξ β) + + (n − 1) (ιξ β) ⊗ (ιξ β) − sym ∇(ιξ β) +
1 β , n+1
where β and ιξ β are defined as in Theorem 1 and sym∇(ιξ β) denotes the symmetric part of ∇(ιξ β). DEFINITION 4. The TW-connections ∇ and ∇ are Ricci equivalent if they are structurally equivalent and have identically equal Ricci tensors. As was mentioned in the previous section, for the special case in which the base manifold M has dimension one, all TW-connections are structurally equivalent. In addition, whenever M has dimension one, E(M) has dimension two and the Ricci tensor of any TW-connection is identically 0. Consequently, all TW-connections are Ricci equivalent for the special case in which the base manifold has dimension one. More generally, Ricci equivalent TW-connections are characterized by sharpening Theorem 1 to the following. THEOREM 3. The TW-connections ∇ and ∇ are Ricci equivalent if and only if there is a unique closed 1-form φ on E(M) such that φ(ξ ) = 0 and ∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ. Proof. If ∇ and ∇ are Ricci equivalent TW-connections, they are structurally equivalent and Theorem 1 implies there exists a unique symmetric (0, 2)-tensor β on E(M) such that Lξ β = 0, (ιξ β)(ξ ) = 0, and ∇ = ∇ + (ιξ β) ⊗ id + id ⊗ (ιξ β) − β ⊗ ξ. Since ∇ and ∇ also have identically equal Ricci tensors, the equation for the difference of their Ricci tensors given by Theorem 2 becomes 1 β = 0. −(n + 1)d(ιξ β) + (n − 1) (ιξ β) ⊗ (ιξ β) − sym ∇(ιξ β) + n+1 Considering the skew-symmetric and symmetric parts, respectively, of this equation yields d(ιξ β) = 0 and 1 β = 0. n+1 From these equations, as well as ∇(ιξ β) = sym ∇(ιξ β) − d(ιξ β), it follows that β = (n + 1)(∇(ιξ β) − (ιξ β) ⊗ (ιξ β)). Thus, setting φ = ιξ β yields a unique closed 1-form such that φ(ξ ) = 0 and (ιξ β) ⊗ (ιξ β) − sym ∇(ιξ β) +
∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ.
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Conversely, if φ is a closed 1-form on E(M) and φ(ξ ) = 0, then ∇φ = sym ∇φ − dφ = sym ∇φ and (Lξ φ)(X) = d(φ(ξ ))(X) + 2 dφ(ξ, X) = 0, for all smooth vector fields X on E(M). Furthermore, if φ is the unique closed 1-form on E(M) satisfying ∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ, then β = (n + 1)(∇φ − φ ⊗ φ) defines a unique symmetric (0, 2)-tensor on E(M) for which Lξ β = 0, ιξ β = φ, and ∇ = ∇ + (ιξ β) ⊗ id + id ⊗ (ιξ β) − β ⊗ ξ. Hence, ∇ and ∇ are structurally equivalent by Theorem 1. Substituting the expressions for β and ιξ β into the equation for the difference in the Ricci tensors of ∇ and ∇ given by Theorem 2 and simplifying shows ∇ and ∇ have identical Ricci ✷ tensors. Therefore, ∇ and ∇ are Ricci equivalent.
4. Gauge Equivalent TW-connections DEFINITION 5. A diffeomorphism g: E(M) → E(M) such that g(ε · a) = g(ε) · a, for all ε in E(M) and real numbers a, and which induces the identity map on the base manifold M is a gauge transformation of E(M). For a TW-connection ∇, a gauge transformation g of E(M), and smooth vector fields X and Y on E(M), the expression g∗−1 (∇g∗ (X) g∗ (Y )) defines a TW-connection since g commutes with the right action of R on E(M). Also, a one-to-one correspondence between the group of gauge transformations of E(M) and the set of smooth maps from M to R can be established by setting g(ε) = ε · (f ◦ π )(ε), for all ε in E(M), where f is a smooth map from M to R and π is the projection map from E(M) to M (Bleecker, 1981). DEFINITION 6. The TW-connections ∇ and ∇ are gauge equivalent if there is a gauge transformation g of E(M) such that ∇ X Y = g∗−1 (∇g∗ (X) g∗ (Y )), for all smooth vector fields X and Y on E(M). The next theorem shows gauge equivalence is a finer equivalence relation than Ricci equivalence and structural equivalence. THEOREM 4. Gauge equivalent TW-connections are Ricci equivalent.
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Proof. If ∇ and ∇ are gauge equivalent TW-connections, there exists a gauge transformation g of E(M) such that ∇ X Y = g∗−1 (∇g∗ (X) g∗ (Y )), for all smooth vector fields X and Y on E(M). If R and R denote the respective curvature tensors of ∇ and ∇, then R(X, Y )Z = g∗−1 (R(g∗ (X), g∗ (Y ))g∗ (Z)), where Z is a smooth vector field on E(M), and it follows that Ricci = g ∗ Ricci. In the previous section, it was noted that there is a (0, 2)-tensor α on M such that Ricci = π ∗ α. Therefore, Ricci = g ∗ Ricci = g ∗ (π ∗ α) = (π ◦ g)∗ α = π ∗ α = Ricci, and the Ricci tensors of gauge equivalent TW-connections are identical. It remains to show that the gauge equivalent TW-connections ∇ and ∇ are structurally equivalent. Let ω be a connection form on E(M). For the gauge transformation g relating ∇ and ∇, g ∗ ω is also a connection form on E(M). Taking X and Y to be smooth vector fields on M, and denoting their respective g ∗ ωhorizontal lifts by X and Y , we have 0 = g ∗ ω(X) = ω(g∗ (X)) and 0 = g ∗ ω(Y ) = ω(g∗ (Y )). Thus, the vector fields g∗ (X) and g∗ (Y ) on E(M) are the ω-horizontal ∗ lifts of X and Y , respectively. For the induced connections ∇ ω and ∇ g ω on M, this implies ∇Xω Y = π∗ (∇g∗ (X) g∗ (Y )) = (π ◦ g −1 )∗ (∇g∗ (X) g∗ (Y )) g∗ω
= π∗ (∇ X Y ) = ∇ X Y since π ◦ g −1 = π . Therefore, ∇ and ∇ induce the same projective structure on M so they are structurally equivalent. ✷ Theorems 3 and 4 suggest the possibility of further refining Theorem 1 for gauge equivalent TW-connections. THEOREM 5. The TW-connections ∇ and ∇ are gauge equivalent if and only if there is a unique exact 1-form φ on E(M) such that φ(ξ ) = 0 and ∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ. Proof. If ∇ and ∇ are gauge equivalent TW-connections, then Theorem 4 implies they are Ricci equivalent. It must be shown that the unique closed 1-form φ given by Theorem 3 is exact for the case of gauge equivalent TW-connections. Recall from the proof of Theorem 3 that the property φ(ξ ) = 0 for a closed 1form φ implies Lξ φ = 0. Hence, there exists a closed 1-form ρ on M such that π ∗ ρ = φ, and the equation in Theorem 3 may be written as ∇ = ∇ + π ∗ ρ ⊗ id + id ⊗ π ∗ ρ − (n + 1)(∇π ∗ ρ − π ∗ ρ ⊗ π ∗ ρ) ⊗ ξ.
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and If ω is a connection form on E(M), X and Y smooth vector fields on M, and X Y their respective ω-horizontal lifts, then this equation becomes = ∇XY + π ∗ ρ(X) Y + π ∗ ρ(Y )X − ∇ XY ∗ Y ) − π ∗ ρ(X)π ∗ ρ(Y ))ξ. − (n + 1)((∇π ρ)(X; Projecting by π∗ to M gives ∇ ωX Y = ∇Xω Y +ρ(X)Y +ρ(Y )X. H. Weyl showed this equation implies ∇ ω and ∇ ω are projectively equivalent; in other words, they have the same geodesics up to a reparameterization (Spivak, 1979). This is equivalent to ∇ ω and ∇ ω belonging to the same projective structure on M (Kobayashi, 1972). The gauge transformation g of E(M) relating ∇ and ∇ provides another means of obtaining the equation showing ∇ ω and ∇ ω are projectively equivalent. Since g may be expressed as g(ε) = ε · (f ◦ π )(ε), for all ε in E(M) and some smooth =X + π ∗ df (X)ξ and g∗ (Y ) = Y + map f from M to R, it follows that g∗ (X) ∗ −1 )ξ . Hence, ∇ XY )) becomes = g∗ (∇g (X) π df (Y g (Y ∗ ∗ 1 Y − 1 π ∗ df (Y )X + π ∗ df (X) n+1 n+1 )) − π ∗ df (∇XY ∗ df (Y ))ξ. ) + 1 π ∗ df (X)π ∗ df (Y + (X(π n+1 Projecting by π∗ to M gives = ∇XY − ∇ XY
∇ ωX Y = ∇Xω Y −
1 1 df (X)Y − df (Y )X, n+1 n+1
which again shows ∇ ω and ∇ ω are projectively equivalent. Setting ρ = −(n + 1)−1 df shows ρ is exact, which in turn implies φ = π ∗ ρ is exact. Conversely, assume there exists an exact 1-form φ satisfying the hypotheses, then ∇ and ∇ are Ricci equivalent by Theorem 3 since an exact 1-form is closed. From the properties dφ = 0 and φ(ξ ) = 0, it follows that Lξ φ = 0, which implies there is a 1-form ρ on M such that φ = π ∗ ρ. Furthermore, ρ must be exact since φ is an exact 1-form satisfying φ(ξ ) = 0. Let f be the smooth map from M to R such that df = ρ. Hence, φ = π ∗ df . Define a gauge transformation g of E(M) by g(ε) = ε · (−(n + 1)(f ◦ π )(ε)), for all ε in E(M). Thus, for smooth vector fields X and Y on E(M), g∗−1 (∇g∗ (X) g∗ (Y )) = ∇X Y + π ∗ df (X)Y + π ∗ df (Y )X − − (n + 1)((∇π ∗ df )(X; Y ) − π ∗ df (X)π ∗ df (Y ))ξ. By assumption, the right-hand side of this equation is merely ∇ X Y since φ = ✷ π ∗ df . Therefore, ∇ and ∇ are gauge equivalent TW-connections. Since gauge equivalence is characterized by an exact 1-form and Ricci equivalence is characterized by a closed 1-form, a relationship to the one-dimensional de Rham cohomology vector space is suggested.
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DEFINITION 7. The vector space H 1 (M, R) = {closed 1-forms on M}{exact 1-forms on M} is the one-dimensional de Rham cohomology vector space of M. A pair of Ricci equivalent TW-connections ∇ and ∇ induce a de Rham cohomology class on M in a natural way. If φ is the unique closed 1-form on E(M) given by Theorem 3, then recalling the proof of Theorem 5 there is a closed 1-form ρ on M such that φ = π ∗ ρ. The induced de Rham cohomology class on M is [ρ]. THEOREM 6. The TW-connections ∇ and ∇ are gauge equivalent if and only if they are Ricci equivalent and the induced de Rham cohomology class on M is 0. Proof. The result follows from Theorems 3 and 5 since the unique closed 1-form φ characterizing a pair of Ricci equivalent TW-connections may be expressed as φ = π ∗ ρ, for a closed 1-form ρ on M, and φ is exact if and only if ρ is exact. ✷ A consideration of the stronger condition that the vector space H 1 (M, R) = 0 yields the following corollary. COROLLARY 7. Ricci equivalence and gauge equivalence are identical if and only if H 1 (M, R) = 0. Since the Poincaré Lemma shows that the one-dimensional de Rham cohomology vector space of a contractible manifold is 0 (Conlon, 1993), Corollary 7 implies Ricci equivalence and gauge equivalence are identical when the base manifold is contractible. In particular, it was noted in the previous section that all TW-connections are Ricci equivalent when the base manifold has dimension one. Therefore, if the base manifold has dimension one and is contractible, Corollary 7 shows that all TW-connections are gauge equivalent. References Bleecker, D. (1981) Gauge Theory and Variational Principles, Addison-Wesley, Reading, MA. Conlon, L. (1993) Differentiable Manifolds: A First Course, Birkhäuser, Boston. Kobayashi, S. (1972) Transformation Groups in Differential Geometry, Springer-Verlag, New York. Roberts, C. W. (1992) The projective connections of T. Y. Thomas and J. H. C. Whitehead on the principal R-bundle of volume elements, PhD Thesis, Saint Louis University, St. Louis, MO. Roberts, C. W. (1995) The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to invariant connections, Differential Geom. Appl. 5, 237–255. Spivak, M. (1979) A Comprehensive Introduction to Differential Geometry II, 2nd edn, Publish or Perish, Wilmington. Thomas, T. Y. (1925) On the projective and equi-projective geometries of paths, Proc. Nat. Acad. Sci. 11, 199–203. Thomas, T. Y. (1926) A projective theory of affinely connected manifolds, Math. Z. 25, 723–733. Whitehead, J. H. C. (1931) The representation of projective spaces, Ann. of Math. 32, 327–360.
Mathematical Physics, Analysis and Geometry 7: 9–46, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Heat Kernel Asymptotics of Zaremba Boundary Value Problem IVAN G. AVRAMIDI Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, U.S.A. e-mail: iavramid@nmt. edu (Received: 30 October 2001; in final form: 18 July 2002) Abstract. The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with discontinuous boundary conditions, which include Dirichlet boundary conditions on one part of the boundary and Neumann boundary conditions on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the asymptotic solution of the heat equation is described in detail and the heat kernel is computed explicitly in the leading approximation. Some of the first nontrivial coefficients of the heat kernel asymptotic expansion are computed explicitly. Mathematics Subject Classifications (2000): 58J35, 58J37, 58J50, 58J32, 35P20, 35K20. Key words: boundary value problem, heat kernel, spectral asymptotics, spectral geometry.
1. Introduction The heat kernel of elliptic partial differential operators acting on sections of vector bundles over compact manifolds proved to be of great importance in mathematical physics. In particular, the main objects of interest in quantum field theory and statistical physics, such as the effective action, the partition function, Green functions, and correlation functions, are described by the functional determinants and the resolvents of differential operators, which can be expressed in terms of the heat kernel. The most important operators appearing in physics and geometry are the second order partial differential operators of Laplace type; such operators are characterized by a scalar leading symbol (even if acting on sections of vector bundles). Within the smooth category this problem has been studied extensively during last years (see, for example, [30, 10]; for reviews see [9, 4, 3] and references therein). In the case of smooth compact manifolds without boundary the problem of calculation of heat kernel asymptotics reduces to a purely computational (algebraic) one for which various powerful algorithms have been developed [1, 43]; this problem is now well understood. In the case of smooth compact manifolds with a smooth boundary and smooth boundary conditions the complexity of the problem depends significantly on the type of the boundary conditions. The clas-
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sical smooth boundary problems (Dirichlet, Neumann, or a mixed combination of those on vector bundles) are the most extensively studied ones (see [12, 13, 34, 2] and the references therein). A more general scheme, so called oblique (or Grubb– Gilkey–Smith) boundary value problem [31, 29, 28], which includes tangential (oblique) derivatives along the boundary, has been studied in [7, 8, 6, 22–24]. In this case the problem is not automatically elliptic; there is a certain strong ellipticity condition on the leading symbol of the boundary operator. This problem is much more difficult to handle, the main reason being that the heat kernel asymptotics are no longer polynomial in the jets of the symbols of the differential operator and the boundary operator. Another class of boundary value problems are characterized by essentially nonlocal boundary conditions, for example, the spectral or Atiyah–Patodi–Singer boundary conditions [30, 32, 11, 38]. All the boundary value problems described above are smooth. A more general (and much more complicated) setting, so called singular boundary value problem, arises when either the symbol of the differential operator or the symbol of the boundary operator (or the boundary itself) are not smooth. In this paper we study a singular boundary value problem for a second order partial differential operator of Laplace type when the operator itself has smooth coefficients but the boundary operator is not smooth. The case when the manifold as well as the boundary are smooth, but the boundary operator jumps from Dirichlet type to Neumann type along the boundary, is known in the literature as Zaremba problem. Such problems often arise in applied mathematics and engineering and there are some exact results available for special cases (two or three dimensions, specific geometry, etc.) [42, 25]. Zaremba problem belongs to a much wider class of singular boundary value problems, i.e. manifolds with singularities (corners, edges, cones, etc.). There is a large body of literature on this subject where the problem is studied from an abstract function-analytical point of view [26, 14–18, 41, 37, 35, 33, 27]. However, the study of heat kernel asymptotics of Zaremba type problems is quite new, and there are only some preliminary results in this area [5, 40, 21, 20]. Moreover, compared to the smooth category the needed machinery is still underdeveloped. We would like to stress that we are interested not only in the asymptotics of the trace of the heat kernel, i.e. the integrated heat kernel diagonal, but also in the local asymptotic expansion of the off-diagonal heat kernel. In this paper we study Zaremba boundary value problem for second-order partial differential operators F of Laplace type acting on sections of a vector bundle V over a smooth compact manifold M of dimension m with the boundary ∂M. The boundary is decomposed as the disjoint union ∂M = 1 ∪ 2 ∪ 0 , so that 1 = 1 ∪ 0 and 2 = 2 ∪ 0 are smooth compact co-dimension one submanifolds with the boundary 0 = ∂ 1 = ∂ 2 , which is a smooth compact co-dimension two submanifold without boundary, ∂0 = ∅. Both the manifold M and its boundary ∂M are assumed to be smooth and the differential operator F to have smooth coefficients. However, the boundary operator B is discontinuous on the boundary, it jumps from the Dirichlet type operator on 1 to Neumann
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type boundary operator on 2 . We will see that to fix the problem completely one has to impose an additional boundary condition along 0 as well (see Section 4 below). Since this problem is not smooth there could be additional logarithmic terms in the asymptotic expansion of the trace of the heat kernel TrL2 exp(−tFB ) as t → 0. However, Seeley [40] has shown recently (confirming the previous conjecture of [5]) that the logarithmic terms do not appear: THEOREM 1. There is asymptotic expansion as t → 0+ in half-integer powers of t only TrL2 exp(−tFB ) ∼
∞
t (k−m)/2 Bk ,
(1)
k=0
with the coefficients Bk given by the integral of local invariants over the manifolds M, 1 , 2 and 0 (0) (1),1 (1),2 Bk = bk + bk + bk + bk(2) . (2) M
1
2
0
This seems to contradict the conclusions of [21], where it has been shown that such an expansion with locally computable coefficients does not exist. The term ‘locally computable’ is confusing though. As we show below the calculation of the coefficients of the asymptotic expansion of the trace of the heat kernel involves the knowledge of some global information, in particular the spectral data of a certain differential operator with mixed boundary conditions on a unit semicircle in the normal bundle to 0 (see Section 8). So, one could say that these coefficients are locally computable in local coordinates on 0 and the normal distance to 0 but are global in the angular coordinate around 0 (see Section 8). Thus the standard asymptotic expansion (1) in powers of t (without logarithmic terms) still exists with coefficients (2) given by integrals over M, 1 , 2 and 0 . The interior coefficients, bk(0) , and the co-dimension one coefficients, bk(1),1 and bk(1),2 , are ‘locally computable’, but the co-dimension two coefficients, bk(2) , are ‘global’ in the angular coordinate (or pseudo-local) and require new methods of calculation (e.g., like the approach of this paper or [40]). They are constructed from the local invariants on 0 . It is the numerical coefficients that are global. Let us stress here that our goal in this paper is not to provide a rigorous construction of the parametrix of the heat equation with all the definitions and estimates, which, for a singular boundary-value problem, is a task that would require a separate paper. For such a treatment the reader is referred to the papers [38, 39, 29, 28, 11, 32] for the smooth case and to [26, 14–18, 41, 37, 35, 33, 27] for the singular case. Here we shall adhere instead to a pragmatic approach and will describe the construction of an asymptotic solution of the heat equation that can be used to calculate explicitly the heat kernel asymptotics. The main object of our investigation is not
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the parametrix but the exact explicit formulas for the coefficients of the asymptotic expansion of the heat kernel. Let us summarize briefly our main results. First of all, we provide the correct formulation of Zaremba type boundary value problem. We find that the boundary conditions on the open sets 1 and 2 are not enough to fix the problem, and an additional boundary condition along the singular set 0 is needed. This additional boundary condition can be considered formally as an ‘extension’ of Dirichlet conditions from 1 to 0 , or an ‘extension’ of Neumann conditions from 2 to 0 . However, strictly speaking the boundary condition on 0 does not follow from the boundary conditions on 1 and 2 and can be chosen rather arbitrarily. In fact, one needs some supplementary ‘physical’ criteria to fix this boundary condition. Second, we describe the geometry of the problem, which involves now some nontrivial geometrical quantities (normal bundle and extrinsic curvatures) that characterize properly the imbedding of a co-dimension two submanifold 0 in M. The higher order coefficients bk(2) are invariants constructed from those geometric quantities. Next, we describe the construction of the asymptotic solution of the heat equation in the interior of the manifold M, in a thin shell close to 1 and 2 , and finally, in a thin strip close to 0 . We use the standard scaling device; the difference is just what coordinates are involved in the scaling. Finally, we find explicit formula for the off-diagonal heat kernel in M0bnd , the thin strip close to 0 , in the leading approximation, and use it to compute the first nontrivial ‘global’ coefficient, b2(2) , of the heat kernel asymptotic expansion. We consider two types of the additional boundary condition along 0 , one being the ‘extension’ of Dirichlet boundary conditions (that we call regular boundary condition), and another being the ‘extension’ of the Neumann (or rather Robin) boundary conditions. We show that the result, i.e. the coefficient b2(2) , does depend on the type of the boundary condition at 0 , i.e. Dirichlet vs Neumann, but does not depend on the parameter of the Robin boundary condition (it will however contribute to the higher order coefficients). Our main result can be summarized in the following THEOREM 2. Let (M, g) be a smooth compact Riemannian manifold of dimension m with the Riemannian metric g with the boundary ∂M = 1 ∪2 ∪0 , where 1 and 2 are disjoint smooth co-dimension one submanifolds with compact closures 1 and 2 with the common boundary 0 = ∂ 1 = ∂ 2 ; 0 being a smooth compact co-dimension two submanifold without boundary. Let V be a smooth Hermitian vector bundle over the manifold M and ∇ be the natural extension of the connection on the vector bundle V by using the Levi-Civita connection. Let Q be a smooth Hermitian endomorphism of the vector bundle V , be a smooth Hermitian endomorphism of the vector bundle V restricted to the boundary, r be the normal geodesic distance to the boundary, ρ be the normal geodesic distance to 0 and s be a real parameter. Let FB be the Laplace type operator F = −g µν ∇µ ∇ν + Q subject to Zaremba boundary conditions ϕ|1 = 0,
(∂r + )ϕ|2 = 0
(3)
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and one of the following √ ( ρϕ)|0 = 0
(4)
or √ (∂ρ − s)( ρϕ)|0 = 0.
(5)
Then the trace of the heat kernel of the Zaremba boundary value problem has the following asymptotic expansion as t → 0+ TrL2 exp(−tFB ) √ −m/2 1/2 π dim V [vol(2 ) − vol(1 )] + dim V vol(M) + t = (4π t) 2 1 1 R− trV Q + dim V K+ dim V +t 6 3 M M ∂M 3/2 π +2 trV + α dim V vol(0 ) + O t , (6) 4 2 where R is the scalar curvature of the metric g, K is the trace of the extrinsic curvature of the boundary, and α is a numerical constant, α = −1 for the boundary condition (4) and α = 7 for the boundary condition (5). This is in agreement with [40, 20]. This paper is organized as follows. In Section 2 the formal description of Zaremba type boundary value problem is given. In Section 3 the general form of the heat kernel asymptotic expansion is described. In Section 4 we describe the relevant geometrical framework. Section 5 is devoted to the construction of the asymptotic solution of the heat equation in the interior of the manifold. In Section 6 we describe the asymptotic solution of the heat equation in a thin strip near 1 with the Dirichlet boundary conditions. Similarly, Section 7 deals with the asymptotic solution of the heat equation in a thin strip near 2 with Neumann boundary conditions. Section 8 is the central section of the paper. Here we construct the local asymptotic solution of the Zaremba boundary value problem in the neighborhood of the singular set 0 in the leading approximation and explicitly calculate the first non-trivial singular heat kernel coefficients. 2. General Setup 2.1. LAPLACE TYPE OPERATORS Let (M, g) be a smooth compact Riemannian manifold of dimension m with a boundary ∂M, equipped with a positive definite Riemannian metric g. Let V be a vector bundle over M, V ∗ be its dual, and End(V ) ∼ = V ⊗ V ∗ be the corresponding bundle of endomorphisms. Given any vector bundle V , we denote by C ∞ (V ) its
14
IVAN G. AVRAMIDI
space of smooth sections. We assume that the vector bundle V is equipped with a Hermitian metric. This naturally identifies the dual vector bundle V ∗ with V , and defines a natural L2 inner product and the L2 -trace using the invariant Riemannian measure d volg on the manifold M. The completion of C ∞ (V ) in this norm defines the Hilbert space L2 (V ) of square integrable sections. We denote by TM and T ∗ M the tangent and cotangent bundles of M. Let a connection, ∇ V : C ∞ (V ) → C ∞ (T ∗ M ⊗ V ), on the vector bundle V be given, which we assume to be compatible with the Hermitian metric on the vector bundle V . The connection is given its unique natural extension to bundles in the tensor algebra over V and V ∗ . In fact, using the Levi-Civita connection ∇ LC of the metric g together with ∇ V , we naturally obtain connections on all bundles in the tensor algebra over V , V ∗ , TM and T ∗ M; the resulting connection will usually be denoted just by ∇. It is usually clear which bundle’s connection is being referred to, from the nature of the section being acted upon. We also adopt the Einstein convention and sum over repeated indices. With our notation, Greek indices, µ, ν, . . . , label the local coordinates x = (x µ ) on M and range from 1 through m, lower case Latin indices from the middle of the alphabet, i, j, k, l, . . . , label the local coordinates xˆ = (xˆ i ) on ∂M (codimension one submanifold) and range from 2 through m, and lower case Latin indices from the beginning of the alphabet, a, b, c, d, . . . , label the local coordinates x˜ = (x˜ a ) on a codimension two submanifold 0 ⊂ ∂M that will be described later and range over 3, . . . , m. Further, we will denote by gˆ and ∇ˆ the induced metric and the corresponding Levi-Civita connection on the codimension one submanifolds 1 and 2 and by g˜ and ∇˜ the induced metric and the corresponding Levi-Civita connection on the codimension two submanifold 0 . Let ∇ ∗ be the formal adjoint of the covariant derivative defined using the Riemannian metric and the Hermitian structure on V and let Q ∈ C ∞ (End(V )) be a smooth Hermitian section of the endomorphism bundle End(V ). DEFINITION 1. form
A partial differential operator F : C ∞ (V ) → C ∞ (V ) of the
F = ∇ ∗ ∇ + Q = −g µν ∇µ ∇ν + Q
(7)
is called Laplace type operator. Alternatively, the Laplace type operators are second-order partial differential operators with positive definite scalar leading symbol of the form σL (F ; x, ξ ) = I|ξ 2 | = Ig µν (x)ξµ ξν .
(8)
Hereafter I denotes the identity endomorphism of the vector bundle V . We will often omit it whenever it does not cause any misunderstanding. Any second-order operator with a scalar leading symbol can be put in the form (7) by choosing the Riemannian metric g, the connection ∇ V on the vector bundle V and the endomorphism Q.
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
15
2.2. BOUNDARY CONDITIONS In the case of manifolds with boundary, one has to impose some boundary conditions in order to make a (formally self-adjoint) differential operator self-adjoint (at least symmetric) and elliptic. As usual, by using the inward geodesic flow, we identify a narrow neighbourhood of the boundary ∂M with a part of R+ ×∂M and define a split of the cotangent bundle T ∗ M = R ⊕ T ∗ ∂M. Let r be the normal geodesic distance to the boundary, so that N = ∂r is the inward unit normal on ∂M and xˆ = (xˆ i ) be the local ˆ coordinates on ∂M. Near ∂M we choose the local coordinates x = (x µ ) = (r, x). Let W = V |∂M be the restriction of the vector bundle V to the boundary ∂M. We define the boundary data map ψ: C ∞ (V ) → L2 (W ⊕ W ) by
ϕ|∂M . (9) ψ(ϕ) = ∇N ϕ|∂M The boundary conditions then read Bψ(ϕ) = 0,
(10)
where B: L2 (W ⊕ W ) → L2 (W ⊕ W ) is the boundary operator, which will be specified later. If the operator B is a tangential differential operator (possibly of order zero), then the boundary conditions are local. Otherwise, for example, when B is a pseudo-differential operator, the boundary conditions are nonlocal. To define the boundary operator one needs a self-adjoint orthogonal projector , that splits the space L2 (W ) in two orthogonal subspaces L2 (W ) = L2 (W ) ⊕ L2⊥ (W ),
(11)
where L2 (W ) = ,L2 (W )
and
L2⊥ = (Id − ,)L2(W ),
(12)
and a self-adjoint operator : L2 (W ) → L2 (W ), such that L2 (W ) = {0}, i.e. , = , = 0. Hereafter Id denotes the identity operator. The boundary operator is then defined by
, 0 B= , (13) Id − , which is equivalent to the following boundary conditions ,(ϕ|∂M ) = 0, (Id − ,)(∇Nϕ |∂M ) + (ϕ|ϕM ) = 0. It is easy to see that the boundary operator B and the operator
Id − , 0 K = Id − B = , − ,
(14) (15)
(16)
16
IVAN G. AVRAMIDI
are complimentary projectors on L2 (W ⊕ W ), i.e. B 2 = B,
K 2 = K,
BK = KB = 0.
(17)
Hence, a section that satisfies the boundary conditions can be parametrized by χ(ϕ) = u(ϕ) ⊕ v(ϕ) ∈ L2 (W ⊕ W ), u(ϕ) ∈ L2⊥ , v(ϕ) ∈ L2 , so that ψ(ϕ) = Kχ(ϕ) =
u(ϕ) −u(ϕ) + v(ϕ)
.
(18)
It is mainly the projector , that specifies the boundary conditions. It is not difficult to see that the boundary operator (13) incorporates all standard types of boundary conditions. Indeed, by choosing , = I and = 0 one gets the Dirichlet boundary conditions, by choosing , = 0, = 0 one gets the Neumann boundary conditions (if is a smooth endomorphism then these are called Robin boundary conditions). More generally, the case when , and are smooth endomorphisms of the bundle W corresponds to the mixed boundary conditions. If is a first order tangential differential operator then we have oblique boundary conditions. Remark. The boundary ∂M could be, in general, a disconnected manifold con sisting of a finite number of disjoint connected parts, ∂M = ni=1 i , with each i being compact connected manifold without boundary, ∂i = ∅ and i ∩ j = ∅ if i = j . Thus one can impose different boundary conditions on different connected parts of the boundary i . This means that the full boundary operator decomposes B = B1 ⊕ · · · ⊕ Bn , with Bi being different boundary operators acting on different bundles. 2.2.1. Nonsmooth Boundary Conditions We always assume the manifold M itself and the coefficients of the operator F to be smooth in the interior of M. If, in addition, the boundary ∂M is smooth, and the boundary operator B is a differential operator with smooth coefficients, then (F, B) is called smooth local boundary value problem. In this paper we are interested in a different class of boundary conditions. Namely, we do not assume the boundary operator to be smooth. Instead, we will study the case when it has discontinuous coefficients. Such problems are often called mixed boundary conditions; to avoid misunderstanding we will not use this terminology. We impose different boundary conditions on connected parts of the boundary, which makes the boundary value problem singular. Roughly speaking, one has a decomposition of a smooth boundary in some parts where different types of the boundary conditions are imposed, i.e. Dirichlet or Neumann. The boundary operator is then discontinuous at the intersection of these parts. The boundary value problems of this type are called Zaremba problem in the literature [14, 15] (see also [42, 25, 5, 21, 20]).
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
17
In this paper we consider the simplest case when there are just two components. We assume that the boundary of the manifold ∂M is decomposed as the disjoint union ∂M = 1 ∪ 2 ∪ 0 ,
(19)
so that the closures 1 = 1 ∪ 0 and 2 = 2 ∪ 0 are smooth compact submanifolds of dimension (m−1) (codimension one submanifolds), with the same boundary 0 = ∂ 1 = ∂ 2 , that is a smooth compact submanifold of dimension (m − 2) (codimension two submanifold) without boundary, i.e. ∂0 = ∅. Let us stress here that when viewed as sets both 1 and 2 are considered to be disjoint open sets, i.e. 1 ∩ 2 = ∅. Let χi : ∂M → R, (i = 0, 1, 2), be the characteristic functions of the sets i ; ˆ = 1 if xˆ ∈ i and χi (x) ˆ = 0 if xˆ ∈ / i . Obviously, χ1 (x)+χ ˆ ˆ ˆ =1 χi (x) 2 (x)+χ 0 (x) for any xˆ ∈ ∂M. Let πi : L2 (W ) → L2 (W ), (i = 0, 1, 2), be the trivial projections ˆ = χi (x)ψ( ˆ x), ˆ i.e. of sections, ψ, of a vector bundle W to i defined by (πi ψ)(x) ˆ = ψ(x) ˆ if xˆ ∈ i and (πi ψ)(x) ˆ = 0 if xˆ ∈ / i . In other words π1 maps (πi ψ)(x) smooth sections of the bundle W to their restriction to 1 , extending them by zero on 2 , and similarly for π2 . Obviously, π1 + π2 + π0 = Id, πi2 = πi , (i = 0, 1, 2), and πi πj = 0 for i = j . In principle, these projections can be used to define the boundary conditions. However, we will not use them in this paper. We will see later that to specify the solution uniquely, i.e. to completely determine the domain of the operator FB , we also need an additional condition which specifies the type of the singularity on 0 . Therefore, the boundary data are ψ(ϕ) = ψ1 (ϕ) ⊕ ψ2 (ϕ) ⊕ ψ0 (ϕ), where ψ1 and ψ2 are defined as above
ϕ|i , (i = 1, 2). ψi (ϕ) = ∇N ϕ|i
(20)
(21)
The boundary data map ψ0 is more involved since 0 is a codimension two submanifold. It turns out that the solutions of the boundary value problem could be singular at 0 and still be in L2 (V ) (and smooth in the interior of M). Thus, the restriction map to 0 is singular, in general, i.e. the data ϕ|0 and ∇N ϕ|0 are not well defined. Instead we shall define the boundary data as follows. Let ρ be the normal geodesic distance to 0 and N = ∂ρ be the unit inward normal vector field to 0 . Then √
( ρϕ)|0 √ . (22) ψ0 (ϕ) = ∇N ( ρϕ)|0 The boundary operator decomposes accordingly B = B1 ⊕ B2 ⊕ B0 ,
(23)
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IVAN G. AVRAMIDI
where Bi are the boundary operators of mixed type (13). We choose B1 and B2 to be Dirichlet and Neumann (Robin) boundary operators
I 0 0 0 B1 = , B2 = , (24) 0 0 I where is a smooth Hermitian endomorphisms of the vector bundle W . As far as the boundary operator B0 is concerned we will study both cases
I 0 B0 = , (25) 0 0 and
B0 =
0 −s
0 I
,
(26)
with s a real parameter. In other words, we have Dirichlet boundary conditions on 1 and Neumann (Robin) boundary conditions on 2 ϕ|1 = 0, (∇N + )ϕ|2 = 0 as well as an additional condition on 0 : √ ( ρϕ)|0 = 0
(27) (28)
(29)
or √ (∇N − s)( ρϕ)|0 = 0.
(30)
DEFINITION 2. The boundary value problem for a Laplace operator F (7) with the boundary conditions (27–30) is called Zaremba problem. Roughly speaking the boundary condition (29) corresponds to the extension of the Dirichlet boundary conditions from 1 to 0 while the condition (30) corresponds to the extension of the Neumann boundary conditions from 2 to 0 (however this should not be taken literally). We will discuss below the origin and the meaning of these boundary conditions and show that the case (29) leads to regular solutions at 0 and that is why will be called regular whereas the case (30) leads to singular solutions and will be called singular. 2.3. SYMMETRY Let us define the antisymmetric bilinear form I (ϕ1 , ϕ2 ) ≡ (F ϕ1 , ϕ2 )L2 (V ) − (ϕ1 , F ϕ2 )L2 (V ) ,
(31)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
19
for any two smooth sections ϕ1 , ϕ2 ∈ C ∞ (V ) of the vector bundle V . By integrating by parts on M one can easily see that this bilinear form depends only on the boundary data I (ϕ1 , ϕ2 ) = (ψ1 (ϕ1 ), J ψ1 (ϕ2 ))L2 (W ⊕W,1 ) + + (ψ2 (ϕ1 ), J ψ2 (ϕ2 ))L2 (W ⊕W,2 ) , where J =
0 I −I 0
(32)
.
(33)
Therefore, it vanishes on sections of the bundle V with compact support disjoint from the boundary ∂M when the boundary data vanish ψ(ϕ1 ) = ψ(ϕ2 ) = 0. This is a simple consequence of the fact that the operator F is formally self-adjoint. A formally self-adjoint operator is essentially self-adjoint if its closure is selfadjoint. This means that the operator is such that: (i) it is symmetric on smooth sections satisfying the boundary conditions, and (ii) there exists a unique selfadjoint extension of it. To prove the latter property one has to study the deficiency indices; however, this will not be the subject of primary interest in the present paper. We check only the first property, i.e. that the operator F is symmetric. For any two sections ϕ1 , ϕ2 ∈ C ∞ (V ) satisfying the boundary conditions (27–30) we easily obtain I (ϕ1 , ϕ2 ) = (ϕ1 , ϕ2 )L2 (W,2 ) − (ϕ1 , ϕ2 )L2 (W,2 ) .
(34)
That is for any Hermitian endormophism ∈ C ∞ (End(W )) the form I (ϕ1 , ϕ2 ) vanishes. Therefore, we immediately obtain PROPOSITION 1. Zaremba boundary value problem is symmetric. Note that this property does not depend on the boundary conditions (29, 30) at the singular codimension two submanifold 0 . 2.4. ELLIPTICITY Roughly speaking ellipticity means invertibility up to a compact operator in appropriate functional spaces (see [39, 41, 28, 32, 30]). This is basically a condition that implies invertibility “locally”. That is why it has three components: (i) in the interior of the manifold M, (ii) at the boundary parts 1 and 2 (codimension one submanifolds) and (iii) at the singular set 0 (codimension two submanifold). It can be formulated as follows. 2.4.1. Interior Ellipticity DEFINITION 3. The operator F is called elliptic if its leading symbol σL (F ; x, ξ ) is nonsingular for any ξ = 0 and any interior point x in M.
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IVAN G. AVRAMIDI
For Laplace type operator (7) defined with a positive-definite nonsingular Riemannian metric we obviously have PROPOSITION 2. Laplace type operator (8) is elliptic.
2.4.2. Codimension One Ellipticity At the boundary we use the coordinates x = (r, x) ˆ and define a split of the cotangent bundle T ∗ M = R ⊕ T ∗ ∂M so that ξ = (ξµ ) = (ω, ζ ) ∈ T ∗ M, where ζ = (ζj ) ∈ T ∗ ∂M and ω ∈ R. Let further λ be a complex number which does not lie on the positive real axis, λ ∈ C − R+ . We consider the leading symbol ˆ w, ζ ) of the operator F , substitute r = 0 and ω → −i∂r and consider σL (F ; r, x, the following ordinary differential equation on a half-line (r ∈ R+ ) ˆ −i∂r , ζ ) − λ]ϕ = 0, [σL (F ; 0, x,
(35)
with an asymptotic condition lim ϕ = 0.
r→∞
(36)
Consider now the general boundary operator of mixed type (13) (with , and being some endomorphisms). Its graded leading symbol is defined by [28, 30]
, 0 ˆ ζ) = . (37) σgL (B; x, 0 (I − ,) DEFINITION 4. The boundary operator B is called elliptic with respect to an elliptic operator F if for each boundary point xˆ in ∂M, each ζ ∈ Txˆ∗ ∂M, each λ ∈ C − R+ , (ζ, λ) = (0, 0), and each f ∈ C ∞ (W ⊕ W ) there is a unique solution ϕ to the equation (35) subject to the condition (36) and satisfying ˆ ζ )[ψ(ϕ) − f ] = 0, σgL (B; x,
(38)
where ψ(ϕ) are the boundary data. It is not difficult to check PROPOSITION 3. The Dirichlet and Neumann boundary operators are elliptic with respect to the Laplace type operator F in ∂M \ 0 . 2.4.3. Codimension Two Ellipticity The question of ellipticity of Zaremba boundary value problem is a subtle one. At the singular codimension two submanifold 0 we use the local coordinates x = (r, y, x) ˜ and define a split of the cotangent bundle T ∗ M = R2 ⊕ T ∗ 0 so
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
21
that ξ = (ξµ ) = (ω, ν, η) ∈ T ∗ M, where η = (ηa ) ∈ T ∗ 0 , and (ω, ν) ∈ R2 . Let further λ be a complex number which does not lie on the positive real axis, λ ∈ C − R+ . We consider the leading symbol σL (F ; r, y, x, ˜ ω, ν, η) of the operator F , substitute r = y = 0, ω → −i∂r , ν → −i∂y , and consider the following ordinary differential equation on a half-plane ((r, y) ∈ R+ × R) ˜ −i∂r , −i∂y , η) − λ]ϕ = 0, [σL (F ; 0, 0, x,
(39)
with an asymptotic conditions lim ϕ = lim ϕ = 0.
r→∞
|y|→∞
(40)
Define the graded leading symbol of the boundary operator B0 (25) and (26) by
I 0 0 0 ˜ η) = , or σgL(B0 ; x, ˜ η) = . (41) σgL (B0 ; x, 0 0 0 I DEFINITION 5. The boundary operator B0 is called elliptic with respect to an elliptic operator F and the elliptic boundary operators B1 and B2 if for each point x˜ in 0 , each η ∈ Tx˜∗ 0 , each λ ∈ C − R+ , (η, λ) = (0, 0), and each f = f1 ⊕ f2 ⊕ f0 , where fi ∈ C ∞ (W ⊕ W, i ); (i = 0, 1, 2), there is a unique solution ϕ to Equation (39) subject to the asymptotic conditions (40) and boundary conditions ˜ η)[ψ(ϕ) − f ] = 0, σgL (B; x,
(42)
where B = B1 ⊕ B2 ⊕ B0 is the total boundary operator and ψ(ϕ) = ψ1 (ϕ) ⊕ ψ2 (ϕ) ⊕ ψ0 (ϕ) are the boundary data defined by (21, 22). We will construct the unique solution to an equivalent problem in the Section 8, establishing: PROPOSITION 4. The boundary operator B0 is elliptic with respect to the Laplace type operator F and the Dirichlet and Neumann boundary operators B1 and B2 . This leads finally to: PROPOSITION 5. The Zaremba boundary value problem is elliptic.
3. Heat Kernel Asymptotics For t > 0 the heat semi-group operator U (t) = exp(−tF ): L2 (V ) → L2 (V ) is well defined. The integral kernel of this operator, called the heat kernel, is defined by the equation (∂t + F )U (t|x, x ! ) = 0
(43)
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IVAN G. AVRAMIDI
with the initial condition U (0|x, x ! ) = δ(x, x ! ),
(44)
where δ(x, x ! ) is the covariant Dirac distribution, the boundary condition Bψ[U (t|x, x ! )] = 0
(45)
and the self-adjointness condition U (t|x, x ! ) = U ∗ (t|x ! , x).
(46)
Hereafter all differential operators as well as the boundary data map act on the first argument of the heat kernel, unless otherwise stated. Let λ be a complex number with a sufficiently large negative real part, Re λ " 0. The resolvent can then be defined by the Laplace transform ∞ dt et λ U (t), (47) G(λ) = 0
and by analytical continuation elsewhere. The heat kernel can be expressed, in turn, in terms of the resolvent by the inverse Laplace transform w+i∞ 1 dλ e−t λ G(λ), (48) U (t) = 2π i w−i∞ where w is a sufficiently large negative real number, w " 0. As it has been done here, we will sometimes omit the space arguments if it does not cause any confusion. It is well known [30] that the heat kernel U (t|x, x ! ) is a smooth function near diagonal of M × M, i.e. for x close to x ! , and has a well defined diagonal value U diag (t|x) = U (t|x, x), and the functional trace
(49)
TrL2 exp(−tF ) =
trV U diag (t),
(50)
M
where trV is the fiber trace and the integration is defined with the help of the usual Riemannian volume element d volg . It is also well known that THEOREM 3. In the smooth category the trace of the heat kernel has an asymptotic expansion as t → 0+ of the form [30] TrL2 exp(−tF ) ∼
∞ k=0
t (k−m)/2 Ak .
(51)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
23
Here Ak are the famous so-called (global) heat-kernel coefficients (sometimes called also Minakshisundaram–Plejel coefficients). They have the following general form [30]: (0) (1) A2k = a2k + a2k , (52) M ∂M (1) a2k+1 , (53) A2k+1 = ∂M
ak(0)
and ak(1) are the (local) interior and boundary heat-kernel coefficients. where The local interior coefficients ak(0) are also called HMDS (Hadamard–Minackshisundaram–De Witt–Seeley) coefficients in the literature. Hereafter the integration over the boundary is defined with the help of the usual Riemannian volume element d volgˆ on ∂M with the help of the induced metric g. ˆ The interior coefficients ak(0) do not depend on the boundary conditions B. The (0) are calculated for Laplace-type operators up to a8(0) even order coefficients a2k [1, 43]. The boundary coefficients ak(1) do depend on both the operator F and the boundary operator B. They are far more complicated because in addition to the geometry of the manifold M they depend essentially on the geometry of the boundary ∂M. For Laplace-type operators they are known for the usual boundary conditions (Dirichlet, Neumann, or mixed version of them) up to a5(1) [12, 13, 34]. For oblique boundary conditions including tangential derivatives some coefficients were recently computed in [36, 7, 8, 6, 22, 23]. However, the Zaremba boundary value problem considered in the present paper is essentially singular. Even if the manifold M, its boundary ∂M and the operator F are all in smooth category, the coefficients of the boundary operator B are discontinuous on 0 , which makes it a singular problem. For such problems THEOREM 4. The asymptotic expansion of the trace of the heat kernel has additional nontrivial logarithmic terms, [27, 15] TrL2 exp(−tFB ) ∼
∞ k=0
t (k−m)/2 Bk + log t
∞
t k/2 Hk .
(54)
k=0
Whereas there are some results concerning the coefficients Bk , until recently almost nothing was known about the coefficients Hk . Since the Zaremba problem is local, or better to say ‘pseudo-local’, all these coefficients have the form (0) (1),1 (1),2 (2) b2k + b2k + b2k + b2k , (55) B2k = M 1 2 0 (1),1 (1),2 (2) b2k+1 + b2k+1 + b2k+1 , (56) B2k+1 = 2 0 1 hk . (57) Hk = 0
24
IVAN G. AVRAMIDI
Here the new feature is the appearance of the integrals over 0 , which complicates the problem even more, since the coefficients now depend on the geometry of the imbedding of the codimension two submanifold 0 in M that could be pretty complicated, even if smooth. The asymptotic expansion of the trace of the heat kernel has been studied recently in [40]. It has been shown there that for the Zaremba type problem considered in the present paper, the logarithmic terms do not appear, i.e. Hk = 0 for any k (see Theorem 1), which confirmed the conjecture of [5]. 4. Geometrical Framework First of all, we need to describe properly the geometry of the problem. Let us fix two small positive numbers ε1 , ε2 > 0. We split the whole manifold in a disjoint union of four different parts: M = M int ∪ M bnd = M int ∪ M1bnd ∪ M2bnd ∪ M0bnd .
(58)
Here M0bnd is defined as the set of points in the narrow strip M bnd of the manifold M near the boundary ∂M of the width ε1 that are at the same time in a narrow strip of the width ε2 near 0 M0bnd = {x ∈ M | dist(x, ∂M) < ε1 , dist(x, 0 ) < ε2 }.
(59)
Further, M1bnd is the part of the thin strip M bnd of the manifold M (of the width εi ) near the boundary ∂M that is near 1 but on the finite distance from 0 , i.e. M1bnd = {x ∈ M | dist(x, 1 ) < ε1 , dist(x, 0 ) > ε2 }.
(60)
Similarly, M2bnd = {x ∈ M | dist(x, 2 ) < ε1 , dist(x, 0 ) > ε2 }.
(61)
Finally, M int is the interior of the manifold M without a thin strip at the boundary ∂M, i.e. M int = M \ (M1bnd ∪ M2bnd ∪ M0bnd ) = {x ∈ M | dist(x, ∂M) > ε1 }.
(62)
We will construct the asymptotic solution of the heat equation on M by using different approximations in different domains. Strictly speaking, to glue them together in a smooth way one should use ‘smooth characteristic functions’ of different domains (partition of unity) and carry out all necessary estimates. What one has to control is the order of the remainder terms in the limit t → 0 and their dependence on ε1 and ε2 . Since our task here is not to prove the form of the asymptotic expansion (54), which is known, but rather to compute explicitly the coefficients of the asymptotic expansion, we will not worry about such subtle details. We will compute the asymptotic expansion as t → 0 in each domain and
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
25
then take the limit ε1 , ε2 → 0. For a rigorous treatment see [14, 15, 27] and the references therein. We will use different local coordinates in different domains. In M int we do not fix the local coordinates; our treatment will be manifestly covariant. In M1bnd we choose the local coordinates as follows. Let {eˆi }, (i = 2, . . . , m), be the local frame for the tangent bundle T 1 and xˆ = (xˆ i ) = (xˆ 2 , . . . , xˆ m ), (i = 2, . . . , m), be the local coordinates on 1 . Let r = dist(x, 1 ) be the normal distance to 1 , (r = 0 being the defining equation of 1 ), and Nˆ = ∂r |1 be the inward pointing unit normal to 1 . Then by using the geodesic flow we get the ˆ for the tangent bundle TM and the local coordinates local frame {N(r, x), ˆ ei (r, x)} bnd x = (r, x) ˆ on M1 . The geometry of 1 is described by the extrinsic curvature K (second fundamental form) ∇ˆ i ej − = Kij N,
j ∇ˆ i N = −Ki ej .
(63)
The coordinate r ranges from 0 to ε1 , 0 r ε1 . The local coordinates in M2bnd are chosen similarly. Finally, in M0bnd we choose the local coordinates as follows. Let {e˜a (x)}, ˜ (a = 3, . . . , m), be a local frame for the tangent bundle T 0 and let x˜ = (x˜ a ) = (x˜ 3 , . . . , x˜ m } be the local coordinates on 0 . Let dist∂M (x, 0 ) be the distance from a point x on ∂M to 0 along the boundary ∂M. Then define y = +dist∂M (x, 0 ) > 0 if x ∈ 1 and y = −dist∂M (x, 0 ) < 0 if x ∈ 2 . In other words, y = 0 on 0 , (r = y = 0 being the defining equations of 0 ), y > 0 on 1 and y < 0 ˜ x) ˜ = ∂y |0 be the unit normal to 0 pointing inside 1 . Then on 2 . Let n( by using the tangential geodesic flow along the boundary (that is normal to 0 ) ˜ for the tangent bundle we first get the local orthonormal frame {n(y, x), ˜ ea (y, x)} ˆ T ∂M. Further, let the unit normal vector field to the boundary N(y, x) ˜ be defined as above. Then by using the normal geodesic flow to the boundary we get the ˜ for the tangent bundle TM and local local frame {N(r, y, x), ˜ n(r, y, x), ˜ ea (r, y, x)} coordinates (r, y, x) ˜ on M0bnd . The geometry of 0 (codimension two manifold) is described by two extrinsic curvatures K and L and an additional vector T : ∇˜ a eb = Kab n + Lab N. ∇˜ a n = −Kab eb + Ta N,
∇˜ a N =
−Lba eb
− Ta n.
(64) (65)
The ranges of the coordinates r and y are: 0 r ε1 and −ε2 < y ε2 . Finally, we introduce the polar coordinates r = ρ cos θ,
y = ρ sin θ.
(66)
The angle θ ranges from −π/2 to π/2 with θ = −π/2 on 1 and θ = π/2 on 2 . To cover the whole M0bnd , ρ should range from 0 to some ε3 (depending on ε1 and ε2 ), 0 ρ ε3 .
26
IVAN G. AVRAMIDI
5. Interior Heat Kernel This is the easiest case. The construction of the heat kernel goes along the same lines as for manifolds without boundary (see, e.g., [19, 30, 1, 9, 4]. The basic case (when the coefficients of the operator F are frozen at a point x0 ) is, in fact, zero-dimensional, i.e. algebraic. By using the normal coordinates at x0 and Fourier transform one easily obtains PROPOSITION 6. The leading order interior heat kernel is
|x − x ! |2 U0int(t|x, x ! ) = (4π t)−m/2 exp − . 4t
(67)
We try to find the fundamental solution of the heat equation near diagonal for small t, i.e. x → x ! and t → 0+ , that, instead of the boundary conditions satisfies asymptotic condition at infinity. This means that effectively one introduces a small expansion parameter ε reflecting the fact that the points x and x ! are close to each other and the parameter t is small. This can be done by fixing a point x0 = x ! in M int , choosing the normal coordinates at this point (with gµν (x ! ) = δµν ), scaling x → x ! + ε(x − x ! ),
y → x ! + ε(y − x ! ),
t → ε 2 t,
(68)
and expanding in a power series in ε. We will label the scaled objects by ε, e.g., U ε . The scaling parameter ε will be considered as a small parameter in the theory and we will use it to expand everything in power (asymptotic) series in ε. At the very end of calculations we set ε = 1. The nonscaled objects, i.e. those with ε = 1, will not have the label ε. Another way of doing this is by saying that we will√expand all quantities in the homogeneous functions of (x − x ! ), (y − y ! ) and t. This construction is standard and we do not repeat it here. One can also use instead a manifestly covariant method [19, 9, 1, 3, 4, 43], which gives a convenient formula for the asymptotics as t → 0+
∞ σ int E1/2 U (t) ∼ exp − t (k−m)/2 ak , (69) 2t k=0 where σ = σ (x, x ! ) = (1/2)[dist(x, x ! )]2 is one half of the square of the geodesic ! distance between x and x ! , E = E(x, x ! ) = g −1/2 (x)g −1/2 (x ! ) det[−∂µx ∂νx σ (x, x ! )] is the corresponding Van Vleck–Morette determinant, g = det gµν , and ak = ak (x, x ! ) are the off-diagonal heat-kernel coefficients (note that odd order coefficients vanish identically, i.e. a2k+1 = 0). These coefficients satisfy certain differential recursion relations which can be solved in form of a covariant Taylor series near diagonal [1]. PROPOSITION 7. The asymptotic expansion of the heat kernel on the diagonal reads ∞ diag int (t) ∼ t (k−m)/2 ak , (70) Udiag k=0
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
27
where diag
ak (x) = ak (x, x).
(71)
This asymptotic expansion can be integrated over the interior of the manifold M . Since both the local interior coefficients ak and the volume element d volg are regular at the boundary, these integrals have well defined limits as ε1 → 0 diag diag trV ak = trV ak . (72) lim+ int
ε1 →0
M int
M
Thus we obtain PROPOSITION 8. The local interior contribution to the global heat kernel coefficients Bk is given by diag
(0) = trV a2k . b2k
(73)
(0) = 0. The As we already noted above all odd order coefficients vanish, b2k+1 (0) (0) explicit formulas for even order coefficients b2k are known up to b8 [1, 43]. The first two coefficients have the well known form:
THEOREM 5. b0(0) = (4π )−m/2 dim V , b2(0) = (4π )−m/2 trV 16 R − Q ,
(74) (75)
where R is the scalar curvature.
6. Dirichlet Heat Kernel In this section we will follow closely the ideas of the paper [2]. For an ellipdiag tic boundary-value problem the diagonal of the heat kernel Ubnd (t) in M1bnd has exponentially small terms, i.e. of order ∼ exp(−r 2 /t), (recall that r is the normal geodesic distance to the boundary) as t → 0+ and r > 0. These terms do not contribute to the asymptotic expansion of the heat-kernel diagonal outside the boundary as t → 0+ . However, they behave like distributions near the bnd boundary, and, ε1 therefore, the integrals over M1 , more precisely, the integrals limε1 →0 1 0 dr(. . .), do contribute to the asymptotic expansion with coefficients being the integrals over 1 . It is this phenomenon that leads to the boundary terms in the heat kernel coefficients. Thus, such terms determine the local boundary contributions bk(1) to the global heat-kernel coefficients Bk . The same applies to the Neumann heat kernel and 2 . The Dirichlet heat kernel U bnd,1 (t|x, x ! ) in M1bnd is constructed as follows. Now we want to find the fundamental solution of the heat equation near diagonal, i.e.
28
IVAN G. AVRAMIDI
for x → x ! and for small t → 0 in the region M1bnd close to the boundary, i.e. for small r and r ! , that satisfies Dirichlet boundary conditions on 1 and asymptotic condition at infinity. We fix a point on the boundary, x0 ∈ 1 , and choose normal coordinates on 1 at this point (with gij (0, xˆ0 ) = δij ). The basic case here (when the coefficients of the operator F are frozen at the point x0 is one-dimensional. The zeroth-order term U0bnd,(1) is defined by the heat equation (∂t + F0 )U0bnd,(1) = 0,
(76)
where F0 = −∂r2 − ∂ˆ 2 ,
(77)
the initial condition ˆ r ! , xˆ ! ) = δ(r − r ! )δ(x, ˆ xˆ ! ), U0bnd,(1) (0|r, x;
(78)
the boundary conditions, U0bnd,(1) |1 = 0,
(79)
and the asymptotic condition ˆ r ! , xˆ ! ) = !lim U0bnd,(1) (t|r, x; ˆ r ! , xˆ ! ) = 0. lim U0bnd,(1) (t|r, x; r →∞
r→∞
(80)
Note that the restriction to the boundary (. . .)|1 applies only to the first argument, i.e. r → 0. The operator F0 is a partial differential operator with constant coefficients. By using the Fourier transform in the boundary coordinates (xˆ − xˆ ! ) it reduces to an ordinary differential operator of second order. Clearly, the 0 part factorizes and the solution to the remaining one-dimensional problem can be easily obtained by using the Laplace transform, for example. PROPOSITION 9. The leading order Dirichlet heat kernel has the form ˆ r ! , xˆ ! ) = K(t|r, x; ˆ r ! , xˆ ! ) − K(t|r, x; ˆ −r ! , xˆ ! ) U0bnd,(1) (t|r, x; where !
!
K(t|r, x; ˆ r , xˆ ) = (4π t)
−m/2
|xˆ − xˆ ! |2 + (r − r ! )2 . exp − 4t
(81)
(82)
Note that in addition to the usual symmetry of the heat kernel, the Dirichlet heat kernel possesses the following ‘mirror symmetry’ ˆ r ! , xˆ ! ) U0bnd,(1) (t|r, x; ˆ r ! , xˆ ! ) = −U0bnd,(1)(t|r, x; ˆ −r ! , xˆ ! ), = −U0bnd,(1) (t| − r, x;
(83)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
29
i.e. it is an odd function of the coordinates r and r ! separately. To construct the whole heat kernel, we again scale the coordinates. But now we include the coordinates r and r ! in the scaling xˆ → xˆ0 + ε(xˆ − xˆ0 ), r → εr, r ! → εr ! ,
xˆ ! → xˆ0 + ε(xˆ ! − xˆ0 ) t → ε 2 t.
(84) (85)
The corresponding differential operators are scaled by 1 1 1ˆ ∂r → ∂r , ∂t → 2 ∂t . ∂ˆ → ∂, ε ε ε Then, we expand the scaled operator Fε in the power series in ε, i.e. F → Fε ∼
∞
ε n−2 Fn ,
(86)
(87)
n=0
where Fn are second-order differential operators with homogeneous symbols. Since the Dirichlet boundary operator does not contain any derivatives and has constant coefficients on 1 it does not scale at all. The subsequent strategy is rather simple. We expand the scaled heat kernel in ε Uεbnd,(1)
∼
∞
ε 2−m+n Unbnd,(1) ,
(88)
n=0
and substitute into the scaled version of the heat equation and the Dirichlet boundary condition on 1 . Then, by equating the like powers in ε one gets an infinite set of recursive differential equations (∂t +
F0 )Ukbnd,(1)
=−
k
bnd,(1) Fn Uk−n ,
k = 1, 2, . . . ,
(89)
n=1
with the boundary conditions ˆ r ! , xˆ ! ) = Ukbnd,(1) (t|r, x; ˆ 0, xˆ ! ) = 0, Ukbnd,(1) (t|0, x;
(90)
and the asymptotic conditions ˆ r ! , xˆ ! ) = !lim Ukbnd,(1) (t|r, x; ˆ r ! , xˆ ! ) = 0. lim Ukbnd,(1) (t|r, x;
r→∞
r →∞
(91)
In other words, we decompose the heat √ kernel into the homogeneous parts with respect to (xˆ − xˆ0 ), (xˆ ! − xˆ0 ), r, r ! and t, i.e. ˆ r ! , xˆ ! ) Ukbnd,(1) (t|r, x; = t (k−m)/2 Ukbnd,(1)(1|t −1/2 r, xˆ ! + t −1/2 (xˆ − xˆ ! ); t −1/2 r ! , xˆ ! ),
(92)
in particular, on the diagonal we have ˆ r, x) ˆ = t (k−m)/2 Ukbnd,(1) (1|t −1/2 r, x; ˆ t −1/2 r, x), ˆ Ukbnd,(1) (t|r, x; and, therefore:
(93)
30
IVAN G. AVRAMIDI
PROPOSITION 10. As t → 0 bnd,(1) Udiag (t|r, x) ˆ ∼
∞
t (k−m)/2 Ukbnd,(1)(1|t −1/2 r, x; ˆ t −1/2 r, x). ˆ
(94)
k=0
To compute the contribution to the asymptotic expansion of the trace of the heat bnd,(1) (t) over M1bnd . One should kernel, we will need to compute the integral of Udiag stress that the volume element should also be scaled d vol(r, x) ˆ → d vol(εr, x) ˆ = d vol(0, x) ˆ ·
∞
εk
k=0
rk gk (x), ˆ k!
(95)
where
ˆ
∂ k d vol(r, x) ˆ = k . gk (x) ∂r d vol(0, x) ˆ r=0
Combining the above equations and changing the variable r = bnd,(1) Udiag (t) M1bnd
(96) √
tξ we obtain
d vol(r, x) ˆ bnd,(1) U (t|r, x; ˆ r x) ˆ d vol(0, x) ˆ diag 1 0 ε1 /√t ∞ k−1 1 bnd,(1) gn (x) t (k−m)/2 ˆ dξ ξ n Uk−n−1 (1|ξ, x; ˆ ξ, x). ˆ ∼ 0 1 n=0 n! k=0 =
ε1
dr
(97)
We note that even if the coefficients Ukbnd,(1) satisfy the asymptotic regularity condition at r → ∞ (91) off-diagonal, the diagonal values of them do not fall off at infinity. They have the following general form ˆ ξ, x) ˆ = Pk (ξ, x) ˆ + Yk(1)(ξ, x), ˆ Ukbnd,(1) (1|ξ, x;
(98)
ˆ are polynomials in ξ and Yk(1)(ξ, x) ˆ are exponentially small, more where Pk (ξ, x) precisely ∼ ξ α exp(−ξ 2 ) with some α, as ξ → ∞ (which corresponds to t → 0). Obviously, the integrals over the polynomial part over M1bnd vanish after taking the asymptotic expansion as t → 0 and the limit ε1 , ε2 → 0. The coefficients Pk constitute simply the ‘interior part’ of the heat kernel and are not essential in computing the boundary contribution. The coefficients Yk(1), in contrary, behave like distributions near 1 . They give the 1 contributions to√the boundary heat ε1 / t (1) . In the limit t → 0 the integral dξ(. . .) becomes kernel coefficients b k 0 ∞ 0 dξ(. . .) plus an exponentially small remainder term. Then in the limit ε1 → 0 we obtain integrals over 1 up to an exponentially small function that we are not interested in. As the result we get
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
31
PROPOSITION 11. The coefficients bk(1) are given by bk(1),1
∞ k−1 1 (1) gn = dξ ξ n trV Yk−n−1 (ξ, x). ˆ n! 0 n=0
(99)
These are the standard boundary heat kernel coefficients for Dirichlet boundary conditions. They are listed, for example, in [12, 13] up to k = 4. The first two have the form: THEOREM 6. b0(1),1 = 0, b1(1),1 = −(4π )−(m−1)/2 dim V 14 , b2(1),1
= (4π )
−m/2
dim V
(100)
1 K, 3
where K is the trace of the extrinsic curvature (second fundamental form) of the boundary.
7. Neumann Heat Kernel The construction of the Neumann heat kernel in M2bnd goes essentially along the same lines except that now the boundary operator, in fact the endomorphism , is not constant and should be also scaled, so that the scaled boundary conditions are [7, 8]
1 ∂r + ε ϕ
= 0, (101) ε 2 where ε ∼
∞
ε k k .
(102)
k=0
The zeroth-order operator F0 is given by the same formula (77) and the zero-order boundary operator is just the standard Neumann one. The basic zero-order problem can again be easily solved giving PROPOSITION 12. The leading order Neumann heat kernel has the form ˆ r ! , xˆ ! ) = K(t|r, x; ˆ r ! , xˆ ! ) + K(t|r, x; ˆ −r ! , xˆ ! ) U0bnd,(2) (t|r, x; with the same kernel K (82).
(103)
32
IVAN G. AVRAMIDI
Note that the Neumann heat kernel has another mirror symmetry ˆ r ! , xˆ ! ) U0bnd,(2) (t|r, x; ˆ r ! , xˆ ! ) = U0bnd,(2)(t|r, x; ˆ −r ! , xˆ ! ), = U0bnd,(2) (t| − r, x;
(104)
i.e. it is an even function of the coordinates r and r ! separately. The construction of the heat kernel goes along the same lines as in Dirichlet case. We have the recursive differential equations (∂t +
F0 )Ukbnd,(2)
=−
k
bnd,(2) Fn Uk−n ,
k = 1, 2, . . . ,
(105)
n=1
with the boundary conditions k−1
bnd,(2) n Uk−n−1 , ∂r Ukbnd,(2) = − 2
2
(106)
n=1
and the asymptotic conditions ˆ r ! , xˆ ! ) = !lim Ukbnd,(2) (t|r, x; ˆ r ! , xˆ ! ) = 0. lim Ukbnd,(2) (t|r, x; r →∞
r→∞
(107)
As we already noted above the restriction to the boundary applies only to the first argument r. One can repeat here everything said at the end of the previous subsection about Dirichlet heat kernel. We have again homogeneity property ˆ r ! , xˆ ! ) Ukbnd,(2) (t|r, x; = t (k−m)/2 Ukbnd,(2)(1|t −1/2 r, xˆ ! + t −1/2 (xˆ − xˆ ! ); t −1/2 r ! , xˆ ! )
(108)
and the following expansion for the diagonal: PROPOSITION 13. As t → 0 bnd,(2) (t|r, x) ˆ Udiag
∼
∞
t (k−m)/2 Ukbnd,(2)(1|t −1/2 r, x; ˆ t −1/2 r, x). ˆ
(109)
k=0
By separating the polynomial and exponentially small parts, ˆ ξ x) ˆ = Pk (ξ, x) ˆ + Yk(2)(ξ, x), ˆ Ukbnd,(2) (1|ξ, x;
(110)
and repeating the arguments at the end of the previous subsection we obtain the 2 contributions to the boundary heat kernel coefficients bk(1),2 PROPOSITION 14. bk(1),2
∞ k−1 1 (2) gn = dξ ξ n trV Yk−n−1 (ξ, x). ˆ n! 0 n=0
(111)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
33
These are the standard boundary heat kernel coefficients for Neumann boundary conditions. They are listed, for example, in [12, 13] up to k = 4. The first two have the form THEOREM 7. b0(1),2 = 0, b1(1),2 = (4π )−(m−1)/2 dim V 14 , b2(1),2 = (4π )−m/2 dim V 13 K + 2 .
(112)
8. Zaremba Heat Kernel This is the most complicated (and the most interesting) case, since here the basic problem with frozen coefficients at a point x˜0 on 0 is two-dimensional. We will limit ourselves in this paper to the leading order and will be actually working in the tangent space R+ × R × Tx˜0 ∂0 , so that the basic problem in M0bnd will be reduced to the problem on the half-plane. As above we denote by r the normal geodesic distance to the boundary ∂M and by y the signed normal geodesic distance to 0 along the boundary and we choose normal coordinates on 0 at the point x˜0 (with gab (0, 0, x˜0 ) = δab ). Then the operator F0 has the form (113) F0 = −∂r2 − ∂y2 − ∂˜ 2 , where ∂˜ 2 = δ ab ∂˜a ∂˜b . This operator acts on the square integrable sections of the vector bundle V in a neighbourhood of the point x˜0 . We extend the operator ap˜ so that it coincides with the initial propriately to L2 (V , R+ , R, Rm−2 ; dr dy dx), operator in the neighborhood of the point x˜0 . By choosing the polar coordinates in the normal bundle described above we obtain the operator 1 1 (114) F0 = −∂ρ2 − ∂p − 2 ∂θ2 − ∂˜ 2 ρ ρ acting on
π π 2 m−2 , R ; ρ dρ dθ dx˜ . (115) L V , R+ , − , 2 2 Note that in the polar coordinates the set 1 corresponds to θ = π/2, the set 2 corresponds to θ = −π/2 and the singular set 0 corresponds to ρ = 0. The zero order inward pointing normal N to the boundary in polar coordinates has the form 1 (116) N0 |1 = ∂r |y>0 = − ∂θ |ρ>0, θ=π/2 , ρ 1 (117) N0 |2 = ∂r |y<0 = ∂θ |ρ>0, θ=−π/2 , ρ (118) N0 |0 = ∂r |y=0 = ∂θ |ρ=0, θ=0 .
34
IVAN G. AVRAMIDI
Also, in the leading order the endormorphism in the Neumann boundary operator does not contribute. Therefore, the Zaremba boundary conditions on 1 and 2 read ϕ|θ=π/2 = 0,
∂θ ϕ|θ=−π/2 = 0,
(ρ > 0).
(119)
Hence the boundary operator is discontinuous, and there is a singularity at the origin ρ = 0. As we already discussed above these boundary conditions do not completely determine the domain of the operator F0 – we need an additional condition along the singular set 0 , i.e. we need to specify the behavior of the solution as ρ → 0+ . Since ϕ must be square integrable,
Rm−2
dx˜
π/2
dθ −π/2
∞
dρρ|ϕ|2 < ∞,
(120)
0
the section ϕ must decrease at infinity faster than ρ −1/2 as ρ → ∞. We require that lim
ρ→∞
√
√ ρϕ = lim ∂ρ ( ρϕ) = 0. ρ→∞
(121)
On another hand, since the volume element has an extra power of ρ, the square integrable section can be singular as ρ → 0. However, this must be an integrable singularity, i.e. ϕ cannot be singular stronger than ρ −1/2 as ρ → 0+ . The type of the singularity should be specified by an additional boundary condition at ρ → 0+ . Since the point ρ = 0 is singular, this boundary condition cannot be imposed arbitrarily. Also it does not follow from the boundary conditions on 1 and 2 . We impose it in one of the following forms √
ρϕ|ρ=0+ = 0,
(122)
or √ (∂p − s)( ρϕ)|ρ=0+ = 0,
(123)
where s is a real parameter. We call the boundary condition (122) regular boundary condition and the boundary condition (123) singular boundary condition. The regular boundary condition corresponds formally to the limit s → +∞ in the singular one. We will see that the heat kernel asymptotics do depend on this boundary condition as well. 8.1. SEPARATION OF VARIABLES Thus, the Zaremba heat kernel in the leading approximation is determined by the fundamental solution of the heat equation for the operator F0 (114) with the
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
35
boundary conditions (119), (121) and (122) or (123). The part due to 0 easily factorizes U0bnd,(0) (t|ρ, θ, x; ˜ ρ ! , θ ! , x˜ ! )
|x˜ − x˜ ! |2 −(m−2)/2 I(t|ρ, θ; ρ ! , θ ! ), exp − = (4π t) 4t
(124)
and for I(t|ρ, θ; ρ ! , θ ! ) we obtain a two-dimensional boundary value problem: the heat equation
1 1 2 2 (125) ∂t − ∂ρ − ∂ρ − 2 ∂θ I(t|ρ, θ; ρ ! , θ ! ) = 0, ρ ρ the initial condition I(0+ |ρ, θ; ρ ! , θ ! ) = √
1 δ(ρ − ρ ! )δ(θ − θ ! ), ρρ !
(126)
the symmetry condition I(t|ρ, θ; ρ ! , θ ! ) = I ∗ (t|ρ ! , θ ! ; ρ, θ),
(127)
and the boundary conditions: I(t|ρ, θ; ρ ! , θ ! )|θ=π/2 = 0, ∂θ I(t|ρ, θ; ρ ! , θ ! )|θ=−π/2 = 0, lim ρρ ! I(t|ρ, θ; ρ ! , θ ! ) = lim ∂ρ ρρ ! I(t|ρ, θ; ρ ! , θ ! ) = 0
(130)
ρρ ! I(t|ρ, θ; ρ ! , θ ! ) ρ=0+ = 0,
(131)
ρ→∞
ρ→∞
(128) (129)
and
or (∂ρ − s)
ρρ ! I(t|ρ, θ; ρ ! , θ ! ) ρ=0+ = 0.
(132)
The existence of a unique solution to this two-dimensional problem is actually equivalent to the ellipticity of the Zaremba boundary value problem. To construct the heat kernel we study first the operator L = −∂θ2
(133)
on L2 (V , [−π/2, π/2]) with the boundary conditions ϕ(θ)|θ=π/2 = 0,
∂θ ϕ(θ)|θ=−π/2 = 0.
It is not difficult to find the spectral resolution of this operator.
(134)
36
IVAN G. AVRAMIDI
PROPOSITION 15. The orthonormal eigenfunctions and eigenvalues of the operator L are
1 π 2 ϕn (θ) = cos n + θ+ , (135) π 2 2 2 (136) λn = n + 12 , where n = 0, 1, 2, . . . . By separating the variables I(t|ρ, θ; ρ ! , θ ! ) =
∞
ϕn (θ)ϕn (θ ! )un (t|ρ; ρ ! )
(137)
n=0
we obtain the equation
1 1 1 2 un (t|ρ; ρ ! ) = 0, ∂t − ∂ρ − ∂ρ + 2 n + ρ ρ 2 with the initial condition 1 un (0+ |ρ; ρ ! ) = √ ! δ(ρ − ρ ! ), ρρ
(138)
(139)
the symmetry condition un (t|ρ; ρ ! ) = un (t|ρ ! ; ρ) and the boundary conditions lim ρρ ! un (t|ρ; ρ ! ) = lim ∂p ρρ ! un (t|ρ; ρ ! ) = 0. ρ→∞
and
ρ→∞
ρρ ! un (t|ρ, θ; ρ ! , θ ! ) ρ=0+ = 0,
or (∂p − s)
ρρ ! un (t|ρ, θ; ρ ! , θ ! ) ρ=0+ = 0.
Let us consider the operator
1 1 1 2 2 . Dn = −∂ρ − ∂ρ + 2 n + ρ ρ 2
(140)
(141)
(142)
(143)
(144)
PROPOSITION 16. The operators Dn , (n = 0, 1, 2, . . .), have the “generalized eigenfunctions” Jν (µρ): Dn Jν (µρ) = µ2 Jν (µρ),
(145)
where µ is a positive real parameter, Jν (z) are Bessel functions of the first kind of order ν, and ν can take one of two values, either ν = (n + (1/2)) or ν = −(n + (1/2)).
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
37
Let us look at the behavior of the generalized eigenfunctions at ρ → 0. In the case n 1 the Bessel functions J−(n+(1/2)) (µρ) behave like ∼ρ −(n+(1/2)) at ρ → 0, which is too singular and violates the integrability condition near boundary. This means that for n 1 we have to choose ν = (n + (1/2)). Note that these are not “true” eigenfunctions, since they are nonnormalizable. Instead, there holds PROPOSITION 17. The generalized eigenfunctions, Jn+(1/2) (µρ), (n = 0, 1, 2, . . .), satisfy the “generalized orthogonality” condition ∞ 1 dµµJn+(1/2) (µρ)Jn+(1/2) (µρ ! ) = √ ! δ(ρ − ρ ! ). (146) ρρ 0 In the contrary, in the case n = 0 both choices, ν = +1/2 or ν = −1/2, are possible, which makes the analysis of the problem more complicated. Therefore, we will treat the cases n 1 and n = 0 separately. 8.2. REGULAR GENERALIZED EIGENFUNCTIONS We consider first the case n 1. We will solve this problem by employing the Hankel transform which is well defined in the class of functions satisfying the conditions imposed above. We define ∞ ! dρ ρJn+(1/2) (µρ)un (t|ρ, ρ ! ). (147) vn (t|µ, ρ ) = 0
Then !
un (t|ρ, ρ ) =
∞
dµ µJn+(1/2) (µρ)vn (t|µ, ρ ! ).
(148)
0
Next, by integrating by parts and using the Equation (145), we compute the Hankel transform ∞ dρ ρJn+(1/2) (µρ)Dn un (t|ρ, ρ ! ) 0 ∞ 2 dρ ρJn+(1/2) (µρ)un (t|ρ, ρ ! ) + =µ 0 ∞ + ρ ∂ρ Jn+(1/2) (µρ) un (t|ρ, ρ ! ) − Jn+(1/2) (µρ)∂ρ un (t|ρ, ρ ! ) 0 .(149) Finally, by taking into account the boundary conditions (141) and (122), (123) and the asymptotic form of the Bessel functions, we obtain ∞ dρ ρJn+(1/2) (µρ)Dn un (t|ρ, ρ ! ) = µ2 vn (t|µ, ρ ! ). (150) 0
Thus, the Hankel transform of the heat Equation (138) is (∂t + µ2 )vn (t|µ, ρ ! ) = 0.
(151)
38
IVAN G. AVRAMIDI
From (139) we also obtain the initial condition vn (0+ |µ, ρ ! ) = Jn+(1/2) (µρ ! ).
(152)
It immediately follows that vn (t|µ, ρ ! ) = e−t µ Jn+(1/2) (µρ ! ), 2
and, therefore, !
un (t|ρ, ρ ) =
∞
dµ µe−t µ Jn+1/2 (µρ)Jn+1/2 (µρ ! ). 2
(153)
(154)
0
This integral can be computed by using the properties of the Bessel functions. We obtain finally PROPOSITION 18. The boundary value problem (138–143) for n = 1, 2, . . . has a unique solution
!
ρ 2 + ρ !2 1 ρρ ! exp − In+1/2 , (155) un (t|ρ, ρ ) = 2t 4t 2t where In+1/2 (z) is the modified Bessel function of first kind. Note that although this solution was obtained without making use of the boundary conditions (142) or (143), it satisfies both of them since it is regular at ρ → 0. 8.3. SINGULAR GENERALIZED EIGENFUNCTION Now let us consider the case n = 0. As we have seen the condition of integrability near boundary does not fix the solution uniquely, since there are two linearly independent integrable solutions, which corresponds to the choices ν = −1/2 and ν = +1/2. The Hankel transform in this case reduces to the standard cosine and sine Fourier transforms. However, we will not use them, but will solve the heat equation directly. Let us single out the allowed singular factor u0 (t|ρ, ρ ! ) = √
1 w(t|ρ, ρ ! ). ! ρρ
(156)
Then, the heat equation (138), the initial condition (139), and the boundary conditions (142) and (143) take the form (∂t − ∂p2 )w(t|ρ, ρ ! ) = 0,
(157)
w(0+ |ρ, ρ ! ) = δ(ρ − ρ ! ), w(t|ρ, ρ ! )|ρ=0 = 0,
(158) (159)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
39
or (∂ρ − s)w(t|ρ, ρ ! )|ρ=0 = 0.
(160)
There is also the usual regularity condition at infinity p → ∞. As we see, w is just the standard one-dimensional heat kernel on the half-axis. By using the Laplace transform we easily obtain the solution of this problem c+i∞ √ 1 1 ! −t λ √ exp − −λ|ρ − ρ ! | + w(t|ρ, ρ ) = dλe 2π i c−i∞ 2 −λ √ √ −λ − s ! exp − −λ(ρ + ρ ) , (161) +√ −λ + s √ √ where c is a sufficiently large negative real constant, i.e. −c > −s, and −λ is defined √ in the complex plane of λ with a cut along the real positive half-axis, so that Re −λ > 0. Notice that the boundary conditions (159) correspond to the limit s → +∞. The limit s → −∞ is not well defined since the constant c depends on s and would have to go to −∞ as well. Next, let us change the variable λ according to √ µ = i −λ, (162) λ = µ2 , where Im µ > 0. In the upper half-plane, Im µ > 0, this change of variables is single-valued and well defined. Under this change the complex λ-plane is mapped onto the upper half µ-plane, and the cut in the complex λ-plane along the positive real axis from 0 to ∞ is mapped onto the whole real axis in the µ-plane. The contour of integration √ in the complex µ-plane is a hyperbola going from (ei3π/4 )∞ through the point −c to (eiπ/4 )∞ . It can be deformed to a contour C that is above all poles of the integrand. It comes from −∞ along the real axis, encircles posible poles on the imaginary axis in the clockwise direction, and goes to +∞ along the real axis. After such a transformation we obtain dµ exp −tµ2 + iµ|ρ − ρ ! | + w(t|ρ, ρ ! ) = C 2π µ − is 2 ! exp −tµ + iµ(ρ + ρ ) . (163) + µ + is This function is an analytic function of s since the contour C is above the pole at −is. Therefore, we can compute it, for example, for s > 0, and then make an analytical continuation on the whole complex s-plane. So, let s > 0. Then the pole −is is in the lower half-plane. Therefore, the contour C can be deformed to just the real axis, i.e. −∞ < µ < ∞. Next, we use the following trick ∞ 1 µ − is = 1 − 2is = 1 − 2s dp eip(µ+is) . (164) µ + is µ + is 0
40
IVAN G. AVRAMIDI
This integral converges since s > 0. Substituting this equation in (163) and evaluating the Gaussian integral over µ, we obtain (ρ + ρ ! )2 (ρ − ρ ! )2 w(t|ρ, ρ ! ) = (4π t)−1/2 exp − + exp − − 4t 4t ∞ (ρ + ρ ! + p)2 − ps , (165) dp exp − − 2s 4t 0 which can be expressed in terms of the complimentary error function w(t|ρ, ρ ! ) = (4π t)
−1/2
(ρ + ρ ! )2 (ρ − ρ ! )2 + exp − − exp − 4t 4t √ 2 √ √ ρ + ρ! ! − 2 π s t exp ts + (ρ + ρ )s erfc √ +s t . 2 t
Here erfc(z) is defined by ∞ 2 2 du e−u . erfc(z) = √ π z
(166)
(167)
The case s < 0 can be analyzied either directly or by the analytical continuation. The direct computation is different since now the pole −is is in the upper half-plane and one has to take into account the residue at this pole. However, the integral along the real axis is also different, so that the sum is the same. In other words, the result for s < 0 has the same analytical form (166). Finally, we obtain PROPOSITION 19. The heat kernel component u0 is u0 (t|ρ, ρ ! )
(ρ + ρ ! )2 1 (ρ − ρ ! )2 + exp − − √ ! exp − = (4π t) 4t 4t ρρ
√ 2 √ √ ρ + ρ! ! √ +s t . − 2 π s t exp ts + (ρ + ρ )s erfc 2 t −1/2
(168)
In the particular case s = 0 we get !
u0 (t|ρ, ρ ) = (4π t)
−1/2
(ρ + ρ ! )2 1 (ρ − ρ ! )2 + exp − . √ ! exp − 4t 4t ρρ
Note that this eigenfunction is singular at ρ → 0 for any finite s.
(169)
41
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
The case s → +∞ corresponds to the regular boundary conditions (142). In this case the solution reads (ρ − ρ ! )2 (ρ + ρ ! )2 ! −1/2 1 √ ! exp − − exp − u0 (t|ρ, ρ ) = (4π t) 4t 4t ρρ 2 !
!2
ρ +ρ ρρ 1 exp − I1/2 . (170) = 2t 4t 2t This solution coincides with the solution (155) for n = 0 obtained by the Hankel transform and is obviously regular. 8.4. OFF - DIAGONAL ZAREMBA HEAT KERNEL Combining our results and using the explicit form of the eigenfunctions ϕn , we obtain LEMMA 1. The boundary value problem (125–132) has the unique solution I(t|ρ, θ; ρ ! , θ ! ) = (4π t)−1 L(t|ρ, ρ ! ) ×
θ + θ! + π θ − θ! + cos + × cos 2 2
ρ 2 + ρ !2 −1 × + (4π t) exp − 4t
! ! ρρ ρρ ! ! ,θ − θ + M ,θ + θ + π , × M 2t 2t
(171)
where
t 1/2 4 (ρ + ρ ! )2 − exp − L(t|ρ, ρ ) = √ 4t π ρρ !
√ √ √ ρ + ρ! − π s t exp ts 2 + (ρ + ρ ! )s erfc √ +s t , 2 t !
(172)
and M(z, γ ) = 2
∞
In+1/2 (z) cos[(n + 12 )γ ].
(173)
n=0
Notice that for the “regular” boundary conditions (131), which correspond to the limit s → +∞, the function L(t|ρ, ρ ! ) vanishes. This series can be evaluated by using the following integral representation of the Bessel function n+1/2 1 z 1 dp e−pz (1 − p 2 )n . (174) In+1/2 (z) = √ π n! 2 −1
42
IVAN G. AVRAMIDI
Substituting this integral in the series and summing over n we obtain 1 z M(z, γ ) = dp e−pz exp 12 (1 − p 2 )zeiγ + 12 iγ + 2π −1 + exp 12 (1 − p 2 )ze−iγ − 12 iγ .
(175)
The remaining integral can be expressed in terms of the error function, so that finally we get √ γ z cos γ , (176) erf 2z cos M(z, γ ) = e 2 where the error function is defined by z 2 2 dp e−p . erf(z) = √ π 0
(177)
By adding the 0 factor we obtain the final result: THEOREM 8. The leading order off-diagonal Zaremba heat kernel has the form ˜ ρ ! , θ ! , x˜ ! ) = L(t|ρ, θ, x; ˜ ρ ! , θ ! , x˜ ! ) + U0bnd,(0) (t|ρ, θ, x; + L(t|ρ, θ, x; ˜ ρ ! , −θ ! − π, x˜ ! ),
(178)
where L(t|ρ, θ, x; ˜ ρ ! , θ ! , x˜ ! )
|x˜ − x˜ ! |2 θ − θ! −m/2 ! L(t|ρ, ρ ) cos + exp − = (4π t) 4t 2 1 + (4π t)−m/2 exp − |x˜ − x˜ ! |2 + ρ 2 + ρ !2 − 2ρρ ! cos(θ − θ ! ) × 4t
! θ − θ! ρρ cos . (179) × erf t 2 An important corollary from this formula are the symmetries of the heat kernel. First of all, we have the usual ‘self-adjointness’ symmetry θ → θ !,
ρ → ρ!.
(180)
Second, we have the ‘periodicity’ symmetries θ → θ + 4π n,
θ ! → θ ! + 4π m,
n, m ∈ Z.
(181)
Finally, there is additional ‘mirror’ symmetry θ → θ, θ ! → −θ ! − π, θ → −θ − π, θ ! → θ !.
(182) (183)
43
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
Note the essential difference of the symmetries of the Zaremba heat kernel versus those of the Dirichlet and Neumann parametrices. The mixed heat kernel is a periodic function of the angles (expected), but not with the period 2π but with the period 4π (not expected). That is why there are two different mirror images, (ρ, −θ − π, x) ˜ and (ρ, −θ + π, x), ˜ of a point with the coordinates (ρ, θ, x). ˜ In other words the double reflection of a point does not bring it back – the double image is not identical with the original point. Denoting by T the transformation θ → −θ − π we have T 4 = Id,
but
T 2 = Id.
(184)
The operator T has four eigenvalues 1, −1, i and −i. Whereas the first two, 1 and −1, are the standard ones, the latter two, i and −i, correspond to some new images. This might have some interesting applications. 8.5. ZAREMBA HEAT TRACE ASYMPTOTICS Now it is easily found: PROPOSITION 20. The diagonal of the leading Zaremba heat kernel is bnd,(0) (t|ρ, θ, x) ˜ Udiag 2
ρ cos2 θ ρ sin θ ρ + erf = (4π t)−m/2 1 − erfc √ − exp − √ t t t √ 2
ρ t 4 exp − − + (1 − sin θ) √ t π ρ
√ 2 √ √ ρ . (185) − πs t exp ts + 2ρs erfc √ + s t t
Next, we compute the integral of the diagonal of the heat kernel over M0bnd
M0bnd
bnd,(0) trV Udiag,0 (t)
=
ε3
π/2
dρ ρ 0
−π/2
bnd,(0) dθ trV Udiag,0 (t)
(186)
for some finite ε3 > ε12 + ε22 > 0. First of all, obviously the integrals over θ of the odd functions in θ vanish identically. So, we only need to consider the even part. Second, since in the limit ε3 → 0 the volume of M0bnd vanishes, the regular part of the heat kernel diagonal does not contribute to the trace either. It is only the singular part of the heat kernel diagonal, which behaves like a distribution near 0 , that contributes to the integral in the limit ε3 → 0.
44
IVAN G. AVRAMIDI
The integral over ρ can be computed exactly. It reads bnd,(0) trV Udiag,0 (t) M0bnd
=
(4π t)
−m/2
0
√ π ε32 π dim V + t − + 2π O( ts) + X(t) , 2 4
(187)
where 2
O(z) = ez erfc(z), and
2 √ π ε32 ε3 πt ε3 + − erfc √ − X(t) = 2 π tε3 exp − t 4 2 t
√ ε3 − 2π t exp ts 2 + 2sε3 erfc √ + ts . t
(188)
(189)
Notice that π ε32 /2 is nothing but the area of the semi-circle of radius ε3 , so that vol(0 )π ε32 /2 = vol(M0bnd ). In the limit ε3 → 0 this term does not contribute to the asymptotics. By using the asymptotic behavior of the error function as z → ∞ 1 2 erfc(z) ∼ √ e−z πz
(190)
we find that the function X(t) is exponentially small, i.e. it is suppressed by the factor ∼ exp(−ε32 /t), as t → 0, and, therefore, does not contribute to the asymptotic expansion of the heat kernel in powers √ of t (54) either. The behavior of the function O( ts) depends on the parameter s. For a finite s in the limit t → 0 we have √ (191) O( ts) = 1 + O(t 1/2 ). √ On another hand, by using (190) we see that for a finite t the function O( ts) vanishes in the limit s → ∞: √ (192) O( ts)|s→∞ = 0. It immediately follows THEOREM 9. The singular heat kernel coefficients bk(2) depend on the type of the additional boundary conditions, i.e. (131) vs. (132), at the singular set 0 . The leading singular heat kernel coefficient b2(2) has the form: 7 dim V b2(2) = (4π )−(m−2)/2 16
(193)
for the singular boundary conditions (finite s) (132), and is equal to 1 dim V b2(2) = −(4π )−(m−2)/2 16
for the regular boundary conditions (131) (s → +∞).
(194)
HEAT KERNEL ASYMPTOTICS OF ZAREMBA BOUNDARY VALUE PROBLEM
45
Notice that the leading singular coefficient does not depend on s explicitly. Acknowledgements I would like to thank Jochen Brüning, Stuart Dowker, Giampiero Esposito, Stephen Fulling, Peter Gilkey, Gerd Grubb, and Werner Müller for stimulating and fruitful discussions. I am also very grateful to Robert Seeley for clarifying discussions of the boundary conditions and sharing the preliminary results. The support by the NSF Block Travel Grant DMS-9988119, by the MSRI, and by the Istituto Italiano per gli Studi Filosofici and the Azienda Autonoma Soggiorno e Turismo, Napoli, is gratefully acknowledged. References 1. 2.
3. 4. 5.
6. 7. 8.
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Avramidi, I. G.: A covariant technique for the calculation of the one-loop effective action, Nuclear Phys. B 355 (1991), 712–754. Erratum: Nuclear Phys. B 509 (1998), 557–558. Avramidi, I. G.: A method for calculating the heat kernel for manifolds with boundary, Yadernaya Fiz. 56 (1993), 245–252, [Russian]; Phys. Atomic Nuclei 56 (1993), 138–142 [English]. Avramidi, I. G.: Green functions of higher-order differential operators, J. Math. Phys. 39 (1998), 2889–2909. Avramidi, I. G.: Covariant techniques for computation of the heat kernel, Rev. Math. Phys. 11 (1999), 947–980. Avramidi, I. G.: Heat kernel asymptotics of non-smooth boundary value problem, New Mexico Tech., 1999; In: M. van den Berg and V. Liskevich (eds), Workshop on Spectral Geometry, Abstracts of Int. Conf., University of Bristol, Bristol, U.K., July 10–15, 2000. Avramidi, I. G. and Esposito, G.: Lack of strong ellipticity in Euclidean quantum gravity, Classical Quantum Gravity 15, (1998), 1141–1152. Avramidi, I. G. and Esposito, G.: Gauge theories on manifolds with boundary, Comm. Math. Phys. 200 (1999), 495–543. Avramidi, I. G. and Esposito, G.: Heat kernel asymptotics of the Gilkey-Smith boundary value problem, In: V. Alexiades and G. Siopsis (eds), Trends in Mathematical Physics, Stud. Adv. Math. 13, Amer. Math. Soc. and International Press, 1999, pp. 15–34. Avramidi, I. G. and Schimming, R.: Algorithms for the calculation of the heat kernel coefficients, In: M. Bordag (ed.), Quantum Field Theory under the Influence of External Conditions, Teubner-Texte Phys. 30, Teubner, Stuttgart, 1996, pp. 150–162. Berline, N., Getzler, E. and Vergne, M.: Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1992. Booss-Bavnbek, B. and Wojciechowski, K. P.: Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, 1993. Branson, T. and Gilkey, P. B.: The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245–272. Branson, T. P., Gilkey, P. B., Kirsten, K. and Vassilevich, D. V.: Heat kernel asymptotics with mixed boundary conditions, Nuclear Phys. B 563 (1999), 603–626. Brüning, J. and Seeley, R.: Regular singular asymptotics, Adv. in Math. 58 (1985), 133–148. Brüning, J. and Seeley, R,. T.: The expansion of the resolvent near a singular stratum of conical type, J. Funct. Anal. 95 (1991), 255–290. Callias, C.: The heat equation with singular coefficients I, Comm. Math. Phys. 88 (1983), 357– 385. Cheeger, J.: On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103–2106.
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Cheeger, J.: Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575–657. 19. De Witt, B. S.: The Spacetime Approach to Quantum Field Theory, In: B. S. De Witt and R. Stora (eds), Relativity, Groups and Topology II, North-Holland, Amsterdam, 1984, pp. 383– 738. 20. Dowker, J. S.: The N ∪ D problem, University of Manchester, 2000, hepth/0007127. 21. Dowker, J. S., Gilkey, P. B. and Kirsten, K.: On properties of the asymptotic expansion of the heat trace for the N/D problem, Internat. J. Math. 12 (2001), 505–517. 22. Dowker, J. S. and Kirsten, K.: Heat-kernel coefficients for oblique boundary conditions, Classical Quantum Gravity 14 (1997), L169–L175. 23. Dowker, J. S. and Kirsten, K.: The a3/2 heat-kernel coefficient for oblique boundary conditions, Classical Quantum Gravity 16 (1999), 1917–1936. 24. Elizalde, E. and Vassilevich, D. V.: Heat Kernel Coefficients for Chern–Simons Boundary Conditions in QED, Classical Quantum Gravity 16 (1999), 813–823. 25. Fabrikant, V. I.: Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer, Dordrecht, 1991. 26. Fedosov, B. V.: Asymptotic formulas for the eigenvalues of the Laplace operator in the case of a polyhedron, Soviet Math. Dokl. 5 (1964), 988–990. 27. Gil, J. B.: Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Temple University (2001), math.AP/0004161. 28. Gilkey, P. B. and Smith, L.: The eta invariant for a class of elliptic boundary value problems, Comm. Pure Appl. Math. 36 (1983), 85–132. 29. Gilkey, P. B. and Smith, L.: The twisted index theorem for manifolds with boundary, J. Differential Geom. 18 (1983), 393–344. 30. Gilkey, P. B.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Chemical Rubber Company, Boca Raton, 1995. 31. Grubb, G.: Properties of normal boundary value problems for elliptic even-order systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), 1–61. 32. Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems, Progr. Math. 65, Birkhäuser, Boston, 1996. 33. Karol’, A. L: Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points, Math. Notes 63(1–2) (1998), 25–32. 34. Kirsten, K.: The a5 heat kernel coefficient on a manifold with boundary, Classical Quantum Gravity 15 (1998), L5–L12. 35. Lesch, M.: Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, TeubnerTexte Math. 136, Teubner, Stuttgart–Leipzig, 1997. 36. McAvity, D. M. and Osborn, H.: Asymptotic expansion of the heat kernel for generalized boundary conditions, Classical Quantum Gravity 8 (1991), 1445–1454. 37. Mooers, E. A.: Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1–36. 38. Seeley, R. T.: Topics in pseudo-differential operators, In: CIME Conference on PseudoDifferential Operators 1968, Edizioni Cremonese, Roma (1969), pp. 169–305. 39. Seeley, R. T.: The resolvent of an elliptic boundary value problem, Amer. J. Math. 91 (1969), 963–983. 40. Seeley, R. T.: Trace Expansions for the Zaremba Problem, Comm. Partial Differential Equations 27 (2002), 2403–2421. 41. Simanca, S. R.: Mixed elliptic boundary value problems, Comm. Partial Differential Equations 12 (1987), 123–200. 42. Sneddon, I. N.: Mixed Boundary Value Problems in Potential Theory, Wiley, New York, 1966. 43. van de Ven, A. E. M.: Index free heat kernel coefficients, Classical Quantum Gravity 15 (1998), 2311–2344.
Mathematical Physics, Analysis and Geometry 7: 47–96, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
47
Tau-functions on Hurwitz Spaces A. KOKOTOV and D. KOROTKIN Department of Mathematics and Statistics, Concordia University, 7141 Sherbrook West, Montreal H4B 1R6, Quebec, Canada. e-mail: {alexey, korotkin}@mathstat.concordia.ca (Received: 22 February 2002; in final form: 18 February 2003) Abstract. We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P1 . A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ‘Wirtinger tau-function’). In higher genus we compute the modulus square of the Wirtinger tau-function. In particular one gets formulas for the isomonodromic tau-functions of semisimple Frobenius manifolds connected with the Hurwitz spaces Hg,N (1, . . . , 1). Mathematics Subject Classification (2000): 32G99. Key words: the Wirtinger projective connection, Hurwitz spaces, the Bergmann kernel.
1. Introduction Holomorphic line bundles over moduli spaces of Riemann surfaces were studied by many researchers during last 20 years (see, e.g., Fay’s survey [3]). In the present paper we consider (flat) holomorphic line bundles over Hurwitz spaces (the spaces of meromorphic functions on Riemann surfaces or, what is the same, the spaces of branched coverings of the Riemann sphere P1 ) and over coverings of Hurwitz spaces. The covariant constant sections (we call them tau-functions) of these bundles are the main object of our consideration. Our work was inspired by a coincidence of the isomonodromic tau-function of a class of 2 × 2 Riemann–Hilbert problems solved in [7] with the heuristic expression which appeared in the context of the string theory and was interpreted as the determinant of the Cauchy–Riemann operator acting in a spinor line bundle over a hyperelliptic Riemann surface (see the survey [8]). To illustrate our results consider, for example, the Hurwitz space Hg,N (1, . . . , 1) consisting of N-fold coverings of genus g with only simple branch points, none of which coincides with infinity. (In the main text we work with coverings having branch points of arbitrary order.) Let L be a covering from Hg,N (1, . . . , 1), we use the branch points λ1 , . . . , λM (i.e. the projections of the ramification points P1 , . . . , PM of the covering L) as
48
A. KOKOTOV AND D. KOROTKIN
local coordinates on the space Hg,N (1, . . . , 1); according to the Riemann–Hurwitz formula M = 2g + 2N − 2. Let λ be the coordinate of the projection of a point P ∈ L to P1 . In a √ neighborhood of a ramification point Pm we introduce the local coordinate xm = λ − λm . Besides the Hurwitz space Hg,N (1, . . . , 1), we shall use the ‘punctured’ Hur (1, . . . , 1), which is obtained from Hg,N (1, . . . , 1) by excluding witz space Hg,N all branched coverings which have at least one vanishing theta-constant. (1, . . . , 1) × C we introduce the connection In the trivial bundle Hg,N dW = d −
M
Am dλm ,
(1.1)
m=1
where d is the external differentiation operator including both holomorphic and antiholomorphic parts; connection coefficients are expressed in terms of the invariant Wirtinger projective connection SW on the covering L as follows: 1 SW (xm )|xm =0 , Am = − 12
m = 1, . . . M.
(1.2)
The connection coefficients Am are holomorphic with respect to λm and well (1, . . . , 1). defined for all coverings L from the ‘punctured’ Hurwitz space Hg,N Connection (1.1) turns out to be flat; therefore, it determines a character of (1, . . . , 1); this character defines a flat holomorphic the fundamental group of Hg,N line bundle TW over Hg,N (1, . . . , 1). We call this bundle the ‘Wirtinger line bundle’ over Hurwitz space; its horizontal holomorphic section we call the Wirtinger taufunction of the covering L. In a trivial bundle U (L0 ) × C, where U (L0 ) is a small neighborhood of a given covering L0 in Hg,N (1, . . . , 1) we can define also the flat connection dB = d − M m=1 Bm dλm , where the coefficients Bm are built from the Bergmann projective connection SB in a way similar to (1.2): 1 SB (xm )|xm =0 . Bm = − 12
The covariant constant section of this line bundle in case of hyperelliptic coverings (N = 2, g > 1) turns out to coincide (see [7] for explicit calculation) with heuristic expression for the determinant of the Cauchy–Riemann operator acting in the trivial line bundle over a hyperelliptic Riemann surface, which was proposed in [8]. This section also appears as a part of isomonodromic tau-function associated to matrix Riemann–Hilbert problems with quasi-permutation monodromies [9]. Its (−1/2)-power coincides with isomonodromic tau-function of a Frobenius manifold corresponding to the Hurwitz space Hg,N (1, . . . , 1) (see [1]). However, since the Bergmann projective connection, in contrast to Wirtinger projective connection, does depend on the choice of canonical basis of cycles on the covering, connection dB can not be globally continued to the whole Hurwitz space, but only to its appropriate covering. We call the corresponding line bundle over this covering the Bergmann line bundle and its covariant constant section – the Bergmann tau-function.
TAU-FUNCTIONS ON HURWITZ SPACES
49
We obtain explicit formulas for the modulus square of the Wirtinger and Bergmann tau-functions in genus greater than 1; in genera 0 and 1 we perform the ‘holomorphic factorization’ and derive explicit formulas for the tau-functions themselves. In genera 1 and 2 (as well as in genus 0) there are no vanishing theta-constants, (1, . . . , 1); therefore, the holomorphic bundle TW is the i.e. Hg,N (1, . . . , 1) = Hg,N bundle over the whole Hurwitz space Hg,N (1, . . . , 1). To write down an explicit formula for the tau-function over the Hurwitz space H1,N (1, . . . , 1), consider a holomorphic (not necessarily normalized) differential v(P ) on an elliptic covering L ∈ H1,N (1, . . . , 1). Introduce the notation fm ≡ fm (0), hk ≡ hk (0), where v(P ) = fm (xm ) dxm near the branch point Pm and v(P ) = hk (ζ ) dζ near the infinity of the kth sheet; ζ = 1/λ, where λ is the coordinate of the projection of a point P ∈ L to P1 . Then the Wirtinger tau-function on H1,N (1, . . . , 1) is given by the formula { N hk }1/6 . (1.3) τW = Mk=1 { m=1 fm }1/12 The analogous explicit formula can be written for coverings of genus 0. The results in genera 0, 1 follow from the study of the properly regularized Dirichlet integral S = 1/2π L |φλ |2 , where eφ |dλ|2 is the flat metric on L obtained ˜ The by projecting down the standard metric |dz|2 on the universal covering L. derivatives of S with respect to the branch points can be expressed through the values of the Schwarzian connection at the branch points; this reveals a close link of S with the modulus of the tau-function. On the other hand, the integral S admits an explicit calculation via the asymptotics of the flat metric near the branch points and the infinities of the sheets of the covering. Moreover, it admits a ‘holomorphic factorization’ i.e. it can be explicitly represented as the modulus square of some holomorphic function, which allows one to compute the tau-function itself. The same tools (except the explicit holomorphic factorization) also work in case of higher genus, when two equivalent approaches are possible. First, one can exploit the Schottky uniformization and introduce the Dirichlet integral corresponding to the flat metric on L obtained by projecting of the flat metric |dω|2 on a fundamental domain of the Schottky group. This approach leads to the expression of the modulus square of the tau-function through the holomorphic function F on the Schottky space, which was introduced in [16] and can be interpreted as the holomorphic determinant of the Cauchy–Riemann operator acting in the trivial line bundle over L. (In the main text we denote this function ¯ directly by det ∂.) The second approach uses the Fuchsian uniformization and the Liouville action corresponding to the metric of constant curvature −1 on L. It gives the following expression for the modulus square of the tau-function: det g g ||−8/(4 +2 ) , (1.4) |τW |2 = e−SFuchs /6 det B β even
50
A. KOKOTOV AND D. KOROTKIN
where det is the determinant of the Laplacian on the L; SFuchs is an appropriately regularized Liouville action which is a real-valued function of the branch points; B is the matrix of b-periods of the branched covering. Existence of explicit holomorphic factorization of our expressions for |τW |2 in genera g = 0, 1 allows to suggest that explicit formulas for τW similar to (1.3) also exist in higher genera. In this paper we use the technical tools developed in [17, 18]. We strongly suspect that in our context it should be possible to avoid the extrinsic formalism of the Dirichlet integrals and Liouville action and, at the least, it should exist a direct way to prove the genus 1 formula (1.3). The paper is organized as follows. In Section 2 after some preliminaries we prove the flatness of the connections dW and dB and introduce the flat line bundles over Hurwitz spaces and their coverings. In Section 2 we find explicitly the tau-functions for genera 0 and 1. In Section 3, using the Schottky and Fuchsian uniformizations, we give the expressions for the modulus square of tau-functions in genus greater than 1. 2. Tau-Functions of Branched Coverings 2.1. THE HURWITZ SPACES Let L be a compact Riemann surface of genus g represented as an N-fold branched covering p: L −→ P1 ,
(2.1)
of the Riemann sphere P . Let the holomorphic map p be ramified at the points P1 , P2 , . . . , PM ∈ L of ramification indices r1 , r2 , . . . , rM respectively (the ramification index is equal to the number of sheets glued at a given ramification point). Let also λm = p(Pm ), m = 1, 2, . . . , M be the branch points. (Following [4], we reserve the name ‘ramification points’ for the points Pm of the surface L and the name ‘branch points’ for the points λm of the base P1 .) We assume that none of the branch points λm coincides with the infinity and λm = λn for m = n. Recall that two branched coverings p1 : L1 → P1 and p2 : L2 → P1 are called equivalent if there exists a biholomorphic map f : L1 → L2 such that p2 f = p1 . Let H (N, M, P1 ) be the Hurwitz space of the equivalence classes of N-fold branched coverings of P1 with M branch points none of which coincides with the infinity. This space can be equipped with natural topology (see [4]) and is a (generally disconnected) complex manifold. Denote by U(L) the connected component of H (N, M, P1 ) containing the equivalence class of the covering L. According to the Riemann–Hurwitz formula, we have 1
g=
M rm − 1 m=1
2
− N + 1,
51
TAU-FUNCTIONS ON HURWITZ SPACES
where g is the genus of the surface L. If all the branch points of the covering L are simple (i.e. all the rm are equal to 2) then U(L) coincides with the space Hg,N (1, . . . , 1) of meromorphic functions of degree N on Riemann surfaces of genus g = M/2 − N + 1 with N simple poles and M simple critical values (see [11]). The space Hg,N (1, . . . , 1) is also called the Hurwitz space ([11]). ˆ Following [1], introduce the set U(L) of pairs g (2.2) L1 ∈ U(L) | a canonical basis {ai , bi }i=1 of cycles on L1 . ˆ The space U(L) is a covering of U(L). The branch points λ1 , . . . , λM of a covering L1 ∈ U(L) can serve as local ˆ coordinates on the space U(L) as well as on its covering U(L). A branched covering L is completely determined by its branch points if in addition one fixes a representation σ of the fundamental group π1 (P1 \ {λ1 , . . . , λM }) in the symmetric group SN . The element σγ ∈ SN corresponding to an element γ ∈ π1 (P1 \{λ1 , . . . , λM }) describes the permutation of the sheets of the covering L if the point λ ∈ P1 encircles the loop γ . One gets a small neighborhood of a given branched covering L moving the branch points in small neighborhoods of their initial positions without changing the representation σ . 2.2. THE BERGMANN AND WIRTINGER PROJECTIVE CONNECTIONS g
Choose on L a canonical basis of cycles {ai , bi }i=1 and thecorresponding basis of holomorphic differentials vi normalized by the conditions ai vj = δij . Let B(P , Q) = dP dQ ln E(P , Q),
(2.3)
where E(P , Q) is the prime form (see [10] or [2]), be the Bergmann kernel on the surface L. The invariant Wirtinger bidifferential W (P , Q) on L is defined by the equality W (P , Q) = B(P , Q) +
g ∂2 2 v (P )v (Q) ln |z=0 , i j g g 4 + 2 i,j =1 ∂zi ∂zj β even g
(2.4)
where B = Bij i,j =1 is the matrix of b-periods of L; β runs through the set of all even characteristics (see [3, 15]). In contrast to the Bergmann kernel, the invariant Wirtinger differential does not depend on the choice of canonical basic cycles {ai , bi }. The invariant Wirtinger bidifferential is not defined if the surface L has at least one vanishing theta-constant. Thus, we introduce the ‘punctured’ space U (L) ⊂ U(L) consisting of equivalence classes of branched coverings with all nonvanishing theta-constants. Unless the g 2 or g > 2 and N = 2 the ‘theta-divisor’
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A. KOKOTOV AND D. KOROTKIN
Z = U(L) \ U (L) forms a subspace of codimension 1 in U(L). If g 2 then the set Z is empty and U (L) = U(L); for hyperelliptic (N = 2) coverings of genus g > 2 a vanishing theta-constant does always exist and, therefore, for such coverings U (L) is empty. The Wirtinger bidifferential has the following asymptotics near diagonal:
1 1 W (P , Q) = + 6 SW (x(P )) + o(1) dx(P ) dx(Q) (2.5) (x(P ) − x(Q))2 as P → Q, where x(P ) is a local coordinate on L. The quantity SW is a projective connection on L; it is called the invariant Wirtinger projective connection. For the Bergmann kernel we have similar asymptotics
1 1 + 6 SB (x(P )) + o(1) dx(P ) dx(Q), (2.6) B(P , Q) = (x(P ) − x(Q))2 where SB is the Bergmann projective connection. The Bergmann and the invariant Wirtinger projective connections are related as follows:
g 12 ∂2 ln |z=0 vi vj . (2.7) SW = SB + g 4 + 2g i,j =1 ∂zi ∂zj βeven As well as the Wirtinger bidifferential itself, the Wirtinger projective connection does not depend on the choice of basic cycles on L while the Bergmann projective connection does. We recall that any projective connection S behaves as follows under the coordinate change x = x(z): 2 dx + R x,z , (2.8) S(z) = S(x) dz where R
x,z
x (z) 3 x (z) 2 − ≡ {x, z} = x (z) 2 x (z)
(2.9)
is the Schwarzian derivative. The following formula for the Bergmann projective connection at an arbitrary point P ∈ L on the Riemann surface of genus g 1 is a simple corollary of expression (2.3) for the Bergmann kernel [2]: P
T H, x(P ) , (2.10) SB (x(P )) = −2 + H where H =
!∗zi (0)fi ;
T =
i,j,k
!∗zi zj zk (0)fi fj fk ;
TAU-FUNCTIONS ON HURWITZ SPACES
53
!∗ is the theta-function with an arbitrary nonsingular odd half-integer characteristic; fi ≡ vi (P )/dx(P ). 2.3. VARIATIONAL FORMULAS Denote by xm = (λ−λm)1/rm the natural coordinate of a point P in a neighborhood of the ramification point Pm , where λ = p(P ). Recall the Rauch formula (see, e.g., [3], formula (3.21) or the classical paper [13]), which describes the variation of the matrix B = bij of b-periods under the variation of conformal structure corresponding to a Beltrami differential µ ∈ L∞ :
µvi vj . (2.11) δµ bij = L
We shall need also the analogous formula for the variation of the Bergmann kernel
1 µ(·)B(·, P )B(·, Q) (2.12) δµ B(P , Q) = 2π i L (see [3], p. 57). Introduce the following Beltrami differential 1 dx¯m |xm | rm −2 1{|xm |ε} µm = − r 2ε m xm dxm
(2.13)
with sufficiently small ε > 0 (where 1{|xm |ε} is the function equal to 1 inside the disc of radius ε centered at Pm and vanishing outside the disc); if rm = 2 this Beltrami differential corresponds to the so-called Schiffer variation). Setting µ = µm in (2.11) and using the Cauchy formula, we get
d rm −2 vi (xm )vj (xm ) 2π i . (2.14) δµm bij = rm (rm − 2)! dxm (dxm )2 xm =0 Observe now that the r.h s. of formula (2.14) coincides with the known expression for the derivative of the b-period with respect to the branch point λm : 1 ∂bij vi (λ(k) )vj (λ(k) ), = 2π ires|λ=λm ∂λm dλ k=1 N
(2.15)
where λ(k) denotes the point on the kth sheet of the covering L which projects to the point λ ∈ P1 . (Only those sheets which are glued together at the point Pm give a nontrivial contribution to the summation at the right-hand side of (2.15).) Thus, we have the following relation for variations of b-periods: ∂λm bij = δµm bij .
(2.16)
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A. KOKOTOV AND D. KOROTKIN
This relation can be generalized for an arbitrary function of moduli. Let Z: Tg → Hg be the standard holomorphic map from the Teichmüller space Tg to Siegel’s generalized upper half-plane. (The Z maps the conformal equivalence class of a marked Riemann surface to the set of b-periods of normalized holomorphic differentials on this surface.) It is well-known that the rank of the map Z is 3g − 3 at any point of Tg \ Tg , where Tg is the (2g − 1)-subvariety of Tg corresponding to hyperelliptic surfaces. Thus, one can always choose some 3g − 3 b-periods as local coordinates in a small neighborhood of any point of Tg \ Tg . Using these coordinates, we get ∂f ∂f δf = δµm bij = , (2.17) δµm ∂bij ∂λm i,j for any differentiable function f on Tg under the condition that the variation in the l.h.s. of (2.17) is taken at a point of Tg \ Tg (i.e. at a nonhyperelliptic surface). Formula (2.15) is well-known in the case of the simple branch point λm (i.e. for rm = 2, see, e.g., [12]). Since we did not find an appropriate reference for the general case, in what follows we briefly outline the proof: Writing the basic differential vi in a neighborhood of the ramification point Pm as vi (xm ) = C0 + C1 xm + · · · + Crm −1 xmrm −1 + O(|xm |rm ) dxm and differentiating this expression with respect to λm , we get the asymptotics 1 1 2 1 ∂ vi (xm ) = C0 1 − + ··· + r m + C1 1 − r m ∂λm rm xm rm xm −1
rm − 1 1 + O(1) dxm . (2.18) + Crm −2 1 − rm xm2 If n = m then in a neighborhood of the ramification point Pn we have the asymptotics ∂ vi (xn ) = O(1) dxn . ∂λm Therefore, the meromorphic differential ∂λm vi has the only pole at the point Pm and its principal part at Pm is given by (2.18). Observe that all the a-periods of ∂λm vi are equal to zero. Thus we can reconstruct ∂λm vi via the first rm − 2 derivatives of the Bergmann kernel:
d rm −2 B(P , xm )vi (xm ) 1 ∂ vi (P ) = . (2.19) ∂λm rm (rm − 2)! dxm (dxm )2 xm =0 To get (2.15) it is enough to integrate (2.19) over the b-cycle bj (whose projection on P1 is independent of the branch points) and use the formula
B(·, xm ) = 2π ivj (xm ). bj
55
TAU-FUNCTIONS ON HURWITZ SPACES
One may apply the same arguments to get the following formula for the derivative of the Bergmann kernel with respect to the branch point λm : 1 ∂ B(P , Q) = −res|λ=λm B(P , λ(k) )B(Q, λ(k) ). ∂λm dλ k=1 N
(2.20)
This formula also follows from (2.12) and (2.17). We shall need also another expression for the derivative of the Bergmann kernel:
∂ 1 B(P , Q) = res|λ=λm B(P , λ(j ) )B(Q, λ(k)) . (2.21) ∂λm dλ j =k To prove it we note that the sum j B(P , λ(j ) ) over all the sheets of covering L gives the Bergmann kernel on the sphere P1 dλ dµ(P ) (λ − µ(P ))2 (here µ(P ) = p(P )), therefore, we have (dλ)2 dµ(P ) dµ(Q) (λ − µ(P ))2 (λ − µ(Q))2 = B(P , λ(j ) ) B(Q, λ(k) ) j
=
k
B(P , λ )B(Q, λ(j ) ) + (j )
j
B(P , λ(j ) )B(Q, λ(k)).
j =k
Now taking the residue at λ = λm and using (2.20), we get (2.21). 2.4. THE BERGMANN AND WIRTINGER PROJECTIVE CONNECTIONS AT THE BRANCH POINTS
Here we prove a property of the Bergmann projective connection on a branched covering which plays a crucial role in all our forthcoming constructions. Introduce the following notation: 1 d rm −2 SB (xm )|xm =0 , m = 1, 2, . . . , M, (2.22) Bm = − 6(rm − 2)! rm dxm where SB (xm ) is the Bergmann projective connection corresponding to the local parameter xm = (λ − λm )1/rm near the ramification point Pm . (The factor −1/6 in (2.22) seems to be of no importance, its appearance will be explained later on.) If we deform covering (2.1) moving the branch points in small neighborhoods of their initial positions and preserving the permutations corresponding to the branch points then the quantity Bm becomes a function of (λ1 , . . . , λM ).
56
A. KOKOTOV AND D. KOROTKIN
THEOREM 1. For any m, n = 1, . . . , M the following equations hold ∂Bm ∂Bn = . ∂λn ∂λm
(2.23)
Proof. We start with the following lemma. LEMMA 1. The function Bm can be expressed via the Bergmann kernel as N 1 B(λ(j ), λ(k) ) , (2.24) Bm = 2res|λ=λm dλ k,j =1;j =k where λ(j ) is the point of the j th sheet of covering (2.1) such that p(λ(j )) = λ. Let H (·, ·) be the nonsingular part of the Bergmann kernel, i.e. 1 + H (x(P ), x(Q)) dx(P ) dx(Q), B(P , Q) = (x(P ) − x(Q))2 as P → Q. To prove the lemma we observe that only those sheets which are glued together at the point Pm give a nontrivial contribution to the summation in (2.24). Now we may rewrite the right hand side of (2.24) as 2 rm 1 j k j +k dxm res|λ=λm H (γ xm , γ xm )γ dλ, 3 dλ j,k=1, j =k where γ = e2πi/rm is the root of unity. In terms of coefficients of the Taylor series of H (xm , ym ) at the point Pm : H (xm , ym ) =
∞ s H (p,s−p)(0, 0) s=0 p=0
p!(s − p)!
xmp yms−p
this expression looks as follows: rm rm −2 H (p,rm−2−p) (0, 0) 1 γ (p+1)k+(rm−p−1)j . 2 3rm p=0 p!(rm − 2 − p)! j,k=1,j
Summing up the geometrical progression, we get (2.24). Using (2.24) and (2.21) we conclude that
∂ 1 ∂Bm (j ) (k) = 2 res|λm B(λ , λ ) ∂λn ∂λn dλ j =k
1 1 (j ) (j ) (k ) (k) B(µ , λ )B(µ , λ ) . = 2res|λ=λm res|µ=λn dλ dµ j =k j =k
(2.25)
TAU-FUNCTIONS ON HURWITZ SPACES
57
To finish the proof we note that the last expression is symmetric with respect to m and n. ✷ The analogous statement is also true for the derivatives of the Wirtinger projective connection. Namely, set 1 d rm −2 SW (xm )|xm =0 , m = 1, 2, . . . , M, (2.26) Am = − 6(rm − 2)! rm dxm where SW (xm ) is the Wirtinger projective connection corresponding to the local parameter xm near the ramification point Pm . The following statement is an easy corollary of Theorem 1. THEOREM 2. For any m, n = 1, . . . , M the following equations hold ∂An ∂Am = . ∂λn ∂λm
(2.27)
Proof. A simple calculation shows that the one-form V=
M
(Am − Bm ) dλm
m=1
is a total differential: 4 d ln . V=− g 4 + 2g βeven
(2.28)
To prove (2.28) it is sufficient to use the heat equation for theta-function 1 ∂ 2  ∂ = , (2.29) ∂bj k 4π i ∂zj ∂zk the formula (2.14) for the derivative of the b-period with respect to the branch point and the link (2.7) between the Wirtinger and Bergmann projective connections. ✷
2.5. THE WIRTINGER AND BERGMANN TAU - FUNCTIONS OF BRANCHED COVERINGS
2.5.1. The Wirtinger Tau-function We recall that U (L) denotes the set of branched coverings from the connected component U(L) L of the Hurwitz space H (N, M, P1 ) for which none of the theta-constants vanishes. Introduce the connection M Am dλm , (2.30) dW = d − m=1
acting in the trivial bundle U (L) × C, where d is the external differentiation (having both ‘holomorphic’ and ‘antiholomorphic’ components); the connection coefficients Am are defined by (2.26).
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A. KOKOTOV AND D. KOROTKIN
Remark 1. If we choose another global holomorphic coordinate λ˜ on P, λ = ˜ (a λ + b)/(cλ˜ + d), where ad − bc = 1, then the connection dW turns into a gauge equivalent connection. Consider, for example, the case of branched coverings with simple branch points (all the rm are equal to 2). Let λ˜ m be the new coordinates of the branch points, λm =
a λ˜ m + b ; cλ˜ m + d
(2.31)
then the gauge transformation of connection dW in local coordinates looks as follows dW −→ G−1 dW G,
(2.32)
where G=
M
(cλ˜ m + d)−1/4 .
(2.33)
m=1
Theorem 2 implies the following statement. THEOREM 3. The connection dW , defined in the trivial line bundle over U (L) in terms of the Wirtinger projective connection by formulas (2.30), (2.26), is flat. The flat connection dW determines a character of the fundamental group of U (L), i.e. the representation (2.34) ρ: π1 U (L) → C∗ . Denote by E the universal covering of U (L); then the group π1 (U (L)) acts on the direct product E × C as follows: g(e, z) = (ge, ρ(g)z), where e ∈ E, z ∈ C, g ∈ π1 (U (L)). The factor manifold E × C/π1(U (L)) has the structure of a holomorphic line bundle over U (L); we denote this bundle by TW . DEFINITION 1. The flat holomorphic line bundle TW equipped with the flat connection dW is called the Wirtinger line bundle over the punctured Hurwitz space U (L). The (unique up to a multiplicative constant) horizontal holomorphic section of the bundle TW is called the Wirtinger τ -function of the covering L and denoted by τW . Taking into account the form (2.32), (2.33) of the gauge transformation of connection dW under conformal transformations on the base λ-plane, we see that
TAU-FUNCTIONS ON HURWITZ SPACES
59
the Wirtinger tau-function τW of a branched covering with simple branch points transforms as follows under conformal transformation (2.31): τW −→
M
(cλ˜ m + d)−1/4 τW .
(2.35)
m=1
One can easily derive the analogous formula in the general case of an arbitrary covering. We notice that • In genera 0, 1 and 2 the ‘theta-divisor’ Z = U(L) \ U (L) is empty. Therefore, in this case the bundle TW is a bundle over the whole connected component U(L) of the Hurwitz space H (N, M, P1 ). • Hyperelliptic coverings (N = 2) fall within this framework only in genera g = 0, 1, 2 since for genus g > 2 one of the theta-constants always vanishes for hyperelliptic curves [10]. • In the case of simple branch points the space U(L) is nothing but the Hurwitz space Hg,N (1, . . . , 1) from ([1, 11]). 2.5.2. The Bergmann Tau-function ˆ Consider now the covering U(L) (the set of pairs (2.2)) of the space U(L). Repeating the construction of the previous subsection for the flat connection dB = d −
M
Bm dλm ,
(2.36)
m=1
ˆ in the trivial line bundle U(L) × C, we get a flat holomorphic line bundle TB ˆ over U(L). (Here the coefficients Bm are defined by formula (2.22), the flatness of connection (2.36) follows from Theorem 1.) DEFINITION 2. The flat holomorphic line bundle TB equipped with the flat ˆ of the connection dB is called the Bergmann line bundle over the covering U(L) 1 connected component U(L) of the Hurwitz space H (N, M, P ). The (unique up to a multiplicative constant) horizontal holomorphic section of the bundle TB is called the Bergmann τ -function of the covering L and denoted by τB . According to the link (2.7) between Wirtinger and Bergmann projective connections, the corresponding tau-functions are related as follows:
−1/(4g−1+2g−2 )  . (2.37) τW = τB β even
In contrast to the Wirtinger tau-function, the Bergmann tau-function does depend upon the choice of canonical basis of cycles on L.
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A. KOKOTOV AND D. KOROTKIN
Consider the case of hyperelliptic (N = 2) coverings. As a by-product of computation of isomonodromic tau-functions for Riemann–Hilbert problems with quasi-permutation monodromies (see [7]), it was found the following expression for the Bergmann tau-function τB on the spaces Hˆ g,2 (1, 1):
2g+2
τB = det A
(λm − λn )1/4 ,
(2.38)
m,n=1; m
where A isthe matrix of a-periods of nonnormalized holomorphic differentials on 2g+2 L: Aαβ = aα λβ−1 dλ/ν, with ν 2 = m=1 (λ − λm ). Expression (2.38) coincides with the empirical formula for the determinant of ¯∂-operator, acting in the trivial line bundle over L, derived in [8]. Due to the term det A, the expression (2.38) is explicitly dependent on the choice of canonical basis of cycles on L. On the other hand, the Wirtinger tau-function, which is independent of the choice of canonical basis of cycles, is defined on hyperelliptic curves only if g 2. Consider the case g = 2 (postponing the cases g = 0, 1 to the next section). Recall the classical Thomae formulas, which express the theta-constants of hyperelliptic curves in terms of branch points. Namely, consider an arbitrary partition of the set of branch points {λ1 , . . . , λ2g+2 } into two subsets: T and T , where the subset T (and also T ) contains g + 1 branch points. To each such partition we can associate an even vector of half-integer characteristics [ηT , ηT ] such that U (λm ) − K, (2.39) BηT + ηT = λm ∈T
where U (P ) is the Abel map, K is the vector of Riemann constants. The number of g+1 even characteristics obtained in this way is given by 12 C2g+2 . If we denote the theta function with characteristics [ηT , ηT ] by θ[βT ], the Thomae formula (see [10]) states that related theta-constant can be computed as follows: (λm − λn ) (λm − λn ). (2.40) !4 [βT ](0) = ±(det A)2 λm ,λn ∈T
λm ,λn ∈T
In genus 2 we have 12 (42 + 22 ) = 10 even characteristics in total; this number coincides with the number 12 C63 of nonvanishing even characteristics for which the Thomae formulas take place. Substitution of Thomae formulas (2.40) and expression (2.38) for τB into (2.37) gives the following formula for the Wirtinger tau-function of a hyperelliptic covering of genus 2: τW =
6
(λm − λn )1/20.
(2.41)
m,n=1,m
The independence of the Wirtinger tau-function of the choice of canonical basis of cycles on L is manifest here.
TAU-FUNCTIONS ON HURWITZ SPACES
61
Remark 2. For higher genus (g > 2) two-fold coverings our definition of Wirtinger tau-function does not work, since some of theta-constants always vanish. However, we can slightly modify formula (2.37), averaging only over the set of nonsingular even characteristics. This leads to the following definition g+1
−4/C2g+2 ∗  . (2.42) τW = τB T
Since the set of all characteristics βT is invariant with respect to any change of canonical basis of cycles, function τW∗ does not depend on the choice of this basis. Substitution of expression (2.38) and Thomae formulas (2.40) into (2.42) leads to the following result:
2g+2
τW∗ =
(λm − λn )1/4(2g+1).
(2.43)
m,n=1, m=n
The main goal of the present paper is the calculation of the Wirtinger and Bergmann tau-functions of an arbitrary covering L. In Section 3 we explicitly calculate them for coverings of genera 0 and 1. For arbitrary coverings of higher genus we are able to calculate only the modulus square of the tau-function (see Section 4). Remark 3. The Bergmann tau-function is closely related to some classes of Frobenius manifolds (see [1]). Let φ be a primary differential (see [1], Theorem 5.1) defining the structure of Frobenius manifold Mφ on the covering Hˆ g,N (1, . . . , 1). The rotation coefficients βmn of the corresponding Darboux– Egoroff metric are independent of φ and can be expressed through the Bergmann kernel on the covering L: B(P , Q) 1 . βmn = 2 dx (P ) dx (Q) m
n
P =Pm , Q=Pn
A simple calculation shows that 1 2 β (λn − λm ) = − 12 Bn , Hn = 2 m=n mn
(2.44)
where Hn is the isomonodromic quadratic Hamiltonian from [1]. Relation (2.44) follows from Equation (2.25) and the properties of the vector fields m ∂λm and λ ∂ on the Frobenius manifold M . φ m m λm Thus the Bergmann tau-function is related as follows to the isomonodromic tau-function from [1]: τB = τI−2 , where τI is the isomonodromic tau-function of the Frobenius manifold Mφ . This enables us to answer the question from [14] concerning the relations between our formulas for the Bergmann tau-function and the G-functions of Frobenius manifolds considered in [14]. The details will appear elsewhere.
62
A. KOKOTOV AND D. KOROTKIN
3. Rational and Elliptic Cases If g = 0 the branched covering L can be biholomorphically mapped to the Riemann sphere P1 . Let z be the natural coordinate on P1 \ ∞. The projective connection SB (xm ) reduces to the Schwarzian derivative SB (xm ) = R z,xm = {z(xm), xm }. Therefore −1 d rm −2 z,xm R |xm =0 . Bm = 6rm (rm − 2)! dxm
(3.1)
If g = 1 the branched covering L can be biholomorphically mapped to the torus with periods 1 and µ; in genus 1 there is only one theta-function with odd character1/2 istic which is the odd Jacobi theta-function θ1 (z|µ) = θ 1/2 (z|µ). Using (2.10) and the heat equation ∂z2 θ1 = 4π i ∂µ θ1 , we get SB (xm ) = −8π i
∂ ln θ1 2 v (xm ) + R z,xm , ∂µ
P v. Now the variational where θ1 ≡ ∂θ1 /∂z|z=0, v = v(xm ) dxm and z = formula (2.14) implies that d rm −2 z,xm 2 ∂ ln θ1 1 − R |xm =0 . (3.2) Bm = 3 ∂λm 6rm (rm − 2)! dxm Our way of calculating of the tau-functions τW and τB is rather indirect. Namely, we shall first compute the module of the tau-function. Since the first term in (3.2) can be immediately integrated, in both cases g = 0 and g = 1 one needs to find a real-valued potential S(λ1 , . . . , λn ) satisfying 1 d rm −2 z,xm ∂S = R |xm =0 , (3.3) ∂λm (rm − 2)! rm dxm where z is the natural coordinate on the universal covering of L (i.e. on the complex plane for g = 1 and the Riemann sphere for g = 0). The solution of Equations (3.3) is given by Theorem 4 below. The function S turns out to coinside with the properly regularized Dirichlet integral
1 |φλ |2 , (3.4) 2π L where eφ |dλ|2 is the flat metric on L obtained by projecting the standard metric |dz|2 from the universal covering. (In case g = 0, when the universal covering is the Riemann sphere, the metric |dz|2 is singular.)
TAU-FUNCTIONS ON HURWITZ SPACES
63
The Dirichlet integral (3.4) can be explicitly represented as the modulus square of holomorphic function of variables λ1 , . . . , λM . The procedure of holomorphic factorization gives us the value of the tau-function itself. The next two subsections are devoted to the calculation of the function S. 3.1. THE FLAT METRIC ON RIEMANN SURFACES OF GENUS 0 AND 1 The asymptotics of the flat metric near the branch points. Compact Riemann surfaces L of genus 1 and 0 have the universal coverings L˜ = C and L˜ = P1 respectively. Projecting from the universal covering onto L the metric |dz|2 , we obtain the metric of the Gaussian curvature 0 on L. (In case g = 0 the obtained metric has singularity at the image of the infinity of P1 ). Let J : L˜ → L be the uniformization map; denote its inverse by U = J −1 . Denote by x a local parameter on L. The projection of the metric |dz|2 on L looks as follows: ¯ eφ(x,x) |dx|2 = |Ux (x)|2 |dx|2 ;
(3.5)
where the function φ satisfies the Laplace equation φx x¯ = 0.
(3.6)
In the case g = 1 the map P → U (P ) may be defined by
P v U (P ) = with any holomorphic differential v on L (not necessarily normalized). In the case g = 0 we choose one sheet of the covering L (we shall call this sheet the first one) and require that U (∞(1)) = ∞, where ∞(1) is the infinity of the first sheet. Choose any sheet of the covering L (this will be a copy of the Riemann sphere P1 with appropriate cuts between the branch points; we recall that it is assumed that the infinities of all the sheets are not the ramification points) and cut out small neighborhoods of all the branch points and a neighborhood of the infinity. In the remaining domain we can use λ as global coordinate. Let φ ext (λ, λ¯ ) be the function from (3.5) corresponding to the coordinate x = λ and φ int (xm , x¯m ) be the function from (3.5) corresponding to the coordinate x = xm . LEMMA 2. The derivative of the function φ ext has the following asymptotics near the branch points and the infinities of the sheets: (1) |φλext (λ, λ)|2 = ((1/rm ) − 1)2 |λ − λm |−2 + O(|λ − λm |−2+1/rm ) as λ → λm , (2) |φλext (λ, λ)|2 = 4|λ|−2 + O(|λ|−3 ) as λ → ∞. (3) In the case g = 0 on the first sheet the last asymptotics is replaced by |φλext (λ, λ)|2 = O(|λ|−6 ) as λ → ∞.
64
A. KOKOTOV AND D. KOROTKIN
Proof. In a small punctured neighborhood of Pm on the chosen sheet we have eφ
int (x
m ,x¯ m )
|dxm |2 = eφ
ext (λ,λ) ¯
|dλ|2 .
(3.7)
This gives the equality eφ
ext (λ,λ) ¯
=
1 φint (xm ,x¯m ) e |λ − λm |2/rm −2 rm2
which implies the first asymptotics. In a neighborhood of the infinity of the chosen sheet we may introduce the coordinate ζ = 1/λ. Denote by φ ∞ (ζ, ζ¯ ) the function φ from (3.5) corresponding to the coordinate w = ζ . Now the second asymptotics follows from the equality eφ
ext (λ,λ) ¯
= eφ
∞ (ζ,ζ¯ )
|λ|−4 .
(3.8)
In the case g = 0 near the infinity of the first sheet we have U (λ) = c1 λ + c0 + c−1
1 + ··· λ
with c1 = 0. So at the infinity of the first sheet there is the asymptotics Uλλ = O(|λ|−3 ). φλext (λ, λ¯ ) = Uλ
✷
The Schwarzian connection in terms of the flat metric. Let x be some local coordinate on L. Set z = U (x); here z is a point of the universal covering (C or P1 ). The system of Schwarzian derivatives R z,x (each derivative corresponds to its own local chart) forms a projective connection on the surface L. In accordance with [5], we call it the Schwarzian connection. LEMMA 3. (1) The Schwarzian connection can be expressed as follows in terms of the function φ from (3.5): R z,x = φxx − 12 φx2 .
(3.9)
(2) In a neighborhood of a branch point Pm there is the following relation between the values of Schwarzian connection computed with respect to coordinates λ and xm : 1 1 1 z,λ 2/rm −2 z,xm − R + (3.10) (λ − λm )−2 . R = 2 (λ − λm ) rm 2 2rm2 (3) Let ζ be the coordinate in a neighborhood of the infinity of any sheet of covering (2.1) (except the first one in the case g = 0), ζ = 1/λ. Then R z,λ =
R z,ζ = O(|λ|−4 ). λ4
(3.11)
65
TAU-FUNCTIONS ON HURWITZ SPACES
Proof. The second and the third statements are just the rule of transformation of the Schwarzian derivative under the coordinate change. The formula (3.9) is well-known and can be verified by a straightforward calculation. ✷ The derivative of the metric with respect to a branch point. In this item we set φ(λ, λ¯ ) = φ ext (λ, λ¯ ). The following lemma describes the dependence of the function φ on positions of the branch points of the covering L. LEMMA 4. Let g = 0, 1. The derivative of the function φ with respect to λ is related to its derivative with respect to a branch point λm as follows: ∂φ ∂Fm ∂φ + = 0, + Fm ∂λm ∂λ ∂λ
(3.12)
where Fm = −
Uλm . Uλ
(3.13)
Proof. We have φ = ln Uλ + ln Uλ ; φλ = Uλλ /Uλ , φλm = Uλλm /Uλ and Uλm Uλλ Uλm Uλλm = + . Uλ Uλ Uλ Uλ λ (We used the fact that the map U depends on the branch points holomorphically.) ✷ LEMMA 5. Let g = 0 or g = 1 and let J be the uniformization map J : CP 1 → L or J : C → L respectively. Denote the composition p ◦ J by R. Then (1) The following relation holds: Fm =
∂R . ∂λm
(3.14)
(2) In a neighborhood of the branch point λl the following asymptotics holds: Fm = δlm + o(1),
(3.15)
where δlm is the Kronecker symbol. (3) At the infinity of each sheet (except the first sheet for g = 0) the following asymptotics holds: Fm (λ) = O(|λ|2 ).
(3.16)
Proof. Writing the dependence on the branch points explicitly we have U (λ1 , . . . , λM ; R(λ1 , . . . , λM ; z)) = z
(3.17)
for any z from the universal covering (P1 for g = 0 or C for g = 1). Differentiating (3.17) with respect to λm we get (3.14).
66
A. KOKOTOV AND D. KOROTKIN
Let z0 = z0 (λ1 , . . . , λM ) be a point from the universal covering such that J (z0 ) = Pm . The map R is holomorphic and in a neighborhood of z0 there is the representation R(z) = λm + (z − z0 )rm f (z, λ1 , . . . , λM )
(3.18)
with some holomorphic function f (·, λ1 , . . . , λM ). This together with the first statement of the lemma give (3.15). Let now z∞ = z∞ (λ1 , . . . , λM ) be a point from the universal covering such that J (z∞ ) = ∞, where ∞ is the infinity of the chosen sheet. Then in a neighborhood of z∞ we have λ = R(z) = g(z; λ1 , . . . , λM )(z − z∞ )−1 with holomorphic g(·, λ1 , . . . , λM ). Using the first statement of the lemma, we get (3.16). ✷ COROLLARY 1. Keep m fixed and define Fn (xn ) ≡ Fm (λn + xnrn ). Then Fn (0) = δnm ;
d dxn
k Fn (0) = 0,
k = 1, . . . , rn − 2.
This immediately follows from formulas (3.14) and (3.18). Formulas (3.12) and (3.15) are analogous to the Ahlfors lemma as it was formulated in [17]. However, they are more elementary, since their proof does not use Teichmüller’s theory. 3.2. THE REGULARIZED DIRICHLET INTEGRAL
We recall that the covering L has N sheets and N = M m=1 (rm − 1)/2 − g + 1 due to the Riemann–Hurwitz formula. To the kth sheet Lk of the covering L there corresponds the function φkext : Lk → R which is smooth in any domain Gkr of the form Gkρ = {λ ∈ Lk : ∀m|λ − λm | > ρ and |λ| < 1/ρ}, where ρ > 0. Here λm are all the branch points which belong to the kth sheet Lk of L. In the case of genus zero the above definition of the domain Gkρ is valid for k = 2, . . . , N. The domain G1ρ in this case should be defined separately: G1ρ = {λ ∈ L1 \ ∞1 : ∀m|λ − λm | > ρ}. (Here, again, λm are all the branch points from the first sheet.) We recall that in the case g = 0 we have singled out one sheet of the covering (the first sheet in our enumeration). The function φkext has finite limits at the cuts (except the endpoints which are the ramification points); at the ramification points and at infinity it possesses the asymptotics listed in Lemma 3.
67
TAU-FUNCTIONS ON HURWITZ SPACES
Let us introduce the regularized Dirichlet integral
1 |φλ |2 dS. 2π L Namely, set Q = ρ
N k=1
Gkρ
|∂λ φkext |2 dS,
(3.19)
¯ where dS is the area element on C1 : dS = |dλ ∧ dλ|/2. According to Lemma 3 there exist the finite limits Sell (λ1 , . . . , λM ) M M (rm − 1)2 1 ρ lim Q + 4N + (1 − rm ) ln rm 2π ln ρ + = 2π ρ→0 r m m=1 m=1 (3.20) in the case g = 1 and Srat (λ1 , . . . , λM ) M (rm − 1)2 1 lim Qρ + 4(N − 1) + 2π ln ρ + = 2π ρ→0 rm m=1 +
M
(1 − rm ) ln rm
(3.21)
m=1
in the case g = 0; the last constant term convenience.
M
m=1 (1
− rm ) ln rm we include for
THEOREM 4. Let S = Srat for g = 0, S = Sell for g = 1. Then for any m = 1, . . . , M 1 d rm −2 z,xm ∂S(λ1 , . . . , λM ) = R |xm =0 , (3.22) ∂λm (rm − 2)! rm dxm where z is the natural coordinate on the universal covering of L (P1 for g = 0 and C for g = 1). Proof. We shall restrict ourselves to the case g = 1. The proofs for g = 0 and g = 1 differ only in details concerning the infinity of the first sheet. Let Qρ be defined by formula (3.19). We have m i ∂ Qρ = ∂λm 2 l=1
r
(l)
|λ(l) −λm |=ρ
|∂λ φ|2 dλ¯ +
N (k)
k=1
Gρ
∂ |∂λ φ|2 dS. ∂λm
(3.23)
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A. KOKOTOV AND D. KOROTKIN
Here the first sum corresponds to those sheets of the covering (2.1) which are glued together at the point Pm ; the upper index (l) signifies that the integration is over a contour lying on the lth sheet. LEMMA 6. There is an equality d rm −2 z,xm 2 R |xm =0 (rm − 2)! rm dxm M d rn 1 1 =− Fm (λn + xn rn )|xn =0 . 1− 2 r (r − 1)! dx n n n n=1
(3.24)
Here xn , xm are the local parameters near Pn and Pm . The summation at the right is over all the branch points of the covering L. Proof. Using (3.9) and the holomorphy of R z,λ with respect to λ, we have 0 =
N k=1
= 2
∂Gkρ
Fm (2φλλ − φλ2 ) dλ
N k=1
+
|λ|=1/ρ
Fm R z,λ dλ +
N k=1 λn ∈Lk
|λ−λn |=ρ
Fm (2φλλ − φλ2 ) dλ.
(3.25)
The asymptotics (3.11) and (3.16) imply that the first sum in (3.25) is o(1) as ρ → 0. The second sum coincides with M 2R z,xn 1 1 Fn (xn ) + 2rn 1 − 2 rn xnrn −1 dxn . (3.26) 2rn −2 1/r r n r x x |x |=ρ n n n n n n=1 Here we have used (3.10); the function Fn is from Corollary 1. Now using Corollary 1 together with Cauchy formula and taking the limit ρ → 0 we get (3.24). ✷ The rest of the proof relies on the method proposed in [17]. Denote by H2 the second term in (3.23). Using (3.12) and the equality Fm λ¯ = 0, we get the relation ∂ |φλ |2 = −(Fm |φλ |2 )λ − (Fmλ φλ¯ )λ ∂λm = −(Fm |φλ |2 )λ − (Fmλ φλ )λ¯ − (Fmλ φλ¯ )λ .
(3.27)
This gives N i 2 ¯ Fm |φλ | dλ − Fmλ φλ dλ + Fmλ φλ¯ dλ¯ H2 = − (k) (k) 2 k=1 ∂G(k) ∂Gρ ∂Gρ ρ
69
TAU-FUNCTIONS ON HURWITZ SPACES
i = − 2 λ p=1 rj
(p)
|λ(p) −λj |=ρ
j
Fm |φλ |2 dλ¯ −
−
(p)
|λ(p) −λj |=ρ
Fmλ φλ dλ +
(p)
|λ(p) −λj |=ρ
¯ Fmλ φλ¯ dλ −
N i Fm |φλ |2 dλ¯ − Fmλ φλ dλ + − 2 k=1 |λ(k)|=1/ρ |λ(k) |=1/ρ ¯ + Fmλ φλ¯ dλ .
(3.28)
|λ(k) |=1/ρ
Let I1n (ρ)
=
I3n (ρ) =
rn (p)
(p) p=1 |λ −λn |=ρ rn (p)
p=1
¯ Fm |φλ | dλ; 2
|λ(p) −λn |=ρ
I2n (ρ)
=
rn (p)
p=1
|λ(p) −λn |=ρ
Fmλ φλ dλ;
Fmλ φλ¯ dλ¯ .
We have I1n (ρ)
= δnm
rn (p) |λ(p) −λn |=ρ
p=1
+
|xn |=ρ 1/rn
|φλ |2 dλ¯ +
1 (rn ) 1 Fn(rn −1) (0)xnrn −1 + F (0)xnrn + (rn − 1)! rn ! n rn +1 ) × + O(|xn |
|φxintn |2
1 − rn φxintn 1 − rn φxint ¯n + + rn −1 rn −1 r r −1 r −1 2 2 n n n rn x¯n xn rn x¯n xnrn rn xn x¯n 2 1 1 −1 rn x¯nrn −1 dx¯n + rn xnrn x¯nrn rn (1/rn − 1)2 (rn ) Fn (0) + = δnm |φλ |2 dλ¯ + 2π i (p) −λ(p) |=ρ (r − 1)! n |λ n p=1 ×
+ 2π i
+
1 − rn F(rn −1) (0)φxintn (0) + o(1) rn (rn − 1)! n
as ρ → 0. We get also I2n (ρ)
=
|xn |=ρ 1/rn
1 rn xn
φ int rn −1 xn
+
1 1 − 1 rn × rn xn
70
A. KOKOTOV AND D. KOROTKIN
1 F(rn −1) (0)xnrn −2 + (rn − 2)! n 1 (rn ) rn −1 rn + F (0)xn + O(|xn | ) dxn (rn − 1)! n 1 1 = −2π i −1 F(rn ) (0) − rn (rn − 1)! n 1 − 2π i φ int (0)Fn(rn −1) (0) + o(1) rn (rn − 2)! xn ×
and
I3n (ρ)
1 Fn(rn −1) (0)xnrn −2 + |xn |=ρ 1/rn (rn − 2)! 1 (rn ) rn −1 rn F (0)xn + O(|xn | ) × + (rn − 1)! n rn −1 1 1 x¯n 1 int φ + − 1 rn dx¯n × rn −1 x¯ n rn x¯n xn rn x¯n (1/rn − 1) (rn ) F (0) + o(1). = 2π i (rn − 1)! n =
We note that I1n
−
I2n
+
I3n
= δnm
rn (p)
p=1
|λ(p) −λn |=ρ
p=1
|λ(p) −λn |=ρ
|φλ |2 dλ¯ +
2 1 1 2π i (rn ) F (0) −1 +2 − 1 + o(1) + (rn − 1)! n rn rn rn |φλ |2 dλ¯ − = δnm (p)
1 2π i n) 1 − 2 F(r − n (0) + o(1). (rn − 1)! rn It is easy to verify that N 2 ¯ Fm |φλ | dλ − k=1
|λ(k) |=1/ρ
|λ(k) |=1/ρ
Fmλ φλ dλ +
|λ(k) |=1/ρ
¯ Fmλ φλ¯ dλ = o(1),
so we get
r m i |φλ |2 dλ¯ − H2 = − 2 l=1 |λ(l) −λ(l) |=ρ m M 1 1 (rn ) 1 − 2 Fn (0) + o(1). − 2π i (r − 1)! rn n n=1
(3.29)
71
TAU-FUNCTIONS ON HURWITZ SPACES
Now Lemma 6, (3.23) and (3.29) imply that d rm −2 z,xm 2π ∂ ρ Q = R |xm =0 + o(1). ∂λm (rm − 2)! rm dxm
(3.30)
To prove Theorem 4 it is sufficient to observe that the term o(1) in (3.30) is uniform with respect to parameters (λ1 , . . . , λM ) belonging to a compact neighborhood of the initial point (λ01 , . . . , λ0M ). ✷ COROLLARY 2. The formulas for functions Sell and Srat can be rewritten as follows: Sell(λ1 , . . . , λM ) =
M rm − 1
2
m=1
Srat (λ1 , . . . , λM ) =
M rm − 1
2
m=1
φ int (xm , x¯m )|xm =0 −
N
φ ∞ (∞(k)),
(3.31)
φ ∞ (∞(k)).
(3.32)
k=1
φ int (xm , x¯m )|xm =0 −
N k=2
Here ∞(k) is the infinity of the kth sheet of covering (2.1); φ ∞ (∞(k) ) = φ ∞ (ζ, ζ¯ )|ζ =0 ; ζ = 1/λ is the local parameter near ∞(k) . Proof. Using the Laplace equation (3.6), the Stokes theorem and the asymptotics from Lemma 2, we get in the case g = 1: ρ
Q
N
N 1 = (φλ φ)λ¯ − φλλ¯ φ dS = φλ φ dλ 2i k=1 ∂Gkρ Gkρ k=1 M
1 1 int 1−rm 1 φxm xm + − 1 xm−rm {φ int + = 2i m=1 |xm |=ρ 1/rm rm rm
+ 2(1 − rm ) ln|xm | − 2 ln rm }rm xmrm −1 dxm +
N 2 ∞ −2 ∞ {φ − 4 ln|λ|} dλ −φζ λ − + λ |λ|=1/ρ k=1 = −π
M
(1 − rm )φ (xm )|xm =0 − 2π int
m=1
− 4N +
M
N
φ ∞ (∞(k) ) −
k=1
M (rm − 1)2 (1 − rm ) ln rm + o(1), 2π ln ρ − 2π r m m=1 m=1
as ρ → 0. This implies (3.31). In case g = 0 we repeat the same calculation, omitting the integrals around the infinity of the first sheet. ✷
72
A. KOKOTOV AND D. KOROTKIN
3.3. FACTORIZATION OF THE DIRICHLET INTEGRAL AND THE TAU - FUNCTIONS OF RATIONAL AND ELLIPTIC COVERINGS Now we are in a position to calculate the Bergmann tau-function itself. For rational coverings the Wirtinger and Bergmann tau-functions trivially coincide, in the elliptic case the expression for the Wirtinger tau-function follows from that for the Bergmann one. We start with the tau-functions of elliptic coverings. THEOREM 5. In case g = 1 the Bergmann tau-function of the covering L is given by the following expression: N 1/6 h 2/3 (3.33) τB = [θ1 (0 | µ)] M k=1(rmk−1)/12 , m=1 fm where v(P ) is the normalized Abelian differential on the torus L; v(P ) = fm (xm ) dxm as P → Pm and fm ≡ fm (0); v(P ) = hk (ζ ) dζ as P → ∞(k) and hk ≡ hk (0); µ is the b-period of the differential v(P ). Proof. It is sufficient to observe that φ int (xm , xm ) = ln U (xm ) + ln U (xm ) = ln|fm (xm )|2 in a neighborhood of Pm and φ ∞ (ζ, ζ ) = ln|hk (ζ )|2 in a neighborhood of ∞(k) and to make use of (3.31) and (3.2).
✷
Now Theorem 5, the link (2.37) between the Bergmann and Wirtinger taufunctions, and the Jacobi formula θ1 = π θ2 θ3 θ4 imply the following corollary COROLLARY 3. The Wirtinger tau-function of the elliptic covering L is given by the formula N 1/6 h (3.34) τW = M k=1(rmk−1)/12 . m=1 fm We notice that the result (3.34) does not depend on normalization of the holomorphic differential v(P ): if one makes a transformation v(P ) → Cv(P ) with an arbitrary constant C, this constant cancels out in (3.34) due to the Riemann–Hurwitz formula. For the rational case the Bergmann and Wirtinger tau-functions coincide. THEOREM 6. In case g = 0 the tau-functions of the covering L can be calculated by the formula N dU 1/6 k=2 ( dζ |ζk =0 ) , (3.35) τW ≡ τB = M dU k (rm −1)/12 m=1 ( dxm |xm =0 )
73
TAU-FUNCTIONS ON HURWITZ SPACES
where xm is the local parameter near the branch point Pm , ζk is the local parameter near the infinity of the kth sheet. (We recall that the map U is chosen in such a way that U (∞(1)) = ∞.) The proof is essentially the same. Remark 4. The fractional powers at the right-hand sides of formulas (3.35) and (3.34) are understood in the sense of the analytical continuation. The arising monodromies are just the monodromies generated by the flat connection dW . It should be noted that the 12th powers of tau-functions (3.35) and (3.34) are single-valued global holomorphic functions on the Hurwitz space U(L). It is instructive to illustrate the formulas (3.35) and (3.33) for the simplest twofold coverings with two (g = 0) and four (g = 1) branch points. 3.3.1. Tau-function of a Two-fold Rational Covering Consider the covering of P1 with two sheets and two branch points λ1 and λ2 . Then g = 0 and λ1 + λ2 1 λ+ + (λ − λ1 )(λ − λ2 ) . (3.36) U (λ) = 2 2 We get {U (x1 ), x1 }x1 =0 =
x12
+ x1 λ1 − λ2 + x12 , x1 x
1 =0
x13 3 2 = λ1 − λ2 x1 + x1 + λ1 − λ2 , x1 = 2 λ2 − λ1 x1 =0
and {U (x2 ), x2 }|x2 =0 =
3 . λ1 − λ2
(3.37)
Now direct integration of Equations (3.37) gives the following result: τW = τB = (λ1 − λ2 )1/4
(3.38)
(up to a multiplicative constant). On the other hand, to apply the general formula (3.35), we find √ √ Ux2 (0) = 12 λ2 − λ1 , Ux1 (0) = 12 λ1 − λ2 ; 1 1 1 λ1 + λ2 − + (1 − ζ2 λ1 )(1 − ζ2 λ2 ) U (ζ2 ) = 2 ζ2 2 ζ2 λ1 + λ2 (λ1 − λ2 )2 + ζ2 + · · · . = 2 16
74
A. KOKOTOV AND D. KOROTKIN
Therefore, our formula (3.35) in this case also gives rise to (3.38). 3.3.2. Tau-functions of Two-fold Elliptic Coverings Consider the two-fold covering L with four branch points: µ2 = (λ − λ1 )(λ − λ2 )(λ − λ3 )(λ − λ4 ).
(3.39)
There are two ways to compute the tau-function on the space of such coverings. On one hand, since the elliptic curve L belongs to the hyperelliptic class, we can apply known formula (2.38) which gives: (λm − λn )1/4 , (3.40) τB (λ1 , . . . , λ4 ) = A m,n=1,...4; m
where A = a dλ/µ is the a-period of the nonnormalized holomorphic differential. On the other hand, to apply the formula (3.33) to this case, we notice that the normalized holomorphic differential on L is equal to v(P ) =
1 dλ ; A µ
the local parameters near Pn are xn = (λm − λn )−1/2 , fm = 2A−1
√
λ − λn . Therefore, hk = (−1)k A−1 ,
k = 1, 2.
n=m
According to the Jacobi formula θ1 = π θ2 θ3 θ4 ; moreover, the genus 1 version of Thomae formulas for theta-constants gives θk4 = ±
A2 (λj − λj2 )(λj3 − λj4 ), (2π i)2 1
where k = 2, 3, 4 and (j1 , . . . , j4 ) are appropriate permutations of (1, . . . , 4). Computing θ1 according to these expressions, we again get (3.40). 3.4. THE WIRTINGER TAU - FUNCTION AND ISOMONODROMIC DEFORMATIONS In [9] it was given a solution to a class of the Riemann–Hilbert problems with quasi-permutation monodromies in terms of Szegö kernels on branched coverings of P1 . The isomonodromic tau-function of Jimbo and Miwa associated to these Riemann–Hilbert problems is closely related to the tau-functions of the branched coverings considered in this paper. Here we briefly outline this link for the genus zero coverings L. So, let L be biholomorphically equivalent to the Riemann sphere P1 with global coordinate z. Introduce the ‘prime-forms’ on the z-sphere and the λ-sphere: z − z0 E(z, z0 ) = √ √ , dz dz0
λ − λ0 E0 (λ, λ0 ) = √ √ . dλ dλ0
(3.41)
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TAU-FUNCTIONS ON HURWITZ SPACES
Define a N × N matrix-valued function I(λ, λ0 ) for λ belonging to a small neighborhood of λ0 : (j ) (k) (λ − λ0 ) z (λ ) z (λ0 ) E0 (λ, λ0 ) = , (3.42) Ij k (λ, λ0 ) = (j ) (j ) E(λ(k), λ0 ) z(λ(k)) − z(λ0 ) where z = dz/dλ. To compute the determinant of the matrix I we use the following identity for two arbitrary sets of complex numbers z1 , . . . , zN , µ1 , . . . , µN :
1 j
N
N
N/2 {zλ (λ(k))zλ (λ(k) × 0 )}
k=1
j
(k)
(j )
) − z(λ(j ) )}{z(λ0 ) − z(λ(k) 0 )} (j )
(k) j,k {z(λ ) − z(λ0 )}
.
This expression is symmetric with respect to interchanging of any two sheets, therefore, it is a single-valued function of λ and λ0 . Moreover, it is nonsingular (and equal to 1) as λ = λ0 , and nonsingular as λ → ∞. Therefore, it is globally nonsingular, thus identically equal to 1. The function I obviously equals to the unit matrix as λ → λ0 . The only singularities of the function I in λ-plane are the branch points λm . These are regular singularities with quasi-permutation monodromy matrices with nonvanishing entries equal to ±1. Therefore, function I(λ), being analytically continued from a small neighborhood of point λ0 to the universal covering of P1 \ {λ1 , . . . , λm }, gives a solution to the Riemann–Hilbert problem with regular singularities at the points λm and quasipermutation monodromy matrices. It is nondegenerate outside of {λm }, equals I at λ = λ0 , and satisfies the equations Am ∂I = I, ∂λ λ − λ m m=1 M
∂I Am =− I ∂λm λ − λm
(3.44)
for some N × N matrices {Am } depending on {λm }. Compatibility of Equations (3.44) implies the Schlesinger system for the functions Am ({λn }). The corresponding Jimbo–Miwa tau-function τJ M ({λm}) is defined by the equations ∂ ln τJ M = 12 res|λ=λm tr(Iλ I −1 )2 . ∂λm
(3.45)
76
A. KOKOTOV AND D. KOROTKIN
The tau-function, as well as the expression tr(Iλ I −1 )2 , is independent of the normalization point λ0 ; taking the limit λ0 → λ in this expression, we get zλ (λ(j ) )zλ (λ(k) ) (λ0 − λ) + O((λ − λ0 )2 ), z(λ(j ) ) − z(λ(k)) as λ0 → λ
Ij k =
Ijj = 1 + o(1) (3.46)
and
1 tr 2
Iλ I −1 (λ)
2
=−
1 (j ) B z(λ ), z(λ(k)) , 2 (dλ) j =k
(3.47)
where B(z, z˜ ) =
dz d˜z (z − z˜ )2
is the Bergmann kernel on P1 . Consider the behavior of expression (3.47) as λ → λm ; suppose that the sheets glued at the ramification point Pm have numbers s and t. Then, since dλ = 2xm dxm , we have as λ → λm ,
1 tr 2
Iλ I −1 (λ)
2
zxm (λ(s))zxm (λ(t )) 1 + O(1) 4(λ − λm ) [z(λ(s)) − z(λ(t ))]2 1 1 + = − (s) 4(λ − λm ) [xm (λ ) − xm (λ(t ))]2 + 16 {z, xm }|xm =0 + O(1) 1 1 1 + {z, xm }|xm =0 + O(1). = − 4(λ − λm ) 4(λ − λm ) 6 = −
Therefore, the definition of isomonodromic tau-function (3.45) gives rise to ∂ ln τJ M 1 = − 24 {z, xm }|xm =0 ; ∂λm
(3.48)
thus, in genus zero we get the following relation between isomonodromic and Wirtinger tau-functions: τJ M = {τW }−1/2 , where τW is given by (3.35). 4. The Case of Higher Genus In this section we calculate the modulus square of the Bergmann and Wirtinger tau-functions for an arbitrary covering of genus g > 1. ˆ In a small neighborhood of L0 we may consider Let L0 be a point of U(L). ˆ the branch points λ1 , . . . , λM as local coordinates on U(L).
TAU-FUNCTIONS ON HURWITZ SPACES
77
The tau-function τB (a section of the Bergmann line bundle) can be considered as a holomorphic function in this small neighborhood of L0 . Its modulus square, |τB |2 is the restriction of a section of the ‘real’ line bundle TB ⊗ TB . To compute |τB |2 we are to find a real-valued potential ln|τB |2 such that ∂ ln|τB |2 = Bm ; ∂λm
m = 1, . . . , M.
(4.1)
If the covering L has genus g > 1 then it is biholomorphically equivalent to the quotient space H/ L, where H = {z ∈ C : z > 0}; L is a strictly hyperbolic Fuchsian group. Denote by πF : H → L the natural projection. The Fuchsian projective connection on L is given by the Schwarzian derivative {z, x}, where x is a local coordinate of a point P ∈ L, z ∈ H, πF (z) = P . We recall the variational formula ([19], see also [3]) for the determinant of the Laplacian on the Riemann surface L:
1 det =− (SB − SF )µ, δµ ln det B 12π i L where B is the matrix of b-periods, SB is the Bergmann projective connection, SF is the Fuchsian projective connection, µ is a Beltrami differential. Since, as we discussed above, the derivation with respect to λm corresponds to the Beltrami differential µm from (2.13), we conclude that d rm −2 1 (SB (xm ) − {z, xm })|xm =0 − 6rm , (rm − 2)! dxm ∂ det . (4.2) = ln ∂λm det B Remark 5. This formula explains the appearance of the factor −1/6 in Definition (2.22) of the connection coefficient Bm . Therefore, the calculation of the modulus of the Bergmann tau-function of the covering L reduces to the problem of finding a real-valued function SFuchs(λ1 , . . . , λM ) such that d rm −2 1 ∂SFuchs = {z, xm }|xm =0 , m = 1, . . . M. (4.3) ∂λm rm (rm − 2)! dxm Another link of |τB |2 with known objects can be established if we introduce the Schottky uniformization of the covering L. Namely, the covering L (of genus g > 1) is biholomorphically equivalent to the quotient space L = D/H, where H is a (normalized) Schottky group, D ⊂ P1 is its region of discontinuity. Denote by πH : D → L the natural projection.
78
A. KOKOTOV AND D. KOROTKIN
Introduce the Schottky projective connection on L given by the Schwarzian derivative {ω, x}, where x is a local coordinate of a point P ∈ L; ω ∈ D; πH (ω) = P . Due to the formula (2.17) and the results of [16] (namely, see Remark 3.5 in [16]), we have ∂ d rm −2 1 ¯ 2 . (4.4) (SB (xm ) − {ω, xm })|xm =0 = ln |det ∂| − 6rm (rm − 2)! dxm ∂λm ¯ Here det ∂¯ is the holomorphic determinant of the family of ∂-operators (this holomorphic determinant can be considered as a nowhere vanishing holomorphic function on the Schottky space; see Theorem 3.4 [16] for precise definitions and an ¯ 2 ). explicit formula for |det ∂| Therefore, the calculation of the modulus square of the Bergmann tau-function of the covering L reduces to the integration of the following system of equations for real-valued function SSchottky: ∂SSchottky d rm −2 1 = {ω, xm }|xm =0 , m = 1, . . . M. (4.5) ∂λm rm (rm − 2)! dxm In the following two subsections we solve, first, system (4.5) and, second, system (4.3). 4.1. THE DIRICHLET INTEGRAL AND THE SCHOTTKY UNIFORMIZATION 4.1.1. The Schottky Uniformization and the Flat Metric on Dissected Riemann Surface The Schottky uniformization. We refer the reader to [18] for a brief review of Schottky groups and the Schottky uniformization theorem. Fix some marking of the Riemann surface L (i.e. a point x0 in L and some system of generators α1 , . . . , αg , β1 , . . . , βg of the fundamental group π1 (L, x0 ) g such that Mi=1 αi−1 βi−1 αi βi = 1). The marked surface L is biholomorphically equivalent to the quotient space D/H, where H is a normalized marked Schottky group, D ⊂ P1 is its region of discontinuity. (A Schottky group is said to be marked if a relation-free system of generators L1 , . . . , Lg is chosen in it. For the normalized Schottky group L1 (ω) = k1 ω with 0 < |k1 | < 1 and the attracting fixed point of the transformation L2 is 1.) Choose a fundamental region D0 for H in D. This is a region in P1 bounded by 2g disjoint Jordan curves c1 , . . . , cg , c1 , . . . , cg with ci = −Li (ci ), i = 1, . . . , g; the curves ci and ci are oriented as the components of ∂D0 , the minus sign means the reverse orientation. Let πH : D → L be the natural projection. Set C i = πH (ci ). g Denote by Ldissected the dissected surface L \ i=1 Ci . The map πH : D0 → Ldissected is invertible; denote the inverse map by G0 .
79
TAU-FUNCTIONS ON HURWITZ SPACES
4.1.2. The Flat Metric on Ldissected ¯ Let x be a local parameter on Ldissected. Define a flat metric eφ(x,x) |dx|2 on Ldissected by ¯ |dx|2 = |dω|2 . eφ(x,x)
(4.6)
Here ω ∈ D0 , πH (ω) = x. Thus, to each local chart with local parameter x there corresponds a function φ(x, x). ¯ We specify the function φ ext (λ, λ¯ ) of local parameter λ by eφ
ext (λ,λ) ¯
|dλ|2 = |dω|2 = |G0 (λ)|2 |dλ|2 .
(4.7)
Here ω ∈ D, πH (ω) = P ∈ L and p(P ) = λ. Introduce also the functions φ int(xm , x¯m ), m = 1, . . . , M and φ ∞ (ζk , ζ¯k ), k = 1, . . . , N corresponding to the local parameters xm near the ramification points Pm and the local parameters ζk = 1/λ near the infinity of the kth sheet. In the intersections of the local charts we have eφ
int (x
m ,x¯ m )
|dxm |2 = eφ
ext (λ,λ) ¯
|dλ|2
(4.8)
and eφ
∞ (ζ ,ζ¯ ) k k
|dζk |2 = eφ
ext (λ,λ) ¯
|dλ|2 .
(4.9)
Choose an element L ∈ H and consider the fundamental region D1 = L(D0 ). Introduce the map G1 : Ldissected → D1 and the metric eφ1 (x,x)|dx|2 on Ldissected corresponding to this new choice of fundamental region. Since G1 (x) = L(G0 (x)), we have ¯ = φ(x, x) ¯ + ln|L (G0 (x))|2 , φ1 (x, x) L (G0 (x)) G (x) ¯ x = φx (x, x) ¯ + [φ1 (x, x)] L (G0 (x)) 0
(4.10) (4.11)
and ¯ x¯ = φx¯ (x, x) ¯ + [φ1 (x, x)]
L (G0 (x)) L (G0 (x))
G0 (x).
(4.12)
The following statements are complete analogs of those from Section 3.1. Lemmas 7 and 8 are evident, to get Lemmas 9, 10 and Corollary 4 one only needs to change the map U : L x → z ∈ L˜ to the map G0 : Ldissected x → ω ∈ D0 in the proofs of corresponding statements from Section 3.1. Since the map G0 , similarly to the map U , depends on the branch points λ1 , . . . , λM holomorphically, all the arguments from Section 3.1 can be applied in the present context. LEMMA 7. The derivative of the function φ ext has the following asymptotics near the branch points and the infinities of the sheets:
80
A. KOKOTOV AND D. KOROTKIN
(1) |φλext (λ, λ)|2 = ((1/rm ) − 1)2 |λ − λm |−2 + O(|λ − λm |−2+1/rm ) as λ → λm , (2) |φλext (λ, λ)|2 = 4|λ|−2 + O(|λ|−3 ) as λ → ∞. Let x be a local coordinate on L. Set R ω,x = {ω, x}, where ω ∈ D, πH (ω) = x. LEMMA 8. (1) The Schwarzian derivative can be expressed as follows in terms of the function φ from (4.6): R ω,x = φxx − 12 φx2 .
(4.13)
(2) In a neighborhood of a branch point Pm there is the following relation between Schwarzian derivatives computed with respect to coordinates λ and xm : 1 1 1 ω,λ 2/rm −2 ω,xm − R + (4.14) (λ − λm )−2 . R = 2 (λ − λm ) rm 2 2rm2 (3) Let ζ be the coordinate in a neighborhood of the infinity of any sheet of covering L, ζ = 1/λ. Then R ω,λ =
R ω,ζ = O(|λ|−4 ). λ4
(4.15)
LEMMA 9. The derivatives of the function φ with respect to λ are related to its derivatives with respect to the branch points as follows: ∂φ ∂Fm ∂φ + = 0, + Fm ∂λm ∂λ ∂λ
(4.16)
where Fm = −
[G0 ]λm . [G0 ]λ
(4.17)
LEMMA 10. Denote the composition p ◦ πH by R. Then (1) The following relation holds: Fm =
∂R . ∂λm
(4.18)
(2) In a neighborhood of the point λl the following asymptotics holds: Fm = δlm + o(1),
(4.19)
where δlm is the Kronecker symbol. (3) At the infinity of each sheet the following asymptotics holds: Fm (λ) = O(|λ|2 ).
(4.20)
81
TAU-FUNCTIONS ON HURWITZ SPACES
COROLLARY 4. Keep m fixed and define Fn (xn ) ≡ Fm (λn + xnrn ). Then d k Fn (0) = 0, k = 1, . . . , rn − 2. Fn (0) = δnm ; dxn 4.1.3. The Regularized Dirichlet Integral Assume that the ramification points and the infinities of sheets do not belong to the cuts Ci . To the kth sheet L(k) dissected of the dissected surface L (we should add some cuts connecting the branch points) there corresponds the function φkext : L(k) dissected → R k which is smooth in any domain r of the form kρ = {λ ∈ L(k) dissected : ∀m|λ − λm | > ρ and |λ| < 1/ρ}, where ρ > 0 and λm are all the branch points from the kth sheet L(k) dissected of Ldissected. The function φkext has finite limits at the cuts (except the endpoints which are the ramification points); at the ramification points and at the infinity it possesses the asymptotics listed in Lemma 7. Introduce the regularized Dirichlet integral
|φλ |2 dS. Ldissected
Namely, set Qρ =
N k=1
kρ
|∂λ φkext |2 dS,
(4.21)
where dS is the area element on C1 : dS = |dλ ∧ dλ¯ |/2. According to Lemma 3 there exists the finite limit
M (rm − 1)2 2 |φλ | dS = lim Qρ + 4N + 2π ln ρ . (4.22) reg ρ→0 r m Ldissected m=1 Now set 1 reg SSchottky(λ1 , . . . , λM ) = 2π
|φλ |2 dS + Ldissected
g L (G0 (λ)) i φ(λ, λ¯ ) k G0 (λ) dλ¯ − + 4π k=2 Ck Lk (G0 (λ))
¯ Lk (G0 (λ)) G0 (λ) dλ + φ(λ, λ) − Lk (G0 (λ)) Ck
82
A. KOKOTOV AND D. KOROTKIN
2 Lk (G0 (λ)) ln|Lk (G0 (λ)| G0 (λ) dλ¯ Lk (G0 (λ)) Ck g ln|lk |2 . 2
+ +
+ (4.23)
k=2
Here Lk are generators of the Schottky group H, the orientation of contours Ck is defined by the orientation of countours ck and the relations Ck = πH (ck ); the value of the function φ(λ, λ¯ ) at the point λ ∈ Ck is defined as the limit limµ→λ φ(µ, µ), ¯ µ = πH (ω) and ω tends to the contour ck from the interior of the region D0 ; lk is the left-hand lower element in the matrix representation of the transformation Lk ∈ PSL(2, C). The summations at the right-hand side of (4.23) start from k = 2 due to the normalization condition for the group H (the terms with k = 1 are equal to zero). Observe that the expression at the right-hand side of (4.23) is real and does not depend on small movings of the cuts Ck (i.e. on a specific choice of the fundamental region D0 ). In particular, we can assume that the contours Ck are {λ1 , . . . , λM }independent. (To see this one should make a simple calculation based on (4.11), (4.12) and the Stokes theorem.) Thus all terms in this expression except the last one are rather natural. The role of the last term will become clear later. The main result of this section is the following theorem. THEOREM 7. For any m = 1, . . . , M the following equality holds 1 d rm −2 ω,xm ∂SSchottky(λ1 , . . . , λM ) = R |xm =0 . ∂λm (rm − 2)! rm dxm
(4.24)
Remark 6. This result seems to be very similar to Theorem 1 from [18]. However, we would like to emphasize that in oppose to [18] we deal here with the Dirichlet integral corresponding to a flat metric. Thus, the following proof does not explicitly use the Teichmüller theory and, therefore, is more elementary than the proof of an analogous result in [18]. Proof. Set i = Qρ + 2 k=2 g
Sρ
φ(λ, λ¯ )
Ck
Ck
+
Lk (G0 (λ))
φ(λ, λ¯ )
−
Lk (G0 (λ))
G0 (λ) dλ¯ −
Lk (G0 (λ)) G (λ) dλ + Lk (G0 (λ)) 0
2 Lk (G0 (λ)) ¯ ln|Lk (G0 (λ)| G0 (λ)) dλ . Lk (G0 (λ) Ck
(4.25)
83
TAU-FUNCTIONS ON HURWITZ SPACES
We recall that the contours Ck are assumed to be {λ1 , . . . , λM }-independent. From now on we write G(λ) and φ instead of G0 (λ) and φ ext . Since φλλ¯ = 0, we have |φλ |2 = (φλ φ)λ¯ . The Stokes theorem and the formulas (4.10), (4.11) give M r N n i φλ φ dλ + φλ φ dλ − Qρ = − (k) 2 n=1 l=1 |λ(l) −λn |=ρ k=1 |λ |=1/ρ g Lk (G(λ)) i G (λ) × φλ φ − φλ + − 2 k=2 Ck Lk (G(λ))
2 dλ. (4.26) × φ + ln|Lk (G(λ))| Here λ(k) denotes the point on the kth sheet of the covering L whose projection to P1 is λ. Denote the first term in (4.26) by − 2i [Tρ ]. Substituting (4.26) into (4.25) and using the equalities Ck d[φ(λ, λ¯ ) ln|L (G(λ))|2 ] = 0 and Ck d[ln2 |Lk (G(λ))|2 ] = 0, we get g i i φ ¯ (λ, λ¯ ) ln|Lk (G(λ))|2 dλ¯ − Sρ = − [Tρ ] − 2 2 k=2 Ck λ i − 2 k=2 g
¯ φ(λ, λ)
Ck
Lk (G(λ)) G (λ) dλ. Lk (G(λ))
(4.27)
LEMMA 11. For the first term in (4.27) we have the asymptotics 2π d rm −2 ω,xm i ∂ [Tρ ] = R |xm =0 + − 2 ∂λm (rm − 2)! rm dxm i + 2 k=1 g
Ck ∪Ck−
Fm (2φλλ − φλ2 ) + [Fm ]λ φλ dλ + o(1),
(4.28)
as ρ → 0. Here Ck− is the contour Ck provided by the reverse orientation, the value of the integrand at a point λ ∈ Ck− is understood as the limit as µ → λ, where µ = πH (ω), ω tends to ck from the interior of the region D0 ; the function Fm is from Lemma 9. Proof. Using Lemma 9, we get M rn ∂ φλ φ dλ ∂λm n=1 l=1 |λ(l) −λn |ρ rm (φλ2 + φφλλ ) dλ − = l=1
|λ(l) −λm |=ρ
84
A. KOKOTOV AND D. KOROTKIN
− =−
rn M
|λ(l) −λm |=ρ
l=1
+
|λ(l) −λn |=ρ
n=1 l=1 rm
|φλ |2 dλ¯ +
rn M |λ(l) −λn |=ρ
n=1 l=1
(Fm φλ + [Fm ]λ )φλ + φ([Fm ]λ φλ + Fm φλλ + [Fm ]λλ ) dλ
Fm |φλ |2 dλ¯ + φλ¯ [Fm ]λ dλ¯ .
(4.29)
For the integrals around the infinities we have the equality N N ∂ φλ φ dλ = Fm |φλ |2 dλ¯ + φλ¯ [Fm ]λ dλ¯ . (4.30) (k) ∂λm k=1 |λ(k) |=1/ρ k=1 |λ |=1/ρ
Applying the Cauchy theorem to the (holomorphic) function [Fm ]λ φλ , we get g [Fm ]λ φλ dλ k=1
Ck ∪Ck−
=−
rn M n=1 l=1
|λ(l) −λn |=ρ
+
N k=1
|λ(k) |=1/ρ
[Fm ]λ φλ dλ.
(4.31)
By (4.29), (4.30) and (4.31) rm i i ∂ [Tρ ] = |φλ |2 dλ¯ − − 2 ∂λm 2 l=1 |λ(l) −λm |=ρ M r N n i + × − (l) (k) 2 n=1 l=1 |λ −λn |=ρ k=1 |λ |=1/ρ
× Fm |φλ |2 dλ¯ − [Fm ]λ φλ dλ + [Fm ]λ φλ¯ dλ¯ i + 2 k=1 g
+
Ck ∪Ck−
[Fm ]λ φλ dλ.
Denote the expression in the large braces by H2 . We claim that r m i i |φλ |2 dλ¯ − − H2 = − 2 2 l=1 |λ(l) −λm |=ρ M 1 1 n) 1 − 2 F(r − 2π i n (0) + o(1), (r − 1)! r n n n=1 where the function Fn is from Corollary 4.
(4.32)
(4.33)
85
TAU-FUNCTIONS ON HURWITZ SPACES
To prove this we set I1n (ρ)
=
I2n (ρ) = I3n (ρ) =
rn |λ(p) −λn |=ρ
p=1 rn p=1
|λ(p) −λn |=ρ
p=1
|λ(p) −λn |=ρ
rn
¯ Fm |φλ |2 dλ; [Fm ]λ φλ dλ; [Fm ]λ φλ¯ dλ¯ .
By Corollary 4 we have I1n (ρ)
= δnm
rn p=1
+
|λ(p) −λn |=ρ
|xn |=ρ 1/rn
|φλ |2 dλ¯ +
1 (rn ) 1 Fn(rn −1) (0)xnrn −1 + F (0)xnrn + (rn − 1)! rn ! n + O(|xn |rn +1 ×
|φxintn |2
1 − rn φxintn 1 − rn φxint ¯n + + rn −1 rn −1 rn rn −1 rn −1 rn 2 2 r r rn xn x¯n x¯n xn x¯n xn n n 2 1 1 −1 rn x¯nrn −1 dx¯n + rn xnrn x¯nrn rn (1/rn − 1)2 (rn ) F (0) + = δnm |φλ |2 dλ¯ + 2π i (p) (rn − 1)! n p=1 |λ −λn |=ρ ×
+ 2π i
+
1 − rn F(rn −1) (0)φxintn (0) + o(1) rn (rn − 1)! n
as ρ → 0. We get also
1 1 F(rn ) (0) − −1 rn (rn − 1)! n 1 φ int (0)Fn(rn −1) (0) + o(1) − 2π i rn (rn − 2)! xn
I2n (ρ) = −2π i
and I3n (ρ) = 2π i
(1/rn − 1) (rn ) F (0) + o(1). (rn − 1)! n
86
A. KOKOTOV AND D. KOROTKIN
We note that I1n
−
I2n
+
I3n
= δnm
rn p=1
|λ(p) −λ
n |=ρ
p=1
|λ(p) −λn |=ρ
|φλ |2 dλ¯ +
2 1 1 2π i (rn ) F (0) −1 +2 − 1 + o(1) + (rn − 1)! n rn rn rn |φλ |2 dλ¯ − = δnm 1 2π i n) 1 − 2 F(r − n (0) + o(1). (rn − 1)! rn It is easy to verify that N k=1
|λ(k) |=1/ρ
Fm |φλ |2 dλ¯ −
|λ(k) |=1/ρ
[Fm ]λ φλ dλ +
|λ(k) |=1/ρ
[Fm ]λ φλ¯ dλ¯
= o(1), so we get (4.33). The function Fm (2φλλ − φλ2 ) is holomorphic outside of the ramification points, the infinities and the cuts. Applying to it the Cauchy theorem and making use of Lemma 8 and the asymptotics from Lemma 10, we get the equality 1 1 1 − 2 Fn(rn ) (0) (rn − 1)! rn n=1 d rm −2 ω,xm 4π i R (xm )|xm =0 + =− (rm − 2)! rm dxm g {Fm (2φλλ − φλ2 )} dλ. +
2π i
M
k=1
Ck ∪Ck−
Summarizing (4.32), (4.33) and (4.34), we get (4.28).
(4.34) ✷
Now we shall differentiate with respect to λm the remaining terms in (4.27). Denote by Lk;m, G;m the derivatives ∂/∂λmLk , ∂/∂λmG. Since φλ is holomorphic with respect to λm , we have [φλ¯ ]λm = 0. Thus, g i ∂ ¯ ln|Lk (G(λ))|2 dλ¯ − φ ¯ (λ, λ) − ∂λm 2 k=2 Ck λ g Lk (G(λ)) i G (λ) dλ φ(λ, λ¯ ) − 2 k=2 Ck Lk (G(λ))
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TAU-FUNCTIONS ON HURWITZ SPACES
i 2 k=2 g
=
φλ
Lk;m(G(λ)) + Lk (G(λ))G;m(λ)
Ck
Lk (G(λ))
dλ +
L (G(λ)) i G (λ) dλ. (Fm φλ + [Fm ]λ ) k + 2 k=2 Lk (G(λ)) g
(4.35)
(We have used the equality φλ¯
∂2 ∂ ln|Lk (G(λ))|2 dλ¯ + φ ln|Lk (G(λ))|2 dλ ∂λm ∂λ∂λm ∂ ∂ 2 ln|Lk (G(λ))| − φλ ln|Lk (G(λ))|2 dλ =d φ ∂λm ∂λm
and Lemma 9.) To finish the proof we have to rewrite the last term at the right-hand side of (4.28) as follows
i Fm (2φλλ − φλ2 ) + [Fm ]λ φλ dλ − 2 Ck ∪Ck
i φλ φλm dλ = 2 Ck ∪Ck−
Lk (G(λ)) i G (λ) × φλ φλm − φλ + = 2 Ck Lk (G(λ)) Lk;m (G(λ)) + Lk (G(λ))G;m(λ) dλ × φλm + Lk (G(λ))
Lk;m (G(λ)) + Lk (G(λ))G;m (λ) Lk (G(λ)) i + φ G (λ) + φλ =− λ m 2 Ck Lk (G(λ)) Lk (G(λ)) Lk (G(λ)) Lk;m(G(λ)) + Lk (G(λ))G;m(λ) G (λ) dλ. (4.36) + Lk (G(λ)) Lk (G(λ)) Collecting (4.27), (4.28), (4.35) and (4.36) and using the equality φλm = we get
G;m(λ) G (λ)
,
2π d rm −2 ω,xm ∂Sρ + o(1) = R |xm =0 − ∂λm (rm − 2)! rm dxm g Lk (G(λ))Lk;m(G(λ)) i G (λ) dλ − − 2 k=2 Ck [Lk (G(λ))]2 g Lk (G(λ)) 2 i G (λ)G;m (λ) dλ − − 2 k=2 Ck Lk (G(λ))
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A. KOKOTOV AND D. KOROTKIN
−i
g k=2
Ck
Lk (G(λ)) G (λ) dλ. Lk (G(λ)) ;m
(4.37)
Since {Lk (ω), ω} ≡ 0, the last two terms in (4.37) cancel (one should beforehand integrate the last term by parts). For the second term we have the equality ([18]):
Lk (G(λ))Lk;m(G(λ)) lk;m i G (λ) dλ = −4π . − 2 Ck [Lk (G(λ))]2 lk To prove Theorem 7 it is sufficient to observe that the term o(1) in (4.37) is uniform with respect to parameters (λ1 , . . . , λM ) belonging to a compact neighborhood of ✷ the initial point (λ01 , . . . , λ0M ). 4.2. THE LIOUVILLE ACTION AND THE FUCHSIAN UNIFORMIZATION 4.2.1. The Metric of Constant Curvature −1 on L and its Dependence upon the Branch Points The covering L is biholomorphically equivalent to the quotient space H/ L, where H = {z ∈ C : z > 0}, L is a strictly hyperbolic Fuchsian group. Denote by πL : H → L the natural projection. Let x be a local parameter on L, introduce the ¯ |dx|2 of the constant curvature −1 on L by the equality metric eχ(x,x) ¯ |dx|2 = eχ(x,x)
|dz|2 , |z|2
(4.38)
where z ∈ H, πL (z) = x. As usually we specify the functions χ ext (λ, λ¯ ), χ int (xm , x¯m ), m = 1, . . . , M and χ ∞ (ζk , ζ¯k ), k = 1, . . . , N setting x = λ, x = xm and x = ζk in (4.38). Set R z,x = {z, x}, where z ∈ H, πL (z) = x. Clearly, Lemmas 7 and 8 still stand with χ ext , R z,x instead of φ ext and R ω,x , whereas Lemma 9 should be reconsidered, since the Fuchsian uniformization map depends upon the branch points nonholomorphically. ¯ |dω|2 of constant curvature −1 on D0 (see the preIntroduce the metric eψ(ω,ω) vious section) by the equation ¯ |dω|2 = eψ(ω,ω)
|dz|2 , |z|2
where πH (ω) = πL (z). Then there is the following relation between the derivatives of the function ψ: ¯ + ψω (ω, ω)F ¯ m (ω, ω) ¯ + [Fm ]ω (ω, ω) ¯ = 0, ψλm (ω, ω)
(4.39)
where F is a continuously differentiable function on D0 ; (the proof of (4.39) is parallel to the one in [18]). We shall now prove the analog of (4.39) and Lemma 9 for the function χ = χ ext .
TAU-FUNCTIONS ON HURWITZ SPACES
89
LEMMA 12. There is the following relation between the derivatives of the function χ: ∂χ(λ, λ¯ ) ∂Fm (λ, λ¯ ) ∂χ(λ, λ¯ ) + = 0, (4.40) + Fm (λ, λ¯ ) ∂λm ∂λ ∂λ where 1 + Fm (λ). (4.41) Fm (λ, λ¯ ) = Fm (G0 (λ), G0 (λ)) G0 (λ) Here Fm = −[G0 ]λm /[G0 ]λ is the function from Lemma 9, Fm is the function from (4.39). Proof. Since ¯
eχ(λ,λ) |dλ|2 = eψ(G0 (λ),G0(λ)) |G0 (λ)|2 |dλ|2 , we have the equality χ(λ, λ¯ ) = ψ(G0 (λ), G0 (λ)) + φ(λ, λ¯ ),
(4.42)
where φ(λ, λ¯ ) = ln|G0 (λ)|2 is the function from (4.7). Differentiating (4.42) with respect to λm via formulas (4.39) and (4.16), after some easy calculations we get (4.40). ✷ Remark 7. Observe that the function Fm does not have jumps at the cycles Ck , whereas the both terms at the right hand side of (4.41) do. This immediately follows from the formulas Lk;m(G+ 0 (λ)) , Fm− (λ) = Fm+ (λ) − + Lk (G0 (λ))[G+ 0 ]λ (λ) + + [G− 0 ]λ (λ) = Lk (G0 (λ))[G0 ]λ (λ) and the formula from [18]: Fm ◦ Lk = Fm Lk + Lk;m . Here the indices + and − denote the limit values of the corresponding functions at the ‘ck ’ and the ‘ck ’ sides of the cycle Ck . LEMMA 13. Fix a number m = 1, . . . , M. Then for any n = 1, . . . , M the following asymptotics holds Fm (λn + xnrn , λ¯ n + x¯nrn ) = δmn + an xnrn −1 + bn x¯n xnrn −1 + cn xnrn + O(|xn |rn +1 )
(4.43)
as xm → 0; here an , bn , cn are some complex constants. At the infinity of the kth sheet of the covering L there is the asymptotics ¯ = Ak λ2 + Bk λ + Ck λ2 λ¯ −1 + O(1) (4.44) Fm (λ, λ) as λ → ∞(k) ; here ∞(k) is the point at infinity of the kth sheet of the covering L; Ak , Bk , Ck are some complex constants.
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A. KOKOTOV AND D. KOROTKIN
Proof. This follows from Corollary 4, asymptotics (4.20) and formula (4.41). ✷ 4.2.2. The Regularized Liouville Action Here we define the regularized integral
reg (|χλ |2 + eχ ) dS L
and calculate its derivatives with respect to the branch points λm . Set Qkρ = {λ ∈ L(k) : ∀m |λ − λm | > ρ and |λ| < 1/ρ}, where Pm are all the ramification points which belong to the kth sheet L(k) of the covering L. To the sheet L(k) there corresponds the function χkext : L(k) → R which is smooth in any domain Qkρ , ρ > 0. The function χkext has finite limits at the cuts (except the endpoints which are the ramification points); at the ramification points and at the infinity it possesses the same asymptotics as the function φkext from the previous section. ext Observe also that the function eχk is integrable on L(k) . Set Tρ =
N k=1
Qkρ
|∂λ χkext |2 dS.
(4.45)
Then there exists the finite limit
reg (|χλ |2 + eχ ) dS L N M 2 (r − 1) ext m eχk dS + 4N + 2π ln ρ . (4.46) = lim Tρ + ρ→0 (k) rm L k=1 m=1 Set SFuchs (λ1 , . . . , λM )
M 1 reg (|χλ |2 + eχ ) dS + (rn − 1)χ int (xn )|xn =0 − = 2π L n=1 −2
N
χ ∞ (ζk )|ζk =0 .
(4.47)
k=1
Now we state the main result of this section. THEOREM 8. For any m = 1, . . . , M the following equality holds 1 d rm −2 z,xm ∂SFuchs(λ1 , . . . , λM ) = R |xm =0 . ∂λm (rm − 2)! rm dxm
(4.48)
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TAU-FUNCTIONS ON HURWITZ SPACES
Proof. Set Qρ =
N
m ∂ i Tρ = ∂λm 2 k=1
r
k=1
Qkρ . Then
|∂λ χ| dλ¯ +
2
|λ(k) −λm |=ρ
Qρ
∂ |∂λ χ|2 dS. ∂λm
(4.49)
By (4.40) the last term in (4.49) can be rewritten as
∂ |∂λ χ|2 dS ∂λ m Qρ
= ((2χλλ − χλ2 )[Fm ])λ¯ − 2(χλ [Fm ]λ¯ )λ + (χλ [Fm ]λ )λ¯ − Qρ
− (χλ¯ [Fm ]λ )λ − (|χλ |2 [Fm ])λ dS
i (2χλλ − χλ2 )Fm dλ + =− 2 ∂Qρ + 2χλ [Fm ]λ¯ dλ¯ + χλ [Fm ]λ dλ + χλ¯ [Fm ]λ dλ¯ + |χλ |2 Fm dλ¯ i n [I + 2I2n + I3n + I4n + I5n ] − 2 n=1 1 M
=−
i ∞,k [J1 + J2∞,k + J3∞,k + J4∞,k + J5∞,k ], − 2 k=1 N
(4.50)
where I1n =
rn l=1
J1∞,k =
|λ(l) −λ
n |=ρ
|λ(k) |=1/ρ
(2χλλ − χλ2 )Fm dλ,
(2χλλ − χλ2 )Fm dλ
and the terms Ipn and Jp∞,k , p = 2, 3, 4, 5 are the similar sums of integrals and integrals with integrands χλ [Fm ]λ¯ dλ¯ , χλ [Fm ]λ dλ, χλ¯ [Fm ]λ dλ¯ and |χλ |2 Fm dλ¯ respectively. It should be noted that the circles |λ − λn | = ρ are clockwise oriented whereas the circles |λ| = 1/ρ are counter-clockwise oriented. Using (4.43), we get z,xn 1 2R (xn ) 1 n + 1− 2 × I1 = rn xn2rn rn xn2rn −2 |xn |=ρ 1/rn × δmn + an xnrn −1 + bn x¯n xnrn −1 + cn xnrn + O(|xn |rn +1 ) rn xnrn −1 dxn 4π i d rn −2 z,xm = −δnm R (0) − (rn − 2)! rn dxn 1 (4.51) − 2π irn 1 − 2 cn + o(1). rn
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A. KOKOTOV AND D. KOROTKIN
In the same manner we get I2n = o(1), I3n
rn − 1 1 int = −2π i an χxn (0) + rn − 1 cn + o(1), rn rn
(4.52)
and
1 − 1 rn cn + o(1), rn r n = δmn |χλ |2 dλ¯ +
I4n = 2π i I5n
l=1
+
|λ(l) −λ|=ρ
1 2π iχxintn (0)
2 − rn 1 an + 2π i − 1 rn cn + o(1). rn rn
(4.53)
Using (4.44), we get also J1∞,k = o(1),
J2∞,k = o(1),
(0) + Bk ) + o(1), J3∞,k = −4π i(Ak χζ∞ k
(4.54)
and J4∞,k = 4π iBk + o(1),
J5∞,k = −4π i(Ak χζ∞ (0) + 2Bk ) + o(1). (4.55) k
Summarizing (4.49–4.55), we have M 2π 1 − rn d rm −2 z,xm ∂ Tρ = R (0) + 2π an χxintn (0) + cn − ∂λm (rm − 2)! rm dxm rn n=1 − 4π
N
(0) + Bk + o(1). Ak χζ∞ k
(4.56)
k=1
To finish the proof we need the following lemma. LEMMA 14. The equalities hold 1 ∂ int χ (xn )|xn =0 = − (an χxintn (0) + cn ) ∂λm rn
(4.57)
∂ ∞ χ (ζk )|ζk =0 = Ak χζ∞ (0) + Bk . k ∂λm
(4.58)
and
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TAU-FUNCTIONS ON HURWITZ SPACES
Proof. We shall prove (4.57); (4.58) can be proved analogously. Since eχ
int (x
n ,x¯ n )
|dxn |2 = eχ
ext (λ,λ) ¯
|dλ|2 ,
we get χ (xn , x¯n ) = χ (λ, λ¯ ) − int
ext
1 1 − 1 2 ln|λ − λn |2 rn rn
(4.59)
and (λ, λ¯ ) = χλintm (xn , x¯n ) + const δmn χλext m
1 . xnrn
(4.60)
By (4.43) and (4.40) we have ¯ (λ, λ) χλext m = − δmn + an xnrn −1 + bn x¯n xnrn −1 + cn xnrn + O(|xn |rn +1 ) × 1 1 1 int χ (xn , x¯n ) + − 1 rn − × rn −1 xn rn xn rn xn rn − 1 1 rn − 1 x¯n − an − bn − cn + O(|xn |). rn xn rn xn
(4.61)
Now substituting (4.61) in (4.60) and comparing the coefficients near the zero ✷ power of xn , we get (4.57). Observe that
∂ eχ dS = 0 ∂λm L due to the Gauss–Bonnet theorem and the term o(1) in (4.56) is uniform with respect to (λ1 , . . . , λM ) belonging to a compact neighborhood of the initial point ✷ (λ01 , . . . , λ0M ). This together with (4.56) and Lemma 14 proves Theorem 8. Remark 8. Consider the functional defined by the right-hand side of (4.47). If we introduce variations δχ which are smooth functions on L vanishing in neighborhoods of the branch points and the infinities then the Euler–Lagrange equation for an extremal of this functional coinsides with the Liouville equation χλλ¯ =
1 χ e . 2
The last equation is equivalent to the condition that the metric eχ |dλ|2 has constant curvature −1.
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4.3. THE MODULUS SQUARE OF BERGMANN AND WIRTINGER TAU - FUNCTIONS IN HIGHER GENUS Now we are in a position to calculate the modulus square of Bergmann (and, therefore, Wirtinger) tau-function. Actually, we shall give two equivalent answers: one is given in terms of the Fuchsian uniformization of the surface L and the determinant of the Laplacian, another one uses the Schottky uniformization and the holomorphic determinant of the Cauchy–Riemann operator in the trivial line bundle over L. Indeed, formula (4.2) and Theorem 8 imply the following statement. THEOREM 9. Let the regularized Liouville action SFuchs be given by formula (4.47). Then we have the following expression for the modulus square |τB |2 of the Bergmann tau-function of the covering L: det . (4.62) det B For the modulus square |τW |2 of the Wirtinger tau-function we have the expression: det g−1 g−2 ||−2/(4 +2 ) . (4.63) |τW |2 = e−SFuchs /6 det B β even |τB |2 = e−SFuchs /6
On the other hand, using formula (4.4) and Theorem 7, we get the following alternative answer. THEOREM 10. Let the regularized Dirichlet integral SSchottky be given by formula (4.23). Then the modulus square of the Bergmann and Wirtinger tau-functions of the covering L can be expressed as follows: ¯ 2, |τB |2 = e−SSchottky /6 |det ∂| g−1 g−2 ¯2 ||−2/(4 +2 ) . |τW |2 = e−SSchottky /6 |det ∂|
(4.64) (4.65)
β even
¯ 2 from [16], Remark 9. Comparing (4.64), (4.62) and formula (3.3) for |det ∂| we get the equality 1 S, 2π where S is the Liouville action from [18]. Whether it is possible to prove this relation directly is an open question. SSchottky − SFuchs =
Remark 10. Looking at the formulas for the tau-functions in genera 0 and 1 (and for genus 2 two-fold coverings), one may believe that the expressions for the tau-functions in higher genus can be also given in pure holomorphic terms, without any use of the Dirichlet integrals and, especially, the Fuchsian uniformization. At the least, the Dirichlet integral should be eliminated from the proofs in genus 0 and 1.
TAU-FUNCTIONS ON HURWITZ SPACES
95
Remark 11. The number of sheets of the covering Hg,N (1, . . . , 1) −→ C(M) \ (or, equivalently, the degree of the Lyashko–Looijenga map) is finite and equals (up to the factor N!) to the Hurwitz number hg,N . Here M = 2g + 2N − 2, C(M) is the Mth symmetric power of C, = i,j {λi = λj }. Due to Remark 4, in case g = 0, 1 the 12th power τW12 of the Wirtinger tau-function gives a global holomorphic function on Hg,N (1, . . . , 1). It would be very interesting to connect the Wirtinger tau-function with the Hurwitz numbers hg,N .
Acknowledgements Our work on this paper was greatly influenced by Andrej Nikolaevich Tyurin; in particular, he attracted our attention to the Wirtinger bidifferential. The authors are also greatly indebted to the anonymous referee; a lot of his proposals and remarks were used here. This work was partially supported by the grant of Fonds pour la Formation de Chercheurs et l’Aide a la Recherche de Quebec, the grant of Natural Sciences and Engineering Research Council of Canada and Faculty Research Development Program of Concordia University. References 1.
2. 3. 4. 5. 6. 7. 8. 9.
10. 11.
Dubrovin, B.: Geometry of 2D topological field theories, In: Integrable Systems and Quantum Groups. Proceedings, Montecatini Terme, 1993, Lecture Notes in Math. 1620, Springer, Berlin, 1996, pp. 120–348. Fay, J. D.: Theta-functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer, 1973. Fay, J. D.: Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc. 96(464) (1992). Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. 90 (1969), 542–575. Hawley, N. S. and Schiffer, M.: Half-order differentials on Riemann surfaces, Acta Math. 115 (1966), 199–236. Jimbo, M., Miwa, M. and Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients, I, Phys. D 2 (1981), 306–352. Kitaev, A. and Korotkin D.: On solutions of Schlesinger equations in terms of theta-functions, Internat. Math. Res. Notices 17 (1998), 877–905. Knizhnik, V. G.: Multiloop amplitudes in the theory of quantum strings and complex geometry, Sov. Phys. Usp. 32(11) (1989), 945–971. Korotkin, D.: Matrix Riemann–Hilbert problems related to branched coverings of CP1, archive math-ph/0106009, In: I. Gohberg, A. F. dos Santos and N. Manojlovic (eds), Operator Theory: Advances and Application, Proceedings of the Summer School on Factorization and Integrable Systems, Algarve, September 6–9, 2000, Birkhäuser, Boston, 2002, to appear. Mumford, D.: Tata Lectures on Theta, Birkhäuser, 1984. Natanzon, S. M.: Topology of 2-dimensional coverings and meromorphic functions on real and complex algebraic curves, Selecta Math. Soviet. 12(3) (1993), 251–291.
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12.
Rauch, H. E.: Weierstrass points, branch points, and moduli of Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 543–560. Rauch, H. E.: A transcendental view of the space of algebraic Riemann surfaces, Bull. Amer. Math. Soc. 71 (1965), 1–39. Strachan, I. A. B.: Symmetries and solutions of Getzler’s equation for Coxeter and extended affine Weyl Frobenius manifolds, math-ph/0205012. Tyurin, A. N.: Periods of quadratic differentials (Russian), Uspekhi Mat. Nauk 33(6(204)) (1978), 149–195. Zograf, P. G.: Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces, Leningrad. Math. J. 1(4) (1990), 941–965. Zograf, P. G. and Takhtajan, L. A.: On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0, Math. USSR-Sb. 60(1) (1988), 143–161. Zograf, P. G. and Takhtajan, L. A.: On the uniformization of Riemann surfaces and on the Weil–Petersson metric on the Teichmüller and Schottky spaces, Math. USSR-Sb. 60(2) (1988), 297–313. Zograf P. G. and Takhtajan, L. A.: Potential of the Weil–Peterson metric on Torelli space, J. Soviet. Math. 52 (1990), 3077–3085.
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Mathematical Physics, Analysis and Geometry 7: 97–117, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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On Equilibria of the Two-fluid Model in Magnetohydrodynamics DIMITRI J. FRANTZESKAKIS1, IOANNIS G. STRATIS2 and ATHANASIOS N. YANNACOPOULOS3
1 Department of Physics, University of Athens, Panepistimiopolis, GR 15784 Zografou, Athens, Greece. e-mail:
[email protected] 2 Department of Mathematics, University of Athens, Panepistimiopolis, GR 15784 Zografou, Athens, Greece. e-mail:
[email protected] 3 Department of Statistics and Actuarial Science, University of the Aegean, GR 82300 Karlovassi, Samos, Greece. e-mail:
[email protected]
(Received: 11 April 2002; in final form: 13 July 2003) Abstract. We show how the equilibria of the two-fluid model in magnetohydrodynamics can be described by the double curl equation and through the study of this equation we study some properties of these equilibria. Mathematics Subject Classifications (2000): 76W05, 35Q35. Key words: Beltrami fields, double curl equation, equilibrium solutions, magnetohydrodynamics, two-fluid model.
1. Introduction The study of ideal magnetohydrodynamics (MHD) is a problem that has occupied the scientific community for nearly four decades as this theory finds many applications ranging from fusion studies to astrophysics (see, e.g., [3, 6, 12]). The equations of ideal MHD present interesting dynamical behaviour and support important and diverse classes of solutions, such as soliton solutions or solutions displaying turbulent behaviour, etc. However, a very helpful insight to the understanding of the dynamics of such systems is provided by the thorough study of the equilibrium solutions (time stationary solutions). In this paper, we undertake a necessary and important first step in the study of the properties of the equilibria of the two-fluid model in ideal MHD: the proof of existence of such states. The two-fluid model of a plasma describes the strong coupling between the magnetic properties of a plasma and its properties as a fluid, in a macroscopic limit [4]. In a recent paper, Yoshida and Mahajan [20], using the twofluid description and neglecting dissipative effects have reduced the determination of certain types of equilibrium solutions to a double curl type equation with constant coefficients. This equation was reduced to appropriate single curl equations
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DIMITRI J. FRANTZESKAKIS ET AL.
and the equilibria were expressed in terms of Beltrami fields (see, e.g., [18]). In this paper, we wish to study a more general type of such equilibrium solutions which may be obtained by the solution of a double curl equation with spatially varying coefficients. There is some related work on the single curl equation with spatially varying coefficients, see, e.g., [7, 13]. This set of equilibria solutions may prove useful in the modelling of plasmas especially in the context of laboratory plasmas. This paper is organized as follows: in Section 2 we derive the double curl equation in the context of the two fluid model. In Section 3 we study the existence of equilibrium solutions using a variational formulation and a fixed point scheme. In Section 4 we show that the considered problem can, under suitable constraints, be reduced to a single curl equation. Finally, in Section 5 we consider some interesting special cases (regarding the relation between the spatial scales of the magnetic and the velocity fields) in which the considered system reduces to simpler forms. 2. Derivation of the Double Curl Equation In this section we present the derivation of the double curl equation in the context of the two fluid model in magnetohydrodynamics. We assume that the plasma is composed of two different fluids, a fluid of electrons (mass m and charge −e) and a fluid of ions (mass M and charge e). The electron fluid has velocity ve and pressure pe and the ion fluid has velocity vi and pressure pi . We assume a neutral plasma with number density of the electrons and ions n. The momentum equation for each of these two fluids takes the form (see, e.g., [12]) e 1 ∂ve + (ve · ∇)ve = − (E + ve × B) − ∇pe , (1a) ∂t m mn e 1 ∂vi + (vi · ∇)vi = (E + vi × B) − ∇pi . (1b) ∂t M Mn We shall use a scaled version of these equations. The following scaled variables are to be employed 2 B λ i ˆ ˆ B = B0 B, ˆ tˆ, p= t= pv ˆ = VA v, x = λi x, VA µ0 √ where VA = B0 / µ0 Mn is the Alfven velocity and λ2i = M/(µ0 n) is a relevant lengthscale. In the scaled variables the equations become ∂ve + (ve · ∇)ve = −E − ve × B − ∇pe , µ ∂t ∂vi + (vi · ∇)vi = E + vi × B − ∇pi , ∂t where the hats have been dropped. In the above equation µ = m/M is a small parameter. In the limit µ → 0 (i.e. in the limit where the electron mass can be
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TWO-FLUID MHD EQUILIBRIA
neglected with respect to the ion mass) the equations become ∂A + ∇φ − ve × B − ∇pe = 0, ∂t 2 ∂A v ∂vi − vi × (∇ × vi ) + ∇ i = − − ∇φ − vi × B − ∇pi , ∂t 2 ∂t where we used the divergence free property of the velocity fields and we have expressed the electric field with the help of a vector potential A and a scalar potential φ in the form E=−
∂A − ∇φ. ∂t
(2)
The next step is to eliminate the electron fluid velocity ve from these equations. The electric current in the plasma is given (in dimensionless variables) by j = vi − ve . By the Maxwell equations the current is related to the magnetic field by j = ∇ ×B. So, the electron velocity field is related to the magnetic field by ve = vi − ∇ × B. Finally, we define the mean velocity of the two fluid system (in dimensionless variables) by V =
vi + µve vi . 1+µ
The two fluid system in the limit µ → 0 thus becomes ∂A − ∇φ + (V − ∇ × B) × B − ∇pe , ∂t 2 V ∂A ∂V − V × (∇ × V ) + ∇ =− − ∇φ − V × B − ∇pi . ∂t 2 ∂t
0=−
We may now eliminate the pressure terms by taking the curl of these equations −
∂B − ∇ × ((V − ∇ × B) × B) = 0, ∂t
∂B ∂ (∇ × V ) − ∇ × (V × (∇ × V )) = − + ∇ × (V × B). ∂t ∂t These equations may be written in symmetric form as ∂ωj − ∇ × (Uj × ωj ) = 0, ∂t
j = 1, 2
in terms of the generalized vorticities ω1 = B,
ω2 = B + ∇ × V
and the effective flows U1 = V − ∇ × B,
U2 = V .
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We may now look for equilibrium solutions of these equations. Such equilibrium solutions are solutions of the equations ∇ × (Uj × ωj ) = 0,
j = 1, 2.
A special class of solutions is in terms of generalized Beltrami fields (or nonlinear force free fields) in the form Uj = θj (x)ωj or in terms of the original fields B = a(x)(V − ∇ × B), B = −∇ × V + b(x)V , where a(x) =
1 , θ1 (x)
b(x) =
1 . θ2 (x)
We may now eliminate the velocity field V and obtain a single equation in terms of the magnetic field only. This is the following double curl equation with nonconstant coefficients 1 − b(x) ∇ × B + ∇ × (∇ × B) + a(x) 1 b(x) B +∇ × B = 0. (3) + 1− a(x) a(x) In the special case of constant coefficients the above equation assumes the form ∇ × (∇ × B) +
b 1 −b ∇ ×B + 1− B = 0. a a
Equations of the same type appear in the modelling of chiral media in electromagnetic theory. (See, e.g., [2] where a similar equation has been treated in the case where the functions a(x) and b(x) are known functions. In our case, these functions are to be determined.)
3. Existence of Equilibrium Solutions In this section we study the existence of equilibrium solutions with spatially varying coefficients. We rewrite the problem in the following form
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∇ × U1 = −U2 + λ1 U1 ∇ × U 2 = U 1 + λ2 U 2 ∇ · U1 = ∇ · U2 = 0 U1 · n = g1 U2 · n = g2 (∇ × U1 ) · n = −g2 + λ10 g1 (∇ × U2 ) · n = g1 + λ20 g2 U1 · n dσ = ai(1) , i U2 · n dσ = ai(2) ,
in , in , in , on ∂, on ∂, on 1− , on 2− ,
(4)
i
where (U1 , U2 ) = (B, U ) and λ1 , λ2 are considered as unknown functions to be determined. is a bounded domain of R3 which is not necessarily simply connected. The domain can be turned into a simply connected one by ‘cutting’ m surfaces i . By n we denote the outer unit normal on the boundary of , ∂. The surfaces i− are the parts of ∂ on which the vector fields gi are incoming. Remark 1. One set of boundary conditions refers to the determination of the normal components of the magnetic field and the velocity on the boundary of the domain . Another set of boundary conditions is needed for the determination of the unknown eigenfunctions λi , i = 1, 2. To obtain this set of boundary conditions we work in complete analogy with [7]: We take the curl of the equations leading to ∇ · (λi Ui ) = 0. For smooth enough λi this set of equations could be interpreted with the help of the ordinary differential equations x˙ = Ui (x) along the orbits of which λi is a constant. From this observation a proper boundary condition may be obtained by specifying the value of λi , say λi0 , on the sets i− which are the subsets of ∂ on which the fields gi are incoming. To deal with the problem of the possible lack of necessary smoothness of λi , Ui , for the above interpretation to be valid we choose instead to reformulate these boundary conditions as written above, where we have used the equations themselves to obtain an equivalent form which is more suitable for our purpose. Finally, the last set of boundary conditions is necessary to complete the formulation of the problem in the case where the domain is not simply connected; i are the surfaces by which we need to ‘cut’ the domain in order to renderit simply connected. The surfaces i have the following properties: (a) 0 = \ m i=1 i is simply-connected, (b) i ∩ j = ø if i = j and (c) the boundary of i , i = 1, . . . , m is contained in ∂. Remark 2. As seen in the above remark we may formally write ∇ · (λi Ui ) = 0. For smooth enough λi this set of equations could be interpreted with the help of the ordinary differential equations x˙ = Ui (x) along the orbits of which λi is a constant. In the case of λi ∈ C 1 this would imply that the equilibrium solutions of the MHD equations satisfying the Beltrami condition with λi = constant will
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be two-dimensional solutions. Eventhough 2D solutions are of great interest in plasma physics (see, e.g., [17, 16, 14, 5, 1]) some care has to be taken in writing and interpreting the equation ∇ ·(λi Ui ) = 0. Since, as shown in this paper (see also [7]), λi ∈ L∞ and is not necessarily in C 1 , the above condition may only be valid in the almost everywhere sense. Thus, the solution will be two-dimensional but there might be sets of measure zero where it is not necessarily two-dimensional, in other words it may be considered as a ‘quasi-two-dimensional’ solution. Before dealing with the above system, we must make the necessary changes in order to turn it into one with homogeneous boundary conditions. In order to achieve that, we must subtract a properly chosen potential field which is responsible for the nonhomogeneous boundary conditions. We write Ui = ui + ∇φi ,
i = 1, 2
and choose φi to be the solution of the following Neumann problem φi = 0
in ,
∂φi = gi on ∂. ∂n The new variables ui , i = 1, 2 will solve the homogeneous system ∇ × u1 = −u2 + λ1 u1 + J¯1 ∇ × u2 = u1 + λ2 u2 + J¯2 ∇ · u1 = ∇ · u2 = 0 u1 · n = 0 u2 · n = 0 (∇ × u1 ) · n = −g2 + λ10 g1 (∇ × u2 ) · n = g1 + λ20 g2 u1 · n dσ = βi(1) , i u2 · n dσ = βi(2) ,
in , in , in , on ∂, on ∂, on ∂, on ∂,
i
where J¯1 = −∇φ2 + λ1 ∇φ1 , and
J¯2 = ∇φ1 + λ2 ∇φ2
(j )
βi
(j )
= ai −
∇φj · n dσ, i
i = 1, . . . , m, j = 1, 2.
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3.1. FUNCTIONAL SET- UP OF THE PROBLEM We now introduce the functional set-up of the problem. Let V be the subspace of L2 ()3 defined by V = {v ∈ L2 ()3 , ∇ · v ∈ L2 ()3 , ∇ × v ∈ L2 ()3 and v · n = 0 on ∂} equipped with the norm v = ( v 20, + ∇ · v 20, + ∇ × v 20, )1/2 . It is useful to note that this space is a Hilbert space [11] which can be identified with V = {v ∈ H 1 ()3 , v · n = 0 on ∂}. On the space V is is useful to define the equivalent norm v V = ( ∇ · v 20, + ∇ × v 20, + PH v 20, )1/2 , where PH is the orthogonal projection of V on the finite-dimensional (of dimension m) subspace H defined by H = {v ∈ V, ∇ · v = 0, ∇ × v = 0}. Furthermore, we have the inequality v 0,
1 v V , c0
∀v ∈ V
for a constant c0 . For more details on the above, see [7–9, 19] and references therein. For the problem we study, the proper functional set-up for (u1 , u2 ) is the product space V × V equipped with the norm v V×V = ( u1 2V + u2 2V )1/2.
3.2. A FIXED POINT SCHEME FOR THE SOLUTION OF THE PROBLEM The solution of the problem will be considered with the use of the following fixed point scheme: • We first assume that (λ1 , λ2 ) are known and we solve the following problem ∇ × u1 = −u2 + λ1 u1 + J1 in , in , ∇ × u2 = u1 + λ2 u2 + J2 in , ∇ · u1 = ∇ · u2 = 0 on ∂, (PROBLEM A) u1 · n = 0 on ∂, u2 · n = 0 u1 · n dσ = βi(1) , i u2 · n dσ = βi(2) , i
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where J1 = −∇φ2 + λ1 ∇φ1 + ∇p1 , J2 = ∇φ1 + λ2 ∇φ2 + ∇p2 and the terms ∇pi , i = 1, 2 have been added so as to ensure the divergence free property of the fields since for general λi that may be used in Problem A it will not be true that ∇ · (λi ui ) = 0. Of course, this property will always be true for the λi that are the eigenfunctions of the problem. • We then solve the problem for λi , i = 1, 2 assuming ui are known. We solve the following problem − λi + ui · ∇λi = 0 in λi = σi
on ∂
(PROBLEM B)
The terms − λi are elliptic terms which are added to regularize the hyperbolic system that the eigenfunctions λi satisfy. The solution of the problem we wish to solve is now the fixed point of the successive solution of Problems A and B. In the next subsections we treat separately these problems and prove the existence of a fixed point in the above scheme. 3.2.1. Solution of Problem A Let us temporarily assume that λ1,2 ∈ L∞ () and that they satisfy the bounds λi ∞ λ∞ . We will first introduce the variational form of Problem A. We may prove the following lemma LEMMA 3.1. The quadruple (u1 , u2 , p1 , p2 ) ∈ V × V × H01 () × H01 () is a solution of Problem A if and only if it is a solution of the following variational problem; ∀(v1 , v2 , w1 , w2 ) ∈ V × V × H01 () × H01 () (∇ × u1 + u2 − λ1 u1 , ∇ × v1 ) + (∇ · u1 , ∇ · v1 ) + (PH u1 , PH v1 ) m βi(1) (qi , PH v1 ), = (J¯1 , ∇ × v1 ) + i=1
(∇ × u2 − u1 − λ2 u2 , ∇ × v2 ) + (∇ · u2 , ∇ · v2 ) + (PH u2 , PH v2 ) m ¯ βi(2) (qi , PH v2 ), = (J2 , ∇ × v2 ) + i=1
(∇p1 , ∇w1 ) = −(J¯1 + λ1 u1 , ∇w1 ), (∇p2 , ∇w2 ) = −(J¯2 + λ2 u2 , ∇w2 ). Proof. The proof of this lemma is a straightforward generalization of the proof of Lemma 8 in [7] and is omitted here for the sake of brevity. 2 We may now use the above variational characterization of Problem A, along with the Lax–Milgram lemma, to guarantee its solution and obtain estimates on the solution. We have the following
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LEMMA 3.2. Problem A admits a unique solution satisfying the estimates ui V C, ∇pi 0, C1,i . Proof. We will use the linear forms li (vi ) = (J¯i , ∇ × vi ) +
m
βj(i) (qj , PH vi )
j
and the bilinear forms a1 (u1 , v1 ) = (∇ × u1 + u2 − λ1 u1 , ∇ × v1 ) + (∇ · u1 , ∇ · v1 ) + + (PH u1 , PH v1 ), a2 (u2 , v2 ) = (∇ × u2 − u1 − λ2 u2 , ∇ × v2 ) + (∇ · u2 , ∇ · v2 ) + + (PH u2 , PH v2 ). The variational problem may then be written in the form a1 (u1 , v1 ) + a2 (u2 , v2 ) = l1 (v1 ) + l2 (v2 ). In order to apply the Lax–Milgram lemma we need to check the boundedness of the form a1 + a2 and its coercivity. The boundedness is straightforward to check using the following estimates |a1 (u1 , v1 )| ∇ × u1 0, ∇ × v1 0, + u2 0, ∇ × v1 0, + + λ1 ∞ u1 0, ∇ × v1 0, + ∇ · u1 0, ∇ · v1 0, + + PH u1 0, PH v1 0, , |a2 (u2 , v2 )| ∇ × u2 0, ∇ × v2 0, + u1 0, ∇ × v2 0, + + λ2 ∞ u2 0, ∇ × v2 0, + ∇ · u2 0, ∇ · v2 0, + + PH u2 0, PH v2 0, from which we may conclude that |a1 (u1 , v1 ) + a2 (u2 , v2 )| C(c0 , λi ∞ ) u V×V v V×V which guarantees the boundedness of the bilinear forms in the space V × V. We now deal with the coercivity property of the forms. We have that a1 (u1 , u1 ) + a2 (u2 , u2 ) = u1 2V + u2 2V + (u2 , ∇u1 ) − − (λ1 u1 , ∇ × u1 ) − (u1 , ∇ × u2 ) − − (λ2 u2 , ∇ × u2 )
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from which we may conclude that |a1 (u1 , u1 ) + a2 (u2 , u2 )| u1 2V + u2 2V − u2 0, ∇ × u1 0, − − u1 0, ∇ × u2 0, − − λ1 ∞ u1 0, ∇ × u1 0, − − λ1 ∞ u1 0, ∇ × u1 0, . Using standard inequalities we may find that |a1 (u1 , u1 ) + a2 (u2 , u2 )| u1 2V + u2 2V −
2 λ∞ u1 V u2 V − ( u1 2V + u2 2V ) c0 c0
from which we may conclude that 1 λ∞ u 2V×V , |a1 (u1 , u1 ) + a2 (u2 , u2 )| 1 − − c0 c0 where λ∞ is the bound for the L∞ () norm of λi (i = 1, 2) and u = (u1 , u2 ). The above result guarantees the coercivity of the form a1 + a2 . We may furthermore show that the linear forms li are bounded as follows m (i) |βj PH vi 0, |li (vi )| Ji 0, ∇ × vi 0, + j
from which we may conclude that |l1 (v1 ) + l2 (v2 )| J¯1 2 + J¯2 2 +
m j =1
βj(1)
2 m 2 12 (2) + βj v V×V . j =1
From a straightforward application of the Lax–Milgram lemma we may thus see that the above problem has a unique solution that satisfies the bound m (1) 2 (2) 2 1 2 ( J¯1 2 + J¯2 2 + ( m j =1 βj ) + ( j =1 βj ) ) ≡ C. u V×V c0 c0 − 1 − λ∞ The proof for the solvability of the equation for p follows similarly, and a possible choice for C1,i , i = 1, 2, is C1,1 = ∇φ2 0, + λ∞ ( ∇φ1 0, + u1 0, ), C1,2 = ∇φ1 0, + λ∞ ( ∇φ2 0, + u2 0, ). This concludes the proof of the lemma. 2 Let us note that the solvability of this problem requires c0 − 1 − λ∞ > 0. This implies that c0 1 and λi ∞ c0 which is a constraint both on the initial data for the ’eigenvalues’ λi as well as on the domain, since the constant c0 depends on the choice of domain (see Lemma 5 in [7]).
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3.2.2. The Solution of Problem B In this section we treat the solution of Problem B, i.e. the solution of the problem of determination of λi with ui given. This problem is decoupled, so the solvability and the bounds for this problem can be obtained straight away from the relevant results of Boulmezaoud and Amari [7] on the single curl equation. In this direction we have LEMMA 3.3. Assume that ui ∈ H 1 ()3 , ∇ · ui = 0, σi ∈ H 3/2(∂), i = 1, 2. Then Problem B has a unique solution λi ∈ H 1 (). Moreover λi ∈ H 2 () and satisfy the following estimates λi ∞ σi ∞,∂ , ∇λi 0, C2 −1/2 , λi 2, C3 −7/4 . Proof. In complete analogy with the proof of Lemma 10 of [7].
2
3.2.3. The Coupled Problem We are now in a position to address the coupled problem that will give us the solution of the double curl problem ∇ × u1 = −u2 + λ1 u1 + J1 on , on , ∇ × u2 = u1 + λ2 u2 + J2 on , ∇ · u1 = ∇ · u2 = 0 (j ) uj · n = βi , i = 1, . . . , m, j = 1, 2,
(5)
i
− λi + (ui + ∇φi ) · ∇λi = 0 λi = σi
in , on ∂
with (u1 , u2 ) ∈ V × V. We define the closed subspace of V V1 = {v ∈ V/∇ · v = 0}. For the coupled system we will work with the closed subspace V1 × V1 of the functional space V × V. We now consider the operator T : (j1 , j2 ) ∈ V1 × V1 → (u1 , u2 ) ∈ V1 × V1 where (u1 , u2 ) is a solution of Problem A with (λ1 , λ2 ) being a solution of problem − λi + (ji + ∇φi ) · ∇λi = 0
in
λi = σ i
on ∂
for i = 1, 2.
(PROBLEM C)
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A fixed point of operator T is a solution of the coupled system. To prove the existence of a fixed point for the operator T we will employ Schauder’s fixed point theorem. We need the following lemma. LEMMA 3.4. The operator T is compact from V1 × V1 into V1 × V1 . Proof. We will first check the continuity of the operator T . Let (j1 , j2 ), (j1∗ , j2∗ ) ∈ V1 × V1 . By (λ1 , λ2 ), (λ∗1 , λ∗2 ) we denote the solutions of Problem C corresponding to these choices of ji . Set (u1 , u2 ) = T (j1 , j2 ), (u∗1 , u∗2 ) = T (j1∗ , j2∗ ) and let us denote by an overbar the difference between the starred and the nonstarred quantities, e.g., u¯ 1 = u1 − u∗1 , etc. We now consider the equations that the differences will satisfy. We begin with the equations for λ¯ i . Subtracting the equations for λi we obtain − λ¯ i + (ji + ∇φi ) · ∇ λ¯ i + j¯i · ∇ λ¯ ∗i = 0 with homogeneous boundary conditions. Multiplying by λ¯ i and integrating over the whole domain we obtain 2 ¯ λ∗i j¯i · ∇ λ¯ i d |λ∗i |∞ j¯1 0, λ¯ i 1, λi 1, =
C0 j¯1 0, λ¯ i 1,
(where we have used the estimates of Lemma 3.3). From the above inequality we obtain the estimate λ¯ i 1, C0 j¯1 0, . On the other hand we have that ∇ × u¯ 1 = −u¯ 2 + λ1 u¯ 1 + λ¯ 1 u∗1 + λ¯ 1 ∇φ1 + ∇ p¯1 , PH u¯ 1 = 0, ∇ · u¯ 1 = 0, ∇ × u¯ 2 = u¯ 1 + λ2 u¯ 2 + λ¯ 2 u∗2 + λ¯ 2 ∇φ2 + ∇ p¯2 , PH u¯ 2 = 0. ∇ · u¯ 2 = 0, We may now use the estimate for the solution of Problem A to estimate these differences. We have that
u ¯ V×V C1 { λ¯ 1 u∗1 + λ¯ 1 ∇φ1 20, + λ¯ 2 u∗2 + λ¯ 2 ∇φ2 20, }1/2.
Using Hölder’s inequality with p = 3, q = 3/2 we have that λ¯ i (∇φi + u∗i ) 0, λ¯ i L6 ∇φi + u∗i L3 C λ¯ i 1, , where we have used the compact embedding H 1 () → L6 () → L3 () and the boundedness of u∗i V . By the above we have that
u ¯ V C( λ¯ 1 21, + λ¯ 2 21, )1/2 C ( j¯1 20, + j¯2 20, )1/2
which guarantees the continuity of the operator T .
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We will now use arguments related to the continuity of T to prove the compactness property. Recall that an operator T : H1 → H2 is called compact if it maps bounded sequences in H1 into sequences in H2 that contain convergent subsequences. Let (j1n , j2n ) be a bounded sequence of V1 × V1 . By the definition of V1 we may extract a subsequence (j1k , j2k ) converging strongly in L2 ()3 × L2 ()3 and weakly in H 1 ()3 × H 1 ()3 . Consider now the sequence (u1k , u2k ) = T (j1k , j2k ). By the continuity inequality we have proved above we see that
T (jk1 ) − T (jk2 ) V×V C jk1 − jk2 V×V which guarantees the convergence of the subsequence. This concludes the proof of compactness of the operator T . 2 Using the above lemma we may now show the existence of a solution of the coupled system. THEOREM 3.1. The coupled system admits a solution satisfying the bounds
ui V×V C ,
λi ∞ C1 , ∇λi 0, C2 −1/2 , λi 2, C3 −7/4 , ∇pi 0, C2 1/2. Proof. We will use Schauder’s fixed point theorem. Consider the ball B of V×V defined by B = {v ∈ V1 × V1 : v V×V R}, where R is chosen in such a way that it is larger than the bound for (u1 , u2 ) provided by Problem A. Then T (B) ⊂ B and since T is a compact operator by Schauder’s fixed point theorem we may conclude that there exists a fixed point for operator T . This fixed point is a solution of the coupled system. A possible choice for the bounds of the solution is m m c0
1 2 βj qj + βj qj J1 0, + J2 0, + C = c0 − 1 − σ∞ j =1
0,
j =1
0,
and C1 = σ∞ where σ∞ = max( σi ∞ ). For the estimate of pi we proceed as follows: By taking the divergence of the equations for ui , using the fact that ∇ · vi = 0 and the equation for λi we find that pi = −∇ · (λi ui + Ji ) = − λi . Multiplying by pi and integrating over the whole of we obtain ∇pi 0, ∇λi 0, C2 1/2 .
(6)
This concludes the proof of the theorem.
2
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3.2.4. The Limit as → 0 In the treatment of the problem we have regularized Problem B by the use of a small elliptic part in the operator. The final technical part of the proof consists of studying the limit as this term goes to zero, that is the limit as → 0. We prove the following lemma LEMMA 3.5. The solution of the coupled system (5) converges weakly to a solution of the system (4) as → 0. Proof. We will consider the coupled problem (5) using the boundary condition
λi0 on i− , σi = 0 on i+ . 1
Since ∇pi 1, C 2 we may deduce that pi → 0 strongly in H01 () as → 0. Furthermore, since ui V and λ ∞ are bounded independently of we may conclude the existence of a sequence (λ1 , λ2 , u1 , u2 ) in L∞ () × L∞ () × V × V such that λi → λi weakly star in L∞ () and ui → ui weakly in H 1 ()3 and strongly in L2 ()3 . Furthermore, by the uniform estimates on λi ∞ we conclude that λi ∞ λ0i i− . From the above considerations we find that the products λi ui converge weakly to λi ui in L2 ()3 . We may further deduce the convergence of ∇pi to 0 strongly in L2 ()3 (by the bound on pi 1, ). We thus conclude that ∇ × u1 = λ1 u1 − u2 + ∇p1 → ∇ × u1 = λ1 u1 − u2 , ∇ × u2 = λ2 u2 + u1 + ∇p2 → ∇ × u2 = λ2 u2 + u1 weakly in L2 ()3 × L2 ()3 . Using the above we may conclude that ui converge weakly to ui in V → H 1 ()3 . The limiting fields Ui = ui +∇φi solve the problem ∇ × U1 = −U2 + λ1 U1 ∇ × U 2 = U 2 + λ2 U 2 ∇ · U1 = ∇ · U2 = 0 U1 · n = g1 U2 · n = g2 U1 · n dσ = ai(1) , i U2 · n dσ = ai(2) .
in , in , in , on ∂, on ∂,
i
We only need to check now the boundary conditions on i− , i = 1, 2. In direct analogy with the method used in the single curl equations we define two functions ¯ such that wi ∈ C 2 ()
wi on i− , wi = 0 on ∂\i− .
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We now multiply ∇ × U1 = λ1 U1 − U2 + ∇p1 , ∇ × U2 = λ2 U2 + U1 + ∇p2 by ∇w1 and ∇w2 respectively and integrate over . Using integration by parts we conclude that (∇ × U1 ) · nw1 dσ ∇ × U1 · n, w1 ≡ ∂ U2 w2 dσ + ∇p1 ∇w1 d, = λ1 U1 · ∇w1 d − ∂ (∇ × U2 ) · nw2 dσ ∇ × U2 · n, w2 ≡ ∂ U1 w1 dσ + ∇p2 ∇w2 d, = λ2 U2 · ∇w2 d +
∂
where we have used the fact that ∇ · (∇ × Ui ) = 0 and ∇ · Ui = 0. We must now estimate the integrals λi Ui · ∇wi d, i = 1, 2. To this end, we set Ui = ui + ∇φi and λˆ i = λi − ui0 where ui0 satisfies the Dirichlet problem ui0 = 0 in , ui0 = σi
on ∂.
The functions λˆ i satisfy the equation − λˆ i + Ui · λˆ i = 0 with homogeneous Dirichlet boundary conditions, which upon multiplied by wi and integrated over the whole domain yields λi Ui · ∇wi d
∇ λˆ i · ∇wi d −
=
i−
wi
∂ λˆ i dσ + ∂n
i−
λ0i gi wi dσ,
where we have used the definition of the functions wi . We thus conclude that ∂ λˆ 1 ˆ dσ + w1 ∇ × U1 · n, w1 = ∇ λ1 · ∇w1 d − ∂n 1− λ01 g1 w1 dσ − g2 w2 dσ + ∇p1 ∇w1 d, +
1−
1−
∇ λˆ 2 · ∇w2 d −
∇ × U2 · n, w2 =
+
2−
λ02 g2 w2 dσ +
2−
∂ λˆ 2 dσ + ∂n 2− U1 w1 dσ + ∇p2 ∇w2 d. w2
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By the weak convergence of ∇ × Ui to ∇ × Ui in L2 ()3 and by the vanishing of the divergence of the curl for all we have that ∇ × Ui · n, wi −→ ∇ × Ui · n, φ for i = 1, 2. We have also that ˆ ∇ λi · ∇wi d λˆ i 1, wi 1, −→ 0, ∇pi ∇wi d pi 1, wi 1, −→ 0,
i−
∂ λˆ i wi dσ −→ 0. ∂n
The last inequality can be proved in the same way as it was done in [7]. So we may conclude that in the limit as → 0 we recover the boundary conditions ∇ × U1 = λ01 g1 − g2 ∇ × U2 = λ02 g2 + g1
on 1− , on 2− .
This concludes the proof of the lemma.
2
The steps in Sections 3.2.1, 3.2.2, 3.2.3 and 3.2.4 guarantee the existence of a solution to Problem A. 4. Reduction to the Single Curl Equation In [20] the existence and uniqueness of equilibrium solutions for the two fluid equations are obtained in the case of constant coefficients. This is done using a factorization of the double curl equation into two single curl equations. This technique reveals some interesting properties of the equilibria in terms of eigenfunctions of the curl operator (Beltrami fields), [15]. As we will see in the case of nonconstant coefficients this factorization is no longer possible (except in the case of special choice of coefficients). THEOREM 4.1. If the coefficients a(x) and b(x) satisfy the constraint 1 + b(x) = C a(x) with C = constant, the double curl equation (3) can be factorized into two equations (curl − + (x))(curl − − (x))B = 0,
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where
+ (x) = b(x) − φ, − (x) = −
1 +φ a(x)
with φ = constant and the constants should satisfy the equation φ 2 − Cφ + 1 = 0. Under the same conditions the solution can be written as a linear combination of two ‘nonlinear’ Beltrami fields. Proof. By a simple algebraic manipulation of (curl − + (x))(curl − − (x))B = 0, we find that the factorized equation will be equivalent to the double curl equation (3) as long as 1 + b(x), a(x) b(x) ,
+ − = 1 − a(x) 1 . ∇ − = −∇ a(x)
+ + − = −
The first and the third equation imply that 1 + φ, a(x)
+ = b(x) − φ,
− = −
where φ is a constant. Substituting these into the second equation we obtain the consistency condition 1 2 + b(x) φ + 1 = 0. φ − a(x) This may be true as long as 1 + b(x) = C, a(x) where C is a constant. In order to be able to write the general solution of this equation in terms of a linear combination of nonlinear Beltrami fields, i.e. in terms of solutions of the equations Lˆ ± G± = (curl − ± (x))G± = 0 the operators Lˆ + and Lˆ − will have to commute. This implies that ∇ + = ∇ −
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which in turn is equivalent to the condition 1/a(x) + b(x) = c which is consistent with the previous condition. 2 In the special case that this consistency condition holds the results of Boulmezaoud and Amari [7] may be used for the treatment of the existence of equilibria. 5. Some Interesting Limiting Situations In this section we present some interesting limiting situations of the double curl equation. 5.1. THE CASE WHERE 1/a(x) = b(x) In this case the equation takes the simpler form 1 b(x) B +∇ ×B =0 ∇ × (∇ × B) + 1 − a(x) a(x) with boundary conditions B · n = 0 on ∂, where n is the normal vector to the boundary ∂. In the case of constant coefficients this equation becomes equivalent to the vector Helmholtz equation b B = 0. − B + 1 − a
5.2. THE CASE OF EQUAL LENGTH SCALES a(x) = b(x) In this case the equation becomes 1 1 − b(x) ∇ × B + ∇ × B = 0. ∇ × (∇ × B) + a(x) a(x) In the case of constant coefficients this equation can be written in terms of the curl of the magnetic field u = ∇ × B as 1 −b u=0 ∇ ×u+ a so that the magnetic field is a Beltrami field. This case corresponds physically to the case where the magnetic field and the velocity field of the fluid have the same spatial scales.
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5.3. REDUCTION OF THE PROBLEM TO AN ITERATIVE SEQUENCE OF POISSON ’ S EQUATIONS In this section we propose a reduction of the problem in the case that the magnetic field and the velocity field are of the same spatial scales. Consider the constant coefficients problem − B + ω2 B = 0, B · n = 0, where we have set ω2 = 1 − b/a. We will assume that ω is small. It is well known that Helmholtz’s equation is analytic with respect to ω. We may thus assume the power series expansion B(x) =
∞ ωm m=0
m!
m (x),
where m are independent of ω. This is in analogy with the low frequency expansion for electromagnetic fields, [10]. Substituting this expansion into the Helmholtz equation we end up with − m = 0, m = 0, 1, − m = −m(m − 1)m−2 , m 2, m · n = 0, m = 0, 1, 2, . . . The potentials 0 , 1 are harmonic functions and m , m 2, solve Poisson’s equation. Using the above expansion we conclude that we may obtain the magnetic fields by iteratively solving a sequence of simple potential problems. This approach to the problem may prove useful for obtaining approximate solutions of the double curl equation, in terms of special functions for special geometries (e.g., cylindrical geometry, spherical geometry, etc.). Such solutions may be used to check numerical algorithms or for other computational purposes. 5.4. EXTENSION TO NON - BELTRAMI FIELDS The equilibrium solutions given by the generalized Beltrami fields in this paper have the property that ∇pe = 0 and pi = −V 2 /2. The property that the electron pressure is a constant everywhere may be rather restrictive. To remedy this situation we propose a way to obtain more general equilibrium solutions to the ideal MHD equations using as starting point the Beltrami equilibrium solutions. To this end we write V = Vb + V1 , B = Bb + B1 ,
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where Vb and Bb are the Beltrami solutions obtained here and V1 , B1 are (small) perturbations to the Beltrami solutions. We look for equilibrium solutions. We substitute this ansatz to the MHD equations and linearize with respect to the fields V1 and B1 . Then the equations become 0 = ∇ × {[v1 − ∇ × B1 − θ1 B1 ] × Bb },
1 0 = ∇ × B1 + ∇ × V 1 − V 1 × V b , θ2 where θ1 and θ2 are the Beltrami proportionality factors which are specified by the solution of the original problem. In the above we used the definition of the fields Bb and Vb . The general solution of the above equation is of the form (V − ∇ × B1 − θ1 B1 ) × Bb = ∇ψ1 , 1 1 B1 + ∇ × V1 − V1 × Vb = ∇ψ2 , θ2 where now ψ1 and ψ2 are ‘arbitrary’ functions of x. It can be seen that for the above fields, the pressure will satisfy pe = ψ1 and pi = −(V 2 /2) + ψ2 . Thus, the solution of the above equations for Bb , Vb and ψ1 , ψ2 given, will provide us with an approximate equilibrium solution of the two-fluid model with prescribed pressure characteristics. However, a quick inspection of the above set of equations shows that the fields ∇ψ1 and ∇ψ2 are constrained by the conditions Bb · ∇ψ1 = Vb · ∇ψ2 = 0, that is the pressure gradients should be orthogonal to the (unperturbed) Beltrami fields. In the regions where the unperturbed fields are two-dimensional (see Remark 2 in Section 3) the surfaces of constant pressure define tori. Other configurations are also possible. The above constraints result from the linearization of the MHD equations. The full nonlinear model is expected to be free from such limitations. We will return to the matter of the fully nonlinear model in separate communications. The above equations are formally of the same type as the ones considered in the main part of this paper, and the ones considered in [2] in a different setting.
Acknowledgements A.N.Y. wishes to acknowledge partial financial support from the Postdoctoral programme of the Hellenic State Scholarship Foundation. I.G.S. and A.N.Y. acknowledge partial financial support from the Special Research Account of the University of Athens (grant no. 70/4/5643).
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References 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Aly, J. J. and Amari, T.: Current sheets in two-dimensional magnetic fields I, II, III, Astronom. Astrophys. 221 (1989), 287–294; 227 (1990), 628–633; 319 (1997), 699–719. Ammari, H. and Nédélec, J.-C.: Small chirality behaviour of solutions to electromagnetic scattering problems in chiral media, Math. Methods Appl. Sci. 21 (1998), 327–359. Biskamp, D.: Nonlinear Magnetohydrodynamics, Cambridge Univ. Press, Cambridge, 1993. Braginskii, S. I.: Transport processes in a plasma, In: M. A. Leontovich (ed.), Reviews of Plasma Physics, Consultants Bureau, New York, 1965, 205–311. Bogoyavlenskij, O. I.: Exact axially symmetric MHD equilibria, C.R. Acad. Sci. Paris Sér. I Math. 331 (2000), 569–574. Boulmezaoud, T. Z.: Etude des champs de Beltrami dans des domaines de R3 bornés et non bornés et applications en astrophysique, PhD Thesis, Univ. P. M. Curie, Paris, 1998. Boulmezaoud, T. Z. and Amari, T.: On the existence of non-linear force-free fields in threedimensional domains, Z. Angew. Math. Phys. 51 (2000), 942–967. Boulmezaoud, T. Z. and Amari, T.: Approximation of linear force-free fields in bounded 3-D domains, Math. Comput. Modelling 31 (2000), 109–129. Boulmezaoud, T. Z., Maday, Y. and Amari, T.: On linear force-free fields in bounded and unbounded three-dimensional domains, Math. Modelling Numer. Anal. (M2AN) 33 (1999), 359–393. Dassios, G. and Kleinman, R.: Low Frequency Scattering, Clarendon Press, Oxford, 2000. Dautray, R. and Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications, Springer, Berlin, 1991. Davidson, P. A.: An Introduction to Magnetohydrodynamics, Cambridge Univ. Press, Cambridge, 2001. Kravchenko, V. V.: On Beltrami fields with nonconstant proportionality factor, J. Phys. A 36 (2003), 1515–1522. Laurence P. and Stredulinsky, E. W.: Two-dimensional magnetohydrodynamic equilibria with prescribed topology, Comm. Pure Appl. Math. 53 (2000), 1177–1200. Moses, H. E.: Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem and applications to electromagnetics and fluid dynamics, SIAM J. Appl. Math. 21 (1971), 114–144. Neukirch, T.: Quasiequilibria: a special class of time dependent solutions of the twodimensional magnetohydrodynamic equations, Phys. Plasmas 2 (1995), 4389–4399. Rappaz, J. and Touzani, R.: On a two-dimensional magnetohydrodynamic problem I: Modelling and analysis, Math. Modelling Numer. Anal. (M2AN) 26 (1991), 347–364. Yoshida, Z.: Application of Beltrami functions in plasma physics, Nonlinear Anal. 30 (1997), 3617–3627. Yoshida, Z. and Giga, Y.: Remarks on spectra of operator rot, Math. Z. 204 (1990), 235–245. Yoshida, Z. and Mahajan, S.: Simultaneous Beltrami conditions in coupled vortex dynamics, J. Math. Phys. 40 (1999), 5080–5091.
Mathematical Physics, Analysis and Geometry 7: 119–149, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Transformation Operators for Sturm–Liouville Operators with Singular Potentials Dedicated to Professor V. A. Marchenko on the occasion of his 80th birthday ROSTYSLAV O. HRYNIV1 and YAROSLAV V. MYKYTYUK2 1 Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv,
Ukraine and Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine. e-mail:
[email protected] 2 Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine. e-mail:
[email protected] (Received: 26 September 2002; in final form: 4 April 2003) Abstract. We construct transformation operators for Sturm–Liouville operators with singular potentials from the space W2−1 (0, 1) and show that these transformation operators naturally appear during factorisation of Fredholm operators of a special form. Some applications to the spectral analysis of Sturm–Liouville operators with singular potentials under consideration are also given. Mathematics Subject Classifications (2000): Primary: 34C20; secondary: 34B24, 34L05, 47A68. Key words: transformation operators, Sturm–Liouville operators, singular potentials.
1. Introduction In the present work we shall study transformation operators (TOs) for Sturm–Liouville (SL) operators generated in a Hilbert space H = L2 (0, 1) by the differential expressions (f ) := −f + qf
(1.1)
with complex-valued distributions q from the space W2−1 (0, 1). (The precise definitions of the SL operators considered and the TOs are given in the next section.) Starting from the works by Povzner [25], Marchenko [21], Gelfand and Levitan [9] TOs have been successfully used in the spectral analysis of SL operators and other classical operators of mathematical physics. In particular, TOs have proved to be an important tool for solution of inverse spectral problems for SL operators (see the original papers [9, 21] and the monographs [20, 22, 26] for extended reference lists) and have been thoroughly studied for the case of regular, i.e., locally integrable, potentials q. Recent development of the theory of Sturm–Liouville and The work was partially supported by the Ukrainian Foundation for Basic Research DFFD under
grant no. 01.07/00172.
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Schrödinger operators with singular (i.e., not locally integrable) potentials [4, 14– 16, 24, 28–30] (see also the books [1] and [2] for general theory and detailed bibliography) has made feasible a thorough spectral analysis for SL operators with potentials from W2−1 (0, 1) and, in turn, led to inverse spectral problems for such class of operators. That posed the problem of existence and properties of TOs for SL operators with singular potentials, which we study in detail in the present article. In the subsequent papers [17, 18] we use the TOs constructed to solve the inverse spectral problems for SL operators with potentials from the space W2−1 (0, 1). Note that inverse spectral problems for SL operators with nonsmooth coefficients (in particular, for SL operators in the impedance form), were treated in different manner in, e.g., [3, 5, 7, 13, 27, 31, 32]. The main aim of this article is two-fold. Firstly, we shall construct the TOs for SL operators in H with singular potentials q ∈ W2−1 (0, 1) and study their properties. Secondly, we shall point out connection between the TOs constructed and a factorisation problem for Fredholm operators of a special form. Although this connection is known in the regular case, in the singular case under consideration it takes a very explicit form and allows a complete description. The organisation of the paper is the following. In the next section we briefly introduce the related concepts and give formulation of the main results. Section 3 is devoted to construction of some special TOs, which are then used in Section 4 to construct TOs for SL operators from the class considered and then to study their dependence on the potential. In Section 5 we establish results on connection of TOs with factorisation of some Fredholm operators, and in Section 6 we give some applications of the TOs to the spectral analysis of singular SL operators. 2. Preliminaries and Formulation of Main Results Throughout the paper we denote by dom T and ker T respectively the domain and kernel of an operator T in a Banach space X, while Wps ([0, 1], X) and Lp ((0, 1), X) will stand for the Sobolev and Lebesgue spaces of X-valued strongly measurable functions on [0, 1]. We shall write Wps [0, 1] and Lp (0, 1) instead of Wps ([0, 1], R) and Lp ((0, 1), R) respectively; in particular, W2−1 (0, 1) is the dual space of W21 [0, 1] with respect to L2 (0, 1). Suppose that q ∈ W2−1 (0, 1); then there exists a function σ ∈ H such that q = σ in the distributional sense, and differential expression (1.1) can be recast in terms of σ as d d +σ − σ f − σ 2 f, σ (f ) := −(f − σf ) − σf = − dx dx where again the derivatives are understood in the sense of distributions. We denote by Tσ a differential operator in H given by Tσ f = σ (f ) on the domain dom Tσ := {f ∈ W21 [0, 1] | f [1] ∈ W11 [0, 1], σ (f ) ∈ H },
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where f [1] := f − σf is the quasi-derivative of a function f . It is shown in [29] that the operator Tσ is closed. We consider a family Tσ,h of restrictions of Tσ to the linear manifolds dom Tσ,h := {f ∈ dom Tσ | f [1] (0) = hf (0)}. Here h is an arbitrary number from the extended complex plane C = C ∪ {∞}, and for h = ∞ the above relation is interpreted as dom Tσ,∞ := {f ∈ dom Tσ | f (0) = 0}. It is easily seen that Tσ,h = Tσ +h,0 for any h ∈ C, so that {Tσ,h | σ ∈ H, h ∈ C} = {Tσ,0 | σ ∈ H } ∪ {Tσ,∞ | σ ∈ H }, and it suffices to consider only the cases h = 0 and h = ∞. We also observe that Tσ,∞ = Tσ +h,∞ for any h ∈ C, so that the operator Tσ,∞ depends on the equivalence class σˆ := σ + C of H /C rather than on σ itself. It turns out that within each of the orbits {Tσ,0 | σ ∈ H } and {Tσ,∞ | σ ∈ H } all the operators are similar to each other. We recall first some definitions. DEFINITION 2.1. We say that closed and densely defined operators A and B in a Banach space X are similar and write A ∼ B if there exists a bounded and boundedly invertible operator U = UA,B (called the transformation operator (TO) for the pair (A, B)) such that AU = U B. Remark 2.2. Our definition of TO slightly differs from the one given in [20]; despite some shortcomings, it is more convenient for our purposes. The set (A, B) of all TOs for a pair (A, B) has the form (A, B) = {U V | V ∈ MB }, where U is a fixed TO and MB denotes the group of all bounded and boundedly invertible operators that commute with B. A TO U for a pair (A, B) is said to be unique up to a scalar factor if (A, B) = {λU | λ ∈ C \ {0}}; according to the above remark this is equivalent to the equality MB = {λI | λ ∈ C \ {0}}. THEOREM 2.3. Suppose that σ ∈ H ; then the following statements are true: (i) Tσ,0 ∼ T0,0 and Tσ,∞ ∼ T0,∞ ; (ii) a transformation operator Uσ,h for the pair (Tσ,h , T0,h ), h = 0, ∞, is unique up to a scalar factor and can be chosen as Uσ,h = I + Kσ,h , where Kσ,h is an integral Volterra operator of Hilbert–Schmidt class of the form x kσ,h (x, t)f (t) dt; (2.1) Kσ,h f (x) = 0
(iii) the operators T0,0 and T0,∞ are not similar.
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Remark 2.4. For the case of Sturm–Liouville operators on semiaxis with σ ∈ C (R+ ) claim (i) of the theorem is proved in [20, Theorem 1.2.1]. 1
The operators Kσ,h and Lσ,h := (I + Kσ,h )−1 − I , h = 0, ∞, have some other nice properties, to formulate which we have to introduce some functional spaces. Suppose that X is a Banach space with norm | · |; we shall also use the notation | · | for the operator norm in the algebra B(X) of all bounded operators in X. We denote by Gp (X), p 1, the set of all strongly measurable (classes of equivalent) functions k on [0, 1]2 with values in B(X) having the property that |k| ∈ Lp ((0, 1)2 ) and that the mappings x −→ k( · , x) ∈ Lp (0, 1), B(X) x −→ k(x, · ) ∈ Lp (0, 1), B(X) , are continuous on the interval [0, 1] (i.e., coincide a.e. with some continuous mappings of [0, 1] into Lp ((0, 1), B(X))). Also Gp (X) will denote the set of all integral operators with kernels from Gp (X). The set Gp (X) becomes a Banach space upon introducing the norm kGp (X) := max{ max k(x, · )Lp ((0,1),B(X)), max k( · , x)Lp ((0,1),B(X))}, x∈[0,1]
x∈[0,1]
while Gp (X) turns into a Banach algebra under the norm KGp (X) := kGp (X) with k being the kernel of K. It is easily seen that Gp (X) ⊂ G1 (X) for all p 1 and that the norm · G1 (X) coincides with the so-called Holmgren norm [8]; thus every operator K ∈ Gp (X), p 1, is continuous in all spaces Lq ((0, 1), X), q 1, see [6, Lemma XX.2.5]. Also every K ∈ G2 (X) acts continuously from L2 ((0, 1), X) into the space C([0, 1], X), and KL2 ((0,1),X)→C([0,1],X) KG2 (X) . Put + := {(x, t) ∈ (0, 1)2 | x > t}, − := {(x, t) ∈ (0, 1)2 | x < t} and − denote by G+ p (X) and Gp (X) the subspaces of Gp (X) consisting of all operators K whose kernels k satisfy the condition k(x, t) = 0 a.e. in − and k(x, t) = 0 − a.e. in + , respectively. G+ p (X) and Gp (X) form closed subalgebras of Gp (X) + − and, moreover, Gp (X) = Gp (X) Gp (X). Finally we observe that every element of G± p (X) is a Volterra operator in the space Lq ((0, 1), X) with any q 1. THEOREM 2.5. For any σ ∈ H and h = 0 or h = ∞ the operators Kσ,h and + Lσ,h = (I + Kσ,h )−1 − I belong to G+ 2 := G2 (C), and the mappings H σ −→ Kσ,h ∈ G+ 2,
H σ −→ Lσ,h ∈ G+ 2
are continuous from H into G+ 2. Consider now the sets of operators K0 := {Kσ,0 | σ ∈ H },
K∞ := {Kσ,∞ | σ ∈ H }.
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It turns out that these sets can be completely described in terms of the factorisation theory of Fredholm operators. We shall briefly recall its main notions, referring the reader to the books [10, 12] for further details. Let S2 denote the ideal of all Hilbert–Schmidt operators in H . Recall that every − operator in S2 is an integral operator; we denote by S+ 2 (S2 ) the set of those K ∈ − + S2 , whose kernels k vanish on (vanish on , respectively). It is obvious that − + − S2 = S+ 2 ⊕S2 and that the operators from S2 and S2 are Volterra ones. Also, the ± ± inclusions G2 (C) ⊂ S2 , G2 (C) ⊂ S2 hold and, moreover, KS2 KG2 (C) . DEFINITION 2.6. We say that an operator I + Q with Q ∈ S2 admits factori− ∈ S− sation (or is factorisable) if there exist operators K + ∈ S+ 2 and K 2 such that I + Q = (I + K + )−1 (I + K − )−1 . Note that an operator I + Q can admit at most one factorisation, so that the operators K ± = K ± (Q) are determined uniquely by Q. We denote by F2 the set of those Q ∈ S2 , for which I + Q is factorisable. The results of [23] imply the following statement. PROPOSITION 2.7. Put G2 := G2 (C); then (i) the set G2 ∩ F2 is open and everywhere dense in G2 ; (ii) for every Q ∈ G2 ∩ F2 the operators K ± (Q) belong to G± 2 and the operatorvalued mappings G2 ∩ F2 Q −→ K ± (Q) ∈ G± 2 are locally uniformly continuous. It turns out that the sets K0 and K∞ can be described as ranges of K + (·) when the argument Q runs through some special sets of operators in F2 . Namely, for φ ∈ L2 (0, 2) we denote by Fφ,0 and Fφ,∞ integral operators in G2 with kernels φ(x + t) + φ(|x − t|) and φ(x + t) − φ(|x − t|) respectively, i.e., 1 φ(x + t) + φ(|x − t|) f (t) dt, Fφ,0 f (x) := 0 1 φ(x + t) − φ(|x − t|) f (t) dt. Fφ,∞ f (x) := 0
We observe that, just like for Tσ,∞ , for any h ∈ C we have Fσ +h,∞ = Fσ,∞ , so that Fσ,∞ depends on the equivalence class φˆ := φ + C in L2 (0, 2)/C rather than on φ itself. Put for h = 0, ∞ Fh := {φ ∈ L2 (0, 2) | Fφ,h ∈ F2 }; then by Proposition 2.7 the sets F0 and F∞ are open in L2 (0, 2).
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THEOREM 2.8. The following equalities hold: K0 = {K + (Fφ,0 ) | φ ∈ F0 },
K∞ = {K + (Fφ,∞ ) | φ ∈ F∞ }.
In other words, for every σ ∈ H the operator Kσ,h can be obtained as a result of factorisation of I + Fφ,h for some φ ∈ L2 (0, 2) and, conversely, for every φ ∈ L2 (0, 2) such that I + Fφ,h is factorisable, the operator I + K + (Fφ,h ) is a TO for some SL operator Tσ,h . Thus Theorem 2.8 states that there exist bijections ∞ : H /C → F∞ /C, given by 0 : H → F0 and and H σ −→ φ0 =: 0 (σ ) ∈ F0 ˆ H /C σˆ −→ φ∞ =: ∞ (σˆ ) ∈ F∞ /C ∞ can be lifted to a mapping ∞ between H and F∞ and, respectively. In fact, moreover, the maps 0 and ∞ can be made more explicit. THEOREM 2.9. The mappings h , h = 0, ∞, are homeomorphic, and, moreover, with φσ,h := h (σ ) it holds x 1 lσ,h (x, t)2 dt, φσ,h (2x) = − σ (x) + 2 0 where lσ,h is the kernel of the integral operator Lσ,h = (I + Kσ,h )−1 − I .
3. Some Special Transformation Operators The aim of this section is to construct some special TOs for first order systems of differential equations on an interval. In the next section these TOs will be used to find TOs for SL operators with singular potentials from the space W2−1 (0, 1). The systems we shall consider here arise in a very natural way. Namely, according to our definition of σ the equality σ (u) = v is to be interpreted as −(u[1] ) − σ u[1] − σ 2 u = v with u[1] = u − σ u, or, in other words, as the first order system d u1 u1 0 −σ (x) −1 . + V (x) = , V (x) := u2 −v σ 2 (x) σ (x) dx u2 Observe that for σ ∈ H the entries of V (x) are integrable functions so that any solution to the above system is absolutely continuous and enjoys the standard uniqueness properties. It is reasonable to expect that TOs for the SL operators with singular potentials could be constructed through the analogous TOs for the d + V in the space H × H . More precisely, we shall seek for bounded operator dx ± operators A such that d ± V A± f = A∓ f dx
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for any function f ∈ W21 [0, 1] × W21 [0, 1] with f(0) = 0. Since V 2 = 0, the upperd d left components of ( dx ∓ V )( dx ± V ) coincide with −±σ , and hence the upper-left components of A± are strongly connected with the TOs for T±σ . In fact, without any technical complication we can consider a more abstract setting. Suppose that X is a Hilbert space with norm | · | and that v(·) is a function from L1 ((0, 1), B(X)). We denote by V an operator from L∞ ((0, 1), X) into L1 ((0, 1), X) given by ‘pointwise multiplication’, (Vf )(x) = v(x)f (x), and by · p the norm in the space Lp ((0, 1), B(X)), p 1. THEOREM 3.1. There exist operators A± ∈ B(L2 ((0, 1), X)) such that the equalities d + V A+ f = A− f , dx (3.1) d − + −V A f = A f dx hold for all f ∈ W21 ([0, 1], X) with f (0) = 0. The operators A± have the form x ± a ± (x, t)f (t) dt, f ∈ H , (3.2) A f (x) = f (x) + 0
and (A± − I ) ∈ G+ 1 (X). The proof of the theorem will rely on several lemmata. Consider first the case where v is smooth, i.e., v ∈ C ∞ ([0, 1], B(X)). Putting B + := 12 (A+ + A− ),
B − := 12 (A+ − A− ),
(3.3)
we rewrite system (3.1) as d (B + f ) + V B − f = B + f , dx d (B − f ) + V B + f = −B − f . dx
(3.4)
In terms of the kernels b± := (a + ± a − )/2 of the operators B ± (assuming that they are continuously differentiable in + ) these equations read (recall that f (0) = 0) x ∂ ∂ + − + b (x, t) + v(x)b (x, t) f (t) dt = 0, ∂x ∂t 0 (3.5) x ∂ ∂ − + − − b (x, t) + v(x)b (x, t) f (t) dt = − v(x) + 2b (x, x) f (x). ∂x ∂t 0
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It is easily seen that relations (3.5) hold for all f ∈ W21 ([0, 1], X) with f (0) = 0 if the kernels b± satisfy the system x + v(ξ )b− (ξ, ξ − x + t) dξ, b (x, t) = − x−t (3.6) x 1 x +t − + b (x, t) = − . v(η)b (η, x + t − η) dη − v x+t 2 2 2 For convenience we assume the kernels a ± and b± to be extended by zero outside the domain + . We find a solution of system (3.6) by successive approximation method in the form b± =
∞
bn± ,
(3.7)
n=1
where the kernels bn± , n ∈ N, satisfy the following recurrent relations 1 x +t − , b1 (x, t) = − v 2 2 x + v(ξ )bn− (ξ, ξ − x + t) dξ, bn (x, t) = − x−t x − (x, t) = − v(η)bn+ (η, x + t − η) dη bn+1
(3.8)
x+t 2
for (x, t) ∈ + and equal zero otherwise. Remark 3.2. Suppose that v ∈ C ∞ ([0, 1], B(X)) and denote C := maxx∈(0,1) |v(x)| + maxx∈(0,1) |v (x)|. The standard induction arguments applied to (3.8) show that the functions bn± are continuously differentiable in + and also yield the following inequalities for all n ∈ N and all (x, t) ∈ + : 1 C 2n−1 2n−2 x , 2 (2n − 2)! 1 C 2n x 2n−1 , |bn+ (x, t)| 2 (2n − 1)!
2n−1
∂ −
b (x, t) , ∂ b− (x, t) (2n − 1)C (1 + x)2n−2 , n n
∂t
∂x (2n − 2)!
2n
∂ +
b (x, t) , ∂ b+ (x, t) (2n)C (1 + x)2n−1 .
∂t n
(2n − 1)!
∂x n
|bn− (x, t)|
This implies that series (3.7) as well as the series obtained after term-by-term differentiation of (3.7) in x or t converge uniformly in the domain + . Henceforth the functions b± and a ± are continuously differentiable in + and the operators A± given by formulae (3.2) satisfy equalities (3.1).
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Our ultimate goal is to construct the operators A± satisfying equalities (3.1) for an arbitrary v ∈ L1 ((0, 1), B(X)), and to this end we shall study convergence of (3.7) in a more suitable norm. LEMMA 3.3. Suppose that v ∈ C ∞ ([0, 1], B(X)). Then the functions bn± , n ∈ N, verify the inequalities x 2n−1 1 bn− (x, ·)1 |v(ξ )| dξ , (2n − 1)! 0 (3.9) x 2n 1 |v(ξ )| dξ . bn+ (x, ·)1 (2n)! 0 Proof. Using recurrent formulae (3.8), we find that x 1 x |v(x/2 + t/2)| dt |v(ξ )| dξ, b1− (x, ·)1 = 2 0 0 x x + dt |v(ξ )| |bn− (ξ, ξ − x + t)| dξ bn (x, ·)1 0 x−t x x dξ |v(ξ )| |bn− (ξ, ξ − x + t)| dt 0 x−ξ x |v(ξ )| bn− (ξ, ·)1 dξ, 0 x x − dt |v(η)| |bn+ (η, x + t − η)| dη bn+1 (x, ·)1
x+t 2
0
x x 2
0
dη |v(η)| 0
x
2η−x
|bn+ (η, x + t − η)| dt
|v(η)| bn+ (η, ·)1 dη.
Inequalities (3.9) follow now by induction if we use the identity x ξ n n+1 1 1 x |v(ξ )| |v(τ )| dτ dξ = |v(τ )| dτ . n! 0 (n + 1)! 0 0
(3.10) 2
The lemma is proved.
LEMMA 3.4. Suppose that v ∈ C ∞ ([0, 1], B(X)); then the functions b+ , b− belong to G1 (X) and b− G1 (X) sinh(v1 ),
b+ G1 (X) cosh(v1 ) − 1.
(3.11)
Remark 3.5. Observe that since the functions b+ and b− are continuous, they belong to the spaces Gp (X) for all p 1; however, inequalities analogous to (3.11) with p > 1 in general do not hold.
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Proof of Lemma 3.4. Integrating relations (3.8) in x and using Fubini’s theorem, we get 1 − b1 (·, t)1 |v(ξ )| dξ, 0 1 x + dx |v(ξ )| |bn− (ξ, ξ − x + t)| dξ bn (·, t)1
t
x−t 1
ξ +t
dξ |v(ξ )|
0
|bn− (ξ, ξ − x + t)| dx
ξ 1
|v(ξ )| bn− (ξ, ·)1 dξ, 0 1 x − (·, t)1 dx |v(η)| |bn+ (η, x + t − η)| dη bn+1
1
x+t 2
t
t
0
|bn+ (η, x + t − η)| dx
η 1
2η−t
dη |v(η)|
|v(η)| bn+ (η, ·)1 dη.
Recalling (3.9) and using (3.10), we easily establish the inequalities bn+ G1 (X)
1 v2n 1 , (2n)!
bn− G1 (X)
1 v2n−1 , 1 (2n − 1)! 2
and relations (3.11) follow.
Now we show that the functions b± ∈ G1 (X) depend continuously on the function v in the L1 ((0, 1), B(X))-norm. ± and LEMMA 3.6. Assume that v, v ∈ C ∞ ([0, 1], B(X)) and let bn± = bn,v ± ± bn = bn, v be the corresponding kernels constructed as above. Then for every n ∈ N and all x ∈ [0, 1] the following inequalities are satisfied: x 2n−2 v − v 1 − − , |v(ξ )| + | v (ξ )| dξ bn (x, ·) − bn (x, ·)1 (2n − 2)! 0 x 2n−1 (3.12) v − v 1 + + bn (x, ·)1 . |v(ξ )| + | v (ξ )| dξ bn (x, ·) − (2n − 1)! 0
bn± results Proof. Applying the arguments of the proof of Lemma 3.3 to bn± and in b1− (x, ·)
− b1− (x, ·)1
x
|v(ξ ) − v (ξ )| dξ, 0
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bn+ (x, ·)1 bn+ (x, ·) −
0
x
|v(ξ ) − v (ξ )|bn− (ξ, ·)1 dξ + x
+ 0
− − (x, ·) − bn+1 (x, ·)1 bn+1
| v (ξ )|bn− (ξ, ·) − bn− (ξ, ·)1 dξ, x
0
|v(η) − v (η)|bn+ (η, ·)1 dη + x
+ 0
| v (η)|bn+ (η, ·) − bn+ (η, ·)1 dη.
Using Lemma 3.3 and the induction assumption for bn− (x, ·) − bn− (x, ·)1 , we get bn+ (x, ·)1 bn+ (x, ·) − x 2n−1 v − v 1 |v(ξ )| dξ + (2n − 1)! 0 ξ 2n−2 v − v 1 x | v (ξ )| dξ |v(u)| + | v (u)| du + (2n − 2)! 0 0 ξ 2n−2 v − v 1 x dξ |v(ξ )| + | v (ξ )| |v(u)| + | v(u)| du (2n − 2)! 0 0 x 2n−1 v − v1 . |v(u)| + | v (u)| du = (2n − 1)! 0 − − (x, ·) − bn+1 (x, ·)1 In the same manner we obtain the required estimate for bn+1 based on that for bn+ (x, ·) − bn+ (x, ·)1 , and the proof by induction is complete. 2
Modifying similarly the arguments of Lemma 3.4 and using Lemma 3.6, we arrive at the following conclusion. b± are the LEMMA 3.7. Suppose that v, v ∈ C ∞ ([0, 1], B(X)) and that b± , corresponding kernels. Then b+ G1 (X) v − v 1 sinh(v1 + v 1 ), b+ − − − v 1 cosh(v1 + v 1 ). b − b G1 (X) v − We now return to the kernels a ± and the corresponding operators A± . Recall that by definition a + = b+ + b− ,
a − = b+ − b− .
Denote by av± (respectively A± v ) the kernels (respectively operators) that correspond to v ∈ C ∞ ([0, 1], B(X)). Since the space G1 (X) is complete, Lemma 3.7 and extension by continuity yield the following statement.
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COROLLARY 3.8. The mappings v → av± extend uniquely to continuous functions L1 (0, 1), B(X) v −→ av± ∈ G1 (X), and for arbitrary v, v ∈ L1 ((0, 1), B(X)) it holds v1 exp(v1 + v 1 ). av± − a v± G1 (X) v − It turns out that the functions av± even for an arbitrary v ∈ L1 ((0, 1), B(X)) still are the kernels of the corresponding transformation operators as stated in Theorem 3.1. Proof of Theorem 3.1. Suppose that v is an arbitrary function from L1 ((0, 1), ∞ B(X)). We fix a sequence (vn )∞ n=1 ⊂ C ([0, 1], B(X)) that converges to v in ± the L1 -norm and denote by Vn and An the corresponding operators of ‘pointwise multiplication’ by vn and transformation operators, respectively. It follows from Corollary 3.8 that there exist operators A± = A± v that are continuous in the space ± Lp ((0, 1), X) for every p ∈ [1, ∞] and such that A± n f → A f in Lp ((0, 1), X) for any f ∈ Lp ((0, 1), X). Take an arbitrary f ∈ W21 ([0, 1], X) with f (0) = 0; then (recall Remark 3.2) we have d ∓ ± Vn A± (3.13) n f = An f . dx Since f ∈ L∞ ((0, 1), X) ∩ L1 ((0, 1), X) and f ∈ L2 ((0, 1), X) ⊂ L1 ((0, 1), X), ± ± we have that A± n f → A f in L∞ ((0, 1), X) and L1 ((0, 1), X) and that An f → ± A f in L1 ((0, 1), X) as n → ∞. Convergence of vn to v in the L1 -norm now ± implies that Vn A± n f → V A f in L1 ((0, 1), X), so that by (3.13) we get ∓ ± (A± n f ) −→ A f ∓ V A f
as n → ∞ in the norm of the space L1 ((0, 1), X). It follows that A± f ∈ W11 ([0, 1], X) and (A± f ) = A∓ f ∓ V A± f. Thus operators A± verify equalities (3.1) and the theorem is proved.
2
The next theorem shows how equalities (3.1) should be modified for an arbitrary function f ∈ W21 ([0, 1], X). THEOREM 3.9. Suppose that v ∈ L1 ((0, 1), B(X)) and f ∈ W21 ([0, 1], X). Then A± f ∈ W11 ([0, 1], X) and d ± V A± f = A∓ f + a ∓ (x, 0)f (0). dx
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TRANSFORMATION OPERATORS
Proof. Consider the sequence fn = φn f , where φn (x) = min{nx , 1} for x ∈ [0, 1] and n ∈ N. Then fn ∈ W21 ([0, 1], X) and fn (0) = 0, so that by virtue of equalities (3.1) we get (A± f ) ± V A± fn = A∓ fn , or
(A± − I )fn
= (A∓ − I )(φn f ) + + (A∓ − I )(φn f ) ∓ V A± fn ,
n ∈ N.
(3.14)
Observe that fn → f in Lp ((0, 1), X) as n → ∞ for all p ∈ [1, ∞] and thus A± fn → A± f in L1 ((0, 1), X) ∩ L∞ ((0, 1), X) and V A± fn → V A± f in L1 ((0, 1), X). Convergence φn f → f in L1 ((0, 1), X) as n → ∞ implies that A∓ (φn f ) → A∓ f in L1 ((0, 1), X). Since f is continuous and a ∓ ∈ G1 (X), we have that 1/n ∓ (A − I )(φn f )(x) = n a ∓ (x, t)f (t) dt −→ a ∓ (x, 0)f (0) 0
in L1 ((0, 1), X). The above reasonings show that the right-hand side of equality (3.14) converges in L1 ((0, 1), X) to (A∓ − I )f + a ∓ (x, 0)f (0) ∓ V A± f as n → ∞. Since the space W11 ([0, 1], X) is complete, the statements of the theorem follow. 2 Now we establish some additional property of the kernels a ± constructed that will be essentially used for Sturm–Liouville operators. a ± = a v± LEMMA 3.10. Suppose that v, v ∈ C ∞ ([0, 1], B(X)) and a ± = av± , denote the corresponding kernels constructed for v and v respectively. Let P be an arbitrary bounded operator in X; then with g(x) = (1 + x) exp(2x) the following inequalities hold: a ± 2G2 (X) P a ± − P 12(P v − P v 22 + v − v 21 P v 22 )g(v1 + v 1 ).
(3.15)
b± , where b± := Proof. It suffices to prove analogous inequalities for P b± − P − a )/2. It follows from (3.6) that ( a ± +
b+ (x, t)| |P b+ (x, t) − P x |P v(ξ ) − P v (ξ )| |b− (ξ, ξ − x + t)| dξ + x−t x |P v (ξ )| |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ. + x−t
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The Cauchy–Schwarz inequality yields the estimate |P b+ (x, t) − P b+ (x, t)|2 x 2C1 |P v(ξ ) − P v (ξ )|2 |b− (ξ, ξ − x + t)| dξ + x−t x |P v (ξ )|2 |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ, + 2C2 x−t
where
C1 := max
1
y∈[0,1] y
C2 := max
y∈[0,1] y
1
|b− (ξ, ξ − y)| dξ, |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ.
Integration in x and t now produces the inequality b+ 2G2 (X) 2C1 P (v − v )22 b− G1 (X) + P b+ − P v 22 b− − b− G1 (X) , + 2C2 P and it remains to estimate C1 and C2 in a suitable way. Using again (3.6) and Fubini’s theorem, we find that 1 ξ 1 − |b (ξ, ξ − y)| dξ dξ |v(η)| |b+ (η, 2ξ − y − η)| dη + y
y
ξ− y
2
y
1 1
dξ v ξ− + 2 y
2
1 v1 b+ G1 (X) 2
+ 12 v1 ,
so that 2C1 v1 b+ G1 (X) + v1 . Analogously we estimate 2C2 as follows: v 1 b+ G1 (X) + v 1 b+ − b+ G1 (X) + v − v 1 . 2C2 v − Combining these inequalities with the estimates of Lemmata 3.4 and 3.7 we arrive b+ 2G2 (X) with the constant 6 instead of 12. The estimate for at (3.15) for P b+ − P b− 2G2 (X) is derived analogously, and the result follows. 2 P b− − P Passing to the limit and completeness of the corresponding spaces justify now the following statement. COROLLARY 3.11. Suppose that v, v ∈ L1 ((0, 1), B(X)) and P ∈ B(X) are such that P v, P v belong to L2 ((0, 1), B(X)). Then for the corresponding kernels a ± = a v± the inclusions P a ± , P a ± ∈ G2 (X) take place, and, moreover, a ± = av± , inequality (3.15) holds.
TRANSFORMATION OPERATORS
133
4. Transformation Operators for Sturm–Liouville Operators In this section we use the results of the previous section to construct the TOs for the pairs of SL operators (Tσ,h , T0,h ), where σ ∈ H and h = 0, ∞. Denote by M2 := B(C2 ) the Banach space of all 2 × 2 matrices with complex entries and put H := L2 ((0, 1), C2 ). For an arbitrary but fixed σ ∈ H the function −σ (x) −1 v(x) := σ 2 (x) σ (x) belongs to L1 ((0, 1), M2 ). Henceforth by Theorems 3.1 and 3.9 there exist operators A± ∈ B(H) of the form x A± f(x) = f(x) + a ± (x, t)f(t) dt, 0 ±
such that a ∈ G1 (C ) and that for any f ∈ W21 ([0, 1], C2 ) the following relation holds: d ± V A± f = A∓ f + a ∓ (·, 0)f(0). (4.1) dx 2
With respect to the natural decomposition H = L2 (0, 1) × L2 (0, 1) the operators A± and the kernels a ± can be represented in the matrix form ± ± ± A11 A± a11 a12 ± ± 12 , a = . (4.2) A = ± ± ± A± a21 a22 21 A22 It follows that aij± ∈ G1 (C); also with 1 0 P = 0 0 ± ∈ G2 (C) depends continwe have P v(·) ∈ L2 ((0, 1), M2 ), which implies that a11 uously on σ ∈ H in view of Corollary 3.11. Taking f = (f, 0)T with f ∈ W21 [0, 1] in (4.1), we get the following equalities: d − + − − σ (A+ 11 f ) = A11 f + A21 f + a11 (·, 0)f (0), dx d + − + + σ (A− (4.3) 11 f ) = A11 f − A21 f + a11 (·, 0)f (0), dx d − − 2 + + σ (A+ 21 f ) = A21 f − σ A11 f + a21 (·, 0)f (0). dx
In particular, for f ∈ W22 [0, 1] with f (0) = f (0) = 0 this gives d d d + + +σ − σ A11 f = + σ (A− 11 f + A21 f ) dx dx dx 2 + = A+ 11 f − σ A11 f,
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
+ i.e., σ (A+ 11 f ) = A11 (0 (f )). To have this equality satisfied for all f ∈ dom Tσ,h with h = 0 or h = ∞, we should modify the operator A+ 11 in a suitable way. For the sake of brevity we put − (·, 0), α1 = a11
+ α2 = a11 (·, 0),
− α3 = a21 (·, 0).
Take m = mh ∈ H , denote by M = Mh an integral operator in H given by x f (x − t)m(t) dt, Mf (x) = 0
and put ± A˜ ± ij = Aij (I + M).
Since (Mf ) = Mf + f (0)m,
(Mf )(0) = 0
for all f ∈ W21 [0, 1], equalities (4.3) result in d − ˜− ˜+ − σ (A˜ + 11 f ) = A11 f + A21 f + (A11 m + α1 )f (0), dx d + ˜+ ˜− + σ (A˜ − 11 f ) = A11 f − A21 f + (A11 m + α2 )f (0), dx d − 2 ˜+ ˜− + σ (A˜ + 21 f ) = A21 f − σ A11 f + (A21 m + α3 )f (0). dx
(4.4)
We are now in a position to prove similarity of the operators Tσ,h and T0,h , h = 0, ∞. We start with the simpler case h = ∞. LEMMA 4.1. The operators Tσ,∞ and T0,∞ are similar. −1 Proof. Take m∞ := −(A+ 11 ) α2 in equalities (4.4); then for all f ∈ dom Tσ,0 we get (recall that f (0) = 0): d d d ˜+ +σ − σ A˜ + + σ (A˜ − f = 11 11 f + A21 f ) dx dx dx 2 ˜+ = A˜ + 11 f − σ A11 f. ˜+ ˜+ It follows that A˜ + 11 f ∈ dom Tσ,∞ and Tσ,∞ A11 f = A11 T0,∞ f , i.e., that ˜+ A˜ + 11 T0,∞ ⊂ Tσ,∞ A11 . ˜+ ˜ + −1 Since A˜ + 11 is a bounded and boundedly invertible operator, for S := A11 T0,∞ (A11 ) we get S ⊂ Tσ,∞ .
(4.5)
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TRANSFORMATION OPERATORS
Fix an arbitrary nonzero λ ∈ C; it is clear that ran(T0,∞ − λI ) = H,
dim ker(T0,∞ − λI ) = 1,
so that also ran(S − λI ) = H,
dim ker(S − λI ) = 1.
(4.6)
Assuming that S = Tσ,∞ , we deduce from (4.5) and (4.6) that dim ker(Tσ,∞ − λI ) > 1. It follows then that there exists a nonzero function f ∈ W21 [0, 1] such that f [1] ∈ W11 [0, 1], f (0) = f [1] (0) = 0, and d f f 0 +V = −λ . f [1] f dx f [1] This is impossible by uniqueness arguments, so that S = Tσ,0 and the lemma is proved. 2 LEMMA 4.2. The operators Tσ,0 and T0,0 are similar. Proof. Since any f ∈ dom T0,0 satisfies f (0) = 0, we find that d ˜− ˜+ − σ A˜ + 11 f = A11 f + A21 f + pf (0) dx with p := A− 11 m + α1 and − d d d +σ − σ A˜ + + σ A˜ 11 f + A˜ + f = 11 21 f + pf (0) dx dx dx + = A˜ 11 f − σ 2 A˜ + 11 f + qf (0) with
q :=
d + σ p + A− 21 m + α3 . dx
We shall prove below that for a suitable choice of m ∈ W21 [0, 1] we get p ∈ ˜+ W21 [0, 1], p(0) = 0, and q ≡ 0. Then A˜ + 11 T0,0 ⊂ Tσ,0 A11 and the arguments similar to those used in the proof of Lemma 4.1 do the rest. To produce the required m, we denote by J the integration operator, x f (t) dt, Jf (x) := 0 −1 and take m in the form m = (A− 11 ) (J r − α1 ) for some r ∈ H . Then p = J r ∈ W21 [0, 1], p(0) = 0, and the equation q = 0 can be recast as − −1 − − −1 r + σ J r + A− 21 (A11 ) J r − A21 (A11 ) α1 + α3 = 0
(4.7)
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
in terms of r. It is clear that − −1 R := σ J + A− 21 (A11 ) J
is a Hilbert–Schmidt operator with lower triangular kernel; therefore R is a Volterra operator in H and (4.7) has a unique solution − −1 r = (I + R)−1 (A− 21 (A11 ) α1 + α3 ) ∈ H.
2
The proof is complete.
The previous lemmata also show that the TO for the pair Tσ,h and T0,h can be + + chosen as A˜ + 11 = A11 + A11 M, which is of the form (2.1) with x + + a11 (x, s)mh (s − t) ds. kσ,h (x, t) = a11 (x, t) + t
Remark 4.3. Observe that although the kernel kσ,h need not be smooth enough for the function Kσ,h f to belong to W21 [0, 1] even if f ∈ C ∞ [0, 1], the function g := (I + Kσ,h )f has quasi-derivative g [1] ∈ W11 [0, 1] for any f ∈ dom T0,h and x x − + [1] a˜ 11 (x, t)f (t) dt + a˜ 21 (x, t)f (t) dt + ph (x)f (0) g (x) = f (x) + 0
with p∞ ≡ 0 and p0 ∈
0
W21 [0, 1]
constructed in the proof of Lemma 4.2.
We show next that the TO I + Kσ,h is unique up to a scalar factor. To this end it suffices to prove that the set of bounded and boundedly invertible operators commuting with T0,h in H consists of nonzero multiples of the identity operator. Denote sλ (x) :=
sin λx , λ
cλ (x) := cos λx,
with λ an arbitrary complex number and s0 (x) ≡ x. Observe that the mappings λ → sλ and λ → cλ are analytic H -valued functions of the argument λ. LEMMA 4.4. Suppose that a bounded and boundedly invertible operator U in H satisfies one of the following conditions: (a) for all λ ∈ C the function sλ is an eigenvector of U ; (b) for all λ ∈ C the function cλ is an eigenvector of U . Then U = cI for some nonzero c ∈ C. Proof. Assume first that condition (a) is satisfied. Then there exists a function g such that U sλ = g(λ)sλ
(4.8)
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TRANSFORMATION OPERATORS
for all λ ∈ C. For a given λ0 ∈ C the relation g(λ) =
(U sλ , sλ0 ) (sλ , sλ0 )
shows that g analytic in a neighbourhood of λ0 consisting of those λ, for which (sλ , sλ0 ) = 0; since λ0 is arbitrary, g is entire. By (4.8) the function g satisfies the inequality U −1 −1 |g(λ)| U for all λ ∈ C; the Liouville theorem now proves that g ≡ c for some nonzero c. Since the system {sλ | λ ∈ C} is complete in H (e.g., the sequence (sπn )n∈N forms a basis of H ), this implies that U = cI as claimed. The case (b) is considered analogously. 2 LEMMA 4.5. The operators T0,0 and T0,∞ are not similar. Proof. Assume that the claim of the lemma is false. Then there exists a bounded and boundedly invertible operator U such that T0,∞ U = U T0,0 . Since for an arbitrary λ ∈ C we have ker(T0,0 − λ2 I ) = lin{sλ },
ker(T0,∞ − λ2 I ) = lin{cλ },
there exists a function f : C → C such that U sλ = f (λ)cλ
(4.9)
for all λ ∈ C. The relation f (λ) =
(U sλ , cλ0 ) , (cλ , cλ0 )
shows that f is analytic in a neighbourhood of any point λ0 ∈ C and hence is entire. Equality (4.9) implies that |f (λ)|
sλ U , cλ
λ ∈ C.
Writing λ = ξ + iη with ξ, η ∈ R, we find that 1 1 sinh 2η sin 2ξ 2 − , | sin λx| dx = 2 2η 2ξ 0 1 1 sinh 2η sin 2ξ + . | cos λx|2 dx = 2 2η 2ξ 0
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
Therefore lim
λ→∞
sλ =0 cλ
and hence f (λ) = o(1),
λ → ∞.
The Liouville theorem now implies f ≡ 0, which contradicts invertibility of U . 2 The contradiction derived shows that T0,0 and T0,∞ are not similar. With all these results in hand, we can prove Theorems 2.3 and 2.5. Proof of Theorem 2.3. Lemmata 4.1 and 4.2 establish similarity of claim (i) and existence of the TO of required form of claim (ii). Uniqueness of claim (ii) follows from Lemma 4.4, and part (iii) is proved in Lemma 4.5. 2 Proof of Theorem 2.5. We consider first the operators Kσ,h , h = 0, ∞. It follows from the proof of Lemmata 4.1 and 4.2 that + I + Kσ,h = A+ 11,σ + A11,σ Mσ,h ,
(4.10)
where the operator Mσ,h acts according to x mσ,h (x − t)f (t) dt, Mσ,h f (x) := 0
with mσ,0 and mσ,∞ defined in the proofs of Lemmata 4.2 and 4.1 respectively. It is easily seen that Mσ,h − Mσ˜ ,h G+2 mσ,h − mσ˜ ,h L2 (0,1), so that the mapping L2 (0, 1) σ −→ Mσ,h ∈ G+ 2 is continuous as soon as such is the mapping L2 (0, 1) σ −→ mσ,h ∈ L2 (0, 1). Assuming this already established and recalling that G+ 2 is a Banach algebra and + depends continuously on σ ∈ H in G , we conclude from the above that A+ 11,σ 2 arguments and representation (4.10) that the mapping L2 (0, 1) σ −→ Kσ,h ∈ G+ 2 is continuous. Thus it remains to show that mσ,0 and mσ,∞ depend continuously on σ .
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For h = ∞ we have −1 + mσ,∞ := (A+ 11,σ ) a11,σ (·, 0),
and the required continuity follows from Corollary 3.8. For h = 0 the function mσ,0 is given by − −1 mσ,0 := (A− 11,σ ) (J rσ − a11,σ (·, 0)),
where rσ := (I + Rσ )−1 pσ ,
− − −1 − pσ := A− 21,σ (A11,σ ) a11,σ (·, 0) + a21,σ (·, 0)
and − −1 Rσ := σ J + A− 21,σ (A11,σ ) J
with J being the integration operator. It follows from Corollary 3.8 that the mappings → pσ ∈ L1 (0, 1), L2 (0, 1) σ − → Rσ ∈ B L1 (0, 1), L2 (0, 1) L2 (0, 1) σ − are continuous. This implies continuity of the mapping L2 (0, 1) σ −→ rσ ∈ L2 (0, 1) as well as the one L2 (0, 1) σ −→ mσ,0 ∈ L2 (0, 1), and the proof for the operators Kσ,h , h = 0, ∞, is complete. + To treat the operators Lσ,h , we observe first that the mapping L: G+ 2 → G2 given by L(K) = (I + K)−1 − I =
∞
(−K)n
(4.11)
n=1
is continuous. In fact, for the kernel k of an operator K ∈ G+ 2 we find that
1
k(x, s)k(s, t) ds
k(x, ·)L2 (0,1)k(·, t)L2 (0,1) k2G2
0
if x > t, so that by induction K 2n G2 K2n G2 /(n − 1)! and thus series (4.11) converges locally uniformly in K ∈ G2 . It follows now that Lσ,h = (I + Kσ,h )−1 − I = L(Kσ,h ) ∈ G+ 2 depends continuously on σ ∈ H , and the theorem is proved. 2
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5. Proof of Theorems 2.8 and 2.9 In this section, we shall study the question how the TOs Kσ,h enter factorisation of some special operators. Note that the relations we establish below are known for regular potentials (e.g., formula (5.2) can be found in [22]); we derive them here not only for the sake of completeness but also to give a precise characterisation of the TOs for the SL operators with singular potentials from W2−1 (0, 1). Recall that for an integral operator K in H with kernel k its associated operator K is defined as the integral operator with kernel k (x, t) = k(t, x). Claims of Theorems 2.8 and 2.9 are basically contained in the following two lemmata. LEMMA 5.1. Suppose that σ ∈ H . Then for h = 0, ∞ the following equality holds: −1 ) = I + Fφ,h , (I + Kσ,h )−1 (I + Kσ,h
in which the function φ = φσ,h ∈ L2 (0, 2) is given by x 1 lσ,h (x, t)2 dt, x ∈ [0, 1], φσ,h (2x) = − σ (x) + 2 0
(5.1)
(5.2)
and lσ,h is the kernel of the operator Lσ,h = (I + Kσ,h )−1 − I . Proof. First we shall prove equality (5.1) for the case h = 0 and under the assumption that σ ∈ C 2 [0, 1]. Put x kσ,0 (x, t) cos λt dt, λ ∈ C, x ∈ [0, 1]. (5.3) y(λ, x) := cos λx + 0
By the definition of the TO I + Kσ,0 the function y satisfies the equation −y (λ, x) + q(x)y(λ, x) = λ2 y(λ, x) and the initial conditions y(λ, 0) = 1,
y (λ, 0) = ay(λ, 0),
where q = σ and a := σ (0). Since I +Lσ,0 is the inverse of I +Kσ,0 , relation (5.3) can be recast as x lσ,0 (x, t)y(λ, t) dt. cos λx = y(λ, x) + 0
It is shown in [22, Ch. I.2] that the kernel lσ,0 of Lσ,0 is twice continuously differentiable in the closure of the domain + = {(x, t) ∈ (0, 1)2 | x > t} and is a unique solution of the partial differential equation = −ltt + q(t)l, −lxx
(x, t) ∈ + ,
subject to the boundary conditions 1 x q(t) dt, l(x, x) = −a − 2 0
lt (x, t)|t =0 − al(x, 0) = 0.
(5.4)
(5.5)
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Denote by I + F the operator of the left-hand side of (5.1). Then I + F := (I + Lσ,0 )(I + L σ,0 ), so that F is an integral operator with kernel 1 lσ,0 (x, s)lσ,0 (t, s) ds. f (x, t) = lσ,0 (x, t) + lσ,0 (t, x) + 0
It follows that the function f is twice continuously differentiable in [0, 1]2 and is symmetric with respect to x and t. We shall use (5.4) and (5.5) to show that f = −ftt so that f (x, t) = φ1 (x + t) + satisfies in + the wave equation −fxx φ2 (x − t) for some φ1 , φ2 ∈ C 2 [0, 1], and then derive from (5.5) that φ1 = φ2 = φσ,0 with φσ,0 of (5.2). The details are as follows. We write l instead of lσ,0 for brevity and observe that for (x, t) ∈ + it holds t l(x, s)l(t, s) ds. (5.6) f (x, t) = l(x, t) + 0
Differentiating (5.6) in x twice and using (5.4)–(5.5) at various points, we obtain t lxx (x, s)l(t, s) ds fxx (x, t) = lxx (x, t) + 0 t t (x, t) + lss (x, s)l(t, s) ds − l(x, s)q(s)l(t, s) ds = lxx 0 0 t lss (x, s)l(t, s) ds + = lxx (x, t) + 0 t l(x, s) ltt (t, s) − lss (t, s) ds + 0 t l(x, s)ltt (t, s) ds + = lxx (x, t) + 0
t + ls (x, s)l(t, s) − l(x, s)ls (t, s) 0 t = lxx (x, t) + l(x, s)ltt (t, s) ds + 0 + lt (x, t)l(t, t)
Differentiation in t gives ftt (x, t)
=
ltt (x, t)
t
+ 0
− l(x, t)ls (t, s)|s=t .
l(x, s)ltt (t, s) ds +
+ [l(x, t)l(t, t)]t + l(x, t)lt (t, s)|s=t , so that in view of (5.4) and (5.5) (x, t) − ftt (x, t) = −q(t)l(x, t) − 2[l(t, t)]t l(x, t) = 0 fxx
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
as stated. Therefore f (x, t) = φ1 (x + t) + φ2 (x − t) for some φ1 , φ2 ∈ C 2 [0, 1]. It follows that ft (x, t)|t =0 = φ1 (x) − φ2 (x), while (5.6) and (5.5) give ft (x, t)|t =0 = lt (x, t)|t =0 + l(x, 0)l(0, 0) = lt (x, t)|t =0 − al(x, 0) = 0. Thus φ1 − φ2 ≡ c for some constant c, and with φ0 := φ1 − c/2 = φ2 + c/2 we conclude that f (x, t) = φ0 (x + t) + φ0 (x − t), i.e., that F = Fφ0 ,0 . To find φ0 , we observe that (5.6) and (5.5) for x = t give x 1 x φ0 (2x) + φ0 (0) = −a − q(s) ds + l(x, s)2 ds, 2 0 0 so that φ0 (0) = −a/2. Recalling that σ is a primitive of q with σ (0) = a, we arrive at the equality x l(x, s)2 ds. φ0 (2x) = − 12 σ (x) + 0
Summarizing, we have shown that F = Fφ0 ,0 with φ0 = φσ,0 given by (5.2) (i.e., proved the theorem) under the additional assumption that σ ∈ C 2 [0, 1]. Suppose now that σ is an arbitrary function in H and take a sequence (σn ) ⊂ C 2 [0, 1] that converges to σ in H . Denoting by φn ∈ L2 (0, 2) the functions as in (5.2) but corresponding to σn , we conclude from Theorem 2.5 that φn converge in L2 (0, 2) to φσ,0 of (5.2). Therefore lim Fφn ,0 − Fφσ,0 ,0 G2 = 0,
n→∞
and, passing to the limit in the equality Lσn ,0 + L σn ,0 + Lσn ,0 Lσn ,0 = Fφn ,0
results in (5.1) for h = 0 and an arbitrary σ ∈ H . The case h = ∞ is treated analogously; the only reservations are that the function cos λx should be replaced with (sin λx)/λ and boundary conditions (5.5) with the ones 1 x q(t) dt, lσ,∞ (x, 0) = 0. lσ,∞ (x, x) = − 2 0 The lemma is proved.
2
LEMMA 5.2. Suppose that φ ∈ F0 . Then there exists a unique σ0 ∈ H such that −1 ) = I + Fφ,0 ; (I + Kσ,0 )−1 (I + Kσ,0
(5.7)
moreover, σ0 depends continuously on φ ∈ F0 . Similarly, for φ ∈ F∞ there exists a unique σˆ ∞ ∈ H /C such that, for σ ∈ σˆ , )−1 = I + Fφ,∞ ; (I + Kσ,∞ )−1 (I + Kσ,∞
moreover, σˆ ∞ depends continuously on φˆ := φ + C ∈ F∞ /C.
(5.8)
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Proof. We shall consider only the case h = 0 as the other one is completely analogous. Denote by F = Fφ,0 an integral operator with kernel f (x, t) = φ(x + t) + φ(|x − t|); then by assumption the operator I + F is factorisable as explained in Section 2. Put K := K+ (F ); since the integral operator F has a symmetric kernel, it is easily seen that K− (F ) = K , i.e., that I + Fφ,0 = (I + K)−1 (I + K )−1 . Multiplying both sides of the above relation by I + K and equating the corresponding kernels, we arrive at the so-called Gelfand–Levitan–Marchenko (GLM) equation x k(x, s)f (s, t) ds = 0, (x, t) ∈ + , (5.9) k(x, t) + f (x, t) + 0
in which k is the kernel of K. Recall that the operator I + F is factorisable if and only if the Gelfand–Levitan–Marchenko equation is soluble for k. Suppose now that the function φ is smooth (e.g., infinitely differentiable). Then it is shown in [22, Ch. II.3] and [20, Ch. II.4] that the solution k of the GLM equation (5.9) is smooth as well and satisfies the partial differential equation (x, t) + q(x)k(x, t), −ktt (x, t) = −kxx
q(x) := 2[k(x, x)]x
and the boundary condition kt (x, t)|t =0 = 0. Moreover, k is the kernel of the TO I + Kσ,0 with σ (x) := 2[k(x, x) − k(0, 0)]. Denote by l the kernel of the integral operator L := (I + K)−1 − I . Then l = lσ,0 , so that by equality (5.2) x l(x, s)2 ds, x ∈ [0, 1]. (5.10) σ (x) = −2φ(2x) + 2 0
In view of Proposition 2.7 the operator K ∈ G2 (C) (and thus L ∈ G2 (C)) depends continuously on φ ∈ F0 , whence the function σ ∈ H of (5.10) is continuous in φ ∈ F0 ∩ C ∞ [0, 2] in the L2 (0, 2)-norm. Take now an arbitrary φ ∈ F0 and put ck (x) := cos( π2 kx);since the sys∞ = tem (ck )k0 is an orthonormal k=0 αk ck with m basis of L2 (0, 2), we have φ ∞ αk := (φ, ck ). Put φm := k=0 αk ck ; then the sequence (φm )m=1 converges to φ in L2 (0, 2) and therefore by Proposition 2.7 the infinitely differentiable functions φm belong to F0 for all m large enough, say for m m0 . Let (σm)∞ m=m0 be the sequence of functions constructed for φm through formula (5.10); in particular, I + Fφm ,0 = (I + Kσm ,0 )−1 (I + Kσm ,0 )−1 .
(5.11)
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From the above said it follows that (σm )∞ m=m0 is a Cauchy sequence in H and hence converges to some function σ ∈ H . By virtue of Theorem 2.5 we can pass to the limit in equation (5.11) to obtain −1 I + Fφ,0 = (I + Kσ,0 )−1 (I + Kσ,0 ) .
The above arguments also imply that the function σ so constructed depends con2 tinuously on φ ∈ F0 . The proof (for the case h = 0) is complete. Proof of Theorem 2.8. The statements of the theorem are corollaries of Lemmata 5.1 and 5.2. 2 Proof of Theorem 2.9. Continuity of the mappings h : σ → φσ,h , h = 0, ∞, follows from Theorem 2.5 and formula (5.2), and that of the inverse mappings from Lemma 5.2. 2
6. Some Applications In this section we demonstrate usefulness of the TOs constructed for the spectral analysis of Sturm–Liouville operators with singular potentials q from the space W2−1 (0, 1). Namely, we shall establish eigenvalue asymptotics, completeness properties of eigenfunctions, and similarity of related operators. Note that some of these results were established by different methods in [28–30]. Assume that q = σ for some complex-valued function σ ∈ H and denote by Tσ,DD the restriction of the operator Tσ,∞ defined in Section 2 by the Dirichlet boundary condition at the point x = 1. Then Tσ,DD has a discrete spectrum [29] that accumulates at +∞; we denote by λ2k , k ∈ N, the eigenvalues of Tσ,DD counted with multiplicities and ordered so that Re λk+1 Re λk and |Im λk+1 | |Im λk | in the case of equality above. Here and in the following, λk are taken from the closed right half-plane. THEOREM 6.1. For the eigenvalues λ2k ordered as explained above we have λk = λk ) belongs to 2 . πk + λk , where the sequence ( Proof. Observe first that for any λ ∈ C the solution to the equation σ y = λ2 y satisfying the boundary condition y(0) = 0 equals x kσ,∞ (x, t)sλ (t) dt; y(x, λ) = sλ (x) + 0
here sλ (x) = (sin λx)/λ for λ = 0 and s0 (x) ≡ x, and kσ,∞ is the kernel of the TO I + Kσ,∞ . Therefore λ2 is an eigenvalue of the operator Tσ,DD if and only if y(1, λ) = 0, and in that case y(x, λ) is a corresponding eigenfunction. In
TRANSFORMATION OPERATORS
145
other words, the spectrum of Tσ,DD coincides with the squared zeros of the entire function 1 sin λt sin λ kσ,∞ (1, t) (λ) := + dt. λ λ 0 Observe also that λ is a zero of the function of multiplicity k 1 if and only ∂ ∂ k−1 y(x, λ), . . . , ( ∂λ ) y(x, λ) form a chain of eigen- and if the functions y(x, λ), ∂λ associated functions of the operator Tσ,DD corresponding to the eigenvalue λ2 , i.e., the multiplicity of a zero λ of coincides with the algebraic multiplicity of the eigenvalue λ2 of Tσ,DD . Thus it suffices to study the asymptotics of zeros of the function (λ) in the half-plane Re λ 0. Since is an odd function, its zeros are symmetric with respect to the origin and we can study the zeros in the whole plane C. Put 1 kσ,∞ (1, t) sin µt dt; Q(µ) := 0
then Q is an entire function of exponential type and lim e−|Im µ| Q(µ) = 0
|µ|→∞
(6.1)
by [22, Lemma 1.3.1]. For any n ∈ Z, denote by n the boundary of the rectangular Rn := {µ = ν + iτ | ν, τ ∈ R, |ν − π n| < π/2, |τ | < 1}. Then inf |sin µ| = c > 0
µ∈n
is independent of n, while sup |Q(µ)| −→ 0 µ∈n
as |n| → ∞ by virtue of (6.1). By Rouche’s theorem each Rn contains exactly one zero of the function for all n large enough. Representing this solution as µn , we see that µn = π n + µn ) −→ 0 sin µn = (−1)n+1 Q(π n + and whence µn → 0 as n → ∞. It follows now that 1 n+1 kσ,∞ (1, t) sin π nt dt + o(| µn |); µn = (−1) √
since { implies
0
2 sin π nt}∞ n=1 that ( µn ) ∈ 2 .
is an orthonormal basis of H and kσ,∞ (1, t) ∈ H , this
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Finally we observe that for all n large enough sin λ/λ and have the same number of zeros (counting multiplicities) inside the regions Pn := {µ ∈ C | |Re µ| π n + π/2, |Im µ| < n}; this easily follows by Rouche’s theorem (see details in [22, Ch. 1.3]). Therefore λn with λn = µn for all n large enough; in particular, we may write λn = π n + 2 2 (λn ) ∈ . The theorem is proved. It follows from Theorem 6.1 that all eigenvalues λ2n but maybe finitely many are simple. We denote by φn , n ∈ N, the system of eigen- and associated functions of the operator Tσ,DD corresponding to the eigenvalues λ2n . For all n large enough of Tσ,DD ) we normalize the eigenfunctions (e.g., such that λ2n is a simple eigenvalue √ [1] φ (0) = 2λ ; observe that for such n we have φn (t) = φn by the condition n n √ (I + Kσ,∞ ) 2 sin λn t. THEOREM 6.2. The system (φn )∞ n=1 forms a Bari basis of H (i.e., a basis that is quadratically close to√an orthonormal one). 2 Proof. Put ψn := √2 sin λn x if n ∈ N is of √such that λn is a simple eigenvalue √ k Tσ,DD and put ψn := 2 sin λn x, ψn+1 = 2x sin λn x, . . . , ψn+k = 2x sin λn x if λ2n = λ2n+1 = · · · = λ2n+k is an eigenvalue of Tσ,DD of algebraic multiplicity k + 1. Then due to the asymptotics of λn established in the previous theorem the system (ψn )∞ n=1 is complete in H , see [19, App. III]. For all n large enough we have φn = (I + Kσ,∞ )ψn ; we can also choose the remaining functions φn accordingly so that the above equality will hold for all n ∈ N. Since I +Kσ,∞ is a homeomorphism, we conclude that the√system (φn ) is complete in H . Put now ψn,0 := 2 sin π nx; then (ψn,0 ) is an orthonormal basis of H and φk − ψk,0 = (I + Kσ,∞ )(ψk − ψk,0 ) + Kσ,∞ ψk,0 . Observe that √ λk cos(π k + λk /2) = O(| λk |) ψk (x) − ψk,0 (x) = 2 2 sin 2 for all k large enough, so that ψk − ψk,0 2 < ∞ on account of the inclusion ( λk ) ∈ 2 . Since Kσ,∞ is a Hilbert–Schmidt operator and (ψk,0 ) is an orthonormal basis of H , we also have Kσ,∞ ψk,0 2 < ∞. Therefore φk − ψk,0 2 < ∞,
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and thus (φn )∞ n=1 is a Bari basis of H (see [11, Ch. VI]).
2
Our final result shows that existence of TOs implies that the operators Tσ,DD with σ ∈ H are similar to rank one perturbations of the potential-free operator T0,DD . THEOREM 6.3. Suppose that σ ∈ H . Then there exists a function p = pσ ∈ H such that Tσ,DD is similar to the operator S given by (Sf )(x) = −f on the domain
2 dom S = f ∈ W2 [0, 1] | f (0) = f (1) +
1
p(t)f (t) dt = 0 .
0
The similarity is performed by the operator I + Kσ,∞ . Proof. Since Tσ,∞ (I +Kσ,∞ ) = (I +Kσ,∞ )T0,∞ and Tσ,DD is a one-dimensional restriction of Tσ,∞ , the operator Tσ,DD is similar to a one-dimensional restriction S of the operator T0,∞ . The domain of S is found from the requirement that (I + Kσ,∞ ) dom S = dom Tσ,DD . Thus every f ∈ dom S belongs to W22 [0, 1] and satisfies the conditions 1 kσ,∞ (1, t)f (t) dt = 0, f (0) = f (1) + 0
and it remains to observe that p(t) := kσ,∞ (1, t) belongs to H .
2
Similar results also hold for the restriction Tσ,DN of the operator Tσ,∞ by the Neumann boundary condition f [1] (1) = hf (1), h ∈ C, at the point x = 1 and for the restrictions Tσ,ND and Tσ,NN of the operator Tσ,0 by the Dirichlet and Neumann boundary conditions respectively at the point x = 1. References 1. 2. 3. 4. 5.
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Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer, New York, 1988. Albeverio, S. and Kurasov, P.: Singular Perturbations of Differential Operators. Solvable Schrödinger Type Operators, Cambridge University Press, Cambridge, 2000. Andersson, L.: Inverse eigenvalue problems for a Sturm–Liouville equation in impedance form, Inverse Probl. 4 (1988), 929–971. Berezanskii, Yu. M. and Brasche, J.: Generalized selfadjoint operators and their singular perturbations, Methods Funct. Anal. Topol. 8(4) (2002), 1–14. Coleman, C. F. and McLaughlin, J. R.: Solution of the inverse spectral problem for an impedance with integrable derivative, I, Comm. Pure Appl. Math. 46 (1993), 145–184; II, Comm. Pure Appl. Math. 46 (1993), 185–212. Dunford, N. and Schwartz, J. T.: Linear Operators. Part III: Spectral Operators, Wiley–Interscience, New York, 1988.
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Freiling, G. and Yurko, V.: On the determination of differential equations with singularities and turning points, Results Math. 41 (2002), 275–290. Friedrichs, K. O.: Perturbation of Spectra in Hilbert Space, Lectures in Appl. Math. 3, Amer. Math. Soc., Providence, RI, 1965. Gelfand, I. M. and Levitan, B. M.: On determination of a differential equation by its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15(4) (1951), 309–360 (in Russian). Gohberg, I., Goldberg, S. and Kaashoek, M.: Classes of Linear Operators, Birkhäuser, Basel, 1987. Gohberg, I. and Krein, M.: Introduction to the Theory of Linear Non-selfadjoint Operators in Hilbert Space, Nauka Publ., Moscow, 1965 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, RI, 1969. Gohberg, I. and Krein, M.: Theory of Volterra Operators in Hilbert Space and its Applications, Nauka Publ., Moscow, 1967 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Monogr. 24, Amer. Math. Soc., Providence, RI, 1970. Hald, O.: Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math. 37 (1984), 539–577. Herczy´nski, J.: On Schrödinger operators with distributional potentials, J. Oper. Theory 21(2) (1989), 273–295. Hryniv, R. O. and Mykytyuk, Ya. V.: 1D Schrödinger operators with singular periodic potentials, Meth. Funct. Anal. Topol. 7(4) (2001), 31–42. Hryniv, R. O. and Mykytyuk, Ya. V.: 1D Schrödinger operators with singular Gordon potentials, Meth. Funct. Anal. Topol. 8(1) (2002), 36–48. Hryniv, R. O. and Mykytyuk, Ya. V.: Inverse spectral problems for Sturm–Liouville operators with singular potentials, Inverse Probl. 19 (2003), 665–684. Hryniv, R. O. and Mykytyuk, Ya. V.: Inverse spectral problem for Sturm–Liouville operators with singular potentials, II. Reconstruction by two spectra, In: V. Kadets and W. Zelazko (eds), Proceedings of the Conference on Functional Analysis and Its Applications Dedicated to the 110th Anniversary of Stefan Banach, Lviv, May 28–31, 2002, North-Holland Math. Studies, Elsevier, 2004 (to appear). Levin, B. Ya.: Distribution of Zeros of Entire Functions, Gostekhizdat, Moscow, 1956 (in Russian); Engl. transl.: Amer. Math. Soc., Providence, RI, 1964. Levitan, B. M.: Inverse Sturm–Liouville Problems, Nauka Publ., Moscow, 1984 (in Russian); Engl. transl.: VNU Science Press, Utrecht, 1987. Marchenko, V. A.: Some questions of the theory of second order differential operators, Dokl. Akad. Nauk SSSR 72(3) (1950), 457–460 (in Russian). Marchenko, V. A.: Sturm–Liouville Operators and Their Applications, Naukova Dumka Publ., Kyiv, 1977 (in Russian); Engl. transl.: Birkhäuser, Basel, 1986. Mykytyuk, Ya. V.: Factorisation of Fredholm operators, 2001, preprint. Neiman-zade, M. I. and Shkalikov, A. A.: Schrödinger operators with singular potentials from the space of multipliers, Mat. Zametki (Math. Notes) 66(5) (1999), 723–733. Povzner, A. Ya.: On differential Sturm–Liouville operators on semiaxis, Math. USSR-Sb. 23(65) (1948), 3–52. Pöschel, J. and Trubowitz, E.: Inverse Spectral Theory, Pure Appl. Math. 130, Academic Press, Orlando, Florida, 1987. Rofe-Beketov, F. S. and Khristov, E. H.: Some analytical questions and the inverse Sturm– Liouville problem for an equation with highly singular potential, Dokl. Akad. Nauk SSSR 185(4) (1969), 768–771 (in Russian); Engl. transl.: Soviet Math. Dokl. 10(1) (1969), 188–192. Savchuk, A. M.: On eigenvalues and eigenfunctions of Sturm–Liouville operators with singular potentials, Mat. Zametki (Math. Notes) 69(2) (2001), 277–285 (in Russian). Savchuk, A. M. and Shkalikov, A. A.: Sturm–Liouville operators with singular potentials, Mat. Zametki (Math. Notes) 66(6) (1999), 897–912 (in Russian).
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Mathematical Physics, Analysis and Geometry 7: 151–185, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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A General Framework for Localization of Classical Waves: II. Random Media ABEL KLEIN1 and ANDREW KOINES2
1 University of California, Irvine, Department of Mathematics, Irvine, CA 92697-3875, U.S.A. e-mail:
[email protected] 2 Orange Coast College, Department of Mathematics, Costa Mesa, CA 92626, U.S.A.
(Received: 26 November 2002) Abstract. We study localization of classical waves in random media in the general framework introduced in Part I of this work. This framework allows for two random coefficients, encompasses acoustic waves with random position dependent compressibility and mass density, elastic waves with random position dependent Lamé moduli and mass density, electromagnetic waves with random position dependent magnetic permeability and dielectric constant, and allows for anisotropy. We show exponential localization (Anderson localization) and strong Hilbert–Schmidt dynamical localization for random perturbations of periodic media with a spectral gap. Mathematics Subject Classifications (2000): 35Q60, 35Q99, 78A99, 78A48, 74J99, 35P99, 47F05. Key words: wave localization, random media, Anderson localization, dynamical localization, Wegner estimate.
1. Introduction In this series of articles we provide a general framework for studying localization of acoustic waves, elastic waves, and electromagnetic waves in inhomogeneous and random media, i.e., the existence of acoustic, elastic, and electromagnetic waves such that almost all of the wave’s energy remains in a fixed bounded region uniformly over time. Our general framework encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lamé moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. In the first article [KK] we developed mathematical methods to study wave localization in inhomogeneous media; as an application we proved localization for local perturbations (defects) of media with a gap in the spectrum and studied midgap eigenmodes. In this second article these methods are applied to prove existence of exponential localization (Anderson localization) and strong Hilbert– This work was partially supported by NSF Grants DMS-9800883 and DMS-0200710.
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Schmidt dynamical localization for classical waves in random media. This phenomenum has been experimentally observed for light waves [WBLR]. Previous results on localization of classical waves in random media [FK3, FK4, FK7, CHT] considered only the case of one random coefficient. Acoustic and electromagnetic waves were treated separately. Elastic waves were not discussed. Our results extend the work of Figotin and Klein [FK3, FK4, FK7] in several ways: (1) We study a general class of classical waves which includes acoustic, electromagnetic and elastic waves as special cases. (2) We allow for two random coefficients (e.g., electromagnetic waves in media where both the magnetic permeability and the dielectric constant are random). (3) We allow for anisotropy in our wave equations. (4) We prove strong Hilbert–Schmidt dynamical localization in random media, using the bootstrap multiscale analysis of Germinet and Klein [GK1] and the generalized eigenfunction expansion of of Klein, Koines and Seifert [KKS] for classical wave operators. Our approach to the mathematical study of localization of classical waves is operator theoretic and reminiscent of quantum mechanics. It is based on the fact that many wave propagation phenomena in classical physics are governed by equations that can be recast in abstract Schrödinger form [Wi, SW, FK4, Kle, KKS, KK]. The corresponding self-adjoint operator, which governs the dynamics, is a first-order partial differential operator, but its spectral theory may be studied through an auxiliary self-adjoint, second-order partial differential operator. These second-order classical wave operators are analogous to Schrödinger operators in quantum mechanics. The method is particularly suitable for the study of phenomena historically associated with quantum mechanical electron waves, especially Anderson localization in random media [FK3, FK4, FK7, Kle] and midgap defect eigenmodes [FK5, FK6, KK]. Physically interesting inhomogeneous and random media give rise to nonsmooth coefficients in the classical wave equations and, hence, in their classical wave operators. Thus we make no assumptions about the smoothness of the coefficients of classical wave operators. Classical waves do not localize in a homogeneous medium; to obtain wave localization an appropriate medium must be fabricated. We start with an underlying periodic medium (a ‘photonic crystal’ in the case of light waves) with a spectral gap. As randomness is added to the medium, we prove that the gap in the spectrum shrinks (possibly closing), and localization occurs in the spectrum at the edges of the gap. A crucial technical result is a Wegner-type estimate for random second-order classical wave operators with two random coefficients. This paper is organized as follows: In Section 2 we review our framework for studying classical waves. In Section 3 we discuss localization of classical waves in random media. We introduce a model for random media, and consider the corresponding random classical wave operators. Exponential localization and strong Hilbert–Schmidt dynamical localization are defined. The connection between localization of a random first-order classical wave operator and localization of the
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two associated random second order classical wave operators is described in Remarks 3.7 and 3.8. We study the effect of randomness on a spectral gap of an underlying periodic medium in Theorem 3.10. The results on localization are stated in Theorems 3.11–3.14. In Section 4 we show that random second order partially elliptic classical wave operators satisfy the requirements for the bootstrap multiscale analysis in Theorem 4.1; the Wegner estimate for random second-order classical wave operators is given in Theorem 4.4. The results on localization are proven using the bootstrap multiscale analysis. 2. Classical Wave Operators We start by reviewing the mathematical framework for classical waves introduced in the prequel [KK], to which we refer for discussion and examples. Many classical wave equations in a linear, lossless, inhomogeneous medium can be written as first order equations of the form: ∂ ψt (x) = D∗ φt (x), ∂t ∂ R(x)−1 φt (x) = −Dψt (x), ∂t
K(x)−1
(2.1)
where x ∈ Rd (space), t ∈ R (time), ψt (x) ∈ Cn and φt (x) ∈ Cm are physical quantities that describe the state of the medium at position x and time t, D is an m×n matrix whose entries are first-order partial differential operators with constant coefficients (see Definition 2.1), D∗ is the formal adjoint of D, and, K(x) and R(x) are n × n and m × m positive, invertible matrices, uniformly bounded from above and away from 0, that describe the medium at position x (see Definition 2.3). In addition, D satisfies a partial ellipticity property (see Definition 2.2), and there may be auxiliary conditions to be satisfied by the quantities ψt (x) and φt (x). The physical quantities ψt (x) and φt (x) then satisfy second-order wave equations, with the same auxiliary conditions: ∂2 ψt (x) = −K(x)D∗ R(x)Dψt (x), ∂t 2 ∂2 φt (x) = −R(x)DK(x)D∗ φt (x). ∂t 2
(2.2) (2.3)
Conversely, given (2.2) (or (2.3)), we may write this equation in the form (2.1) by introducing an appropriate quantity φt (x) (or ψt (x)), which will then satisfy Equation (2.3) (or (2.2)). The wave equation (2.1) may be rewritten in abstract Schrödinger form: −i
d t = Wt , dt
(2.4)
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where t = W=
ψt φt
and −iK(x)D∗ 0
0 iR(x)D
.
(2.5)
The (first order) classical wave operator W is formally (and can be defined as) a self-adjoint operator on the Hilbert space H = L2 (Rd , K(x)−1 dx; Cn ) ⊕ L2 (Rd , R(x)−1 dx; Cm ),
(2.6)
where, for a k × k positive invertible matrix-valued measurable function S(x), we set L2 (Rd , S(x)−1 dx; Ck ) = {f : Rd → Ck ; f, S(x)−1 f L2 (Rd , dx;Ck ) < ∞}. The auxiliary conditions to the wave equation are imposed by requiring the solutions to Equation (2.4) to also satisfy ⊥ t , t = PW
(2.7)
⊥ denotes the orthogonal projection onto the orthogonal complement of where PW the kernel of W. The solutions to Equations (2.4) and (2.7) are of the form ⊥ 0 , t = eit W PW
0 ∈ H .
(2.8)
The energy density at time t of a solution ≡ t (x) = (ψt (x), φt (x)) of the wave equation (2.1) is given by E (t, x) = 12 {ψ(x)t , K(x)−1 ψt (x)Cn + φt (x), R(x)−1 φt (x)Cm }.
(2.9)
The wave energy, a conserved quantity, is thus given by E = 12 t 2H
for any t.
(2.10)
Note that (2.8) gives the finite energy solutions to the wave equation (2.1). It is convenient to work on L2 (Rd , dx; Ck ) instead of the weighted space 2 L (Rd , S(x)−1 dx; Ck ). To do so, note that the operator VS , given by multiplication by the matrix S(x)−1/2 , is a unitary map from the Hilbert space L2 (Rd , S(x)−1 dx; = (VK ⊕ VR )W(V ∗ ⊕ V ∗ ), we have Ck ) to L2 (Rd , dx; Ck ), and if we set W K R √ √ 0√ −i K(x)D∗ R(x) = √ , (2.11) W 0 i R(x)D K(x) a formally self-adjoint operator on L2 (Rd , dx; Cn ) ⊕ L2 (Rd , dx; Cm ). In addition, if S− I S(x) S+ I with 0 < S− S+ < ∞, as it will be the case in this article, it turns out that if ϕ = VS ϕ, then the functions ϕ(x) and ϕ (x) share the same decay and growth properties (e.g., exponential or polynomial decay).
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Thus it will suffice for us to work on L2 (Rd , dx; Ck ), and we will do so in the remainder of this article. We set H (k) = L2 (Rd , dx; Ck ).
(2.12)
Given a closed densely defined operator T on a Hilbert space H , we will denote its kernel by ker T and its range by ran T ; note ker T ∗ T = ker T . If T is selfadjoint, it leaves invariant the orthogonal complement of its kernel; the restriction of T to (ker T )⊥ will be denoted by T⊥ . Note that T⊥ is a self-adjoint operator on the Hilbert space (ker T )⊥ = PT⊥ H , where PT⊥ denotes the orthogonal projection onto (ker T )⊥ . DEFINITION 2.1. A constant coefficient, first order, partial differential operator D from H (n) to H (m) (CPDO(1) n,m ) is of the form D = D(−i∇), where, for a d-component vector k, D(k) is the m × n matrix D(k) = [D(k)r,s ] r=1,...,m ; s=1,...,n
D(k)r,s = ar,s · k, ar,s ∈ Cd .
(2.13)
We set D+ = sup{ D(k) ; k ∈ Cd , |k| = 1},
(2.14)
so D(k) D+ |k| for all k ∈ Cd . Note that D+ is bounded by the norm of the matrix [|ar,s |] r=1,...,m . s=1,...,n
Defined on D(D) = {ψ ∈ H (n) : Dψ ∈ H (m) in distributional sense},
(2.15)
∞ d n a CPDO(1) n,m D is a closed, densely defined operator, and C0 (R ; C ) (the space of infinitely differentiable functions with compact support) is an operator core for D. We will denote by D∗ the CPDO(1) m,n given by the formal adjoint of the matrix in (2.13).
DEFINITION 2.2. A CPDO(1) n,m D is said to be partially elliptic if there exists a (1) ⊥ CPDOn,q D (for some q), satisfying the following two properties: D⊥ D∗ = 0, D∗ D + (D⊥ )∗ D⊥ [(− ) ⊗ In ],
(2.16) (2.17)
with > 0 being a constant. ( = ∇ · ∇ is the Laplacian on L2 (Rd , dx); In denotes the n × n identity matrix.) If D is partially elliptic, we have H (n) = ker D⊥ ⊕ ker D,
(2.18)
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and D∗ D + (D⊥ )∗ D⊥ = (D∗ D)⊥ ⊕ ((D⊥ )∗ D⊥ )⊥ .
(2.19)
Note that D is elliptic if and only it is partially elliptic with D⊥ = 0. Note also that ∗ a CPDO(1) n,m D may be partially elliptic with D not being partially elliptic [KKS, Remark 1.1]. DEFINITION 2.3. A coefficient operator S on H (n) (COn ) is a bounded, invertible operator given by multiplication by a coefficient matrix: an n×n matrix-valued measurable function S(x) on Rd , satisfying S− In S(x) S+ In ,
with 0 < S− S+ < ∞.
(2.20)
DEFINITION 2.4. A multiplicative coefficient, first order, partial differential operator from H (n) to H (m) (MPDO(1) n,m ) is of the form A=
√
√ RD K
1
on D(A) = K − 2 D(D),
(2.21)
where D is a CPDO(1) n,m , K is a COn , and R is a COm . (We will write AK,R for A whenever it is necessary to make explicit the dependence on the on the medium, i.e., on the coefficient operators. D does not depend on the medium, so it will be omitted in the notation.) √ √ ∗ KD∗ R an An MPDO(1) n,m A is a closed, densely defined operator with A = −1/2 ∞ C0 (Rd ; Cn ) is an operator core for A. The following MPDO(1) m,n . Note that K quantity will appear often in estimates: (2.22)
A ≡ D+ R+ K+ . DEFINITION 2.5. A first-order classical wave operator (CWO(1) n,m ) is an operator of the form 0 −iA∗ on H (n+m) ∼ (2.23) WA = = H (n) ⊕ H (m) , iA 0 ∗ where A is an MPDO(1) n,m . If either D or D is partially elliptic, WA will also be ∗ called partially elliptic. If both D and D are partially elliptic, WA will be called doubly partially elliptic.
Note that our definition of a first-order classical wave operator is more restrictive than the one used in [KKS]. Our definition of partial ellipticity is also different from [KKS], where partially eliptic corresponds to our doubly partially elliptic – see [KKS, Remark 1.2].
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Remark 2.6. The usual first-order classical wave operators are doubly partially elliptic, including the operators corresponding to electromagnetic waves (Maxwell equations), acoustic waves, and elastic waves (see [KK, p. 100]). But there are examples of first-order classical wave operators which are partially elliptic but not doubly partially elliptic (see [KKS, Remark 1.1]). The Schrödinger-like equation (2.4) for classical waves with the auxiliary condition (2.7) may be written in the form: −i
∂ t = (WA )⊥ t , ∂t
t ∈ (ker WA )⊥ = (ker A)⊥ ⊕ (ker A∗ )⊥ ,
(2.24)
with WA a CWO(1) n+m as in (2.23). Its solutions are of the form t = eit (WA )⊥ 0 ,
0 ∈ (ker WA )⊥ ,
(2.25)
which is just another way of writing (2.8). Since ∗ A A 0 2 , (WA ) = 0 AA∗
(2.26)
if t = (ψt , φt ) ∈ H (n) ⊕ H (m) is a solution of (2.24), then its components satisfy the second-order wave equations (2.2) and (2.3), plus the auxiliary conditions, which may be all written in the form ∂2 ψt = −(A∗ A)⊥ ψt , with ψt ∈ (ker A)⊥ , 2 ∂t ∂2 φt = −(AA∗ )⊥ φt , with φt ∈ (ker A∗ )⊥ . ∂t 2 The solutions to (2.27) and (2.28) may be written as 1/2 1/2 ψt = cos t(A∗ A)⊥ ψ0 + sin t(A∗ A)⊥ η0 , ψ0 , η0 ∈ (ker A)⊥ , 1/2 1/2 φt = cos t(AA∗ )⊥ φ0 + sin t(AA∗ )⊥ ζ0 , φ0 , ζ0 ∈ (ker A∗ )⊥ ,
(2.27) (2.28)
(2.29) (2.30)
with a similar expression for the solutions of (2.28). The operators (A∗ A)⊥ and (AA∗ )⊥ are unitarily equivalent (see [KK, Lemma A.1]): the operator U defined by −1/2
U ψ = A(A∗ A)⊥
ψ
for ψ ∈ ran(A∗ A)⊥ , 1/2
(2.31)
extends to a unitary operator from (ker A)⊥ to (ker A∗ )⊥ , and (AA∗ )⊥ = U (A∗ A)⊥ U ∗ .
(2.32)
In particular, t = (ψt , φt ) is the solution of (2.24) given in (2.25) if and only if ψt and φt are the solutions (2.29) and (2.30) of (2.27) and (2.28) with η0 = U φ0 and ζ0 = U ∗ ψ0 .
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In addition, if 1 IA U= √ iU 2
IA , −iU
with IA the identity on (ker A)⊥ ,
(2.33)
U is a unitary operator from (ker A)⊥ ⊕ (ker A)⊥ to (ker A)⊥ ⊕ (ker A∗ )⊥ , and we have the unitary equivalence: 1/2 1/2
(2.34) U∗ (WA )⊥ U = (A∗ A)⊥ ⊕ −(A∗ A)⊥ . Thus the operator (A∗ A)⊥ contains full information about the spectral theory of the operator (WA )⊥ . In particular 1/2 1/2 (2.35) σ ((WA )⊥ ) = σ (A∗ A)⊥ ∪ −σ (A∗ A)⊥ , and to find all eigenvalues and eigenfunctions for (WA )⊥ , it is necessary and sufficient to find all eigenvalues and eigefunctions for (A∗ A)⊥ . For if (A∗ A)⊥ ψω2 = ω2 ψω2 , with ω = 0, ψω2 = 0, we have i i (2.36) (WA )⊥ ψω2 , ± Aψω2 = ±ω ψω2 , ± Aψω2 . ω ω Conversely, if (WA )⊥ (ψ±ω , φ±ω ) = ±ω(ψ±ω , φ±ω ), with ω = 0, it follows that (see [KKS, Proposition 5.2]) (A∗ A)⊥ ψ±ω = ω2 ψ±ω
i and φ±ω = ± Aψ±ω . ω
(2.37)
DEFINITION 2.7. A second-order classical wave operator on H (n) (CWO(2) n ) is (1) ∗ an operator W = A A, with A an MPDOn,m for some m. (We write WK,R = A∗K,R AK,R .) If D in (2.21) is partially elliptic, the CWO(2) n will also be called partially elliptic. Note that a first-order classical wave operator WA is partially elliptic if and only if one of the two second-order classical wave operators A∗ A and AA∗ is partially elliptic. It is doubly elliptic if both A∗ A and AA∗ are partially elliptic. DEFINITION 2.8. A classical wave operator (CWO) is either a CWO(1) n or a . If the operator W is a CWO, we call W a proper CWO. CWO(2) ⊥ n Remark 2.9. A proper classical wave operator W has a trivial kernel by construction, so 0 is not an eigenvalue. But 0 is in the spectrum of W⊥ [KKS, Theorem A.1], so W⊥ and W have the same spectrum and essential spectrum.
3. Wave Localization in Random Media The form of the wave equation (2.1) is given by a constant coefficient, first order, partial differential operator D from H (n) to H (m) ; the properties of the medium are
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encoded in the coefficients matrices K(x) and R(x). Random media is modeled by random coefficients matrices. In this article we study random perturbations of an underlying periodic medium, i.e., a medium specified by periodic coefficients matrices (recall Definition 2.3) K0 (x) and R0 (x) with the same period q (i.e., K0 (x) = K0 (x + qj ) and R0 (x) = R0 (x + qj ) for all j ∈ Zd – we take q ∈ N without loss of generality). We use the following model for random media: ASSUMPTION 3.1 (Random medium). The random medium is modeled by random matrix-valued functions Kg (x) = Kg,ω (x) and Rg (x) = Rg,ω (x) of the form ωi ui (x), (3.1) Kg,ω (x) = γg,ω (x) K0 (x), with γg,ω (x) = 1 + g i∈Zd
Rg,ω (x) = ζg,ω (x)R0 (x),
with ζg,ω (x) = 1 + g
ωi vi (x),
(3.2)
i∈Zd
where (i) K0 (x) and R0 (x) are n × n and m × m periodic coefficient matrices with period q ∈ N. (ii) ui (x) = u(x − i) and vi (x) = v(x − i) for i ∈ Zd , where u and v are real valued measurable functions on Rd with support in the cube centered at the origin with side r < ∞, with ui (x) U+ , (3.3) 0 U− U (x) = i∈Zd
0 V− V (x) =
vi (x) V+
(3.4)
i∈Zd
for a.e. x ∈ Rd , where U± and V± are constants such that 0 < Z− Z+ < ∞,
with Z± = max{U± , V± }.
(3.5)
(iii) ω = {ωi ; i ∈ Zd } is a family of independent, identically distributed random variables taking values in the interval [−1, 1], whose common probability distribution µ has a bounded density ρ > 0 a.e. in [−1, 1]. (iv) g, the disorder parameter, satisfies 0g<
1 . Z+
(3.6)
Remark 3.2. The use of the same random variables in (3.1) and (3.2) models the fact that the medium itself is what is random. This randomness in the medium is modeled by random coefficient matrices, which are not independent since a change in the medium leads to changes in both coefficient matrices.
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Remark 3.3. The results in this article are also valid for random coefficient matrices Kg,ω (x) and Rg,ω (x) of the form −1 K0 (x), Kg,ω (x) = γg,ω
−1 Rg,ω (x) = ζg,ω R0 (x).
(3.7)
The modifications in the proofs are obvious. This is the form used in [FK3, FK4] for acoustic and electromagnetic waves. It follows from Assumption 3.1 that for a.e. ω the coefficient matrices Kg,ω (x) and Rg,ω (x) satisfy (2.20) with Kg,ω,± = Kg,± ≡ K0,± (1 ± gU+ ), Rg,ω,± = Rg,± ≡ R0,± (1 ± gV+ ).
(3.8) (3.9)
Thus multiplication by Kg,ω (x) and Rg,ω (x) yield coefficient operators Kg,ω and Rg = Rg,ω as in Definition 2.3, for a.e. ω. For later use, we set
g = D+ Rg,+ Kg,+ , U± , δ± (g) = 1 ∓ gU+ V± . η± (g) = 1 ∓ gV+
Kg = (3.10) (3.11) (3.12)
The periodic operators associated with the coefficient matrices K0 (x) and R0 (x) will carry the subscript 0, i.e., W0 = WA0 , W0 = A∗0 A0 . (3.13) A0 = R0 D K0 , Similarly, we write (for a.e. ω) Ag,ω = Rg,ω D Kg,ω ,
Wg,ω = WAg,ω ,
Wg,ω = A∗g,ω Ag,ω . (3.14)
We also set Wg,ω,∗ = Ag,ω A∗g,ω ,
(3.15)
and recall (2.32). DEFINITION 3.4. By a random classical wave operator we will always mean either Wg,ω (first order) or Wg,ω (second order) as in (3.14), with the random coefficient matrices satisfying Assumption (3.1) Note that Wg,ω,∗ is also a random second-order classical wave operator. Random classical wave operators are random operators (see Appendix; a random operator is a mapping ω → Hω from a probability space to self-adjoint operators on a Hilbert space such that the mappings ω → f (Hω ) are strongly
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measurable for all bounded measurable functions f on R). In addition, they are qZd -ergodic. It is a consequence of ergodicity that there exist nonrandom sets g and g , such that σ (Wg,ω ) = g and σ (Wg,ω ) = g with probability one. In addition, the decompositions of σ (Wg,ω ) and σ (Wg,ω ) into pure point spectrum, absolutely continuous spectrum and singular continuous spectrum are also independent of the choice of ω with probability one [KM1, CL]. (These sets are related by (2.34).) g,ω to denote a random classical operator of either first or second We will use W g . order; its almost sure spectrum will be denoted by Random classical wave operators may exhibit the phenomenum of localization. We give two definitions: the first, spectral localization, in its stronger form, exponential localization, is sometimes called Anderson localization; the second is a stronger form of dynamical localization introduced in [GK1]. DEFINITION 3.5 (Exponential localization). The random classical wave operator g,ω has g,ω exhibits spectral localization in an interval I if I ∩ g = ∅ and W W only pure point spectrum in I ∩ g with probability one. It exhibits exponential localization in I if it exhibits spectral localization in I and, with probability one, all the eigenfunctions corresponding to eigenvalues in I are exponentially decaying (in the sense of having exponentially decaying local L2 -norms). g,ω exhibits strong HSDEFINITION 3.6. The random classical wave operator W dynamical localization in an interval I if I ∩ g = ∅ and for any bounded region and all p 0 we have
g,ω )EW (I )χ 2 < ∞. (3.16) E sup |X|p/2 f (W g,ω 2 |||f |||1
(The supremum is taken over Borel functions f of a real variable with |||f ||| = supt ∈R |f (t)|; EH ( ) denotes the spectral projection of the self-adjoint operator H ;
B 2 denotes the Hilbert–Schmidt norm of the operator B.) Remark 3.7. In view of (2.32) and (2.34), spectral localization of a random second-order classical wave operator Wg,ω in a compact interval I ⊂ (0, ∞) is equivalent to spectral localization of Wg,ω,∗ in I , and also equivalent to spectral localization of the random first order √ classical wave operator Wg,ω in one (and then both) of the compact intervals ± I . The same is true for exponential localization, in view of (2.36), (2.37), and the interior estimate of [KK, Lemma 3.4]. Remark 3.8. In view of (2.26), strong HS-dynamical localization of both random second-order classical wave operators Wg,ω and Wg,ω,∗ in a compact interval I ⊂ (0, ∞) is equivalent to strong HS-dynamical localization of √ the random first order classical wave operator Wg,ω in both compact intervals ± I . It follows from Remarks 3.7 and 3.8 that it suffices to prove localization for random second-order classical wave operators.
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To create an environment which favors localization, we follow the strategy first introduced in [FK1] and subsequently used in [FK2, FK3, FK4, KSS, CHT]: We start with an underlying periodic medium. The spectrum associated with a periodic medium has band gap structure and may have a gap in the spectrum. We assume the existence of a spectral gap for the underlying periodic medium. We randomize this periodic medium with a gap in the spectrum, prove that the gap shrinks but does not close if the disorder is not too large, and show that exponential localization and strong HS-dynamical localization occurs in a vicinity of the edges of the gap. ASSUMPTION 3.9 (Gap in the spectrum). There is a gap in the spectrum of the periodic second-order classical wave operator W0 . More precisely, there exist a, b ∈ σ (W0 ), 0 < a < b, such that (a, b) ∩ σ (W0 ) = ∅.
(3.17)
When randomness is added to the medium, the spectrum of the corresponding classical wave operator changes. The following theorem gives information on what happens to a spectral gap. THEOREM 3.10 (Location of the spectral gap). Let Wg,ω be a random secondorder classical wave operator satisfying Assumption 3.9. There exists g0 , with 1/4 a 1 1 , 1− max U+ V+ b U+ /4U− V+ /4V− 1 1 1 b b , −1 , − 1 , (3.18) g0 min Z+ U+ a V+ a and increasing, Lipschitz continuous real valued functions a(g) and −b(g) on the interval [0, 1/Z+ ), with a(0) = a, b(0) = b, and a(g) b(g), such that: (3.19) (i) g ∩ [a, b] = [a, a(g)] ∪ [b(g), b]. (ii) If g < g0 we have a(g) < b(g) and (a(g), b(g)) is a gap in the spectrum g of the random operator Wg,ω . Moreover, we have a(1 + gU+ )U− /U+ (1 + gV+ )V− /V+ a(g)
a (1 − gU+ )(1 − gV+ )
(3.20)
and b(1 − gU+ )(1 − gV+ ) b(g)
b (1 + gU+
)U− /U+ (1
+ gV+ )V− /V+
In addition, if 0 g1 < g2 < g0 we have 1 (δ (g ) + η− (g2 ))(a(g1 ) + a(g2 )) 2 − 2 a(g2 ) − a(g1 ) 12 (δ+ (g2 ) + η+ (g2 ))(a(g1 ) + a(g2 )), g2 − g1
. (3.21)
(3.22)
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+ η− (g2 ))(b(g1 ) + b(g2 )) b(g1 ) − b(g2 ) 12 (δ+ (g2 ) + η+ (g2 ))(b(g1 ) + b(g2 )). g2 − g1
163
1 (δ (g ) 2 − 2
(3.23)
(iii) If g0 < 1/Z+ we have a(g) = b(g) for all g ∈ [g0 , 1/Z+ ), and the random classical wave operator Wg,ω has no spectral gap inside the gap (a, b) of the periodic classical wave operator W0 , i.e., we have [a, b] ⊂ g . Theorem 3.10 is proven in Section 5. Localization for continuous random operators is usually proved by a multiscale analysis, e.g., [HM, CH, Klo1, FK3, FK4, KSS, GD, CHT, Kle, DS, GK1, GK4, GK5]. (But note that the fractional moment method [AM, ASFH] has just been extended to the continuum [AENSS].) In this article we use the most recent and powerful version, the bootstrap multiscale analysis introduced in [GK1]. It can be applied in all cases where a multiscale analysis has been used, and it yields both exponential localization and strong HS-dynamical localization. (It gives a lot more, see [GK3, GK6].) A random second order partially elliptic classical wave operator Wg,ω will be shown (Theorem 4.1) to satisfy all the requirements of the bootstrap multiscale analysis in each compact interval I ⊂ (0, ∞). Thus, to prove exponential localization and strong HS-dynamical localization for Wg,ω in some interval centered at E ∈ g \{0}, it suffices to verify the initial length-scale estimate of the bootstrap multiscale analysis [GK1, Equation (3.3)] at E. We will show that if the random second-order partially elliptic classical wave operator Wg,ω has a gap in the spectrum, the random perturbation creates localization near the edges of the gap for g < g0 , where g0 is given in Theorem 3.10. To prove the initial length-scale estimate for the multiscale analysis (as originally done in [FK1]), we need low probability to have spectrum near an edge of the gap in finite but large volume. This can achieved either by hypotheses on the probability distribution µ in Assumption 3.1(iii), which produce classical tails at the edge of the gap, or by postulating the existence of Lifshitz tails at the edges of the gap. Lifshitz tails were originally proved for random Schrödinger operators at the bottom of the spectrum (e.g., [PF, Section 10], [CL, Section VI.2]). Holden and Martinelli [HM] used the Lifshitz tails estimate to obtain the initial length-scale estimate for the Fröhlich–Spencer multiscale analysis at the bottom of the spectrum for random Schrödinger operators. The best estimates on the size of the interval of localization at the bottom of the spectrum at low disorder have been obtained from Lifshitz tails by Klopp [Klo3]. Klopp [Klo2] proved that for a random perturbation of a periodic Schrödinger operator, there are Lifshitz tails at an edge of a spectral gap if and only if the density of states of the periodic operator is nondegenerate at the same edge of the spectral gap. (This nondegeneracy has not been established for arbitrary edges of spectral gaps.) Najar [Na] extended Klopp’s results to random acoustic operators with constant compressibility and smooth mass density. If Lifshitz tails are present at an
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ABEL KLEIN AND ANDREW KOINES
edge of a spectral gap, the Holden–Martinelly argument can be used to obtain the initial length-scale estimate for the bootstrap multiscale analysis. But the existence of Lifshitz tails at the edges of spectral gaps of has not been established for the random classical wave operators studied in this article. We state our results with hypotheses on the probability distribution µ, as in [FK3, FK4]. The following two theorems achieve low probability of extremal values for the random variables in different ways. The results are formulated for the left edge of the gap, with similar results holding at the right edge. We use the notation of Theorem 3.10. THEOREM 3.11 (Localization at the edge). Let Wg,ω be a random second-order partially elliptic classical wave operator satisfying Assumption 3.9. Suppose the probability distribution µ in Assumption 3.1(iii) satisfies µ{(1 − γ , 1]} Kγ η
for all 0 γ 1,
(3.24)
where K < ∞ and η > d/2. Then, for any g < g0 there exists δ(g) > 0, depending only on the constants d, g, q, K0,± , R0,± , , U± , V± , r, ρ ∞ , K, η, a, b, such that the random classical wave operator Wg,ω exhibits exponential localization and strong HS-dynamical localization in the interval [a(g) − δ(g), a(g)]. THEOREM 3.12 (Localization in a specified interval). Let Wg,ω be a random second-order partially elliptic classical wave operator satisfying Assumption 3.9. Let g < g0 , and fix a1 and a2 such that a < a1 < a2 < a(g) and a(g) − a1 b(g) − a(g). Then there exists p1 > 0, depending only on the constants d, g, q, K0,± , R0,± , , U± , V± , r, K, η, a, b, on the fixed a1 , a2 , and on a fixed upper bound on ρ ∞ , such that if g1 ,1 p1 , (3.25) µ g where g1 is defined by a(g1 ) = a1 , the random classical wave operator Wg,ω exhibits exponential localization and strong HS-dynamical localization in the interval [a2 , a(g)]. Theorems 3.11 and 3.12 can be extended to the situation when the gap is totally filled by the spectrum of the random classical wave operator, establishing the existence of a subinterval of the original gap where the random classical wave operator exhibits localization. Note that the extension of Theorem 3.12 says that we can arrange for localization in as much of the gap as we want. Recall that if g0 < 1/Z+ (see (3.18) for a necessary condition), then (a, b) ⊂ g for g ∈ [g0 , 1/Z+ ), i.e., the gaps closes. THEOREM 3.13 (Localization at the meeting of the edges). Let Wg,ω be a random second-order partially elliptic classical wave operator satisfying Assumption 3.9. Suppose the probability distribution µ in Assumption 3.1(iii) satisfies µ{(1 − γ , 1]},
µ{[−1, −1 + γ )} Kγ η
for all 0 γ 1,
(3.26)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
165
where K < ∞ and η > d. Suppose also that g0 < 1/Z+ . Then there exist 0 < < (1/Z+ ) − g0 and δ > 0, depending only on the constants d, g, q, K0,± , R0,± , , U± , V± , r, ρ ∞ , K, η, a, b, such that the random classical wave operator Wg,ω exhibits exponential localization and strong HS-dynamical localization in the interval [a(g0 ) − δ, a(g0 ) + δ] for all g ∈ [g0 , g0 + ). THEOREM 3.14 (Localization in a specified interval in the closed gap). Let Wg,ω be a random second-order partially elliptic classical wave operator satisfying Assumption 3.9. Suppose g0 < 1/Z+ , and fix a1 , a2 , b1 and b2 such that a < a1 < a2 < a(g0 ) = b(g0 ) < b2 < b1 < b. For any g ∈ [g0 , 1/Z+ ) there exist p1 , p2 > 0, depending only on the constants d, g, q, K0,± , R0,± , , U± , V± , r, K, η, a, b, on the given a1 , a2 , b1 , b2 , and on a fixed upper bound on ρ ∞ , such that if g1 g2 µ ,1 p1 and µ −1, − p2 , (3.27) g g where g1 and g2 are defined by a(g1 ) = a1 and b(g2 ) = b1 (notice 0 < g1 , g2 < g0 g), the random classical wave operator Wg,ω exhibits exponential localization and strong HS-dynamical localization in the interval [a2 , b2 ]. Theorems 3.13 and 3.14 are proved similarly to Theorems 3.11 and 3.12, respectively, taking into account both edges of the gap. 4. The Multiscale Analysis and Localization The analysis requires finite volume random classical wave operators. Throughout this paper we use two norms in Rd and Cd : 12 d |xi |2 , |x| =
(4.1)
i=1
x = max{|xi |, i = 1, . . . , d}.
(4.2)
By L (x) we denote the open cube in R , centered at x with side L > 0: L d , L (x) = y ∈ R ; y − x < 2 d
(4.3)
˘ L (x) the half-open/half-closed cube, i.e., by L (x) the closed cube, and by L L d ˘ L (x) = y ∈ R ; − yi − xi < , i = 1, 2, . . . , d . (4.4) 2 2 We will identify a closed cube L (x) with a torus in the usual way, We set χx,L = χL (x) , where χ denotes the characteristic function of a set ⊂ Rd .
(4.5)
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ABEL KLEIN AND ANDREW KOINES
Since we will work with an underlying periodic medium with period q ∈ N, we restrict ourselves to cubes L (x) with x ∈ Zd and L ∈ 2qN. We set (n) = H(n)L (x) = L2 (L (x), dx; Cn ). Hx,L
(4.6)
(n) A CPDO(1) n,m D defines a closed densely defined operator Dx,L from Hx,L (m) to Hx,L with periodic boundary condition; an operator core is given by ∞ Cper (L (x), Cn ), the infinitely differentiable, periodic Cn -valued functions on L (x). If the CPDO(1) n,m D is partially elliptic, then the restriction Dx,L is also partially elliptic, in the sense that Equations (2.16) and (2.17) hold for Dx,L , (D⊥ )x,L , and x,L . ( x,L is the Laplacian on L2 (L (x), dx) with periodic boundary condition.) This can be easily seen by using the Fourier transform; here the use of periodic (n) . boundary condition plays a crucial role. We also have (2.18) and (2.19) with Hx,L We fix a random second order classical wave operator Wg,ω as in (3.14). Given d d ω ∈ RZ , we define ωx,L = ωL (x) ∈ RZ by
ωx,L,i = ωi
˘ L (x) ∩ Zd , for each i ∈
ωx,L,i = ωx,L,i+Lj
for all i, j ∈ Zd .
We set Ag,ω,x,L = Ag,ω,L (x) =
Rg,ωx,L Dx,L Kg,ωx,L
(4.7)
(4.8)
−1/2
(n) . on D(Ag,ω,x,L ) = Kg,ωx,L D(Dx,L ), a closed, densely defined operator on Hx,L (n) The finite volume random classical wave operator Wg,ω,x,L on Hx,L is now defined by
Wg,ω,x,L = Wg,ω,L (x) = A∗g,ω,x,L Ag,ω,x,L .
(4.9)
(Wg,ω,x,L is a “periodic restriction” of Wg,ω to L (x) with periodic boundary condition.) We have the equivalent of (2.31), (2.32), etc. We write Rg,ω,x,L (z) = (Wg,ω,x,L − z)−1
(4.10)
for the finite volume resolvent. The multiscale analysis works with the decay of the finite volume resolvent from the center of a cube to its boundary, or more precisely, to its boundary belt. We set q +r , (4.11) q˜ = min q ∈ qN; q 2 where r is given in Assumption 3.1(ii). Given a cube L (x), we set L ϒL (x) = y ∈ Zd ; y − x = − q˜ , 2
(4.12)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
and define its (boundary) belt by q (y); ϒ˜ L (x) =
167
(4.13)
y∈ϒL (x)
it has the characteristic function x,L = χϒ˜ L (x) = χy,q
a.e.
(4.14)
y∈ϒL (x)
Note that |ϒL (x)| d
d−1 L . q
(4.15)
The following theorem shows that random second-order partially elliptic classical wave operators satisfy the requirements for the bootstrap multiscale analysis of [GK1]. Note that we use the finite volume operators defined in (4.9), and the boundary belt defined in (4.13), i.e., with x,L as in (4.14). THEOREM 4.1. A random second-order partially elliptic classical wave operator is a qZd -ergodic random operator satisfying the requirements for the bootstrap multiscale analysis in any compact interval I0 ⊂ (0, ∞), i.e., it satisfies Assumptions SLI (Simon–Lieb inequality), EDI (eigenfunction decay inequality), IAD (independence at a distance), NE (number of eigenvalues), SGEE (strong generalized eigenfunction expansion), and W (Wegner’s estimate) of [GK1] in I0 . The constants γI0 in Assumption SLI and γ˜I0 in Assumption EDI are given by γI0 = γ˜I0 = supE∈I0 γE , with √ 6 d 100d 2 1/2
g 2E + 2 g , (4.16) γE = q q the kernel polynomial decay where g is given in (3.10). In addition, it satisfies √ estimate of [GK2, Theorem 2] with = (3 7)/(32 g ) for a.e. ω (note that 2 = 0). Remark 4.2. Partial ellipticity is required for Assumption SGEE. Assumptions NE and W require that either Wg,ω or Wg,ω,∗ is partially elliptic. The other assumptions and the kernel polynomial decay estimate do not require partial ellipticity. Remark 4.3. It follows from Theorem 4.1 that the results of [GK3] on the Anderson metal-insulator transport transition apply to random second-order partially elliptic classical wave operators. We have already proven most of Theorem 4.1. Taking into account (3.8) and (3.9), Assumptions SLI, EDI, and NE follow from Lemmas 3.8, 3.9, and 3.3 in
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ABEL KLEIN AND ANDREW KOINES
[KK], respectively. (We used slightly different finite volume operators in [KK], where we used a boundary belt with q˜ = q. But the proofs of Lemmas 3.8 and 3.9 in [KK] still apply with the definitions used in this article due to our choice of q˜ in (4.11).) IAD is true by Assumption 3.1 and the definition of finite volume operators in (4.9); note that = 0. Assumption SGEE is proven in [KKS], in the stronger form of the trace estimate given in [GK1, Equation (2.36)]. The kernel polynomial estimate is just a special case of [GK2, Theorem 2]. Assumption W follows from the following theorem. The constants r, Z± , K0,− , R0,− , and the probability density ρ are as in Assumption (3.1); is the constant in (2.17). THEOREM 4.4 (Wegner estimate). Let Wg,ω be a random second-order classical wave operator, with either Wg,ω or Wg,ω,∗ partially elliptic. Then for all E > 0, cubes = L (x) with x ∈ Zd and L ∈ 2qN, and 0 η E, we have P{dist(σ (Wg,ω,x,L), E) η} Qg ρ ∞ E (d/2)−1ηL2d ,
(4.17)
where Qg =
nCd (2 + r)d (K0,− R0,− )−d/2 , gZ− (1 − gZ+ )d+1
(4.18)
with Cd a constant depending only on the dimension d. Theorem 4.4 is proven in Section 6. In view of Theorems 4.1 and 4.4, to prove exponential localization and strong HS-dynamical localization for Wg,ω in some interval centered at E ∈ g \{0}, it suffices to verify the initial length scale estimate for the bootstrap multiscale analysis [GK1, Equation (3.3)] at E. To state this estimate, we need a definition, which we state in the context of this article. DEFINITION 4.5. Let Wg,ω be a random second-order classical wave operator. Given θ > 0, E > 0, x ∈ Zd , and L ∈ 6qN, we say that the cube L (x) is / σ (Wg,ω,x,L ) and (θ, E)-suitable for Wg,ω if E ∈
x,L Rg,ω,x,L(E)χx,L/3 x,L
1 . Lθ
(4.19)
The following theorem summarizes the results of [GK1] that will be used to prove Theorems 3.11–3.14. THEOREM 4.6. Let Wg,ω be a random second-order partially elliptic classical wave operator. Let E0 ∈ g \{0}. Given θ > 2d, there exists a finite scale L = L(d, q, K0,± , R0,± , , U± , V± , r, ρ ∞ , g, E0 , θ), bounded for E0 in compact subintervals of (0, ∞), such that, if we can verify at some finite scale L L that P{L (0) is (θ, E0 )-suitable for Wg,ω } > 1 −
1 , 841d
(4.20)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
169
then there exists δ0 = δ0 (d, q, K0,± , R0,± , , U± , V± , r, ρ ∞ , g, E0 , θ, L) > 0, such that the random classical wave operator Wg,ω exhibits exponential localization and strong HS-dynamical localization in the interval [E0 − δ0 , E0 + δ0 ]. In addition, we have the conclusions of [GK1, Theorems 3.4, 3.8 and 3.10, Corollaries 3.10 and 3.12]. Proof of Theorem 3.11. In view of Theorem 4.6 it suffices to prove that for all g < g0 and θ > 2d we have lim P{L (0) is (θ, a(g))-suitable for Wg,ω } = 1.
L→∞
(4.21)
We fix g < g0 and θ > 2d. Given L ∈ 6qZd we use Theorem 3.10 and, for large L, define gL ∈ (0, g) by log L 2 , (4.22) a(gL ) = a(g) − κ L where κ > 0 will be specified later. We define the event gL Zd d ˘ for all i ∈ L (0) ∩ Z . EL = ω ∈ R ; ωi g
(4.23)
It follows from Theorem 3.10, Lemma 5.1 and [KK, Theorem 4.3] that (a(gL ), b(g)) ⊂ R\σ (Wg,ω,0,L)
for all ω ∈ EL .
(4.24)
Hence it follows from [KK, Theorem 3.6] that for large L we have −2 2d−1 κ log L e−C2 ((κ log L)/L)L
0,L Rg,ω,0,L (a(g))χ0,L/3 0,L C1 L L C1 L2d+1 = (4.25) (κ log L)2 LκC2 for all ω ∈ EL , where C1 and C2 are finite, strictly positive constants depending only on d, q, K0,± , R0,± , U± , V± , g, a, b. It follows that EL ⊂ {ω ∈ RZ ; L (0) is (θ, a(g))-suitable for Wg,ω } d
(4.26)
for all L sufficiently large if κ > (θ + 2d + 1)/C2 . Fixing κ, denoting by EL the complementary event to EL , and using (3.24), (3.22), and (4.22), we have that η gL d d g − gL ,1 KL (4.27) P(EL ) L µ g g η a(g) − a(gL ) (4.28) KLd ag(δ− (g) + η− (g)) η (κ logLL )2 −→ 0 (4.29) KLd ag(δ− (g) + η− (g))
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ABEL KLEIN AND ANDREW KOINES
as L → ∞ since η > d2 .
2
Proof of Theorem 3.12. Let g < g0 , fix a < a1 < a2 < a(g), with a(g) − a1 b(g) − a(g), and define g1 ∈ (0, g) by a(g1 ) = a1 using Theorem 3.10. We use the notation of the the previous proof with aL = a1 and gL = g1 for all L. As before, it follows from [KK, Theorem 3.6] that for sufficiently large L we have
0,L Rg,ω,0,L (E)χ0,L/3 0,L C1 L2d−1 (a2 − a1 )−1 e−C2
√
a2 −a1 L
for all E ∈ [a2 , a(g)] and ω ∈ EL , where C1 and C2 are the same constants as in (4.25). Thus, given θ > 0, we have d {ω ∈ RZ ; L (0) is (θ, E)-suitable for Wg,ω } (4.30) EL ⊂ E∈[a2 ,a(g)]
for sufficiently large L. We also have, using (3.25), that g1 d ,1 Ld p1 . P(EL ) L µ g
(4.31)
We now fix θ > 2d, and pick L0 ∈ 6qZd , suficiently large so (4.30) holds for this θ and L0 L, where L is given in Theorem 4.6, and take p1 <
1 . 841d Ld0
(4.32)
We then have (4.20) with L = L0 for all E ∈ [a2 , a(g)], so Theorem 3.12 follows from Theorem 4.6. 2 Proof of Theorem 3.13. It proceeds in the same way as the proof of Theorem 3.11, but taking into account both edges of the gap. Since we will use [KK, Theorem 3.6] for an energy in the middle of a gap, we will need η > d instead of η > d/2 as in Theorem 3.11. We verify (4.20) instead of (4.21). We fix g ∈ [g0 , 1/Z+ ) and θ > 2d. Recall a(g) = a(g0 ) = b(g0 ) = b(g). Given L ∈ 6qZd we use Theorem 3.10 and, for large L, define gL± ∈ (0, g0 ) by log L , L log L , b(gL+ ) = a(g0 ) + κ L
a(gL− ) = a(g0 ) − κ
where κ > 0 will be specified later. We define the event g+ g− d ˘ L (0) ∩ Zd . FL = ω ∈ RZ ; − L ωi L for all i ∈ g g
(4.33) (4.34)
(4.35)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
171
It follows from Theorem 3.10, Lemma 5.1 and [KK, Theorem 4.3] that (a(gL− ), b(gL+ )) ⊂ R\σ (Wg,ω,0,L)
for all ω ∈ FL .
(4.36)
Hence it follows from [KK, Theorem 3.6] that for large L we have −1 2d−1 κ log L e−C2 ((κ log L)/L)L
0,L Rg,ω,0,L (a(g0 ))χ0,L/3 0,L C1 L L 2d C1 L = (4.37) κ(log L)LκC2 for all ω ∈ FL , where C1 and C2 are finite, strictly positive constants depending only on d, q, K0,± , R0,± , U± , V± , g, a, b. It follows that FL ⊂ {ω ∈ RZ ; L (0) is (θ, a(g0 ))-suitable for Wg,ω } d
(4.38)
for all L sufficiently large if κ > (θ + 2d)/C2 . L the complementary event to FL , and using (3.26), Fixing κ, denoting by F (3.22), (3.23), (4.33), and (4.34), we have that − gL+ gL d , 1 + µ −1, − (4.39) P(FL ) L µ g g g − gL+ η g − gL− η d + (4.40) KL g g η a(g0 ) − a(gL− ) KLd + + g − g 0 η a(δ− (g0 ) + η− (g0 )) g0 η b(gL+ ) − a(g0 ) (4.41) + g − g0 + a(δ− (g0 ) + η− (g0 )) η κ logLL 2KLd . (4.42) g − g0 + η a(δ− (g0 ) + η− (g0 )) g0 We now fix θ > 2d and κ > (θ + 2d)/C2 , and pick L0 ∈ 6qZd , suficiently large so (4.38) holds for this θ and L0 L, where L is given in Theorem 4.6, and 2KLd0 η g0
2κ logL0L0 a(δ− (g0 ) + η− (g0 ))
η <
1 , 841d
(4.43)
what can be done since η > d. If we now set 1 , − g0 , ε = min a(δ− (g0 ) + η− (g0 )) Z+
κ logL0L0
(4.44)
we have (4.20) with L = L0 and E = a(g0 ) for all g ∈ [go , g0 + ε), so Theorem 3.13 follows from Theorem 4.6. 2
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ABEL KLEIN AND ANDREW KOINES
Proof of Theorem 3.14. The proof is similar to the proof of Theorem 3.12, but taking into account both edges of the gap, as in the proof of Theorem 3.13. 2
5. The Location of the Spectral Gap In this section we prove Theorem 3.10. We proceed as in [FK3, Theorem 3], but we must take into consideration two random coefficients. To do so, we make use of the unitary equivalence between the operators (Wg,ω, )⊥ and (Wg,ω,∗, )⊥ , and use [KK, Theorem 4.3]. We start by approximating the spectrum of the random operator by spectra of periodic operators. If k, n ∈ N, we say that k n if n ∈ kN and that k ≺ n if k n and k = n. Let us fix g as in (3.6) and set Tg = {τ = {τi , i ∈ Zd }; −g τi g} = [−g, g]Z ,
(5.1)
Tg(n) = {τ ∈ T ; τi+nj = τi for all i, j ∈ Zd },
(5.2)
d
and Tg(∞) =
n ∈ N,
Tg(n) .
(5.3)
nq
For τ ∈ Tg we let Kτ (x) = γτ (x)K0 (x),
with γτ (x) = 1 +
τi ui (x),
(5.4)
τi vi (x),
(5.5)
i∈Zd
Rτ (x) = ζτ (x)R0 (x),
with ζτ (x) = 1 +
i∈Zd
Aτ =
Rτ D Kτ ,
Wτ = A∗τ Aτ .
(5.6)
The following lemma shows that the (nonrandom) spectrum of the random classical wave operator Wg,ω is determined by the spectra of the periodic classical wave operators Wτ , τ ∈ Tg(∞) . The analogous result for random Schrödinger operators was proven in [KM2, Theorem 4]. It was extended to certain random classical wave operators in in [FK3, Lemma 19] and [FK4, Lemma 27]. LEMMA 5.1. Let Wg,ω be a random second-order classical wave operator. Its spectrum g is given by g =
(∞) τ ∈Tg
σ (Wτ ).
(5.7)
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A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
Proof. Let g denote the right-hand side of (5.7). We start by showing that σ (Wτ ) ⊂ g
for all τ ∈ Tg ,
(5.8)
which implies that g ⊂ g .
(5.9)
Let { n ; n = 0, 1, 2, . . .} be a sequence in 2N such that 0 = 2q and n ≺ n+1 for each n = 0, 1, 2, . . .. Given τ ∈ Tg , we specify τn ∈ Tg( n ) by requiring (τn )i = τi for i ∈ [− n /2, n /2)d ∩ Zd . We set Rn = (Wτn + 1)−1 , R = (Wτ + 1)−1 . We will show that Rn → R strongly, which implies (5.8), as in [FK3, Lemma 45]. To do so, note that (5.10) Rn = ( Kτn D∗ Rτn D Kτn + 1)−1 −1/2 ∗ −1 −1 −1/2 = Kτn (D Rτn D + Kτn ) Kτn , and similarly for R. Note that we have uniform (in n) bounds on the operator , Rτn , and (D∗ Rτn D + Kτ−1 )−1 . In addition, it is easy to see that norms of Kτ−1 n n → Kτ−1/2 , Kτ−1 → Kτ−1 , and Rτn → Rτ , the convergence being in the Kτ−1/2 n n )−1 → strong operator topology. Thus it suffices to show that (D∗ Rτn D + Kτ−1 n ∗ −1 −1 (D Rτ D + Kτ ) strongly. But this follows from the preceding remarks, the relation )−1 − (D∗ Rτ D + Kτ−1 )−1 (D∗ Rτn D + Kτ−1 n = (D∗ Rτn D + Kτ−1 )−1 (D∗ (R − Rτn )D + n ))(D∗ Rτ D + Kτ−1 )−1 , + (K − Kτ−1 n
(5.11)
)−1 are bounded with norms and the fact that the operators D(D∗ Rτn D + Kτ−1 n uniformly bounded in n and, hence, also their adjoints. To prove the opposite inclusion to (5.9), we introduce the countable sets (N) = Tg(N) ∩ QZ , Tg,Q d
N = 0, 1, 2, . . . , ∞.
(5.12)
(∞) , the Since any τ ∈ Tg(∞) can be approximated uniformly by a sequence τn ∈ Tg,Q previous argument shows that σ (Wτ ) for all τ ∈ Tg(∞) , (5.13) σ (Wτ ) ⊂ (∞)
τ ∈Tg,Q
which implies that σ (Wτ ). g = (∞) τ ∈Tg,Q
(5.14)
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Thus (5.7) follows if we prove that σ (Wτ ) ⊂ g
(∞) for all τ ∈ Tg,Q .
(5.15)
Note that a.e. ω ∈ ≡ [−1, 1]Z . Let { n ∈ N; n = 0, 1, 2, . . .} be such that (q ) 0 = 2q and n ≺ n+1 for each n = 0, 1, 2, . . .. For each n, q q, and τ ∈ Tg,Q we consider the event n,q ,τ = ω ∈ ; there is xω = xn,q ,τ ,ω ∈ q Zd such that 1 (5.16) max |gωi − τi | d+1 ; n i∈(xω +[− n /2, n /2)d )∩Zd d
notice P(n,q ,τ ) = 1. We take the countable intersection =
∞ n=0
q q
(5.17)
n,q ,τ ,
(q ) τ ∈Tg,Q
) = 1. We will show that so we have P( (∞) , and ω ∈ σ (Wτ ) ⊂ σ (Wg,ω ) for all τ ∈ Tg,Q
(5.18)
so (5.15) follows. (q ) (∞) , n ∈ N, and let , say τ ∈ Tg,Q for some q q. Let ω ∈ So let τ ∈ Tg,Q (n) xω = xn,q ,τ ,ω be as in (5.16). We set ω(n) = {ωi = ωi−xω ; i ∈ Zd }, and notice that σ (Wg,ω(n) ) = σ (Wg,ω ). We have the following inequalities for the matrices norms:
(Rg,ω(n) (x) − Rτ (x))χ0, n −r (x)
R0,+ V+ χ0, n −r (x), n
(Kg,ω(n) (x)−1 − Kτ (x)−1 )χ0, n −r (x)
U+ χ0, n −r (x), n K0,− (1 − gU+ )2
(5.19)
(5.20)
(Kg,ω(n) (x)−1/2 − Kτ (x)−1/2 )χ0, n −r (x)
1/2
K0,+ U+ (1 + gU+ ) χ0, n −r (x). n K0,− (1 − gU+ )5/2
(5.21)
Using these inequalities we can proceed as before to show that lim (Wg,ω(n) + I )−1 = (Wτ + I )−1
n→∞
(5.22)
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A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
in the strong operator topology, an hence that σ (W (τ )) ⊂
∞
σ (Wg,ω(n) ) = σ (Wg,ω ).
2 (5.23)
n=0
Given real numbers k, h, with |k|, |h| < 1/Z+ , we set Kk (x) = K0 (x)(1 + kU (x)) A(k, h) = Rh D Kk , W (k, h) = A(k, h)∗ A(k, h),
and
Rh (x) = R0 (x)(1 + hV (x)), (5.24) ∗
W∗ (k, h) = A(k, h)A(k, h) .
LEMMA 5.2. Let W (k, h) be as in (5.24), and let = (x0 ) for some x0 ∈ Rd and q. The positive self-adjoint operator (W (k, h) )⊥ has compact resolvent, so let µ1 (k, h) µ2 (k, h) . . . be its eigenvalues, repeated according to their (finite) multiplicity. Then each µj (h) ≡ µj (h, h), j = 1, 2, . . . , is a Lipschitz continuous, strictly increasing function of h, with + η− (g))(µj (h1 ) + µj (h2 )) µj (h2 ) − µj (h1 ) 12 (δ+ (g) + η+ (g))(µj (h1 ) + µj (h2 )) h2 − h1
1 (δ (g) 2 −
(5.25)
for any h1 , h2 ∈ (−g, g), 0 < g < 1/Z+ , where δ± (g) and η± (g) are given in (3.11) and (3.12). Proof. Let 0 < g < 1/Z+ , −g h1 < h2 g. We have Rh2 (x) − Rh1 (x) = (h2 − h1 )U (x)R0 (x) 0,
(5.26)
so W (k, h2 ) W (k, h1 )λ and, hence, each µj (k, h) is an increasing function of h for fixed k. It also follows from (5.26) that (h2 − h1 )V (x) (5.27) Rh2 (x) = Rh1 (x) 1 + 1 + h2 V (x) and
(h2 − h1 )V (x) , Rh1 (x) = Rh2 (x) 1 − 1 + h1 V (x)
(5.28)
which gives us Rh1 (x)(1 + η− (g)(h2 − h1 )) Rh2 (x) Rh1 (x)(1 + η+ (g)(h2 − h1 ))
(5.29)
Rh2 (x)(1 − η+ (g)(h2 − h1 )) Rh1 (x) Rh2 (x)(1 − η− (g)(h2 − h1 ))
(5.30)
and
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ABEL KLEIN AND ANDREW KOINES
with η± (g) as in (3.12). From (5.29) we get that for each k we have (1 + η− (g)(h2 − h1 ))W (k, h1 ) W (k, h2 ) (1 + η+ (g)(h2 − h1 ))W (k, h1 ) ,
(5.31)
so it follows from the min-max principle that for all j = 1, 2, . . . , (1 + η− (g)(h2 − h1 ))µj (k, h1 ) µj (k, h2 ) (1 + η+ (g)(h2 − h1 ))µj (k, h1 ), (5.32) i.e., η− (g)µj (k, h1 )
µj (k, h2 ) − µj (k, h1 ) η+ (g)µj (k, h1 ). h2 − h1
(5.33)
Similarly, using (5.30) we get η− (g)µj (k, h2 )
µj (h2 , h2 ) − µj (h2 , h1 ) η+ (g)µj (k, h2 ). h2 − h1
(5.34)
η− (g)µj (k, h2 )
µj (k, h2 ) − µj (k, h1 ) η+ (g)µj (k, h1 ). h2 − h1
(5.35)
Thus
Since the operators (W (k, h) )⊥ and (W∗ (k, h) )⊥ are unitarily equivalent, the µj (k, h) are also the egenvalues of (W∗ (k, h) )⊥ , so the above argument gives δ− (g)µj (k2 , h)
µj (k2 , h) − µj (k1 , h) δ+ (g)µj (k1 , h), k2 − k1
(5.36)
where −g k1 < k2 g. Since µj (h2 , h2 ) − µj (h1 , h1 ) = (µj (h2 , h2 ) − µj (h2 , h1 )) + (µj (h2 , h1 ) − µj (h1 , h1 )) = (µj (h2 , h2 ) − µj (h1 , h2 )) + (µj (h1 , h2 ) − µj (h1 , h1 )),
(5.37) (5.38)
we may use (5.35) and (5.36) with (5.37), repeat the procedure with (5.38) instead of (5.37), and take the average of the bounds to obtain (5.25). The properties of the 2 functions µj (h) follow. The following lemma follows immediately from [KK, Theorem 4.3], Lemmas 5.1 and 5.2, and the min-max principle. We write W (h) for W (h, h) as in (5.24).
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
177
LEMMA 5.3. Let Wg,ω be a random second-order classical wave operator. For all sequences { n ∈ N; n = 0, 1, 2, . . .}, with 0 = 2q and n ≺ n+1 for each n = 0, 1, 2, . . . , we have g =
∞
σ (W (h)) =
h∈[−g,g]
σ (W (h) n (0) ).
(5.39)
h∈[−g,g] n=0
In particular, g is increasing in g. We are now ready to prove Theorem 3.10. As g is increasing in g, we expect the gap to shrink as we increase g until it either disappears at some g0 , or it remains open for all allowed g. Thus we define 1 (5.40) ; g ∩ (a, b) = (a, b) . g0 = sup g ∈ 0, Z+ Let { n ; n = 0, 1, 2, . . .} be as in Lemma 5.3, h ∈ [−g, g], and let µ(n) 1 (h) (h) . . . be the nonzero eigenvalues of W (h) , where = µ(n) n n (0), ren 2 (n) peated according to their (finite) multiplicity; notice limj →∞ µj (h) = ∞. By Lemma 5.2 each µ(n) j (h) is a strictly increasing continuous function of h, hence it follows from Lemma 5.3 that g =
∞
σ (W (h)n ) =
n=0 h∈[−g,g]
∞ ∞
(n) [µ(n) j (−g), µj (g)].
(5.41)
n=0 j =1
In particular, g is a countable union of disjoint closed intervals, so for g < g0 we can define a(g) and b(g) by (3.19). Since g is increasing in g ∈ [0, 1/Z+ ) by Lemma 5.3, it follows that a(g) and −b(g) are increasing functions in [0, g0 ). For each n let jn = max{j ; µ(n) j (0) a},
(5.42)
so using Assumption 3.9 and [KK, Equation (4.1) in Theorem 4.3], we have jn + 1 = min{j ; µ(n) j (0) b}.
(5.43)
If g < g0 , it follows from the definition of jn , Assumption 3.9 and [KK, Theorem 4.3], that µjn (−g) and −µjn +1 (g) are both increasing in n, and a(g) = lim µjn (g),
(5.44)
b(g) = lim µjn +1 (−g).
(5.45)
n→∞ n→∞
Thus, given 0 g1 < g2 < g0 , we can conclude from (5.25) that + η− (g2 ))(a(g1 ) + a(g2 )) a(g2 ) − a(g1 ) 12 (δ+ (g2 ) + η+ (g2 ))(a(g1 ) + a(g2 )), g2 − g1
1 (δ (g ) 2 − 2
(5.46)
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ABEL KLEIN AND ANDREW KOINES
+ η− (g2 ))(b(g1 ) + b(g2 )) b(g1 ) − b(g2 ) 12 (δ+ (g2 ) + η+ (g2 ))(b(g1 ) + b(g2 )), (5.47) g2 − g1 which are exactly (3.22) and (3.23). The Lipschitz continuity of a(g) and b(g) follows and, hence, they are absolutely continuous functions. Their a.e. derivatives can be estimated from (5.46) and (5.47): a (h) δ+ (h) + η+ (h), (5.48) δ− (h) + η− (h) a(h) b (h) δ+ (h) + η+ (h). (5.49) δ− (h) + η− (h) − b(h) Using the abolute continuity, we may integrate over h obtaining g2 g2 a(g2 ) (δ− (h) + η− (h)) dh log (δ+ (h) + η+ (h)) dh (5.50) a(g1 ) g1 g1 1 (δ (g ) 2 − 2
and
g2 g1
b(g1 ) (δ− (h) + η− (h)) dh log b(g2 )
g2
(δ+ (h) + η+ (h)) dh.
(5.51)
g1
Performing the integrations, we obtain (3.20) and (3.21), from which (3.18) follows. If g0 < 1/Z+ , we must have limg↑g0 a(g) = limg↑g0 b(g). This follows from (5.41), (5.44) and (5.45), since by (5.25) each µ(n) j (h) is a locally Lipschitz continuous functions of h ∈ (−1/Z+ , 1/z+ ), uniformly in n. Thus, if g ∈ [g0 , 1/Z+ ) it follows that [a, b] ⊂ g ; we set a(g) = b(g) = limg↑g0 a(g). Theorem 3.10 is proven. 6. The Wegner Estimate In this section we prove Theorem 4.4. We proceed as in [FK3, Theorem 23], but we must take into consideration two random coefficients. To do so, we make use of the unitary equivalence between the operators (Wg,ω, )⊥ and (Wg,ω,∗, )⊥ . We assume that Wg,ω is the partially elliptic operator without loss of generality. We start by picking κ ∈ (1, (1/g)(1/Z+ )), say 1 + gZ+ . (6.1) κ= 2gZ+ We rewrite γg,ω and ζg,ω in the form ωˆ i ui , (6.2) γg,ω = γˆ + g i∈Zd
ζg,ω = ζˆ + g
i∈Zd
ωˆ i vi ,
(6.3)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
179
where γˆ = 1 − κg
ui
1 − gU+ > 0, 2
(6.4)
vi
1 − gV+ > 0, 2
(6.5)
i∈Zd
ζˆ = 1 − κg
i∈Zd
and ωˆ i = ωi + κ ∈ [κ − 1, κ + 1] for each i ∈ Zd . We fix = L (x) with x ∈ Zd and L ∈ 2qN. The finite volume operators operators (Wg,ω, )⊥ and (Wg,ω,∗, )⊥ are unitarily equivalent by [KK, Lemma A.1]. Since (Wg,ω, )⊥ has compact resolvent by [KK, Proposition 3.2], so does (Wg,ω,∗, )⊥ , and they have the same eigenvalues, say {λg,ω,n }n∈N . We will denote by {ψg,ω,n }n∈N and {ϕg,ω,n }n∈N the corresponding orthonormal eigenfunctions for (Wg,ω, )⊥ and (Wg,ω,∗, )⊥ , respectively. Note that they can may chosen so they are measurable functions of ω. d (j ) Given j ∈ Zd , we set ε (j ) ∈ RZ by εi = δj,i . Note that Kg,ω+sε(j) (x) and Rg,ω+t ε(j) (x) are coefficient matrices for |s|, |t| sufficiently small; the corresponding classical wave operators will be denoted by Wg,ω (s, t; j ), etc. The remarks of the previous paragraph still apply. Note that we can choose each λg,ω,n (s, t; j ) ˘ ∩ Zd , we have jointly analytic in s and t. If j ∈ ∂ ∂ ∂ λg,ω,n (s, t; j )|(s,t )=(0,0) + λg,ω,n (s, t; j )|(s,t )=(0,0) λg,ω,n = ∂ ωˆ j ∂s ∂t ∂ Wg,ω,∗, (s, t; j )|(s,t )=(0,0) ϕg,ω,n + = ϕg,ω,n , ∂s ∂ Wg,ω, (s, t; j )|(s,t )=(0,0) ψg,ω,n + ψg,ω,n , ∂t = ϕg,ω,n , Wg,ω,∗, (gu j K0 ) ϕg,ω,n + + ψg,ω,n , Wg,ω, (gvj R0 ) ψg,ω,n , where we used (4.7), with uj +Li , vj = vj +Li , u j = i∈Zd
(6.6)
(6.7)
i∈Zd
and Wg,ω,∗, (K1 ) and Wg,ω, (R1 ) the finite volume operators defined by Wg,ω,∗, (K1 ) = Rg,ω D K1 D∗ Rg,ω , Wg,ω, (R1 ) = Kg,ω D∗ R1 D Kg,ω .
(6.8) (6.9)
Since (Wg,ω, )⊥ 0 has compact resolvent, we may define Ng,ω, (E) = tr χ(−∞,E] ((Wg,ω, )⊥ ),
(6.10)
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ABEL KLEIN AND ANDREW KOINES
the number of eigenvalues of (W )⊥ that are less than or equal to E. If E 0, we have NW (E) = 0, and if E > 0, NW (E) is the number of eigenvalues of Wg,ω, (or (Wg,ω, )⊥ ) in the interval (0, E]. Notice that Ng,ω, (E) is the distribution function of the measure ng,ω, (dE) given by (6.11) h(E)ng,ω, (dE) = tr(h((Wg,ω, )⊥ )), for positive continuous functions h of a real variable. Note also that Ng,ω, (E) = Y (E − λg,ω,n ),
(6.12)
n∈N
where Y (x) is the Heaviside function. Let f be a positive, continuous function on the real line with f (0) = 0, and ˘ ∩ Zd . We have, using (6.6), that j ∈ ∂ Ng,ω, (E)f (E) dE − ∂ ωˆ j ∞ ∂ Y (E − λg,ω,n )f (E) dE = − ∂ ωˆ j n=1 ∞ ∂ λg,ω,n δ(E − λg,ω,n )f (E) dE = ∂ ωˆ j n=1 =
∞
f (λg,ω,n )
n=1
∂ λg,ω,n ∂ ωˆ j
∞ = {f (Wg,ω,∗, )ϕg,ω,n , Wg,ω,∗, (gu j K0 ) ϕg,ω,n + n=1
+ f (Wg,ω, )ψg,ω,n , Wg,ω, (gvj R0 ) ψg,ω,n } = tr{Wg,ω,∗, (gu j K0 )f (Wg,ω,∗, )} + tr{Wg,ω, (gvj R0 )f (Wg,ω, )}. (6.13)
The last step used the fact that f (0) = 0. Thus ∂ ωˆ i Ng,ω, (E)f (E) dE − ∂ ω ˆ i d ˘ i∈∩Z
= tr{Wg,ω,∗, ((γg,ω − γˆ )K0 )f (Wg,ω,∗, )} + + tr{Wg,ω, ((ζg,ω − ζˆ )R0 )f (Wg,ω, )} = tr{Wg,ω,∗, f (Wg,ω,∗, )} − tr{Wg,ω,∗, (γˆ K0 )f (Wg,ω,∗, )} + + tr{Wg,ω, f (Wg,ω, )} − tr{Wg,ω, (ζˆ R0 )f (Wg,ω, )}.
(6.14)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
181
We have, for any ω ∈ [−1.1]Z (and hence also for ω ), that −1 −1 ωˆ i ui γˆ + g γˆ γg,ω = γˆ d
i∈Zd
1+
(κ − 1)gU− (1 − gZ+ )U− 1+ , 1 − κgU− 2Z+
(6.15)
and similarly, ζˆ −1 ζg,ω 1 +
(1 − gZ+ )V− . 2Z+
(6.16)
Since f 0, we obtain tr{Wg,ω,∗, (γˆ K0 )f (Wg,ω,∗, )} (1 − gZ+ )U− −1 tr{Wg,ω,∗, f (Wg,ω,∗, )}, 1+ 2Z+ tr{Wg,ω, (ζˆ R0 )f (Wg,ω, )} (1 − gZ+ )V− −1 tr{Wg,ω, f (Wg,ω, )}. 1+ 2Z+
(6.17)
(6.18)
In addition, using the unitary equivalence between (Wg,ω, )⊥ and (Wg,ω,∗, )⊥ , we get tr{Wg,ω,∗, f (Wg,ω,∗, )} = tr{Wg,ω, f (Wg,ω, )}.
(6.19)
It follows from (6.14)–(6.19) that tr{Wg,ω, f (Wg,ω, )} ∂ (2Z+ + (1 − gZ+ )Z− )2 ωˆ i Ng,ω, (E)f (E) dE , − 2(1 − gZ+ )Z+ Z− ∂ ωˆ i d
(6.20)
˘ i∈∩Z
where we used (1 − gZ+ )V− −1 (1 − gZ+ )U− −1 + 1− 1+ 1− 1+ 2Z+ 2Z+ =
1−gZ+ 1−gZ+ U− V− 2Z+ 2Z+ + + + 1 + 1−gZ U− 1 + 1−gZ V− 2Z+ 2Z+ 1−gZ+ (U− + V− ) 2Z+ 1−gZ+ + V 1 + 2Z+ U− 1 + 1−gZ − 2Z+ 1−gZ+ Z − 2(1 − gZ+ )Z+ Z− 2Z+
1+
2 1−gZ+ Z − 2Z+
=
(2Z+ + (1 − gZ+ )Z− )2
.
(6.21)
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ABEL KLEIN AND ANDREW KOINES
For given j ∈ Zd we set ω(j ) = {ωi ; i ∈ Zd \{j }}, and denote the corresponding ˘ ∩ Zd , expectation by E(j ) . We have, for j ∈ ∂ Ng,ω, (E)f (E) dE E − ∂ ωˆ j κ+1 ∂ (j ) Ng,ω, (E)f (E) dE ρ(ωˆ j − κ) dωˆ j − = E ∂ ωˆ j κ−1 (j ) |Ng,{ω(j) ,ωj =−1}, (E) − Ng,{ω(j) ,ωj =1}, (E)|f (E) dE ρ ∞ E −d/2 d 2nCd (Kg,− Rg,− )
ρ ∞ L (6.22) E d/2 f (E) dE, where we used [KK, Lemma 3.3] in the last step. Cd is a constant depending only on d, and is the constant in (2.17). Now let n¯ g, (dE) = E(ng,ω, (dE)). For functions f as above, it now follows from (6.11), (6.20), and (6.22), that Ef (E)n¯ g, (dE) = E{tr{Wg,ω, f (Wg,ω, )}} 2d (6.23) E d/2 f (E) dE, C ρ ∞ L where C = 2nCd (2 + r)d (κ + 1)
(2Z+ + (1 − gZ+ )Z− )2 (Kg,− Rg,− )−d/2 2(1 − gZ+ )Z+ Z−
36nCd (2 + r)d (K0,− R0,− )−d/2 . d+1 gZ− (1 − gZ+ )
(6.24)
We can now conclude that n¯ g, (dE) is absolutely continuous with n¯ g, (dE) C ρ ∞ E (d/2)−1L2d dE
for E 0.
(6.25)
The estimate (4.17) now follows by a standard argument: P{dist(σ (Wg,ω, ), E) < η} ng,ω, (dE) 1 P (E−η,E+η) n¯ g, (dE) 2d/2 C ρ ∞ E (d/2)−1ηL2d , (E−η,E+η)
for all E > 0 and 0 η E. Theorem 4.4 is proven.
(6.26)
A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: II
183
Appendix: Measurability of Random Classical Wave Operators In this appendix we prove measurability for the random classical wave operators Wg,ω and Wg,ω . We also prove measurability for Wg,ω,∗ . We recall that a random operator is a mapping ω → Hω from a probability space to self-adjoint operators on a Hilbert space, such that the mappings ω → f (Hω ) are strongly measurable for all bounded measurable functions f on R. It suffices to require weak measurability. (See [KM1, CL].) PROPOSITION A.1. If the random medium satisfies Assumption 3.1, then Wg,ω , Wg,ω , and Wg,ω,∗ are random operators. Proof. We start by showing that Wg,ω is a random operator. To do so, we prove that (Wg,ω ∓i)−1 is strongly measurable. It then follows from the resolvent identity, continuity of the resolvent, and a connectedness argument that (Wg,ω − z)−1 is strongly measurable for all nonreal z, and hence Wg,ω is a random operator by [KM1, Theorem 3]. Note that we may write (A.1) Wg,ω = Sg,ω WD Sg,ω , where Sg,ω = Kg,ω ⊕ Rg,ω and WD is given by (2.23) with A = D. Thus −1/2 −1 −1 −1/2 (WD ∓ iSg,ω ) Sg,ω , (Wg,ω ∓ i)−1 = Sg,ω
(A.2)
−1 −1 ) is strongly measurable. so it suffices to show that (WD ∓ iSg,ω Let λ > 0; using the resolvent identity we get −1 −1 ) = (WD ∓ iλ)−1 ∓ (WD ∓ iSg,ω −1 −1 −1 ) (λ − Sg,ω )(WD ∓ iλ)−1 , ∓ i(WD ∓ iSg,ω
(A.3)
hence −1 −1 −1 ) (1 ± i(λ − Sg,ω )(WD ∓ iλ)−1 ) = (WD ∓ iλ)−1 . (WD ∓ iSg,ω
(A.4)
If λ > (min{K− , R− })−1 , we have −1 −1 )(WD ∓ iλ)−1 1 − λ−1 Sg,ω
(λ − Sg,ω
1 − λ−1 (min{K− , R− })−1 < 1,
(A.5)
and hence −1 −1 −1 ) = (WD ∓ iλ)−1 (1 ± i(λ − Sg,ω )(WD ∓ iλ)−1 )−1 . (WD ∓ iSg,ω
(A.6)
−1 −1 ) follows. The strong measurability of (WD ∓ iSg,ω We have proved that Wg,ω is a random operator. It follows that W2g,ω is also a / [0, ∞). Thus random operator since (W2g,ω − z)−1 is strongly measurable if z ∈ 2 Wg,ω and Wg,ω,∗ are random operators in view of (2.26).
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ABEL KLEIN AND ANDREW KOINES
Acknowledgements The authors thanks Maximilian Seifert for many discussions and suggestions. A. Klein also thanks Alex Figotin, François Germinet, and Svetlana Jitomirskaya for enjoyable discussions.
References [AM] [ASFH] [AENSS] [CL] [CH] [CHT]
[DS] [FK1] [FK2] [FK3] [FK4] [FK5] [FK6] [FK7] [GD] [GK1] [GK2] [GK3] [GK4] [GK5]
Aizenman, M. and Molchanov, S.: Localization at large disorder and extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), 245–278. Aizenman, M., Schenker, J., Friedrich, R. and Hundertmark, D.: Finite-volume criteria for Anderson localization, Comm. Math. Phys. 224 (2001), 219–253. Aizenman, M., Elgart, A., Naboko, S., Schenker, J. and Stolz, G.: In preparation. Carmona, R. and Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990. Combes, J. M. and Hislop, P. D.: Localization for some continuous, random Hamiltonian in d-dimension, J. Funct. Anal. 124 (1994), 149–180. Combes, J. M., Hislop, P. D. and Tip, A.: Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 381–428. Damanik, D. and Stollman, P.: Multi-scale analysis implies strong dynamical localization, Geom. Funct. Anal. 11 (2001), 11–29. Figotin, A. and Klein, A.: Localization phenomenon in gaps of the spectrum of random lattice operators, J. Statist. Phys. 75 (1994), 997–1021. Figotin, A. and Klein, A.: Localization of electromagnetic and acoustic waves in random media. Lattice Model, J. Statist. Phys. 76 (1994), 985–1003. Figotin, A. and Klein, A.: Localization of classical waves I: Acoustic waves, Comm. Math. Phys. 180 (1996), 439–482. Figotin, A. and Klein, A.: Localization of classical waves II: Electromagnetic waves, Comm. Math. Phys. 184 (1997), 411–441. Figotin, A. and Klein, A.: Localized classical waves created by defects, J. Statist. Phys. 86 (1997), 165–177. Figotin, A. and Klein, A.: Midgap defect modes in dielectric and acoustic media, SIAM J. Appl. Math. 58 (1998), 1748–1773. Figotin, A. and Klein, A.: Localization of light in lossless inhomogeneous dielectrics, J. Opt. Soc. Amer. A 15 (1998), 1423–1435. Germinet, F. and De Bièvre, S.: Dynamical localization for discrete and continuous random Schrödinger operators, Comm. Math. Phys. 194 (1998), 323–341. Germinet, F. and Klein, A.: Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), 415–448. Germinet, F. and Klein, A.: Decay of operator-valued kernels of functions of Schrödinger and other operators, Proc. Amer. Math. Soc. 131 (2003), 911–920. Germinet, F. and Klein, A.: A characterization of the Anderson metal-insulator transport transition, Duke Math. J., to appear. Germinet, F. and Klein, A.: Explicit finite volume criteria for localization in continuous random media and applications, Geom. Funct. Anal., to appear. Germinet, F. and Klein, A.: High disorder localization for random Schrödinger operators through explicit finite volume criteria, Markov Process. Related Fields, to appear.
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[GK6] Germinet, F. and Klein, A.: The Anderson metal-insulator transport transition, Contemp. Math., to appear. [HM] Holden, H. and Martinelli, F.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator, Comm. Math. Phys. 93 (1984), 197–217. [Kle] Klein, A.: Localization of light in randomized periodic media, In: J.-P. Fouque (ed.), Diffuse Waves in Complex Media, Kluwer, Dordrecht, 1999, pp. 73–92. [KK] Klein, A. and Koines, A.: A general framework for localization of classical waves: I. Inhomogeneous media and defect eigenmodes, Math. Phys. Anal. Geom. 4 (2001), 97–130. [KKS] Klein, A., Koines, A. and Seifert, M.: Generalized eigenfunctions for waves in inhomogeneous media, J. Funct. Anal. 190 (2002), 255–291. [Klo1] Klopp, F.: Localization for continuous random Schrödinger operators, Comm. Math. Phys. 167 (1995), 553–569. [Klo2] Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators, Duke Math. J. 98 (1999), 335–396. [Klo3] Klopp F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians, Ann. Inst. H. Poincaré 3 (2002), 711–737. [KM1] Kirsch, W. and Martinelli, F.: On the ergodic properties of the spectrum of general random operators, J. Reine Angew. Math. 334 (1982), 141–156. [KM2] Kirsch, W. and Martinelli, F.: On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys. 85 (1982), 329–350. [KSS] Kirsch, W., Stolz, G. and Stollman, P.: Localization for random perturbations of periodic Schrödinger operators, Random Oper. Stochastic Equations 6 (1998), 241–268. [Na] Najar, H.: Asymptotic of the integrated density of states of random acoustic operators, C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), 191–194. [PF] Pastur, L. and Figotin, A.: Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Heidelberg, 1992. [SW] Schulenberger, J. and Wilcox, C.: Coerciveness inequalities for nonelliptic systems of partial differential equations, Arch. Rational Mech. Anal. 88 (1971), 229–305. [WBLR] Wiersma, D., Bartolini, P., Lagendijk, A. and Righini, R.: Localization of light in a disordered medium, Nature 390 (1997), 671–673. [Wi] Wilcox, C.: Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal. 22 (1966), 37–78.
Mathematical Physics, Analysis and Geometry 7: 187–192, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Forces along Equidistant Particle Paths P. COULTON and G. GALPERIN Department of Mathematics, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL 61920, U.S.A. e-mail: {cfprc, cfgg}@eiu.edu (Received: 28 October 2002; in final form: 7 May 2003) Abstract. Two particles on the sphere leave the equator moving due south and travel at a constant and equal speed along a geodesic colliding at the south pole. An observer who is unaware of the curvature of the space will conclude that there is an attractive force acting between the particles. On the other hand, if particles travel at the same speed (initially parallel) along geodesics in the hyperbolic plane, then the particle paths diverge. Imagine two particles in the hyperbolic plane that are bound together at a constant distance with their center of mass traveling along a geodesic path at a constant velocity, then the force due to the curvature of the space acts to break the bond and increases as a quadratic function of the velocity. We consider this problem for the sphere and the hyperbolic plane and we give the exact formula for the apparent force between the particles. Mathematics Subject Classifications (2000): 53Axx, 70Exx, 85-XX. Key words: geodesic, curvature, relativity.
1. Introduction In this paper we wish to study the apparent force on particles traveling in a two dimensional space with constant sectional curvature. It is well known that geodesics in positively curved space will tend to converge and that particles traveling along geodesic paths in negatively curved space will tend to diverge. Let M denote a 2-dimensional surface of constant nonzero curvature with Riemannian metric g(, ). We will assume that the mass m is constant unless stated otherwise. Let σ (s) denote a geodesic path in the manifold M such that σ (0) = x0 and d/dsσ (s)|s=0 = u0 , where the magnitude of u0 is 1. Recall that the speed of a geodesic path is always a unit. We define an inertial path, µ(t), as a path along a geodesic such that the speed is constant and equal to the initial speed v0 . In other words: d d µ(t) = σ (v0 t), dt dt where σ (t) is a geodesic. We say that a path is a constant speed path if it has no tangential acceleration component. Let λ(t) denote a constant speed path. The external force required for the particle to follow this path is given by F=m
d2 λ(t) . dt 2
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Let p1 and p2 denote particles, each of mass m traveling along constant speed paths λ1 (t) and λ2 (t) respectively, such that the two particles move at a constant speed v at a distance d/2 from the central inertial path. We will say that such a pair of particles along with their respective paths are coupled. The force required to keep each on its path is the coupling force. This is precisely the tension in some imaginary connecting rod. The coupling force is positive when paths are convergent and negative when the paths are divergent. Assume that the direction from the midpoint to the path denoted by λ1 (t) is in the positive direction, then the coupling force is defined by 2 d λ1 (t) d2 λ2 (t) − , Fc = m dt 2 dt 2 where the force may be assumed to be defined by the tension of the connecting rod at the midpoint. We observe that if the force on the particles is only in the direction of the connecting rod and the forces at the midpoint due to the motion is equal and in opposite directions, then the midpoint will continue to move along the geodesic at a constant speed. This is expressed by the Lagranian equation applied at the midpoint: 2 d λ1 (t) d2 λ2 (t) + = 0. m dt 2 dt 2 We also observe that the energy is given by m dλ2 dλ2 m dλ1 dλ1 · + · , E= 2 dt dt 2 dt dt and that the energy is minimized whenever dλ1 d2 λ1 dλ2 d2 λ2 dE =m · 2 +m · 2 = 0. dt dt dt dt dt We conclude that the motion is possible provided that the velocity and the acceleration are perpendicular at each point. This fact follows from the definition of motion in this problem. We will prove the following: THEOREM. Let M be a manifold with constant nonzero sectional curvature K. Then the coupling force between two coupled particles of mass m moving at speed v at a distance d/2 from a central inertial path is given by Fc = 2mκλ(t )v 2 , where κλ(t ) is the geodesic curvature for the path λ(t) and where √ √ Kd κλ(t ) = K tan 2
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in the case of positive curvature, and √ √ −Kd , κλ(t ) = − −K tanh 2 in the case of negative curvature. Note that this force is essentially a function of the geodesic curvature and thus the geometry of the given curve and the manifold. 2. Proof of the Theorem The geodesic curvature is the signed magnitude of the arc-length parameter derivative of a unit tangent vector along the given curve, where the sign is determined by the orientation of the curve with respect to the manifold. The case of positive sectional curvature is straight forward. Recall that geodesic curvature κρ for a circle of radius ρ in on the 2-sphere is given by √ √ κρ = K cot(ρ K). Observe that if the particles in question move at unit speed, then the second derivative with respect to the arc-length is the geodesic curvature. If d2 λ(t) = κρ , dt 2
then
d2 λ(vt) = v 2 κρ . dt 2
The radius of our curve with respect to the north pole of the sphere is ρ = (rπ − d)/2. This yeilds √ √ √ d K d K rπ K − = tan , κρ = cot 2 2 2 √ where K = 1/r. We conclude that the magnitude of the force on each particle is given by F = mv 2 κλ(t ), where λ(t) represents a curve of constant speed at a constant distance d/2 from some geodesic. This proves the theorem in the case of positive curvature. We next consider the case of a space form M with sectional curvature K = −1/k 2 . The Minkowski space is defined by R2,1 = {(x, y, z) | x, y, z ∈ R}, such that the distance between points P = (x1 , y1 , z1 ) and Q = (x2 , y2 , z2 ) is given by ρ(P , Q) = (x1 − x2 )2 + (y1 − y2 )2 − (z1 − z2 )2 .
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Figure 1. The hyperbolic case.
Then the hyperbolic plane H 2 (−k) is defined by the set locus x2 + y2 +
1 = z2 , k2
subject to the induced metric. We obtain a coordinate system for the hyperbolic plane in the xy-plane under the projection (see Figure 1) (x, y, z) −→ (x, y). A parametric equation for a curve of constant distance from the y-axis is given by σ (t) =
α t α 2 + t 2 + 1/k 2
.
Two curves of this type, i.e. symmetric with the y-axis, will give a pair of inertial paths that are equidistant for all t. However, it is convenient to do our calculations at the origin of the coordinate system. It suffices to translate the curve σ (t) back to the origin using an appropriate isometry in hyperbolic space: α
√ 2 2 k α +1 0 −αk t 0 1 √ 0 λ(t) = . 2 2 k α +1 −αk 0 α 2 + t 2 + 1/k 2
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At t = 0, we obtain √ −αk 2 / k 2 α 2 + 1 d2 λ (0) = 0 , dt 2 √ k/ k 2 α 2 + 1 and
2 2 d γ (0) = √ αk . dt 2 k2α2 + 1
The velocity for this curve is not constant but it does take an extremum at t = 0 and the curve has unit speed at t = 0. Thus the second derivative at t = 0 gives the geodesic curvature. We conclude that κλ(t ) = k tanh(kd/2),
where α =
1 sinh(dk/2). k
Therefore, the coupling force is √ √ Fc = −2mv 2 −K tanh(d −K/2). √ If |K|d is sufficiently small, we may approximate by the coupling force by |Fc | ≈ 2W |K|d, where W is the kinetic energy. This formula holds in both cases. 3. Conclusion Particles that are bound together by chemical or nuclear forces may be studied as though they were connected by a massless rod. Force fields in three space, i.e. gravitation, electric and magnetic fields, have characteristics of both positive and negative curvature, depending on the charge of the particles and the motion relative to the field. Observe that the coupling force in each case is related to the kinetic energy and is nearly a linear function of the distance for small distances. From a relativistic point of view, we see that even when the velocity is bounded, the kinetic energy is not. As the kinetic energy increases so does the coupling force. In the case of a negatively curved manifold, the coupling force necessary to keep the particles together must be applied inward, but if the kinectic energy is large enough and the absolute value of the curvature is increasing, then the typical chemical or nuclear binding forces between particles may be broken. High energy particles are sometimes injected into relatively large electric fields or gravitational fields to separate elementary particles. The coupling force relation in this paper should give a reasonable estimate of the force of separation (or combination) in such cases. Finally, note that the force required to keep a particle moving along a path at constant speed is proportional to the geodesic curvature. One must compute the
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geodesic curvature to solve the problem on a general surface, assuming no rotation about the center of mass. Acknowledgement The authors would like to thank the referee for comments which helped to simplify the proof of the theorem. Reference 1.
Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. II, Interscience, New York, 1969.
Mathematical Physics, Analysis and Geometry 7: 193–221, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Three Term Recursion Relation for Spherical Functions Associated to the Complex Projective Plane INÉS PACHARONI and JUAN A. TIRAO CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina. {pacharon, tirao}@mate.uncor.edu (Received: 27 February 2003) Abstract. The aim of this paper is to prove a three term recursion relation for a sequence of matrix w) on G = SU(3) built up of + 1 spherical functions of a given type valued functions (g, πn, , associated to the complex projective plane G/K, K = S(U(2) × U(1)). The three term recursion relation that constitutes our main result, Theorem 5.2, together with the fact that the functions w) are eigenfunctions of all differential operators on G which are left invariant under G and (g, right invariant under K, provides for each ∈ N0 a solution of a matrix valued extension of the Bochner’s problem to G. In fact by restriction to an Abelian Iwasawa subgroup of G, for each (t, w) of matrix valued polynomial functions on t which satisfies ∈ N0 , we obtain a sequence H a three term recursion relation and such that they are eigenfunctions of a second order differential (t, w) satisfies both conditions explicitly asked for by operator on 0 < t < 1. Thus each sequence H Bochner. Mathematics Subject Classifications (2000): Key words:
1. Introduction The general theory of scalar valued spherical functions of arbitrary type, associated to a pair (G, K) with G a locally compact group and K a compact subgroup, goes back to Godement and Harish-Chandra. In [9], attention is focused on the underlying matrix valued spherical functions defined as a solution of an integral identity, see Definition 2.1. These two notions are related by the operation of taking traces. When G is a Lie group the general theory, see [9, 3], gives for a fixed irreducible representation (π, V ) of K a family of matrix valued functions that are eigenfunctions of a system of left invariant differential operators defined on the Lie group G. These spherical functions in fact take values in the set of linear maps from V into itself. Partially supported by CONICET grant PIP655-98.
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In [4] one finds a detailed elaboration of this theory when the symmetric space G/K is the complex projective plane. In this case we have G = SU(3) and K = S(U(2) × U(1)). In particular one constructs, out of several spherical functions of a given type πn, , n ∈ Z and ∈ N0 , a sequence of ( + 1) × ( + 1)-matrix valued (t, w), 0 < t < 1, w max{0, −n}, such that as functions polynomial functions H of the spectral parameter w they satisfy a three term recursion relation of the form (t, w) = Aw H (t, w − 1) + Bw H (t, w) + Cw H (t, w + 1), tH
(1)
where Aw , Bw , Cw are matrices independent of t. On the other hand it is also (t, w) satisfy a differential equation of the form proved that the functions H (t, w)T , (t, w)T = H DH
(2)
where D is a second order differential operator in the variable t whose coefficients depend on t (and not on w). Here is a diagonal matrix with entries that depend on w but not on t. In 1929, S. Bochner [1] solved the problem of determining all families of scalar valued orthogonal polynomials that are eigenfunctions of some arbitrary but fixed second order differential operator. From this point of view one can interpret the results (1) and (2) as an instance of a matrix valued solution to Bochner’s problem. In [4] the three term recursion relation (1) was conjectured for all spherical functions of type πn, for n 0 and any 0. In [5] explicit formulae for the coefficient matrices Aw , Bw and Cw were given as well as a sketch of the way in which the tensor product of certain representations of G could be used to get this result. The aim of this paper is to give a proof of (1). Our strategy is to work with a (g, w) on G built up sequence of ( + 1)2 × ( + 1)-matrix valued functions of + 1 spherical functions of a given type πn, , which parallels the construc(t, w). The three term recursion relation that constitutes our main result, tion of H Theorem 5.2, as is given by w w w (g, w − 1) + B (g, w) + C (g, w + 1) (3) (g, w) = A φ(g)ψ(g) generalizes (1) for spherical functions on G of arbitrary type. It is important to stress that this relation is valid on G, and not just on a one-dimensional submanifold is of G. On the other hand from Proposition 2.3(iii) it follows that for each w, an eigenfunction of all differential operators on G which are left invariant under G (g, w) provides an extension to and right invariant under K. Thus the sequence G of the matrix valued version of the Bochner’s problem. Moreover, (1) follows easily from (3) by restriction, see Proposition 5.3. (t, w) are polynomials in t and are seen in [4, Section 7], to The functions H satisfy orthogonality relations. Nevertheless it is important to notice that the classical argument that gives from here a three term recursion relation cannot be applied directly. It may also be important to repeat a remark from [4]: the polynomial
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matrix valued functions considered in Section 12 do not satisfy all the conditions of the theory in [2]. Besides it is not at all clear how to go from the left-hand side of (1) to the left-hand side of (3). An example that was very useful in the development of our work is the following which corresponds to the spherical functions of one-dimensional types. For each one-dimensional representation π = πn,0 , of K the functions H (t, w), with w max{0, −n}, associated in [4] to the irreducible spherical functions (g, w) of type πn,0 are given by (−1)w (n,1) P (1 − 2t), if n 0, w 0 w+1 w H (t, w) = (−1)w+n t −n (−n,1) (1 − 2t), if n < 0, w −n, P w + n + 1 w+n (α,β)
here Pj (z) are the Jacobi polynomials in the interval [−1, 1]. These polynomials satisfy the following three term recursion relation (α,β)
(2j + α + β + 1)(2j + α + β)(2j + α + β + 2)zPj
(z)
(α,β)
= 2(j + α)(j + β)(2j + α + β + 2)Pj −1 (z)− (α,β)
− (2j + α + β + 1)(α 2 − β 2 )Pj
(z)+ (α,β)
+ 2(j + 1)(j + α + β + 1)(2j + α + β)Pj +1 (z). From this we obtain tH (t, w) = Aw H (t, w − 1) + Bw H (t, w) + Cw H (t, w + 1),
(4)
where Aw , Bw and Cw are suitable constants. Moreover, appealing to property (ii) in Proposition 2.3, for all g ∈ G we get φ(g)ψ(g)(g, w) = Aw (g, w − 1) + Bw (g, w) + Cw (g, w + 1), (5) where φ(g) is the spherical function (g, 1) of type π−1,0 , and ψ(g) is the spherical function (g, 0) of type π1,0 , which is an instance of the three term recursion relation (1). 2. Background The aim of this section is to collect the necessary material to obtain a three term recursion relation for the spherical functions associated to the pair (G, K) = (SU(3), S(U(2) × U(1)).
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2.1. THE LIE ALGEBRA OF SU(3) t
The Lie algebra of G is g = {X ∈ gl(3, C) : X = −X , tr X = 0}. Its complexification is gC = sl(3, C). The Lie algebra k of K can be identified with u(2) and its complexification kC with gl(2, C). The following matrices form a basis of g. i 0 0 i 0 0 H1 = 0 −i 0 , H2 = 0 i 0 , 0 0 0 0 0 −2i 0 1 0 0 i 0 Y2 = i 0 0 , Y1 = −1 0 0 , 0 0 0 0 0 0 0 0 1 0 0 i Y4 = 0 0 0 , Y3 = 0 0 0 , −1 0 0 i 0 0 0 0 0 0 0 0 Y5 = 0 0 1 , Y6 = 0 0 i . 0 −1 0 0 i 0 Let h be the Cartan subalgebra of gC of all diagonal matrices. The corresponding root space structure is given by 0 1 0 0 0 0 1 0 0 X−α = 1 0 0 , Hα = 0 −1 0 , Xα = 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X−β = 0 0 0 , Hβ = 0 1 0 , Xβ = 0 0 1 , 0 0 0 0 1 0 0 0 −1 0 0 1 0 0 0 1 0 0 X−γ = 0 0 0 , Hγ = 0 0 0 , Xγ = 0 0 0 , 0 0 0 1 0 0 0 0 −1 where α(x1 E11 + x2 E22 + x3 E33 ) = x1 − x2 , β(x1 E11 + x2 E22 + x3 E33 ) = x2 − x3 , γ (x1 E11 + x2 E22 + x3 E33 ) = x1 − x3 . We have Xα = 12 (Y1 − iY2 ), X−α = − 12 (Y1 + iY2 ),
Xβ = 12 (Y5 − iY6 ),
Xγ = 12 (Y3 − iY4 ),
X−β = − 12 (Y5 + iY6 ),
X−γ = − 12 (Y3 + iY4 ). We take {α, β, γ } as the set of positive roots of (gC , h). Then λα = 13 (2α + β) and λβ = 13 (α + 2β) are the corresponding fundamental weights.
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2.2. SPHERICAL FUNCTIONS Let G be a locally compact unimodular group and let K be a compact subgroup denote the set of all equivalence classes of complex finite-dimensional of G. Let K let ξδ denote the character of irreducible representations of K; for each δ ∈ K, δ, d(δ) the degree of δ, i.e. the dimension of any representation in the class δ, and χδ = d(δ)ξ δ . We shall choose once and for all the Haar measure dk on K normalized by K dk = 1. We shall denote by V a finite-dimensional vector space over the field C of complex numbers and by End(V ) the space of all linear transformations of V into V . is a DEFINITION 2.1 ([9, 3]). A spherical function on G of type δ ∈ K continuous function on G with values in End(V ) such that (i) (e) = I . (I = identity transformation). (ii) (x)(y) = K χδ (k −1 )(xky) dk, for all x, y ∈ G. PROPOSITION 2.2 ([9, 3]). If : G → End(V ) is a spherical function of type δ then: (i) (kgk ) = (k)(g)(k ), for all k, k ∈ K, g ∈ G. (ii) k → (k) is a representation of K such that any irreducible subrepresentation belongs to δ. Spherical functions of type δ arise in a natural way upon considering representations of G. If g → U (g) is a continuous representation of G, say on a finite-dimensional vector space E, then
χδ (k −1 )U (k) dk P (δ) = K
is a projection of E onto P (δ)E = E(δ); E(δ) consists of those vectors in E, the linear span of whose K-orbit splits into irreducible K-subrepresentations of type δ. The function : G → End(E(δ)) defined by (g)a = P (δ)U (g)a,
g ∈ G, a ∈ E(δ)
is a spherical function of type δ. In fact, if a ∈ E(δ) we have (x)(y)a = P (δ)U (x)P (δ)U (y)a
χδ (k −1 )P (δ)U (x)U (k)U (y)a dk = K
−1 χδ (k )(xky) dk a. = K
If the representation g → U (g) is irreducible then the associated spherical function is also irreducible. Conversely, any irreducible spherical function on a compact group arises in this way.
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If G is a connected Lie group it is not difficult to prove that any spherical function : G → End(V ) is differentiable (C ∞ ), and moreover that it is analytic. Let D(G) denote the algebra of all left invariant differential operators on G and let D(G)K denote the subalgebra of all operators in D(G) which are invariant under all right translation by elements in K. In the following proposition (V , π ) will be a finite-dimensional representation of K such that any irreducible subrepresentation belongs to the same class δ ∈ K. PROPOSITION 2.3 ([9, 3]). A function : G → End(V ) is a spherical function of type δ if and only if (i) is analytic. (ii) (kgk ) = π(k)(g)π(k ), for all k, k ∈ K, g ∈ G, and (e) = I . (iii) [D](g) = (g)[D](e), for all D ∈ D(G)K , g ∈ G.
2.3. IRREDUCIBLE REPRESENTATIONS OF GL(3, C) We recall here some basic facts about the representation theory of GL(n, C), which can be found, for example, in [10, §67]. The equivalence classes of finite-dimensional irreducible holomorphic representations of GL(n, C) are parameterized by the n-tuples of integers m = (m1 , . . . , mn )
such that m1 · · · mn .
We denote by Vm the space of a representation in the class m. A highest weight vector in Vm is a vector 0 = v ∈ Vm invariant under the upper triangular subgroup N of GL(n, C). Since the subgroup of all diagonal matrices (ex1 , . . . , exn ) normalizes N and the subspace VmN of all N-invariant vectors in Vm is one-dimensional, it follows that the diagonal subgroup acts on VmN by a character, namely for v ∈ VmN we have (ex1 , . . . , exn )v = em1 x1 +···+mn xn v. We identify GL(n − 1, C) with the subgroup of GL(n, C) in the following way GL(n − 1, C) 0 GL(n − 1, C) . 0 1 When we restrict the representation m of GL(n, C) to GL(n − 1, C) it decomposes as the direct sum of representations of GL(n − 1, C) in the classes k = (k1 , . . . , kn−1 ) such that m1 k1 m2 k2 · · · kn−1 mn . Each of the representations of GL(n − 1, C) is contained in this decomposition exactly once.
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of GL(3, C) in the In particular when n = 3 the space Vm of a representation class m = (m1 , m2 , m3 ) decomposes as the direct sum Vm = Vk of irreducible representations of GL(2, C), where the sum is over all classes k such that m1 k1 m2 k2 m3 .
In turn each one of these representations Vk decomposes as the direct sum Vk = Vs of one-dimensional irreducible representations of GL(1, C) in the classes s = (s), such that k1 s k2 . Taking one nonzero vector from each one of these subspaces Vs , we get a basis for the entire space Vm . It is clear that each one of these vectors is uniquely determined, up to a scalar, by the following triangle of integers µ=
m2
m1 k1
m3 k2
,
s where m1 k1 m2 k2 m3
and
k1 s k2 .
We denote by vµ a basis vector corresponding to the triangle µ. It is easy to see that each basis vector vµ is a weight vector of the Cartan subalgebra of gl(3, C) of all diagonal matrices (x1 , x2 , x3 ) of weight x1 s + x2 (k1 + k2 − s) + x3 (m1 + m2 + m3 − k1 − k2 )vµ . The basis {vµ }µ taking above is known as a Gelfand–Cetlin basis of the module Vm . Let W = C3 denote the canonical irreducible GL(3, C) module, and let {e1 , e2 , e3 } be the canonical basis of C3 . Then e1 is a dominant vector of weight (1, 0, 0). Let V = Vm be any irreducible GL(3, C) module with m = (m1 , m2 , m3 ). We are interested in the decomposition of the tensor product V ⊗ W as a direct sum of GL(3, C) irreducible submodules. The following proposition is a special case of the so called Pieri’s formula, see [10, §77]. PROPOSITION 2.4. For V = Vm , m = (m1 , m2 , m3 ) we have V ⊗ W V σ1 ⊕ V σ2 ⊕ V σ3 , where V σ1 , V σ2 , V σ3 are irreducible GL(3, C) modules of parameters σ1 = (m1 + 1, m2 , m3 ), σ3 = (m1 , m2 , m3 + 1).
σ2 = (m1 , m2 + 1, m3 ),
Remark. The irreducible modules on the right-hand side whose parameters do not satisfy the conditions m1 m2 m3 have to be omitted.
(m1 , m2 , m3 )
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In other words the module V σ2 appears in the decomposition of V ⊗ W if and only if m1 = m2 , and V σ3 appears if and only if m2 = m3 . Our Lie group K = S(U(2) × U(1)) is isomorphic to U(2) under the map A 0 → A. 0 a Let us recall that the identity representation π1 of U(2) in C2 , as well as the -symmetric power of it π : A → A , A ∈ U(2), of dimension +1 are irreducible. Moreover the representations πn, of U(2) defined by πn, (A) = (det A)n A ,
n ∈ Z, ∈ Z0
give a complete set of representatives of elements in U(2). Then by composing πn, with the above isomorphism we get an irreducible representation of K which we shall still call πn, . We shall refer to (n, ) as the type of πn, . On the other hand the reader can easily see that if Vk = Vk1 ,k2 is an irreducible GL(2, C) submodule of the GL(3, C) module Vm , m = (m1 , m2 , m3 ), then Vk is also a K submodule of type (n, ) given by = k1 − k2 ,
n = k1 + 2k2 − m1 − m2 − m3 .
(6)
In particular let W1 be the irreducible GL(2, C) submodule of W = V(1,0,0) of dimension 1, i.e. W1 = Ce3 . Then W1 = V0,0 as a K-module is of type (−1, 0). 3. Multiplication Formulas The tensor product Vk ⊗ W1 is an irreducible GL(2, C) module of parameters (k1 , k2 ) + (0, 0) = (k1 , k2 ). The GL(3, C) projection Pj : Vm ⊗ W → V σj
(for j = 1, 2, 3) σ
maps Vk ⊗W1 onto the trivial module or onto the GL(2, C) submodule Vk1j,k2 of V σj . For any vµ in the Gelfand–Cetlin basis of Vm corresponding to the triangle m2 m3 m1 µ= k1 k2 k we have vµ ⊗ e3 = v1 + v2 + v3 ∈ V σ1 ⊕ V σ2 ⊕ V σ3 , where the vectors vj are weight vectors in V σj and belong to the GL(2, C) subσ modules Vk1j,k2 . Thus the corresponding triangles of v1 , v2 , v3 are respectively m1 + 1
m2 k1
,
k k2
k1 k
k1
.
m3 k2
k m3 + 1
m2
m1
m2 + 1
m1
m3 k2
,
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We note that the vector vµ ⊗ e3 is of weight (k, k1 + k2 − k, m1 + m2 + m3 + 1 − k1 − k2 ) σ
and each Vk1j,k2 is an irreducible K-module of type (k1 +2k2 −m1 −m2 −m3 −1, ) = (n − 1, ). It is well known (see [7, p. 32]) that there exists a basis {vi }i=0 of Vk such that π˙ (Hα )vi = ( − 2i)vi , π˙ (Xα )vi = ( − i + 1)vi−1 , π˙ (X−α )vi = (i + 1)vi+1 ,
(v−1 = 0), (v+1 = 0),
(7)
where n and are given by (6). Therefore we can normalize the basis of Vk , taken from the Gelfand–Cetlin basis of Vm , in such a way that (7) holds. LEMMA 3.1. Let us consider a U(2) invariant inner product on Vk . Then the basis {vi }i=0 described above is an orthogonal basis such that 2
v0 2 .
vi = i Proof. Let ∗ denote the adjoint operator in End(Vk ) corresponding to an invariant ˙ )∗ = −π(Y ˙ ) for all Y ∈ g we have inner product on Vk . Since π(Y ˙ α ), π˙ (Hα )∗ = i π˙ (H1 )∗ = −i π˙ (H1 ) = π(H ∗ ∗ 1 π˙ (X−α ) = − 2 (π˙ (Y1 ) + i π˙ (Y2 )) = − 12 (−π˙ (Y1 ) + i π˙ (Y2 )) = π˙ (Xα ). Since π˙ (Hα )∗ = π˙ (Hα ) and the vi ’s are eigenvectors corresponding to different eigenvalues of π˙ (Hα ) they are orthogonal to each other. Now the proof will be completed by induction on 0 i . The statement is clearly true for i = 0. Let us assume that the assertion is true for some 0 i − 1. Then (i + 1) vi+1 , vi+1 = π˙ (X−α )vi , vi+1 = vi , π˙ (Xα )vi+1 = ( − i) vi , vi . Thus −1 vi+1 , vi+1 = i+1
v0 , v0 = v0 , v0 . i i+1
2
PROPOSITION 3.2. Let {vi }i=0 be a basis Vk , k = (k1 , k2 ), such that (7) holds, and equip Vm with a G-invariant inner product such that v0 = 1. Similarly take on W the G-invariant inner product such that e3 = 1. Let ai = ai (m, k) be defined by v0 ⊗ e2 = a1 v0σ1 + a2 v0σ2 + a2 v0σ3 ∈ V σ1 ⊗ V σ2 ⊗ Vσ3 ,
(8)
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σ
with aj > 0 and v0 j = 1. Let vi j ∈ V σj be defined by vi ⊗ e3 = a1 viσ1 + a2 viσ2 + a3 viσ3 . σ
σ
Then {vi j }i=0 (j = 1, 2, 3) is a basis of an irreducible GL(2, C) module Vk j contained in V σj such that (7) holds. Hence σj
vi = . i σ
Remark. If Pj (v0 ⊗ e3 ) = 0 we take aj = 0 and we do not define vi j . Proof. Since Pj is in particular a GL(2, C) morphism and e3 is GL(2, C) inσ variant from (8) it follows that each v0 j is a GL(2, C) dominant vector of weight (k1 , k2 ). On the other hand we have i i i i (v0σ1 ) + a2 X−α (v0σ2 ) + a3 X−α (v0σ3 ) = X−α (v0 ⊗ e3 ) = i!vi ⊗ e3 a1 X−α = i!(a1 viσ1 + a2 viσ2 + a3 viσ3 ). σ
σ
i (v0 j ) = i!vi j for j = 1, 2, 3. This completes the proof of the Therefore X−α proposition. 2
THEOREM 3.3. Let be the irreducible spherical function of type (n, ) = (k1 + 2k2 − m1 − m2 − m3 , k1 − k2 ) associated to the G module Vm and the K submodule Vk = Vk1 ,k2 . Let φ be the spherical function of type (−1, 0) associated to the G module W . Let σj be the spherical functions of type (n − 1, ) associated to the G modules V σj , (j = 1, 2, 3). Then φ(g)(g) = a12 σ1 (g) + a22 σ2 (g) + a32 σ3 (g). Proof. Let ui =
−1/2 i
σ
vi and let ui j =
−1/2 i
σ
(9) σ
vi j . Then {ui }0 and {ui j }0 are,
σ
respectively, orthonormal bases of Vk and Vk j for j = 1, 2, 3. We recall now that if is the spherical function associated to the G module Vm and the K submodule Vk then by definition (g)a = P (ga) for all g ∈ G and all a ∈ Vk , where P is the K-projection of Vm onto Vk . Therefore if {ui }0 is an orthonormal basis of Vk we have ij (g) = (g)uj , ui = P (guj ), ui = guj , ui . Now on the one hand we have g(uj ⊗ e3 ), ui ⊗ e3 = guj , ui ge3 , e3 ,
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THREE TERM RECURSION RELATION
and on the other hand we get g(uj ⊗ e3 ), ui ⊗ e3 = a1 guσj 1 + a2 guσj 2 + a3 guσj 3 , a1 uσi 1 + a2 uσi 2 + a3 uσi 3 = a12 guσj 1 , uσi 1 + a22 guσj 2 , uσi 2 + a32 guσj 3 , uσi 3 . Therefore φ(g)ij (g) = a12 σij1 (g) + a22 σij2 (g) + a32 σij3 (g). This completes the proof of the theorem.
2
We rewrite (9) making explicit the dependence on the parameters m = (m1 , m2 , m3 ) and k = (k1 , k2 ). Then, up to equivalences of spherical functions, we have φ(g)m,k (g) = a12 (m, k)m+e1 ,k (g) + a22 (m, k)m+e2 ,k (g) + + a32 (m, k)m+e3 ,k (g),
(10)
where e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 = (0, 0, 1). In the following section we shall prove the following theorem. THEOREM 3.4. The constants ai (m, k) defined in Proposition 3.2 are given by (m1 − k1 + 1)(m1 − k2 + 2) , (m1 − m2 + 1)(m1 − m3 + 2) (k1 − m2 )(m2 − k2 + 1) , a22 (m, k) = (m1 − m2 + 1)(m1 − m3 + 1) (k1 − m3 + 1)(k2 − m3 ) . a32 (m, k) = (m1 − m3 + 2)(m2 − m3 + 1)
a12 (m, k) =
3.1. THE DUAL PICTURE Let V = Vm be an irreducible GL(n, C) module associated to m = (m1 , . . . , mn ). Then it is easy to see that V ∗ is associated to the parameters (−mn , . . . , −m1 ). In particular let W ∗ denote the GL(3, C) module dual to W . Thus W ∗ has parameters (0, 0, −1). Then from Proposition 2.4 we obtain the following proposition. PROPOSITION 3.5. If V = Vm then V ⊗ W ∗ V τ1 ⊕ V τ2 ⊕ V τ3 , where V τ1 , V τ2 , V τ3 are irreducible GL(3, C) modules of parameters τ1 = (m1 − 1, m2 , m3 ), τ3 = (m1 , m2 , m3 − 1).
τ2 = (m1 , m2 − 1, m3 ),
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THEOREM 3.6. Let m,k be the irreducible spherical function associated to the G module Vm and the K submodule Vk1 ,k2 . Let ψ be the spherical function of type (1, 0) associated to the G module W ∗ . Then up to equivalences of spherical functions we have ψ(g)m (g) = c12 (m, k)m−e1 ,k (g) + c22 (m, k)m−e2 ,k (g) + + c32 (m, k)m−e3 ,k (g),
(11)
where cj (m, k) = a4−j (−m3 , −m2 , −m1 , −k2 , −k1 ), j = 1, 2, 3. More explicitly (m1 − k2 + 1)(m1 − k1 ) , (m1 − m2 + 1)(m1 − m3 + 2) (m2 − k2 )(k1 − m2 + 1) , c22 (m, k) = (m1 − m2 + 1)(m2 − m3 + 1) (k2 − m3 + 1)(k1 − m3 + 2) . c32 (m, k) = (m1 − m3 + 2)(m2 − m3 + 1)
c12 (m, k) =
To prove this theorem we can repeat all the arguments given in Section 4, but we give here a short proof which depend on the notion of the dual of a spherical function. then the function If : G → End(V ) is a spherical function of type π ∈ K ∗ ∗ ∗ −1 t : G → End(V ) defined by (g) = (g ) is a spherical function of type π ∗ , where π ∗ denotes the contragredient representation associated to π (see [4]). In particular ψ(g) = φ ∗ (g), because both spherical functions are associated to the same G module and are of the same K type. ˇ = (−m3 , −m2 , −m1 ), simiNow given a parameter m = (m1 , m2 , m3 ) let m larly given a parameter k = (k1 , k2 ) we put kˇ = (−k2 , −k1 ). Therefore we can see that if m,k is the spherical function associated to the G module Vm and the K submodule Vk then ∗m,k = m, ˇ kˇ is the spherical function associated to the G module Vmˇ and the K submodule Vkˇ . Proof of Theorem 3.6. We start from the following identity established in Theorem 3.3 φ(g)m,k (g) = a12 (m, k)m1 ,k (g) + a22 (m, k)m2 ,k (g) + + a32 (m, k)m3 ,k (g), where m1 = (m1 + 1, m2 , m3 ), m3 = (m1 , m2 , m3 + 1).
m2 = (m1 , m2 + 1, m3 ),
(12)
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THREE TERM RECURSION RELATION
By taking ∗ on both sides of (12) we obtain 2 2 ψ(g)m, ˇ kˇ (g) = a1 (m, k)m ˇ 1 ,kˇ (g) + a2 (m, k)m ˇ 2 ,kˇ (g) +
+ a32 (m, k)mˇ 3 ,kˇ (g). ˇ − e3 , m ˇ2 =m ˇ − e2 and m ˇ3 =m ˇ − e1 . Now if we change m ˇ1 =m Notice that m ˇ and k by kˇ we obtain by m ˇ m−e ,k (g) + a22 (m, ˇ m−e ,k (g) + ˇ k) ˇ k) ψ(g)m,k (g) = a12 (m, 3 2 2 ˇ ˇ k)m−e + a3 (m, ˇ ,k (g). 1
ˇ Now the explicit expresions for ˇ k). Then (18) follows with cj (m, k) = a4−j (m, 2 cj (m, k) follow from Theorem 3.4.
4. The Constants of the Multiplication Formulas The goal of this section is to give explicit expressions for the constants a12 , a22 , a32 , appearing in Theorem 3.4 in terms of the parameters k = (k1 , k2 ) and m = (m1 , m2 , m3 ). Let V = Vm be an irreducible GL(3, C) module and equipped V with a G invariant inner product. Let v ∈ V be a highest weight vector such that v = 1. The vector v is of weight (m1 , m2 , m3 ) and it is associated to the triangle m2
m1 m1
m3 m2
.
m1 As we mentioned before as a GL(2, C) module, V decomposes as the direct sum of the submodules Vk such that m1 k1 m2 k2 m3 , all of these with multiplicity one. A highest weight vector vµ in Vk is of weight (k1 , k2 , m1 + m2 + m3 − k1 − k2 ) and corresponds to the triangle µ=
m2
m1 k1
m3 k2
.
k1 Let vµ = 1. The constants a1 , a2 , a3 were defined in a such a way that vµ ⊗ e3 = a1 vµσ1 + a2 vµσ2 + a3 vµσ3 ∈ V σ1 ⊗ V σ2 ⊗ V σ3 , σ
with aj > 0 and vµj = 1.
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PROPOSITION 4.1. Let v be a GL(3, C) highest weight vector in V and let v σj ∈ V ⊗ W be the vectors defined by v σ1 = v ⊗ e1 , v σ2 = E21 v ⊗ e1 − (m1 − m2 )v ⊗ e2 , if m1 = m2 , v σ3 = (E21 E32 − (m2 − m3 )E31 )v ⊗ e1 − (m1 + 1 − m3 )E32 v ⊗ e2 + + (m2 − m3 )(m1 + 1 − m3 )v ⊗ e3 , if m2 = m3 . Then v σj , (j = 1, 2, 3) are dominant vectors in V ⊗ W of weights (m1 + 1, m2 , m3 ), (m1 , m2 + 1, m3 ), (m1 , m2 , m3 + 1), respectively. Proof. From the definition it is clear that v σj = 0 for j = 1, 2, 3. That v σ1 is a vector of the specified weight it follows from the fact that v is a vector of weight (m1 , m2 , m3 ) and that e1 is a vector of weight (1, 0, 0). Similarly v σ2 is a vector of weight (m1 , m2 + 1, m3 ) because E21 v is a vector of weight (m1 − 1, m2 + 1, m3 ) and e2 is a vector of weight (0, 1, 0). In the same way one verifies that v σ3 is a vector of weight (m1 , m2 , m3 + 1). To prove that the v σj are dominant it is enough to verify that they are killed by 2 E12 and E23 . This can be check by a straightforward computation. We introduce now the following elements of the complex universal enveloping algebra U (g) of g, which can be found in [10, §68]. ∇32 = E32 , ∇23 = E23 ,
∇31 = (E11 − E22 + 2)E31 + E21 E32 , ∇13 = (E11 − E22 + 1)E13 + E23 E12 .
We start by observing some elementary properties of these elements. LEMMA 4.2. We have (i) (ii) (iii) (iv) (v)
∇31 ∇32 = ∇32 ∇31 , ∇13 ∇23 = ∇23 ∇13 , ∗ ∗ = ∇13 and ∇32 = ∇23 , ∇31 E12 ∇32 = ∇32 E12 , E12 ∇31 = (∇31 − 2E31 )E12 .
PROPOSITION 4.3. Let v ∈ Vm be a GL(3, C) highest weight vector, and let v σj ∈ V ⊗ W, (j = 1, 2, 3), be the vectors defined in Proposition 4.1. Then v ⊗ e3 =
1 ∇31 (v σ1 ) − (m1 + 1 − m2 )(m1 + 2 − m3 ) 1 ∇32 (v σ2 ) + − (m1 + 1 − m2 )(m2 + 1 − m3 ) 1 v σ3 . + (m1 + 2 − m3 )(m2 + 1 − m3 )
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THREE TERM RECURSION RELATION
Proof. The vector v belongs to the GL(2, C) submodule Vm1 ,m2 . Then we have that v ⊗ e3 = w1 + w2 + w3 ∈ V σ1 ⊕ V σ2 ⊕ V σ3 , where the vectors wj are weight vectors in V σj of weight (m1 , m2 , m3 + 1), which σ belongs to the GL(2, C) submodules Vm1j ,m2 . Moreover, since v and e3 are GL(2, C) dominant the same happens with w1 , w2 and w3 . Lemmas 4.2(iv) and (v) imply that ∇31 (v σ1 ) and ∇32 (v σ2 ) are GL(2, C) dominant vectors in V σ1 and V σ2 of weights (m1 , m2 ), respectively. On the other hand an easy computation gives that ∇31 (v σ1 ) = 0 and ∇32 (v σ2 ) = 0. Therefore we have w1 = x∇31 (v σ1 ),
w2 = y∇32 (v σ2 ),
w2 = zv σ3 ,
for some constants x, y, z. Now it is straightforward to verify that 1 , (m1 + 1 − m2 )(m1 + 2 − m3 ) 1 , y=− (m1 + 1 − m2 )(m2 + 1 − m3 ) 1 . z= (m1 + 2 − m3 )(m2 + 1 − m3 )
x=
2
PROPOSITION 4.4. Let v ∈ Vm be a GL(3, C) highest weight vector with
v = 1 and let v σj be defined as in Proposition 4.1. Then we have
v σ1 2 = 1,
v σ2 2 = (m1 − m2 )(m1 + 1 − m2 ),
v σ3 2 = (m2 − m3 )(m1 + 1 − m3 )(m2 + 1 − m3 )(m1 + 2 − m3 ). Proof. Recall that we have equipped V and W = C3 with G-invariant inner products. The canonical basis of C3 is an orthonormal basis with respect to this inner product. The inner product in V ⊗ W is such that v ⊗ w, v ⊗ w = v, v w, w . Therefore we obtain
v σ1 2 = v ⊗ e1 , v ⊗ e1 = v, v e1 , e1 = 1. We have v σ2 = E21 v ⊗ e1 − (m1 − m2 )v ⊗ e2 , then v σ2 , v σ2 = = = =
E21 v, E21 v + (m1 − m2 )2 v, v v, E12 E21 v + (m1 − m2 )2 v, (E11 − E22 )v + (m1 − m2 )2 (m1 − m2 ) + (m1 − m2 )2 .
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Finally, for the vector v σ3 we have
v σ3 2 = (E21 E32 − (m2 − m3 )E31 )v 2 + (m1 + 1 − m3 )2 E32 v 2 + + (m2 − m3 )2 (m1 + 1 − m3 )2 . We also get
E32 v 2 = v, E23 E32 v = v, (E22 − E33 )v = (m2 − m3 ). On the other hand we have
(E21 E32 − (m2 − m3 )E31 )v 2 = E32 v, E12 E21 E32 v − − 2(m2 − m3 ) v, E13 E21 E32 v + + (m2 − m3 )2 v, E13 E31 v . Since v is a GL(3, C) dominant vector we have E12 E32 v = E32 E12 v = 0 and
E13 E32 v = E32 E13 v + E12 v = 0.
We also have that E32 v is a vector of weight (m1 , m2 − 1, m3 + 1). Thus
(E21 E32 − (m2 − m3 )E31 )v 2 = E32 v, (E11 − E22 )E32 v + 2(m2 − m3 ) v, E23 E32 v + + (m2 − m3 )2 v, (E11 − E33 )v = (m1 − m2 + 1)(m2 − m3 ) + 2(m2 − m3 )2 + (m2 − m3 )2 (m1 − m3 ) = (m2 − m3 )(m1 − m3 + 1)(m2 − m3 + 1). Therefore
v σ3 2 = (m2 − m3 )(m1 − m3 + 1)(m2 − m3 + 1) + + (m1 + 1 − m3 )2 (m2 − m3 ) + (m2 − m3 )2 (m1 + 1 − m3 )2 = (m2 − m3 )(m1 − m3 + 1)(m2 − m3 + 1)(m1 + 2 − m3 ). This completes the proof of the proposition.
2
4.1. THE OPERATORS µ The proof of Theorem 3.4 is based on the action on V σ1 and V σ2 of certain elements 1 and 2 in U (g), which transform the GL(3, C) highest weight vectors v σ1 ∈ V σ1 and v σ2 ∈ V σ2 into GL(2, C) highest weight vectors in Vkσ11,k2 and Vkσ12,k2 , respectively. For m2 m3 m1 k1 k2 µ= k1 we define, as in [10, §68, Theorem 5], m1 −k1 m2 −k2 ∇32 . µ = ∇31
(13)
THREE TERM RECURSION RELATION
209
THEOREM 4.5. Let Vm be a GL(3, C) module and let Vk be a GL(2, C) irreducible submodule. If v ∈ Vm is a GL(3, C) highest weight vector, then µ (v) ∈ Vk is a GL(2, C) highest weight vector associated to the triangle µ. Proof. It is easy to see that µ (v) is a vector of weight (k1 , k2 , m1 + m2 + m3 − k1 − k2 ). To prove that it is a GL(2, C) dominant vector we need to see that it is annihilated by E12 . This follows from Lemmas 4.2(iv) and (v). Now it is clear that µ (v) is associated to the triangle µ. From Theorem 4.6 below it will follow that 2 µ (v) is a nonzero vector. To compute the constants of the multiplication formulas we need the norms of the vectors µ (v) with respect to a G invariant inner product in Vm . Given a linear operator a, as usual, we shall denote [a]n = a(a − 1) · · · (a − n + 1). THEOREM 4.6. Let v ∈ Vm be a GL(3, C) highest weight vector. Then
µ (v) 2 = (m1 − k1 )!(m2 − k2 )![m1 − k2 + 1]m1 −k1 × × [m1 − m3 + 1]m1 −k1 [m1 − m2 ]m1 −k1 [m2 − m3 ]m2 −k2 v 2 . A proof of this theorem can be found in [10, Chapter X], where an explicit realization of the GL(n, C) irreducible modules is used. An abstract proof in our case, is included for completeness in Appendix. 4.2. COMPUTATION OF THE CONSTANTS The aim of this section is to give the proof of Theorem 3.4. THEOREM 3.4. In terms of the parameters m = (m1 , m2 , m3 ) and k = (k1 , k2 ) we have (m1 − k1 + 1)(m1 − k2 + 2) , a12 = a12 (m, k) = (m1 − m2 + 1)(m1 − m3 + 2) (k1 − m2 )(m2 − k2 + 1) , a22 = a22 (m, k) = (m1 − m2 + 1)(m2 − m3 + 1) (k1 − m3 + 1)(k2 − m3 ) . a32 = a32 (m, k) = (m1 − m3 + 2)(m2 − m3 + 1) Proof. Let v ∈ Vm be a GL(3, C) highest weight vector such that v = 1. By Proposition 4.3 we have that 1 ∇31 (v σ1 ) − v ⊗ e3 = (m1 + 1 − m2 )(m1 + 2 − m3 ) 1 ∇32 (v σ2 ) + − (m1 + 1 − m2 )(m2 + 1 − m3 ) 1 v σ3 . + (m1 + 2 − m3 )(m2 + 1 − m3 )
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The operator µ defined in (13) has the property that µ (v) = vµ is a GL(2, C) highest weight vector in the submodule Vk ⊆ Vm . Moreover since e3 ∈ W is a lowest weight vector of weight (0, 0, 1) it follows that µ (v ⊗ e3 ) = µ (v) ⊗ e3 . Then we obtain 1 µ ∇31 (v σ1 ) − µ (v) ⊗ e3 = (m1 + 1 − m2 )(m1 + 2 − m3 ) 1 µ ∇32 (v σ2 ) + − (m1 + 1 − m2 )(m2 + 1 − m3 ) 1 µ (v σ3 ). + (m1 + 2 − m3 )(m2 + 1 − m3 ) This expression is exactly the decomposition of the vector vµ ⊗ e3 in the direct sum V σ1 ⊕ V σ2 ⊕ V σ3 . The constants a1 , a2 , a3 were defined in a such a way that vµ ⊗ e3 = a1 vµσ1 + a2 vµσ2 + a3 vµσ3 ∈ V σ1 ⊕ V σ2 ⊕ V σ3 , σ
with aj > 0 and vµj = 1. Therefore we observe that 1
µ ∇31 (v σ1 ) 2 , (m1 + 1 − m2 )2 (m1 + 2 − m3 )2 µ (v) 2 1
µ ∇32 (v σ2 ) 2 , a22 = (m1 + 1 − m2 )2 (m2 + 1 − m3 )2 µ (v) 2 1
µ (v σ3 ) 2 . a32 = (m1 + 2 − m3 )2 (m2 + 1 − m3 )2 µ (v) 2 a12 =
By Theorem 4.6 we have
µ (v) 2 = (m1 − k1 )!(m2 − k2 )![m1 − k2 + 1]m1 −k1 × × [m1 − m3 + 1]m1 −k1 [m1 − m2 ]m1 −k1 [m2 − m3 ]m2 −k2 . The vector v σ1 ∈ V σ1 is GL(3, C) dominant of weight (m1 + 1, m2 , m3 ), and µ (v σ1 ) µ ∇31 (v σ1 ) =
m1 +1−k1 m2 −k2 µ = ∇31 where ∇32 .
Then by Theorem 4.6, replacing m1 by m1 + 1, we obtain µ (v σ1 ) 2 = (m1 + 1 − k1 )!(m2 − k2 )![m1 − k2 + 2]m1 +1−k1 ×
× [m1 − m3 + 2]m1 +1−k1 [m1 + 1 − m2 ]m1 +1−k1 × × [m2 − m3 ]m2 −k2 . Therefore
µ ∇31 (v σ1 ) 2 = (m1 + 1 − k1 )(m1 + 2 − k2 )(m1 + 2 − m3 ) ×
µ (v) 2 × (m1 + 1 − m2 ).
(14) (15) (16)
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By replacing in (14) we get a12 =
(m1 − k1 + 1)(m1 − k2 + 2) . (m1 − m2 + 1)(m1 − m3 + 2)
We proceed in a similar way with a22 . The vector v σ2 is a GL(3, C)-dominant vector of weight (m1 , m2 + 1, m3 ). Let m1 −k1 m2 +1−k2 (v σ2 ). Thus by Theorem 4.6 we have = ∇31 ∇32 , then µ ∇32 (v σ2 ) = (v σ2 ) 2 = (m1 − k1 )!(m2 + 1 − k2 )![m1 − k2 + 1]m1 −k1 ×
× [m1 − m3 + 1]m1 −k1 [m1 − m2 − 1]m1 −k1 × × [m2 + 1 − m3 ]m2 +1−k2 v σ2 2 . Then by Proposition 4.4 we get
µ ∇32 (v σ2 ) 2 = (m2 + 1 − k2 )(k1 − m2 )(m2 + 1 − m3 )(m1 + 1 − m2 ).
µ (v) 2 Finally, by replacing in (15) we obtain the expression for a22 . The vector v σ3 is a G-dominant vector of weight (m1 , m2 , m3 + 1). Thus by Theorem 4.6 we have
µ (v σ3 ) 2 = (m1 − k1 )!(m2 − k2 )![m1 − k2 + 1]m1 −k1 [m1 − m3 ]m1 −k1 × × [m1 − m2 ]m1 −k1 [m2 − m3 − 1]m2 −k2 v σ3 2 . By using Proposition 4.4 we get
µ (v σ3 ) 2 = (k1 − m3 + 1)(k2 − m3 )(m2 + 1 − m3 )(m1 + 2 − m3 ).
µ (v) 2 By replacing this expression in (16) we finish the proof of the theorem.
2
5. The Three Term Recursion Relation We shall start from the identities (10) and (11) established in Theorems 3.3 and 3.6. Since the parameter k = (k1 , k2 ) will be the same in all places it will not be explicit: φ(g)m (g) = a12 (m)m+e1 (g) + a22 (m)m+e2 (g) + a32 (m)m+e3 (g), (17) ψ(g)m (g) = c12 (m)m−e1 (g) + c22 (m)m−e2 (g) + c32 (m)m−e3 (g). (18) From (17) and (18) we obtain φ(g)ψ(g)m (g) = a12 (m)[c12 (m1 )m (g) + c22 (m1 )m1 −e2 (g) + c32 (m1 )m1 −e3 (g)] + + a22 (m)[c12 (m2 )m2 −e1 (g) + c22 (m2 )m (g) + c32 (m2 )m2 −e3 (g)] + + a32 (m)[c12 (m3 )m3 −e1 (g) + c22 (m3 )m3 −e2 (g) + c32 (m3 )m (g)], (19) where we put mj = m + ej , j = 1, 2, 3.
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At this point it is convenient to make clear when two spherical functions m,k (g) and m ,k (g) are equivalent. PROPOSITION 5.1. The spherical functions m,k (g) and m ,k (g) of the pair (G, K) are equivalent, if and only if m = m + j (1, 1, 1) and k = k + j (1, 1) for some j ∈ Z. Proof. First of all, an irreducible spherical function of (G, K) is characterized by the eigenvalues [2 ](e) and [3 ](e) corresponding to the generators 2 and 3 of the center Z(g) of the universal enveloping algebra U (g) (see [4]). Moreover if p = m1 − m2 and q = m2 − m3 then the corresponding eigenvalues for m,k (g) coincide with the infinitesimal character χλ+δ of Vm evaluated at 2 and 3 . Here λ = pλα + qλβ , λα and λβ being the fundamental weights corresponding to the set of simple roots {α, β}, δ = α + β, and χλ+δ = (λ + δ) · γ where γ : Z(g) → U (h)W is the Harish-Chandra isomorphism of Z(g) onto the invariants in U (h). Now let p = m1 − m2 , q = m2 − m3 and λ = p λα + q λβ . If the spherical functions m,k (g) and m ,k (g) are equivalent then χλ+δ = χλ +δ , and this implies that λ + δ = w(λ + δ) for some w ∈ W , (see Theorem 5.62 in [8]). But then w = 1 and λ = λ because λ + δ and λ + δ are strictly dominant weights. Thus p = p and q = q. If we put j = m1 − m1 = m2 − m2 = m3 − m3 we have on one hand m = m + j (1, 1, 1). On the other hand, since the types of m,k (g) and m ,k (g) must be the same we have n = k1 + 2k2 − m1 − m2 − m3 = k1 + 2k2 − m1 − m2 − m3 and = k1 − k2 = k1 − k2 . Therefore (k1 − k1 ) + 2(k2 − k2 ) = 3j and k1 − k1 = k2 − k2 which give k = k + j (1, 1). The proposition is proved. 2 We introduce now a better parametrization for the irreducible spherical functions. Given integral tuples m = (m1 , m2 , m3 ) and k = (k1 , k2 ) subject to m1 k1 m2 k2 m3 , let w = m1 − k1
and
k = m2 − k2 .
Then it is easy to see that the map m,k → (w, k, n, ) gives a one to one correspondence between the set of equivalence classes of irreducible spherical functions of (G, K) and the set {(w, k, n, ) ∈ Z4 : 0 k , 0 w, 0 w + n + k}. In terms of these new parameters we have (w + 1)(w + + 2) , a12 (m, k) = a12 (w, k, n, ) = (w + − k + 1)(2w + + n + k + 2) ( − k)(k + 1) , a22 (m, k) = a22 (w, k, n, ) = (w + − k + 1)(w + n + 2k + 1) (w + + n + k + 1)(w + n + k) . a32 (m, k) = a32 (w, k, n, ) = (2w + + n + k + 2)(w + n + 2k + 1)
213
THREE TERM RECURSION RELATION 2 ˇ we also obtain ˇ k) (m, From cj2 (m, k) = a4−j 2 (w + n + k, − k, −n − , ). cj2 (m, k) = cj2 (w, k, n, ) = a4−j
Then to rewrite (19) in term of these new parameters we shall omit the parameters (n, ) since all the spherical functions involved, except φ and ψ, are of the same type. We also put bj2 (w, k) = cj2 (w, k, n − 1, ) and aj2 = aj2 (w, k) = aj2 (w, k, n, ), for short. More explicitly, (w + + 1)w , (2w + + n + k + 1)(w + − k + 1) k( − k + 1) , b22 (w, k) = (w + n + 2k)(w + − k + 1) (w + n + k)(w + + n + k + 1) . b32 (w, k) = (w + n + 2k)(2w + + n + k + 1) b12 (w, k) =
Then (19) reads φ(g)ψ(g)(w, k; g) = (a12 b12 (w + 1, k) + a22 b22 (w, k + 1) + a32 b32 (w, k))(w, k; g) + + a12 b22 (w + 1, k)(w + 1, k − 1; g) + a12 b32 (w + 1, k)(w + 1, k; g) + + a22 b12 (w, k + 1)(w − 1, k + 1; g) + a22 b32 (w, k + 1)(w, k + 1; g) + (20) + a32 b12 (w, k)(w − 1, k; g) + a32 b22 (w, k)(w, k − 1; g). Remark. The above identity holds for 0 k , 0 w, 0 w + n + k, even when some spherical functions in the right-hand side of (20) were not defined, because in such cases the coefficients vanish. Now we shall encode the set of identities (20) in a set of three term recursion relations by introducing the necessary notation. (w, g) of + 1 spherical For w max{0, −n} we define a column vector functions of type (n, ) as follows (w, g) = (w, n, ; g) = ((w, 0, n, ; g), . . . , (w, , n, ; g))T . We also define the following ( + 1) × ( + 1) matrices Aw =
a32 (w, k)b12 (w, k)Ekk +
k=0
Bw =
−1
a22 (w, k)b12 (w, k + 1)Ek,k+1 ,
k=0
(a12 (w, k)b12 (w + 1, k) + a22 (w, k)b22 (w, k + 1) +
k=0
+ a32 (w, k)b32 (w, k))Ekk +
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+
a32 (w, k)b22 (w, k)Ek,k−1
+
k=1
Cw =
−1
a22 (w, k)b32 (w, k + 1)Ek,k+1 ,
k=0
a12 (w, k)b32 (w + 1, k)Ekk +
k=0
a12 (w, k)b22 (w + 1, k)Ek,k−1 .
k=1
We recall that given two square matrices M and P the tensor product matrix M ⊗ P is the matrix obtained by blowing up each entry Mij of M to the matrix Mij P . Now let w = Aw ⊗ I, A
w = Bw ⊗ I, B
w = Cw ⊗ I, C
where I denotes the ( + 1) × ( + 1) identity matrix. Then from (20) we obtain the following result. THEOREM 5.2. For each type (n, ), for all integers w max{0, −n} and all g ∈ G we have w w w (w − 1, g) + B (w, g) + C (w + 1, g). (w, g) = A φ(g)ψ(g) w is a nonsingular matrix. Moreover C 0 = 0; this is useful to interpret the above identity Remark. If w = 0 we have A since we have not defined (−1, g). When n < 0 and w = −n the function (w − 1, g) has not defined the first component (w − 1, 0, n, ; g), but the first column of Aw is zero, giving perfect sense to the identity. Proof. An ( + 1)2 × ( + 1) matrix V will be seen as an ( + 1)-column vector V = (V0 , . . . , V )t of ( + 1) × ( + 1) matrices Vk . If M is an ( + 1) × ( + 1) = M ⊗ I , then matrix and M (w, g))k = (M
Mkj (w, j ; g).
j =0
In particular if Aij , Bij and Cij denote, respectively, the ij -entries of the matrices Aw , Bw and Cw we have w (w − 1, g))k = Akk (w − 1, k; g) + Ak,k+1 (w − 1, k + 1; g), (A w (w, g))k = Bk,k−1 (w, k − 1; g) + Bk,k (w, k; g) + (B (w, k + 1; g), + Bk,k+1 w (w + 1, g))k = Ck,k−1 (w + 1, k − 1; g) + Ck,k (w + 1, k; g). (C Therefore, using (20) for any 0 k , we obtain w w w (w, g)k . (w − 1, g) + B (w, g) + C (w + 1, g))k = φ(g)ψ(g) (A
THREE TERM RECURSION RELATION
215
To prove the last statement it is enough to observe that w = det Cw = det C
a12 (w, k)b32 (w + 1, k) = 0.
k=0
The proof of the theorem is completed.
2
It is of interest to see how we can write the above theorem when we restrict the spherical functions to the Abelian subgroup A of G of all matrices of the form cos s 0 sin s a(s) = , 0 1 0 − sin s 0 cos s for any s ∈ R. Let M be the centralizer of A in K. Then M consists of all elements of the form ir 0 0 e −2ir 0 , m(r) = 0 e 0 0 eir for any r ∈ R. then (a(s)) comIf is a spherical function on G of type π = π(n,) ∈ K mutes with π(m) for all m ∈ M. On the other hand there exits a basis {v0 , . . . , v } of Vπ such that m(r) · vj = eir(−2j −n) vj , j = 0, . . . , . Therefore we have that (a(s)) diagonalizes in such a basis for each s ∈ R. Let (0 (a(s)), . . . , (a(s))) be the corresponding diagonal matrix. In the open subset {a(s) ∈ A : 0 < s < π/2} of A we introduce the coordinate t = cos2 (s) and define the vector valued function F (t) = (0 (a(s)), . . . , (a(s))) associated to the spherical function . If (g) = (w, k; g) then we shall also put F (t) = F (w, k; t). (w, g) In a similar way, for w max{0, −n}, corresponding to the function we consider the ( + 1) × ( + 1) matrix valued function F (t, w) whose rows are given by the vectors F (w, k; t) for k = 0, 1, . . . , . More explicitly (w, t) = (Fij (g, t)) F
with Fij (w, t) = j (w, i; a(s)).
PROPOSITION 5.3. For each fixed type (n, ), for all integers w max{0, −n} and all 0 < t < 1 we have (w − 1, t) + Bw F (w, t) + Cw F (w + 1, t). (w, t) = Aw F tF
(21)
Proof. We recall that φ(g) is the spherical function of type (−1, 0) associated to the G-module W = C3 and that ψ(g) is the spherical function of type (1, 0) associated to W ∗ . A direct computation gives φ(a(s)) = ψ(a(s)) = cos s.
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Therefore from the identity (20), for g = a(s), we obtain tF (w, k; t) = (A)kk F (w − 1, k; t) + (A)k,k+1 F (w − 1, k + 1; t) + + (B)k,k−1 F (w, k − 1; t) + (Bw )k,k F (w, k; t) + + (B)k,k+1 F (w, k + 1; t) + + (C)k,k−1 F (w + 1, k − 1; t) + (A)kk F (w + 1, k; t). 2
This is nothing but the equality of the kth rows of the identity (21).
(t) = F (w, t) with the functions Finally we want to relate these functions F H (t) = H (t, w) in [4] and [5] associated to the spherical functions. In [4, Section 4] we considered the function H (g) = (g)π (g)−1 , associated to a spherical and its restriction H (t) = H (a(s)) where function (g) of type π = πn, ∈ K, 2 t = cos (s). There we viewed the diagonal matrix H (t) as a column vector. Then it is easy to verify that 1/2 0 t n/2 t , F (t) = t H (t) 0 1 where the exponent denotes the th symmetric power of the matrix. Explicitly t 1/2 0 is a diagonal matrix whose j th entry is t (−j )/2 , with 0 j . 0 1 In [5, Section 3], for 0, w max{0, −n} we also defined the matrix (t) = H (t, w) whose rows are given by the vectors H (w, k; t) valued function H for 0 k . Then 1/2 t 0 n/2 . F (t) = t H (t) 0 1 By multiplying both sides of (21) on the right by t −n/2
t 1/2 0
0 1
we obtain
(t, w − 1) + Bw H (t, w) + Cw H (t, w + 1), (t, w) = Aw H tH which is exactly the three term recursion relation established in [5, Theorem 3.7]. 2 Appendix For completeness we include here the proof of Theorem 4.6. To reach this goal we need some preparatory material. We start by considering the particular case when m1 = k1 . PROPOSITION 6.1. Let v ∈ Vm be a GL(3, C) highest weight vector. Then for any n ∈ N we have n v 2 = n!
E32
(m2 − m3 )!
v 2 = n![m2 − m3 ]n v 2 . (m2 − m3 − n)!
THREE TERM RECURSION RELATION
217
Proof. It is not difficult to prove, by induction on n, that the following identity holds n−1 n n E32 = E32 E23 + n(E22 − E33 − n + 1)E23 . E23
(22)
Now for n ∈ N we have n−1 n E32 v = 0. E23
(23)
In fact for n = 1 it is true, since v is GL(3, C) dominant. If we assume that it holds for n 1 then, by using (22), we obtain n+1 n n−1 n−1 n E23 E32 v = E23 (E32 E23 + n(E22 − E33 − n + 1)E23 )E32 v n−1 n−1 n n E32 v + n(E22 − E33 − n − 1)E23 E32 v = 0. = E23 E32 E23
From (22) and (23) it follows that n−1 n−1 n−1 n n n E23 E32 v = E32 E32 E32 v + n(E22 − E33 − n + 1)E23 E32 v n−1 n−1 E32 v = n(E22 − E33 − n + 1)E23 n−1 n−1 E32 v, = n(m2 − m3 − n + 1)E23 n−1 n−1 E32 v is (m1 , m2 , m3 ). Now we get by using that the weight of the vector E23 n−1 n−1 n n n v 2 = v, E23 E32 v = n(m2 − m3 − n + 1) v, E23 E32 v
E32 n−1 v 2 . = n(m2 − m3 − n + 1) E32
Thus the proposition follows by induction on n.
(24) 2
PROPOSITION 6.2. Let v ∈ Vm be a GL(3, C) highest weight vector. Let a = b v. Then we have m1 − k1 , b = m2 − k2 and w = E32 a a ∇31 w .
µ (v) 2 = [m1 − m2 + b + 1]a w, E13
Proof. We have b a (v), ∇13 µ (v) . µ (v), µ (v) = ∇32 a µ (v) we first observe that To compute ∇13
E12 (E11 − E22 + 1)E13 = (E11 − E22 − 1)E12 E13 = (E11 − E22 − 1)E13 E12 . a µ (v) = ((E11 − E22 + 1)E13 )a µ (v) since E12 µ (v) = 0. Now by Then ∇13 induction on a we have a . ((E11 − E22 + 1)E13 )a = [E11 − E22 + 1]a E13
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Therefore we obtain a a ∇13 µ (v) = ((E11 − E22 + 1)E13 )a µ (v) = [E11 − E22 + 1]a E13 µ (v) a = [m1 − m2 + b + 1]a E13 µ (v), a µ (v) is (m1 , m2 − b, m3 + b). Finally because the weight of the vector E13 a µ (v), µ (v) = [m1 − m2 + b + 1]a w, E13 13 µ (v) a a w . = [m1 − m2 + b + 1]a w, E13 ∇31
2
LEMMA 6.3. In U (g) we have (i) (ii) (iii) (iv)
n−1 n n E13 E21 = E21 E13 − nE13 E23 . n−1 n n = E32 E13 + nE32 E12 . E13 E32 n−1 n n + E31 E13 . E13 E31 = n(E11 − E33 − n + 1)E13 n+1 n E13 ∇31 ≡ 0 mod(U (g)E12 + U (g)E13 ).
Proof. (i) Since E˙ 21 (E13 ) = E23 and E13 commute we have n−1 n ) = nE13 E23 . E˙ 21 (E13
Therefore n−1 n n − E13 E21 = nE13 E23 . E21 E13
This proves (i), and in the same way we obtain (ii). (iii) For n = 1 the statement is clearly true. We assume that the identity holds for n 1 and we get n+1 n−1 n E31 = nE13 (E11 − E33 − n + 1)E13 + E13 E31 E13 E13 n+1 n n = n(E11 − E33 − n − 1)E13 + (E11 − E33 )E13 + E31 E13 n+1 n = (n + 1)(E11 − E33 − n)E13 + E31 E13 .
This completes the proof of (iii). (iv) For n = 0 the assertion is obvious. By induction on n 0 we assume that n−1 n ∇31 = 0 mod(U (g)E12 + U (g)E13 ). We have E13 n+1 n n+1 n−1 ∇31 = E13 ((E11 − E22 + 2)E31 + E21 E32 ∇31 ) E13 n+1 n−1 n+1 n−1 E31 ∇31 + E13 E21 ∇31 E32 , (25) = (E11 − E22 − n + 1)E13
where we have used that E32 ∇31 = ∇31 E32 (see Lemma 4.2(i)). By (iii) we have n+1 n−1 E31 ∇31 (E11 − E22 − n + 1)E13 n−1 n = (n + 1)(E11 − E22 − n + 1)(E11 − E33 − n)E13 ∇31 + n+1 n−1 ∇31 + (E11 − E22 − n + 1)E31 E13 ≡ 0 mod(U (g)E12 + U (g)E13 ).
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THREE TERM RECURSION RELATION
By (i) we have n+1 n−1 n+1 n−1 n−1 n E21 ∇31 E32 = E21 E13 ∇31 E32 − (n + 1)E23 E13 ∇31 E32 . E13
Now we note that E12 E32 = E32 E12
and
E13 E32 = E32 E13 + E13 .
Therefore the left ideal U (g)E12 + U (g)E13 is invariant under right multiplication by E32 . Then by the inductive hypothesis we obtain n+1 n−1 E21 ∇31 E32 ≡ 0 mod(U (g)E12 + U (g)E13 ). E13 n+1 n By replacing in (25) we obtain E13 ∇31 ≡ 0 mod(U (g)E12 + U (g)E13 ), and this completes the proof of (iv). 2
PROPOSITION 6.4. Let v ∈ Vm be a GL(3, C) highest weight vector. Let b v. Then w = E32 n n ∇31 w = n![m1 − m3 + 1]n [m1 − m2 ]n w 2 . w, E13
Proof. We have n−1 n n n ∇31 = E13 ((E11 − E22 + 2)E31 + E21 E32 )∇31 E13 n−1 n−1 n n = (E11 − E22 + 2 − n)E13 E31 ∇31 + E13 E21 ∇31 E32 .
By Lemmas 6.3(i) and (iii) we get n−1 n−1 n n ∇31 = n(E11 − E22 + 2 − n)(E11 − E33 − n + 1)E13 ∇31 + E13 n−1 n−1 n n ∇31 + E21 E13 ∇31 E32 − + (E11 − E22 + 2 − n)E31 E13 n−1 n−1 ∇31 E32 . − nE23 E13 b is of weight (m1 , m2 − b, m3 + b) and we note that E12 w = The vector w = E32 b b v = 0. Then by E12 E23 v = 0, and by Lemma 6.3(ii) we obtain E13 w = E13 E32 n−1 n−1 n n Lemma 6.3(iv) we have E13 ∇31 w = 0 and E13 ∇31 E32 w = 0. Therefore n n ∇31 w = n(m1 − m2 + b + 2 − n) × E13 n−1 n−1 ∇31 w − × (m1 − m3 − b − n + 1)E13 n−1 n−1 ∇31 E32 w. − nE23 E13
Then n n ∇31 w w, E13 n−1 n−1 ∇31 w − = n(m1 − m2 + b + 2 − n)(m1 − m3 − b − n + 1) w, E13 n−1 n−1 ∇31 E32 w . − n E32 w, E13
(26)
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Recall that by (24) we have
E32 w 2 = (b + 1)(m2 − m3 − b) w 2 . Now we shall prove the proposition by induction on n. For n = 1, by using (26) we have that w, E13 ∇31 w = (m1 − m2 + b + 1)(m1 − m3 − b) w 2 − E32 w 2 = (m1 − m2 )(m1 − m3 + 1) w 2 . Let us assume that the identity in the statement of the proposition is true for n − 1 and any b. Then we have n−1 n−1 ∇31 w = (n − 1)![m1 − m3 + 1]n−1 [m1 − m2 ]n−1 w 2 w, E13
and n−1 n−1 ∇31 E32 w E32 w, E13 = (n − 1)![m1 − m3 + 1]n−1 [m1 − m2 ]n−1 E32 w 2 = (n − 1)![m1 − m3 + 1]n−1 [m1 − m2 ]n−1 (b + 1)(m2 − m3 − b) w 2 .
Thus by replacing in (26) we obtain n n ∇31 w = n![m1 − m3 + 1]n−1 [m1 − m2 ]n−1 × w, E13 × ((m1 − m2 + b + 2 − n)(m1 − m3 − b − n + 1) − − (b + 1)(m2 − m3 − b)) w 2 = n![m1 − m3 + 1]n−1 [m1 − m2 ]n−1 × × (m1 − m3 − n + 2)(m1 − m2 − n + 1) w 2 = n![m1 − m3 + 1]n [m1 − m2 ]n w 2 .
This completes the proof of the proposition.
2
Finally we are in position to give the proof of Theorem 4.6. THEOREM 4.6. Let v ∈ Vm be a GL(3, C) highest weight vector. Then
µ (v) 2 = (m1 − k1 )!(m2 − k2 )![m1 − k2 + 1]m1 −k1 × × [m1 − m3 + 1]m1 −k1 [m1 − m2 ]m1 −k1 [m2 − m3 ]m2 −k2 v 2 . m2 −k2 v. By Proposition 6.2 we have Proof. Let w = E32 m1 −k1 m1 −k1 ∇31 w .
µ (v) 2 = [m1 − k2 + 1]m1 −k1 w, E13
By using Proposition 6.4 we obtain m1 −k1 m1 −k1 ∇31 w = (m1 − k1 )![m1 − m3 + 1]m1 −k1 [m1 − m2 ]m1 −k1 w 2 . w, E13
THREE TERM RECURSION RELATION
221
By Proposition 6.1 we have
w 2 = (m2 − k2 )![m2 − m3 ]m2 −k2 v 2 . Finally putting these things together the theorem follows.
2
Acknowledgements It is a pleasure to thank Prof. F. A. Grünbaum for introducing us to the bispectral problem and for his continuous encouragement. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Bochner, S.: Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730–736. Duran, A. and Van Assche, W.: Orthogonal matrix polynomials and higher order recurrrence relations, Linear Algebra Appl. 219 (1995), 261–280. Gangolli, R. and Varadarajan, V. S.: Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergeb. Math. Grenzgeb. 101, Springer-Verlag, Berlin, 1988. Grünbaum, F. A., Pacharoni, I. and Tirao, J.: Matrix valued spherical functions associated to the complex protective plane, J. Funct. Anal. 188 (2002), 350–441. Grünbaum, F. A., Pacharoni, I. and Tirao, J.: A matrix valued solution to Bochner’s problem, J. Phys. A 34 (2001), 10647–10656. Grünbaum, F. A., Pacharoni, I. and Tirao, J.: Spherical functions associated to the three dimensional hyperbolic space, Internat. J. Math. 13(7) (2002), 727–784. Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. Knapp, A.: Lie Groups beyond an Introduction, Progr. Math., Birkhäuser, Boston, 1996. Tirao, J.: Spherical functions, Rev. Un. Mat. Argentina 28 (1977), 75–98. Zelobenko, D. P.: Compact Lie Groups and Their Representations, Transl. Math. Monographs, Amer. Math. Soc., Providence, RI, 1973.
Mathematical Physics, Analysis and Geometry 7: 223–237, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities To Alain Chenciner in his 60th birthday RICARDO URIBE-VARGAS Collège de France, 11, Pl. Marcelin–Berthelot, 75005 Paris, France. e-mail:
[email protected] (Received: 20 March 2003; in final form: 7 November 2003) Abstract. We prove that the vertices of a curve γ ⊂ Rn are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k + 2)-vertex theorem for convex curves in the Euclidean space R2k . We obtain a very practical formula to calculate the vertices of a curve in Rn . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L: C ∞ (S1 ) → C ∞ (S1 )). Mathematics Subject Classifications (2000): 51L15, 53A04, 53A07, 53C99, 53D05, 53D12, 58K35, 47B25. Key words: caustic, Lagrangian manifold, singularity, space curve, Sturm theory, vertex.
Introduction The geometry of curves is a classical subject which relates geometrical intuition with analysis and topology. Sturm theory and oscillatory properties of solutions have clear geometrical interpretation in terms of geometry of the curve. In particular, to estimate the possible number of special points, e.g., flattenings (points at which the last torsion vanishes) for different types of closed curves is an important problem involving topological, symplectic and analytic methods. In [2], V. I. Arnold pointed out that “most of the facts of the differential geometry of submanifolds of Euclidean or of Riemannian space may be translated into the language of contact (or symplectic) geometry and may be proved in this more general setting. Thus we can use the intuition of Euclidean or Riemannian geometry to guess general results of contact (or symplectic) geometry, whose applications to the problem of ordinary differential geometry provide new information in this classical domain.” The classical four-vertex theorem, [15], asserts that a closed convex plane curve has at least 4 critical points of the curvature. It is easy to construct an arbitrarily small perturbation of the Euclidean metric of the plane in a neighbourhood of the unit circle C such that the curve C has only two critical points
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of the curvature in the obtained Riemannian plane. The validity of the four-vertex theorem, in the particular case of a Riemannian metric, may be re-established if the vertices of a curve are defined in the following way: For each point of our curve, consider the geodesic issuing from that point perpendicularly to the curve and in the direction of the inward normal. The point of intersection of such a geodesic with an infinitely close geodesic normal is said to be a conjugate point (along the original normal). All these conjugate points form the caustic of the original curve (the envelope of the family of geodesic normals). The points of our curve corresponding to the singular points of the caustic are called the vertices of the curve (the generic singular points of the caustic are semi-cubic cusps). So the four-vertex theorem in the Riemannian case asserts that: The caustic of a generic closed convex curve has at least four cusps (counted geometrically). If the curve is nongeneric, then multiplicities must be counted. For instance, the level sets f = c > 0 of the function f (x, y) = x 2 + y 2 + α(x 2 − y 2 ) + 2βxy + x 3 − 3xy 2 , with 0 < α 2 + β 2 < 0.1, are convex curves near the origin. For c sufficiently small, the curve f = c has 4 vertices, while for c > 0.05, the curve f = c has 6 vertices. For some intermediate value c = c0 , the curve f = c0 has 5 vertices, one of them with multiplicity 2. EXAMPLE. The caustic of a circle is a single (singular!) point, with infinite multiplicity: all points of the circle are vertices. We give a formula to calculate the vertices of a curve in Rn and a new proof of a higher dimensional 4-vertex theorem ([20] Theorem 1, below) applying Sturm theory and the theory of Lagrangian singularities. We show the relations between the vertices of curves, the singularities of caustics and Sturm theory of fundamental systems of solutions of disconjugate homogeneous linear differential operators L: C ∞ (S1 ) → C ∞ (S1 ). 1. Statement of Results on Vertices In the sequel Rn will denote a Euclidean space and we assume that the derivatives of order 1, . . . , n − 1, of our curves are linearly independent at any point (this is true for generic curves). DEFINITION 1. Let M be a d-dimensional submanifold of Rn , considered as a complete intersection: M = {x ∈ Rn : g1 (x) = · · · = gn−d (x) = 0}. We say that k is the order of contact of a curve γ : t → γ (t) ∈ Rn with the submanifold M, or that γ and M have k-point contact, at a point γ (t0 ), if each function g1 ◦ γ , . . . , gn−d ◦ γ has a zero of multiplicity at least k at t = t0 , and at least one of them has a zero of multiplicity k at t = t0 .
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Remark. If one needs to make this definition more invariant, one could denote the image of γ by and then write that the order of contact at a point is the minimum of the multiplicities of zero among the functions of the form g| : → R, at that point, where g: Rn → R belongs to the generating ideal of M and we assume that 0 is a regular value of g. EXAMPLE. A smooth curve in Rn has 2-point contact with its tangent line (at the point of tangency) for the generic points of the curve. The curve y = x 3 has 3-point contact with the line y = 0, at the origin: the equation x 3 = 0 has a root of multiplicity 3. By convention, the k-dimensional affine subspaces of the Euclidean space Rm+1 will be considered as k-dimensional spheres of infinite radius. DEFINITION 2. For k = 1, . . . , n − 1, a k-osculating sphere at a point of a curve in the Euclidean space Rn is a k-dimensional sphere having at least (k + 2)-point contact with the curve at that point. For k = n − 1 we will simply write osculating hypersphere. EXAMPLE. A generic plane curve and its osculating circle have 3-point contact at an ordinary point of the curve. DEFINITION 3. A vertex of a curve in Rn is a point at which the curve has at least (n + 2)-point contact with its osculating hypersphere. EXAMPLE. A noncircular ellipse in the plane R2 has 4 vertices. They are the points at which the ellipse intersects its principal axes. DEFINITION 4. An embedded closed curve in Rn (or in RP n ) is called convex if it intersects any hyperplane (or projective hyperplane, respectively) at no more than n points, taking multiplicities into account. EXAMPLE. A closed plane curve is convex if it intersects any straight line in at most two points, taking multiplicities into account. EXAMPLE. For n = 2k, the generalized ellipse, defined as the image of the embedding t → (cos t, sin t, cos 2t, sin 2t, . . . , cos kt, sin kt), is convex in R2k . Indeed, the number of intersection points of this curve with the hyperplane given by the equation a0 +
k (aj xj + bj yj ) = 0, j =1
is equal to the number of zeroes of the trigonometric polynomial
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which is at most 2k. The following theorem was proved in [10] and [20]. THEOREM 1. Any closed convex curve in R2k has at least 2k+2 vertices (counted geometrically). This theorem holds for convex curves in the 2k-sphere S2k ([20]) and also holds for convex curves in Lobachevskian 2k-space 2k ([21]). A more general theorem was proved in [22] for a class of curves invariant under conformal transformations and containing the class of convex curves in R2k (respectively in S2k and in 2k ). Vertices of curves in Euclidean spaces and flattenings of curves in projective (or affine) spaces are related to Sturm theory. In Section 4, we give a new proof of Theorem 1 based on Sturm theory. This new proof allows us to give a formula to calculate the vertices of a curve in Rn as the zeroes of a determinant: THEOREM 2. The vertices of any curve γ : S1 → Rn (or γ : R → Rn ), γ : s → (ϕ1 (s), . . . , ϕn (s)) are given by the solutions s ∈ S1 (or s ∈ R) of the equation det(R1 , . . . , Rn , G) = 0, where Ri (respectively G) is the column vector defined by the first n + 1 derivatives of ϕi (of g = γ 2/2, respectively, where γ 2 := γ , γ ). COROLLARY 1 (see also [23]). The vertices of any curve γ : S1 → Rn (or γ : R → Rn ), γ : s → (ϕ1 (s), . . . , ϕn (s)) correspond to the flattenings of the curve : S1 → Rn+1 (or : R → Rn+1 ), γ 2(s) . : s −→ ϕ1 (s), . . . , ϕn (s), 2 Remark. This means that the vertical projection of a curve γ ⊂ Rn to the paraboloid ‘of revolution’ z = 12 (x12 + · · · + xn2 ) sends the vertices of the curve γ onto the flattenings of its image. Remark. Another formula for calculation of the vertices of a curve (unfortunately not very practical) was found in [11]. For curves in R3 , such formula appears in [16], Exercise 6.8. A curve in the Euclidean plane, has a vertex if and only if the radius of its osculating circle is critical. In higher dimensional spaces we have the following theorem (announced in [20] and proved in Section 3).
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THEOREM 3. The vertices of a smoothly immersed curve in the Euclidean space Rn are critical points of the radius of the osculating hypersphere. Remark. For n > 2, the converse is not always true. For example, all the points of the circular helix t → (cos t, sin t, t) are critical points of the radius of the osculating hypersphere. However it has no vertex. A more generic example is given by the curve t → (a cos t, b sin t, t) which has no vertex for any a, b ∈ R \ {0} such that |a 2 − b2 | < 1/3. Proof of remark. Apply our formula of Theorem 1 to obtain b cos t 1 1/2(b2 − a 2 ) sin 2t + t −a sin t 2 2 (b − a ) cos 2t + 1 −a cos t −b sin t 0 = 0, −b cos t 0 −2(b2 − a 2 ) sin 2t a sin t a cos t b sin t 0 −4(b2 − a 2 ) cos 2t which gives ab(1 − 3(b2 − a 2 ) cos 2t) = 0. For |a 2 − b2 | < 1/3, this equation has no solution t ∈ R. 2 The (noncircular) ellipse is the simplest closed convex curve in the plane having the minimum number of vertices: 4. DEFINITION 5. A generalized ellipse in R2k is the convex curve given by the following parametrization ([7]): θ −→ (a1 cos θ, b1 sin θ, a2 cos 2θ, b2 sin 2θ, . . . , ak cos kθ, bk sin kθ). Since generalized ellipses are the convex curves of lower degree in R2k , one would expect (from the ‘topological economy principle in algebraic geometry’) that they have the minimum number of vertices, i.e. 2k + 2, as it is the case for the noncircular ellipse in the 2-plane. However, the following example shows that generic generalized ellipses in R2k can have more than 2k + 2 vertices. EXAMPLE. The generalized ellipse in R4 , γ (θ) = (a1 cos θ, b1 sin θ, a2 cos 2θ, b2 sin 2θ), with a22 = b22 and a1 b1 a2 b2 = 0 has 8 vertices. If a22 = b22 then γ is a spherical curve and all its points are thus vertices. Denote Ck = cos kθ and Sk = sin kθ. THEOREM 4. Consider the generalized ellipse in R2k given by γ (θ) = (a1 C1 , b1 S1 , a2 C2 , b2 S2 , . . . , ak Ck , bk Sk ), with a1 b1 a2 b2 · · · ak bk = 0. Then, for even k, γ can have 2k + 4, 2k + 8, . . . , 4k or an infinity of vertices depending on the values of the parameters aj and bj , for j (k/2) + 1. For odd k, γ can have 2k + 2, 2k + 6, . . . , 4k or an infinity of vertices depending on the values of the parameters aj and bj , for j (k + 1)/2.
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Remark. In the space of parameters aj and bj , there is a hypersurface which separates the domains with different number of vertices. The points of this hypersurface correspond to generalized ellipses having vertices with multiplicity > 1. To prove that Theorem 1 is sharp, we will construct a convex curve in R2k having the minimum number of vertices, i.e. 2k + 2. Consider the generalized ellipse of Theorem 3 with coefficients a1 = b1 = · · · = ak = bk = 1 and denote it by γ0 . Obviously γ0 is a spherical curve and all its points are vertices. In order to obtain the desired convex curve, we will perturb γ0 in the “radial direction”: THEOREM 5. For ε = 0 sufficiently small, the curve γε = (1 + ε cos(k + 1)θ)γ0 (θ) has exactly 2k + 2 vertices. Theorems 4 and 5 are proved in Section 5. 2. Some Background on Lagrangian Singularities In this section, we give the basic definitions and facts from Lagrangian singularity theory, that we will need later, focused on the normal map introduced below (for a deep study, see [3] or [4]). The reader having familiarity with Lagrangian singularity theory (and the normal map) can go directly to Section 3. A symplectic structure on a manifold M is a closed differentiable 2-form ω, nondegenerate on M, also called symplectic form. A manifold equipped with a symplectic structure is called a symplectic manifold. EXAMPLE. The total space of the cotangent bundle π : T ∗ Rn → Rn of Rn is a symplectic manifold. A submanifold of a symplectic manifold (M 2n , ω) is called Lagrangian if it has dimension n and the restriction to it of the symplectic form ω is equal to 0. EXAMPLE. Let N be any submanifold in the Euclidean space Rn and let L be the n-dimensional manifold formed by the covectors v, · at the end-points of the normal vectors v to N. Then L is a Lagrangian submanifold of the symplectic space T ∗ Rn . A fibration of a symplectic manifold is called Lagrangian fibration if all the fibers are Lagrangian submanifolds.
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EXAMPLE. The cotangent bundle T ∗ V → V of any manifold V is a Lagrangian fibration. The standard action 1-form λ = p dq vanishes along the fibers. Thus its differential ω = dλ, which is the 2-form defining the standard symplectic structure on T ∗ V , also vanishes. Consider the inclusion i: L → E of an immersed Lagrangian submanifold L in the total space of a Lagrangian fibration π : E → B. The restriction of the projection π to L, that is π ◦ i: L → B is called a Lagrangian map. Thus, a Lagrangian map is a triple L → E → B, where the left arrow is a Lagrangian immersion and the right arrow a Lagrangian fibration. The normal map. Consider the set of all vectors normal to a submanifold N in the Euclidean space Rn . Associate to each vector its end point. To the vector v based at the point q associate the point q + v. This Lagrangian map of the n-dimensional manifold of normal vectors to N into the n-dimensional Euclidean space Rn is called normal map. The Lagrangian submanifold L in T ∗ Rn is formed by the covectors v, · at the end points of the normal vectors v. The set of critical values of a Lagrangian map is called its caustic. EXAMPLE. The focal set or evolute of a submanifold of positive codimension in Euclidean space Rn is defined as the envelope of the family of normal lines to the submanifold. The caustic of the normal map associated to that submanifold coincides with its focal set. A Lagrangian equivalence of two Lagrangian maps is a symplectomorphism of the total space transforming the first Lagrangian fibration to the second, and the first Lagrangian immersion to the second. Caustics of equivalent Lagrangian maps are diffeomorphic. Generating families. Consider the total space of the standard Lagrangian fibration R2n → Rn , (p, q) → q with the form dp ∧ dq. Let F (x, q) be the germ, at the point (x0 , q0 ), of a family of smooth functions of k variables x = (x1 , . . . , xk ) which depends smoothly on the parameters q. Suppose (x0 , q0 ) = 0 and (a) ∂F ∂x has rank k at (x0 , q0 ). (b) the map (x, q) → ∂F ∂x (x0 , q0 ), q0 ) of the set Then the germ at the point ( ∂F ∂q ∂F ∂F = 0, p = , LF = (p, q) : ∃x : ∂x ∂q is the germ of a smoothly immersed Lagrangian submanifold of R2n . The family germ F is said to be a generating family of LF and of its Lagrangian map πF : (q, p) → q.
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It turns out that the germ of each Lagrangian map is Lagrange equivalent to the germ of the Lagrangian map πF for a suitable Lagrangian generating family F . Such generating family is said to be a Lagrangian representative family of the corresponding germ of Lagrangian map. The equivalence classes of germs of Lagrangian maps are called Lagrangian singularities. The classification of Lagrangian singularities of manifolds in general position of dimension n 10 is given in [3]. 3. Proof of Theorem 3 and Study of the Focal Set of a Curve in Terms of Its Normal Map Proof of Theorem 3. We will assume that the derivatives of order 1, . . . , n − 1, of our curve, γ : R → Rn , are linearly independent at any point (which is true for generic curves). A generating family of the normal map associated to the curve is given by F : Rn × R → R, Fq (s) = F (q, s) = 12 q − γ (s) 2 . The caustic of this Lagrangian map is the focal set of the curve. We shall write
(i) = {(q, s) ∈ Rn × R : ∂s F (q, s) = 0, . . . , ∂si F (q, s) = 0},
i 1.
According to this notation the manifolds Rn × R ⊃ (1) ⊃ (2) ⊃ . . . ⊃ (i) ⊃ . . . are embedded one inside the other. Hence the kernels of the differentials of the restrictions of F to these embedded submanifolds are also embedded one inside another. The set (1) consists of the pairs (q, s) such that q is the centre of some hypersphere of Rn tangent to γ at s (this means that q lies in the normal hyperplane to γ at s). One can see from the equations defining (i), i n, that it consists of the pairs (q, s) such that q is the centre of some hypersphere having at least (i + 1)point contact with γ at s and that these centres form an affine (n − i)-dimensional subspace of Rn . So (n) is the curve formed by the pairs (q(s), s) such that q(s) is the centre of an osculating hypersphere at γ (s). If γ (s) is not a flattening of γ then the osculating hypersphere at γ (s) is unique and hence the value of F at the point (q(s), s) of
(n) is one half of the square of the radius of the osculating hypersphere at γ (s). The condition for a point p = γ (s0 ) to be a vertex is equivalent to the fact that the first n + 1 derivatives of F with respect to s vanish at s = s0 . Hence to each point (q(s), s) in (n + 1) corresponds a vertex γ (s) of the curve. The set (i + 1) is a smooth submanifold of (i) characterised by the fact that at its points the kernel of (F| (i) )∗ is tangent to (i). Hence the derivative of the restriction of F to the curve (n) has rank 1 at all points of (n), except at the
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points of (n +1) (where it is 0). That is, a point belonging to (n +1) is a critical point of the restriction of F to the curve (n). So a vertex is a critical point of the radius of the osculating hypersphere. 2 Remarks on the focal set (caustic). The centres of the osculating hyperspheres at the vertices of γ are the points q ∈ Rn for which there exists a solution s of the (n + 1)-system of equations Fq (s) = 0, Fq (s) = 0, .. . Fq(n+1) (s) = 0. For a fixed s, the first equation gives the normal hyperplane to the curve at the point γ (s). The first two equations give a codimension 1 subspace of the normal hyperplane to the curve at the point γ (s). Following this process we obtain (for a generic curve) a complete flag at each nonflattening point of the curve. This complete flag is the osculating flag of the focal curve (which is formed by the centres of the osculating hyperspheres and is determined by the first n equations). In particular, the osculating hyperplane of the focal curve at the point q(s) is the normal hyperplane to the curve γ at the point γ (s). As the point moves along the curve γ , the corresponding flag (starting with the codimension 2 subspace) generates a hypersurface which is stratified in a natural way by the components of the flag. This stratified hypersurface is a component of the focal set of the curve γ (the other component being the curve γ itself). The 1-dimensional stratum is the focal curve of γ : it is generated by the 0-dimensional subspace of the flag, that is, by centre of the osculating hypersphere at the moving point. The equation Fqn+1 (s) = 0 gives a finite number of isolated points at which the focal curve is singular (it has a cusp). These points correspond to the vertices of γ . As we explained above, the focal set is the caustic of the normal map defined by the generating family F (q, s) = 12 q − γ (s) 2 . Thus – according to Arnold’s classification of singularities of caustics (see [3] or [4]) – the vertices of a curve in Rn correspond to a Lagrangian singularity An+1 of the normal map.
4. Proof of the (2k + 2)-Vertex Theorem by Sturm Theory We begin this section with some definitions and results of Sturm theory, taken from [7] and [12]. A set of functions {ϕ1 , . . . , ϕ2k+1 } with ϕi : S1 → R is a Chebyshev system if 2 = 0 any linear combination a1 ϕ1 + · · · + a2k+1 ϕ2k+1 , ai ∈ R, with a12 + · · · + a2k+1 1 has at most 2k zeroes on S .
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EXAMPLE 1. The set of functions {1, cos θ, sin θ} is a Chebyshev system. Remark. Any convex closed curve θ → (ϕ1 (θ), . . . , ϕ2k (θ)) in R2k defines a Chebyshev system: {1, ϕ1 , . . . , ϕ2k }. DEFINITION 6. A linear homogeneous differential operator L: C ∞ (S1 ) → C ∞ (S1 ) is called disconjugate if it has a fundamental system of solutions for the equation Lg = 0 which are defined on the circle and form a Chebyshev system. EXAMPLE 2. The operator L = ∂(∂ 2 + 1) is disconjugate. The Chebyshev system {1, cos θ, sin θ} is a fundamental system of solutions for it. EXAMPLE 3. Any convex curve γ : θ → (ϕ1 (θ), . . . , ϕ2k (θ)) in R2k defines a (2k + 1)-order disconjugate operator Lγ defined by Lγ g = det(R1 , . . . , R2k , G), where Ri (respectively G) is the column vector defined by the first 2k+1 derivatives of ϕi (of g, respectively). The Chebyshev system {1, ϕ1 , . . . , ϕ2k } is a fundamental system of solutions of the equation Lγ g = 0. EXAMPLE 4. The generalized ellipse ([7]) γ : θ → (a1 cos θ, b1 sin θ, a2 cos 2θ, b2 sin 2θ, . . . , ak cos kθ, bk sin kθ), defines, up to a constant factor, the (2k + 1)-order disconjugate operator Lγ = ∂(∂ 2 + 1) · · · (∂ 2 + n2 ). Some proofs of 4-vertex type theorems are (implicitly) based on the following theorem due to Hurwitz ([13]) which generalize a theorem of Sturm ([19]): ∞ 1 HURWITZ THEOREM. Any function f ∈ C (S ) whose Fourier series begins with the harmonics of order N, f = kN ak cos kθ + bk sin kθ, has at least 2N zeroes.
In fact any function f ∈ C ∞ (S1 ) without harmonics up to order n is orthogonal to the solutions of the equation ∂(∂ 2 + 1) · · · (∂ 2 + n2 )ϕ = 0, and such solutions form a Chebyshev system. The following theorem generalizes Hurwitz’s theorem. STURM–HURWITZ THEOREM ([7, 12]). Let f : S1 → R be a C ∞ function such that S1 f (θ)ϕi (θ) dθ = 0, {ϕi }i=1,...,2k+1 being a Chebyshev system. Then f has at least 2k + 2 sign changes.
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COROLLARY 2 ([12]). Any function in the image of a disconjugate operator (f = Lg, where g ∈ C ∞ (S1 ) is any function) of order 2k + 1 has at least 2k + 2 sign changes. Proof of the (2k + 2)-vertex theorem in R2k . Let γ : θ → (ϕ1 (θ), . . . , ϕ2k (θ)) be a convex curve in R2k . Consider the family of functions on the circle F : R2k × S1 → R defined by Fq (θ) = 12 q − γ (θ) 2 . In the proof of Theorem 3 we saw that the centres of the osculating hyperspheres at the vertices of γ are the points q ∈ R2k for which there exists a solution θ of the following system of 2k + 1 equations: Fq (θ) = 0, Fq (θ) = 0, .. . Fq(2k+1) (θ) = 0. The focal curve θ → q(θ) (consisting of the centres of the osculating hyperspheres) is determined by the first 2k equations. The last equation is the condition on this curve determining the vertices. Write g = γ 2 /2. Using the fact that −F = γ · q −
γ 2 q2 − , 2 2
the preceding system of equations can be written as γ · q − g = 0, γ · q − g = 0, .. . γ (2k+1) · q − g (2k+1) = 0. This means that the vector (q, −1) in R2k+1 is orthogonal to the 2k + 1 vectors (γ , g ), (γ , g ), .. . (γ (2k+1), g (2k+1)). So the vertices of γ are given by the zeros of the determinant of the matrix whose lines are these 2k + 1 vectors. This determinant is equal to det(R1 , . . . , R2k , G)
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where Ri (respectively G) is the column vector defined by the first 2k+1 derivatives of ϕi (of g = γ 2 /2, respectively). This determinant is the image of the function g = γ 2 /2 under the operator Lγ (see Example 3). So Corollary 2 implies that this determinant has at least 2k + 2 sign changes. This proves the theorem. 2 Proof of Theorem 2. In the above proof of the (2k+2)-vertex theorem for convex curves in R2k , the convexity of the curve and the parity of the dimension were used only in the last step. So the determinant obtained in the proof gives a formula to 2 calculate the vertices of a curve in Rn . This proves Theorem 2.
5. On the Number of Vertices of Generalized Ellipses We will give two examples and then we will prove Theorem 4. EXAMPLE 1. The generalized ellipse in R4 , γ (θ) = (a1 cos θ, b1 sin θ, a2 cos 2θ, b2 sin 2θ), with a22 = b22 and a1 b1 a2 b2 = 0 has 8 vertices. If a22 = b22 then γ is a spherical curve and all its points are thus vertices. Proof. Denote Ck = cos kθ, Sk = sin kθ and g = 2(a12 C12 + b12 S12 + a22 C22 + b22 S22 ). Theorem 2 and Examples 3 and 4 of Section 4 imply that the vertices of γ correspond to the roots of the equation ∂(∂ 2 + 1)(∂ 2 + 22 )g = 0. The trigonometric identity a 2 cos2 θ + b2 sin2 θ = 12 (a 2 + b2 + (a 2 − b2 ) cos 2θ) allows us to write g = (a12 − b12 )C2 + (a22 − b22 )C4 + a12 + b12 + a22 + b22 . The operator ∂ kills the constant terms (i.e. the harmonics of order zero), and the operator (∂ 2 + 22 ) kills the second-order harmonics. Moreover ∂C4 = −4S4 , ∂S4 = 4C4 and (∂ 2 + k 2 )C4 = (k 2 − 42 )C4 . Thus ∂(∂ 2 + 1)(∂ 2 + 22 )g = ∂(∂ 2 + 1)(∂ 2 + 22 )(a22 − b22 )C4 = K(a22 − b22 )S4 , where K = −4(1 − 42 )(22 − 42 ) is a nonzero constant. Thus the vertices of γ correspond to the solutions of the equation K(a22 − b22 )S4 = 0, i.e. γ has 8 vertices 2 for a22 = b22 and all its points are vertices for a22 = b22 . We keep the notation Ck = cos kθ and Sk = sin kθ.
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EXAMPLE 2. The generalized ellipse in R6 , γ (θ) = (a1 cos θ, b1 sin θ, a2 cos 2θ, b2 sin 2θ, a3 cos 3θ, b3 sin 3θ), 3 with i=1 ai bi = 0 may have 8, 12 or an infinity of vertices, depending on the values of the parameters a2 , b2 , a3 , b3 . In particular, if a22 = b22 and a32 = b32 then γ has 12 vertices, and if a22 = b22 and a32 = b32 then γ has 8 vertices. If a22 = b22 and a32 = b32 then γ is a spherical curve and all its points are thus vertices. Proof. As in Example 1, the vertices of γ are the roots of the equation given by ∂(∂ 2 + 1)(∂ 2 + 22 )(∂ 2 + 32 )g = 0, where g = (a12 − b12 )C2 + (a22 − b22 )C4 + (a32 − b32 )C6 +
3 (ai2 + bi2 ). i=1
The operator ∂(∂ 2 + 1)(∂ 2 + 22 )(∂ 2 + 32 ) kills the harmonics of orders zero, one, two and three. Thus ∂(∂ 2 + 1)(∂ 2 + 22 )(∂ 2 + 32 )g = K2 (a22 − b22 )S4 + K3 (a32 − b32 )S6 , where K2 and K3 are nonzero constants.
2
We recall Theorem 4: THEOREM 4. Consider the generalized ellipse in R2k γ (θ) = (a1 C1 , b1 S1 , a2 C2 , b2 S2 , . . . , ak Ck , bk Sk ), with a1 b1 a2 b2 · · · ak bk = 0. Then, for even k, γ can have 2k + 4, 2k + 8, . . . , 4k or an infinity of vertices depending on the values of the parameters aj and bj , for j (k/2) + 1. For odd k, γ can have 2k + 2, 2k + 6, . . . , 4k or an infinity of vertices depending on the values of the parameters aj and bj , for j (k + 1)/2. Proof. As in Examples 1 and 2, the vertices of γ are the roots of the equation given by ∂(∂ 2 + 1)(∂ 2 + 22 ) · · · (∂ 2 + k 2 )g = 0, where g=
k i=1
(ai2 − bi2 )C2i +
k
(ai2 + bi2 ).
i=1
The operator ∂(∂ 2 + 1)(∂ 2 + 22 ) · · · (∂ 2 + k 2 )
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kills the harmonics from the order zero until order k. Thus, for even k, ∂(∂ 2 + 1)(∂ 2 + 22 ) · · · (∂ 2 + k 2 )g = Ki (ai2 − bi2 )S2i , i k2 +1
where Ki is a nonzero constant, for i (k/2) + 1. For odd k Ki (ai2 − bi2 )S2i , ∂(∂ 2 + 1)(∂ 2 + 22 ) · · · (∂ 2 + k 2 )g = i(k+1)/2
where Ki is a nonzero constant, for i (k + 1)/2. Theorem 4 is proved.
2
Proof of Theorem 5. Applying our formula of Theorem 2 we obtain that the number of vertices of the curve γε (θ) = (1 + ε cos(k + 1)θ)γ0 (θ) is given by the number of solutions θ ∈ S1 of an equation of the form ε∂(∂ 2 + 1)(∂ 2 + 22 ) · · · (∂ 2 + k 2 ) cos(k + 1)θ + ε 2 f (θ, ε) = 0, i.e. εK sin(k + 1)θ + ε 2 f (θ, ε) = 0, where K = −(k + 1)
k
(−(k + 1)2 + i 2 ) = 0 i=1
is a constant and the function f (θ, ε) is bounded for |ε| < 1. Thus for ε = 0 small enough this equation has exactly 2k + 2 solutions. 2
Acknowledgements The author is grateful to V. I. Arnold and to M. Kazarian for helpful discussions, questions and remarks. References 1. 2. 3.
4. 5.
Anisov, S. S.: Convex Curves in RP n , Proc. Steklov Inst. Math. 221 (1998), 3–39. Arnold, V. I.: Contact Geometry and Wave Propagation, Enseign. Math. 34 (1989). Arnold, V. I., Varchenko, A. N. and Gussein-Zade, S. M.: Singularities of Differentiable Maps, Vol. 1, Birkhäuser, 1986. (French version: Singularités des applications différentiables, Vol. 1, Mir, Moscow, 1986. Russian version: Nauka, 1982.) Arnold, V. I.: Singularities of Caustics and Wave Fronts, In: Math. Appl. (Soviet series) 62, Kluwer, Dordrecht, 1991. Arnold, V. I.: Sur les propriétés des projections Lagrangiennes en géométrie symplectique des caustiques, In: Cahiers Math. Decision 9320, CEREMADE, 1993, pp. 1–9. Rev. Mat. Univ. Complut. Madrid 8(1) (1995), 109–119.
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6. 7. 8. 9. 10. 11. 12.
13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
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Arnold V. I.: The geometry of spherical curves and the algebra of quaternions, Russian Math. Surveys 50(1) (1995), 1–68. Arnold, V. I.: On the number of flattening points of space curves, Amer. Math. Soc. Transl. 171 (1995), 11–22. Arnold, V. I.: Topological problems of the theory of wave propagation, Russian Math. Surveys 51(1) (1996), 1–47. Arnold, V. I.: Towards the Legendrian Sturm theory of space curves, Functional Anal. Appl. 32(2) (1998), 75–80. Barner, M.: Über die Mindestanzahl stationärer Schmiegeebenen bei geschlossenen strengkonvexen Raumkurven, Abh. Math. Sem. Univ. Hamburg 20 (1956), 196–215. Fukui, T. and Nuño-Ballesteros, J. J.: Isolated roundings and flattenings of submanifolds in Euclidean spaces, Preprint. Guieu, L., Mourre, E. and Ovsienko, V. Yu.: Theorem on six vertices of a plane curve via the Sturm theory, In: V. I. Arnold, I. M. Gelfand, M. Smirnov and V. S. Retakh (Eds), Arnold– Gelfand Mathematical Seminars, Birkhäuser, Boston, 1997, pp. 257–266. Hurwitz, A.: Über die Fourierschen Konstanten integrierbarer Funktionen, Math. Ann. 57 (1903), 425–446. Kazarian, M.: Nonlinear version of Arnold’s theorem on flattening points, C.R. Acad. Sci. Paris Sér. I 323(1) (1996), 63–68. Mukhopadhyaya, S.: New methods in the geometry of a plane arc I, Bull. Calcutta Math. Soc. 1 (1909), 31–37. Porteous, I. R.: Geometric Differentiation, Cambridge Univ. Press, 1994. Romero-Fuster, M. C.: Convexly-generic curves in R3 , Geom. Dedicata 28 (1988), 7–29. Sedykh, V. D.: The theorem about four vertices of a convex space curve, Functional Anal. Appl. 26(1) (1992), 28–32. Sturm, J. C. F.: Mémoire sur les équations différentielles du second ordre, J. Math. Pures Appl. 1 (1836), 106–186. Uribe-Vargas, R.: On the higher dimensional four-vertex theorem, C.R. Acad. Sci. Paris Sér. I 321 (1995), 1353–1358. Uribe-Vargas, R.: On the (2k +2)-vertex and (2k +2)-flattening theorems in higher dimensional Lobatchevskian space, C.R. Acad. Sci. Paris Sér. I 325 (1997), 505–510. Uribe-Vargas, R.: Four-vertex theorems in higher dimensional spaces for a larger class of curves than the convex ones, C.R. Acad. Sci. Paris Sér. I 330 (2000), 1085–1090. Uribe-Vargas, R.: On polar duality, Lagrange and Legendre singularities and stereographic projection to quadrics, Proc. London Math. Soc. 87(3) (2003), 701–724.
Mathematical Physics, Analysis and Geometry 7: 239–287, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Thermal Ionization Dedicated, in admiration and friendship, to Elliott Lieb and Edward Nelson, on the occasion of their 70th birthdays JÜRG FRÖHLICH and MARCO MERKLI Theoretical Physics ETH-Hönggerberg, CH-8093 Zürich, Switzerland. e-mail: {juerg, merkli}@itp.phys.ethz.ch (Received 26 June 2003) Abstract. In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature, we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic) temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of W ∗ -dynamical systems, and the proofs are based on Mourre’s theory of positive commutators and a new virial theorem. Results on the so-called standard form of a von Neumann algebra play an important role in our analysis. Mathematics Subject Classifications (2000): 82C10, 47A10, 46L55. Key words: open quantum system, black-body radiation, CCR algebra, virial theorem, positive commutators, Mourré estimate, standard form of von Neumann algebras, Fermi Golden Rule, Liouville operator.
1. Introduction In this paper, we study an idealized model describing an atom or molecule consisting of static nuclei and electrons coupled to black-body radiation. Our aim is to show that when the quantized radiation field is in a thermal state corresponding to a sufficiently high positive temperature, and under suitable conditions on the interaction Hamiltonian, including infrared and ultraviolet cutoffs and a small value of the coupling constant, the atom or molecule will always be ionized in the limit of very large times. This process is called thermal ionization. Thus, a very dilute gas of atoms or molecules in intergalactic space and subject to the 3K thermal background radiation of the universe will eventually be transformed into a very dilute plasma of nuclei and electrons. If the temperature of the black-body radiation is small, as compared to a typical atomic ionization energy, then an atom initially prepared in an excited bound state will start to emit light and relax towards its ground-state. After a time much longer Supported by an NSERC Postdoctoral Fellowship and by ETH-Zürich. Presented address:
Department of Mathematics, McGill University, Canada. e-mail:
[email protected]
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than its relaxation time, it will be stripped of its electrons in very unlikely events where an atomic electron is hit by a high-energy photon from the thermal background radiation. The life time of the groundstate of an isolated atom interacting with black body radiation at inverse temperature β, before it is ionized, is expected to be exponentially large in β. A precise description of the temporal evolution of such an atom is difficult to come by; but the claim that it will eventually be ionized, is highly plausible. To most physicists, this result must look obvious. Unfortunately a complete proof of it is likely to be very involved. The main purpose of this paper is to present some partial results, thermal ionization at sufficiently high temperatures for simplified models, supporting this picture. If the temperature of electromagnetic radiation is strictly zero then an atom initially prepared in a bound state of maximal energy well below its ionization threshold can be shown to always relax to a groundstate by emitting photons; (for a proof of this statement in some slightly idealized models see [FGS]). This result and our complementary result on thermal ionization provide some qualitative understanding of two fundamental irreversible processes in atomic physics: relaxation to a ground state, and ionization by thermal radiation. Next, we describe the physical system analyzed in this paper somewhat more precisely; (for further details see Section 2.1). It is composed of a subsystem with finitely many degrees of freedom, the ‘atom’ (or ‘molecule’), and a subsystem with infinitely many degrees of freedom, the ‘radiation field.’ The space of pure state vectors of the atom is a separable Hilbert space, Hp ; (where the subscript p stands for ‘particle’). Mixed states of the atom are described by density matrices, ρ, where ρ is a nonnegative, trace-class operator on Hp of unit trace. The expectation value of a bounded operator A on Hp in the state ρ is given by ωρp (A) := tr ρA.
(1)
Before the ‘atom’ or particle system is coupled to the radiation field the time evolution of a bounded operator A on Hp in the Heisenberg picture is given by αt (A) := eit Hp Ae−it Hp , p
(2)
where Hp is the particle Hamiltonian, which is a selfadjoint operator on Hp whose spectrum is bounded from below by a constant E > −∞. To be specific, we may think of Hp as being the Hilbert space Hp = Cn ⊕ L2 (R3 , d3 x),
(3)
and the Hamiltonian Hp as the operator Hp = diag(E0 = E, E1 , . . . , En−1 ) Cn ⊕(−) L2 (R3 ,d3 x) ,
(4)
describing a one-electron atom (with a static nucleus) with n boundstates of energies E0 , E1 , . . . , En−1 < 0 and scattering states of arbitrary energies k 2 ∈ [0, ∞) spanning the subspace L2 (R3 , d3 x) of Hp . Thus, the point spectrum of Hp is given by the eigenvalues {E0 , E1 , . . . , En−1 } and the continuous spectrum of Hp covers
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[0, ∞), has constant (infinite) multiplicity and is absolutely continuous. Just in order to keep things simple, we shall usually assume that n = 1. The bounded operators on a Hilbert space H form a von Neumann algebra denoted by B(H). A convenient algebra of operators encoding the kinematics of the ‘atom’ or particle system is the algebra Ap := B(Hp ). The ‘radiation field’ is described by a free, massless, scalar Bose field ϕ on physical space R3 , a ‘phonon field.’ For purposes of physics, it would be preferable to replace ϕ by the free electromagnetic field. In our entire analysis, this replacement can be made without any difficulties – at the price of slightly more complicated notation. A convenient algebra of operators to encode the kinematics of the radiation field is a C ∗ -algebra Af which can be viewed as a time-averaged version of the algebra of Weyl operators generated by ϕ and its conjugate momentum field π . The time evolution of operators in Af , in the Heisenberg picture, before the field is coupled to the particle system, is given by the free-field time f evolution αt , which is a one-parameter group of ∗automorphisms of Af . A one-parameter group {αt |t ∈ R} defined on a C ∗ -algebra A is a ∗automorphism group of A iff (αt (A))∗ = αt (A∗ ), for all A ∈ A, αt (A) ∈ A, (5) αt (A)αt (B) = αt (AB), for all A, B ∈ A, αt (αs (A)) = αt +s (A), for all A ∈ A, t, s ∈ R. αt =0 (A) = A, Since we work on a time-averaged Weyl algebra, the free field time evolution is f norm continuous, i.e., t → αt (A) is a continuous map from R to Af . General states of the radiation field can be described as states on the algebra Af , i.e., as positive, linear functionals, ω, on Af normalized such that ω(1) = 1. A convenient algebra of operators to encode the kinematics of the system composed of the ‘atom’ and the ‘radiation field’ is the C ∗ -algebra, A, given by (6) A = Ap ⊗ Af . The time evolution of operators in A, before the two subsystems are coupled to each other, is given by p f (7) αt,0 := αt ⊗ αt . A regularized interaction coupling the two subsystems can be introduced by choosing a bounded, selfadjoint operator V ( ) ∈ A, where the superscript ( ) indicates that a regularization has been imposed on an interaction term, V , in such a way that V ( ) = O(1/ ). We define the regularized, interacting time evolution ( ) | t ∈ R} of the algebra A of the coupled system as a ∗automorphism group {αt,λ given by the norm-convergent Schwinger–Dyson series t ∞ ( ) n (iλ) dt1 · · · αt,λ (A) = αt,0 (A) +
tn−1
··· 0
n=1
0
dtn [αtn ,0 (V ( ) ), [αtn−1 ,0 (V ( ) ), . . . , [αt1 ,0 (V ( ) ), αt,0 (A)] · · ·]],
(8)
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for an arbitrary operator A ∈ A. In Equation (8), λ is a coupling constant, and the interaction term V is chosen in accordance with conventional models describing electrons coupled to the quantized radiation field. We are interested in analyzing the time evolution of the coupled system in some states ω of physical interest, i.e., in understanding the time-dependence of expectation values ( ) ω(αt,λ (A)),
A ∈ A,
(9)
in the limit where the regularization is removed, i.e., → 0, and for large times t. The states ω of interest are states ‘close to’ (technically speaking, normal with respect to) a reference state of the form f
ωρ,β := ωρp ⊗ ωβ ,
(10) f
where ωρp is given by a density matrix ρ on Hp , see Equation (1), and ωβ is the thermal equilibrium state of the radiation field at temperature T = (kB β)−1 , where f f kB is Boltzmann’s constant. Technically, ωβ is defined as the unique (αt , β)-KMS state on the algebra Af ; it is invariant under (or ‘stationary’ for) the free-field time f evolution αt . If the density matrix ρ describes an arbitrary statistical mixture of bound states of Hp , but ρ vanishes on the subspace L2 (R3 , d3 x) of Hp , then ωρ,β is stationary for the free time evolution αt,0 defined in Equation (7). However, it is not an equilibrium (KMS) state for αt,0 . In fact, because Hp has continuous spectrum, there are no equilibrium (KMS) states on A for the time evolution αt,0 . Given the algebra A and a reference state ωρ,β on A, as in Equation (10), the GNS construction associates with the pair (A, ωρ,β ) a Hilbert space H , a ∗representation πβ of A on H , and a vector ρ ∈ H , cyclic for the algebra πβ (A), such that ωρ,β (A) = ρ , πβ (A) ρ ,
(11)
for all A ∈ A. The closure of the algebra πβ (A) in the weak operator topology is a von Neumann algebra of bounded operators on H which we denote by Mβ . This algebra depends on β, but is independent of the choice of the density matrix ρ. The states ω on A of interest to us are given by vectors ψ ∈ H in such a way that ω(A) = ψ, πβ (A)ψ ,
A ∈ A.
(12)
We shall see that there exists a selfadjoint operator L( ) λ on H generating the time evolution of the coupled system, in the sense that ( )
( )
( ) (A)) = eit Lλ πβ (A)e−it Lλ , πβ (αt,λ
(13)
for A ∈ A; L( ) λ is called the (regularized) Liouvillian. Clearly, ( )
( )
( ) (K) := eit Lλ Ke−it Lλ , σt,λ
K ∈ Mβ ,
(14)
THERMAL IONIZATION
243
defines a ∗automorphism group of time translations on Mβ . For an interesting class of models, we shall show that ( )
s-lim eit Lλ =: eit Lλ →0
(15)
exists, for all t, and defines a unitary one-parameter group on H . It then follows from (14) and (15) that σt,λ (K) := eit Lλ Ke−it Lλ
(16)
defines a one-parameter group of ∗automorphisms on the von Neumann algebra Mβ . The pair (Mβ , σt,λ ) defines a so-called W ∗ -dynamical system. If the couf pling constant λ vanishes then a state ωρ,β = ωρp ⊗ ωβ , where the density matrix ρ vanishes on the subspace L2 (R3 , d3 x) ⊂ Hp corresponding to the continuous spectrum of Hp and commutes with Hp , is an invariant state for σt,0 , in the sense that ωρ,β (σt,0(K)) := ρ , σt,0 (K) ρ = ωρ,β (K),
(17)
for all K ∈ Mβ . The main result proven in this paper can be described as follows: For an interesting class of interactions, V , for an arbitrary inverse temperature 0 < β < ∞, and for all real coupling constants λ with 0 < |λ| < λ0 (β), where λ0 (β) depends on the choice of V , and on β as λ0 (β) ∼ eβE0 , where E0 < 0 is the ground state energy of the particle system, there do not exist any states ω on Mβ close, in the sense of Equation (12), to a reference state ωρ,β , as in Equation (10), which are invariant under the time evolution σt,λ on Mβ , (in the sense that ω(σt,λ(K)) = ω(K), for K ∈ Mβ ). In other words, we show that, under the hypotheses described above, there are no time-translation invariant states of the coupled system of asymptotic temperature T = (kB β)−1 > 0. It will turn out that this result is a consequence of the following one: For a certain canonical definition of the Liouvillian Lλ of the coupled system, and under the hypotheses sketched above, Lλ does not have any eigenvectors ψ ∈ H, in particular, ker Lλ = {0}. This result will be proven with the help of Mourre’s theory of positive commutators applied to Lλ and a new virial theorem. As a corollary of our results it follows that, for an arbitrary vector ψ ∈ H and an arbitrary compact operator K on H , ψ, eit Lλ Ke−it Lλ ψ −→ 0,
(18)
as time t → ∞, (at least in the sense of ergodic means). This means that the survival probability of an arbitrary bound state of the atom coupled to the quantized radiation field in a thermal equilibrium state at positive temperature tends to zero, as time t → ∞. Heuristically, this can be understood by using Fermi’s Golden Rule.
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One may wonder how the quantum-mechanical motion of an electron looks like, after it has been knocked off the atom by a high-energy boson, i.e., after thermal ionization. We cannot give an answer to this question in this paper, because we are not able to analyze appropriately realistic models yet. But it is natural to expect that this motion will be diffusive, furnishing an example of ‘quantum Brownian motion.’ Progress on this question would be highly desirable. Organization of the paper. In Section 2, we define the model, and state our main result on thermal ionization, Theorem 2.4, which follows from spectral properties of the Liouvillian proven in our key technical theorem, Theorem 2.3. In Section 3, we state two general virial theorems, Theorems 3.2 and 3.3, we present a result on regularity of eigenfunctions of Liouvillians, Theorem 3.4, and explain some basic ideas of the positive commutator method. The proof of Theorem 2.3 (spectrum of Liouvillian) is given in Section 4. It consists of two main parts: verification that the virial theorems are applicable in the particular situation encountered in the analysis of our models (Subsection 4.2), and proof of a lower bound on a commutator of the Liouvillian with a suitable conjugate operator (Subsections 4.3, 4.4). In Section 5, we establish some technical results on the invariance of operator domains and on certain commutator expansions that are needed in the proofs of the virial theorems and of the theorem on regularity of eigenfunctions. Proofs of the latter results are presented in Section 6. In Section 7, we describe some results on unitary groups generated by vector fields which are needed in the definition of our ‘conjugate operator’ Aap in the positive commutator method. The last section, Section 8, contains proofs of several propositions used in earlier sections of the paper. 2. Definition of Models and Main Results on Thermal Ionization In Section 2.1, we introduce our model and use it to define a W ∗ -dynamical system (Mβ , σt,λ ). Our main results on thermal ionization are described in Section 2.2. 2.1. DEFINITION OF THE MODEL ( ) Starting with the algebra A and a (regularized) dynamics αt,λ on it, we intro( ) duce a reference state ωρ0 ,β , and consider the induced (regularized) dynamics σt,λ on πβ (A), where (H , πβ , ρ0 ) denotes the GNS representation corresponding to ( ) tends to a ∗automorphism group, σt,λ , of (A, ωρ0 ,β ). We show that, as → 0, σt,λ the von Neumann algebra Mβ , defined as the weak closure of πβ (A) in B(H ). We determine the generator, Lλ , of the unitary group, eit Lλ , on H implementing σt,λ; Lλ is called a Liouvillian. The relation between eigenvalues of Lλ and invariant normal states on Mβ will be explained later in this section (see Theorem 2.2). We will sometimes write simply L instead of Lλ , for λ = 0.
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THERMAL IONIZATION
2.1.1. The Algebra Af We introduce a C ∗ -algebra suitable for the description of the dynamics of the free field, and, as we explain below, for the description of the interacting dynamics. Let W = W(L20 ) be the Weyl CCR algebra over L20 := L2 (R3 , d3 k) ∩ L2 (R3 , |k|−1 d3 k), i.e., the C ∗ -algebra generated by the Weyl operators, W (f ), for f ∈ L20 , satisfying W (−f ) = W (f )∗ ,
W (f )W (g) = e−iIm f,g /2 W (f + g).
Here ·, · denotes the inner product of L20 . The latter relation implies the CCR W (f )W (g) = e−iIm f,g W (g)W (f ).
(19)
The expectation functional for the KMS state of an infinitely extended free Bose field in thermal equilibrium at inverse temperature β is given by 1 2 f 2 3 g −→ ωβ (W (g)) = exp − |g(k)| d k , 1 + β|k| 4 R3 e −1 which motivates the choice of the space L20 (as opposed to g ∈ L2 (R3 )). The free field dynamics on W is given by the ∗automorphism group αtW (W (f )) = W (ei|k|t f ).
(20)
It is well known that for f = 0, t → αtW (W (f )) is not a continuous map from R to W, but t → ω(αtW (W (f ))) is continuous for a large (weak∗ dense) class of states ω on W. An interacting dynamics is commonly defined using a Dyson series expansion, hence we should be able to give a sense to time integrals over αtW (a), for a ∈ W. Because of the lack of norm-continuity of the free dynamics, such an integral cannot be interpreted in norm sense, but only in a weak hence representation dependent way. In order to give a representation independent definition of the (coupled) dynamics, we modify the algebra in such a way that the free dynamics becomes norm-continuous. The idea is to introduce a time-averaged Weyl algebra, generated by elements given by ds h(s)αsW (a), (21) a(h) = R
for functions h in a certain class, and a ∈ W (if h is sharply localized at zero, the integral approximates a ∈ W). The free dynamics is then given by f W W ds h(s)αs (αt (a)) = ds h(s − t)αsW (a). αt (a(h)) = R
R
∗
We now construct a C -algebra whose elements, when represented on a Hilbert space, are given by (21), where the integral is understood in a weak sense.
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Let P be the free algebra generated by elements {a(h) | a ∈ W, hˆ ∈ C0∞ (R)}, where ˆ denotes the Fourier transform. Taking the functions h to be analytic (i.e., having a Fourier transform in C0∞ ) allows us to construct KMS states w.r.t. the free dynamics, as we explain below. We equip the algebra P with the star operation ¯ and introduce the seminorm defined by (a(h))∗ = (a ∗ )(h), W (22) p(a(h)) = sup dt h(t)π(αt (a)) , π∈Rep W
R
where the supremum extends over all representations of W. The integral on the r.h.s. of (22) is understood in the weak sense (t → π(αtW (a)) is weakly measurable for any π ∈ Rep W), and the norm is the one of operators acting on the representation Hilbert space. It is not difficult to verify that N = {a ∈ P | p(a) = 0} is a two-sided ∗ideal in P. We can therefore build the quotient ∗algebra P/N consisting of equivalence classes [a] = {a + n | a ∈ P, n ∈ N}, on which p defines a norm [a] = p(a),
[a] ∈ P/N,
having the C ∗ property [a]∗ [a] = [a]2 . The C ∗ -algebra Af of the field is defined to be the closure of the quotient in this norm, ·
Af = P/N . Notice that every πW ∈ Rep W induces a representation πf ∈ Rep Af according to πf (a(h)) = dt h(t)πW (αtW (a)). The algebra Af can be viewed as a timeaveraged version of the Weyl algebra. The advantage of Af over W is that the free field dynamics on Af , defined by f
αt (a(h)) = a(ht ),
ht (x) = h(x − t),
(23) f
is a norm-continous ∗automorphism group, i.e., αt (a) − a → 0, as t → 0, for any a ∈ Af . There is a one-to-one correspondence between (β, αtW )-KMS states ωβW on W f f and (β, αt )-KMS states ωβ on Af , given by the relation f
ωβ (a1 (f1 ) · · · an (fn )) (a1 ) · · · αtW (an )). = dt1 · · · dtn f1 (t1 ) · · · fn (tn )ωβW (αtW n 1
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THERMAL IONIZATION β
f
If (H, πW , ) is the GNS representation of (W, ωβW ) then the one of (Af , ωβ ) is β given by (H, πf , ), where β
πf (a1 (f1 ) · · · an (fn )) β (a0 ) · · · αtW (an )). = dt1 · · · dtn f1 (t1 ) · · · fn (tn )πW (αtW n 1
(24) β
It follows that any unitary group implementing the free dynamics relative to πW β implements it in the representation πf , and conversely. ( ) 2.1.2. The Algebra A and the Regularized Dynamics αt,λ
The C ∗ -algebra A describing the ‘observables’ of the combined system is the tensor product algebra A = Ap ⊗ Af .
(25)
Here Ap = B(Hp ) is the C ∗ -algebra of all bounded operators on the particle Hilbert space ⊕ 2 He de, (26) Hp = C ⊕ L (R+ , de; H) ≡ C ⊕ R+
where de is the Lebesgue measure on R+ , H is a (separable) Hilbert space, and the r.h.s. is the constant fibre direct integral with He ∼ = H, e ∈ R+ . An element in Hp is given by ψ = {ψ(e)}e∈{E}∪R+ , where ψ(E) ∈ C, and ψ(e) ∈ H, e ∈ R+ . Hp is a Hilbert space with inner product ψ(e), φ(e) H de. ψ, φ = ψ(E)φ(E) + R+
p
Let αt denote the ∗automorphism group on Ap given by αt (A) = eit Hp Ae−it Hp , p
where Hp is a selfadjoint operator on Hp , which is diagonalized by the direct integral decomposition of Hp : ⊕ e de, for some E < 0. (27) Hp = E ⊕ R+
The domain of definition of Hp is given by ⊕ 2 2 He de e ψ(e)H de < ∞ . D(Hp ) = C ⊕ ψ ∈ R+
(28)
R+
The dense set C0∞ (R+ ; H) ≡ C0∞ consists of all elements ψ ∈ Hp s.t. the support, supp(ψ R+ ), is a compact set in the open half-axis (0, ∞), and s.t. ψ is
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JÜRG FRÖHLICH AND MARCO MERKLI
infinitely many times continuously differentiable as an H-valued function. Clearly, C0∞ ⊂ D(Hp ), and since eit Hp leaves C0∞ invariant, it follows that C0∞ is a core for Hp . It is sometimes practical to identify C ∼ = Cϕ0 , and we say that ϕ0 is the eigenfunction of Hp corresponding to the eigenvalue E. EXAMPLE. This model is inspired by considering a block-diagonal Hamiltonian Hp on the Hilbert space C ⊕ L2 (R3 , d3 x), with Hp C = E < 0, Hp L2 (R3 , d3 x) = −. Passing to a diagonal representation of the Laplacian (Fourier transform), we have the following identifications, using polar coordinates: Hp = C ⊕ L2 (R3 , d3 k) = C ⊕ L2 (R+ × S 2 , |k|2 d|k| × d) = C ⊕ L2 (R+ , |k|2 d|k|; L2 (S 2 , d)) = C ⊕ L2 (R+ , dµ; H), 2 where we set H = L2 (S 2 , d), and make √ the change of variables |k| = e, so that dµ(e) = µ(e) de, with µ(e) = (1/2) e. To arrive at the form (26), (27) of Hp , Hp , we use the unitary map U : L2 (R+ , dµ; H) → L2 (R+ , de; H), given by √ ψ −→ U ψ = µψ.
If Hp is the operator of multiplication by e on L2 (R+ , dµ; H), then its transform, U Hp U −1 , is the operator of multiplication by e on L2 (R+ , de; H). We define the noninteracting time-translation ∗automorphism group of A (the free dynamics) by p
f
αt,0 := αt ⊗ αt . Given = 0, set 1 {(W ( gα ))(h ) − (W ( gα ))(h )∗ } ∈ A, Gα ⊗ V ( ) := 2i α
(29)
where the sum is over finitely many indices α, with Gα = G∗α ∈ B(Hp ), gα ∈ L20 , for all α, and where h is an approximation of the Dirac distribution localized at 2 2 zero. To be specific, we can take h (t) = (1/ )e−t / . For any value of the real coupling constant λ, the norm-convergent Dyson series αt,0 (A) + t n (iλ) dt1 · · · + n1
=:
( ) (A), αt,λ
0
tn−1
dtn [αtn ,0 (V ( ) ), [· · · [αt1 ,0 (V ( ) ), αt,0 (A)] · · ·]]
0
(30)
where A ∈ A, defines a ∗automorphism group of A. The multiple integral in (30) is understood in the product topology coming from the strong topology of B(Hp ) and the norm topology of Af .
249
THERMAL IONIZATION
( ) as a regularized dynamics, in the sense that it has a limit, One should view αt,λ as → 0, in suitably chosen representations of A; (this is shown below). The functions gα ∈ L20 are called form factors. Using spherical coordinates in R3 , we often write gα = gα (ω, ), where (ω, ) ∈ R+ × S 2 . In accordance with the direct integral decomposition of Hp , the operators Gα are determined by integral kernels. For ψ = {ψ(e)} ∈ Hp , we set Gα (E, e )ψ(e ) de , if e = E, Gα (E, E)ψ(E) + R + (31) (Gα ψ)(e) = Gα (e, e )ψ(e ) de , if e ∈ R+ . Gα (e, E)ψ(E) + R+
The families of bounded operators Gα (e, e ): He → He , with HE = C, have the following symmetry properties (guaranteeing that Gα is selfadjoint): Gα (E, E) ∈ R, Gα (E, e)∗ = Gα (e, E), Gα (e, e )∗ = Gα (e , e),
∀e ∈ R+ , ∀e, e ∈ R+ .
Here, ∗ indicates taking the adjoint of an operator in B(H, C) or B(H). Remarks. (1) The map Gα (E, e): He → C is identified (Riesz) with an element α (e) ∈ He , so that Gα (E, e)ψ(e) = α (e), ψ(e) He . Then Gα (E, e)∗ : C → He is given by Gα (E, e)∗ z = zα (e), for all z ∈ C. Consequently, the above symmetry condition implies that Gα (e, E)z = zα (e). (2) Assuming the strong derivatives w.r.t. the two arguments (e, e ) ∈ R2+ of Gα (·, ·) exist, we have that ∂1,2 Gα (e, e ) are operators H → H. Similarly, one introduces higher derivatives. We assume that all derivatives occuring are bounded operators on H. For Gα (·, ·) ∈ C n (R+ × R+ , B(H)), it is easily verified that the above symmetry conditions imply that (∂1n1 ∂2n2 Gα (e, e ))∗ = ∂1n2 ∂2n1 Gα (e , e),
(32)
for any n1,2 0, n1 + n2 n, where ∗ is the adjoint on B(H). Similar statements hold for Gα (E, e), Gα (e, E). The interaction is required to satisfy the following three conditions: (A1) Infrared and ultraviolet behaviour of the form factors: for any fixed ∈ S 2 , gα (·, ) ∈ C 4 (R+ ), and there are two constants 0 < k1 , k2 < ∞, s.t. if ω < k1 , then |∂ωj gα (ω, )| < k2 ωp−j ,
for some p > 2,
(33)
uniformly in α, j = 0, . . . , 4 and ∈ S 2 . Similarly, there are two constants 0 < K1 , K2 < ∞, s.t. if ω > K1 , then |∂ωj gα (ω, )| < K2 ω−q−j ,
for some q > 72 .
(34)
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JÜRG FRÖHLICH AND MARCO MERKLI
(A2) The map (e, e ) → Gα (e, e ) is C 3 (R+ × R+ , B(H)), and we have
R+
R+
dee−m1 ∂1m2 Gα (e, E)2H < ∞, m de de e−m1 (e )−m1 ∂1m2 ∂2 2 Gα (e, e )2B(H) < ∞,
(35) (36)
R+
for all integers m1,2 , m1,2 0, s.t. m1 +m1 +m2 +m2 = 0, 1, 2, 3. Moreover, de de eGα (e, e )2B(H) < ∞. (37) R+
R+
(A3) The Fermi Golden Rule condition. Define a family of bounded operators on Hp by gα (ω, )Gα . (38) F (ω, ) = α
There is an 0 > 0, s.t. for 0 < < 0 , we have that ∞ ω2 p¯ 0 p0 F (ω, ) dω d βω F (ω, )∗ p0 e −1 (Hp − E − ω)2 + 2 −E S2 γp0 , (39) for some strictly positive constant γ > 0. Here p0 is the orthogonal projection onto the eigenspace C of Hp (see (26), (27)), and p¯0 = 1 − p0 is the projection onto L2 (R+ , de; H). Remarks. (1) Since E < 0 we have that γ ∼ eβE decays exponentially in β, for large β. (2) Recalling that Gα (E, e) is identified with α (e) ∈ He , see Remark (1) after (31) above, we can rewrite the l.h.s. of (39) as ω2 dω d de βω × 2 e − 1 (e − E − ω)2 + 2 (−E,∞)×S R+ g¯α (ω, ) α (e), α (e) H gα (ω, ), × α,α
and this expression has the limit ω2 dω d βω g¯α (ω, ) α (E + ω), α (E + ω) H gα (ω, ), e −1 (−E,∞)×S 2 α,α
as → 0, because α (e) is continuous in e. Consequently, (39) is satisfied if this integral is strictly positive.
251
THERMAL IONIZATION
2.1.3. The Reference State ωρ0 ,β Let ρ0 be a strictly positive density matrix on Hp , i.e., ρ0 > 0, tr ρ0 = 1, and f f denote by ωρp0 the state on Ap given by A → tr ρ0 A. Let ωβ be the (αt , β)-KMS state on Af and define the reference state f
ωρ0 ,β = ωρp0 ⊗ ωβ . The GNS representation (H, πβ , ρ0 ) corresponding to (A, ωρ0 ,β ) is explicitly known. It has first been described in [AW]; (we follow [JP] in its presentation). The representation Hilbert space is H = Hp ⊗ Hp ⊗ F ,
(40)
where F is a shorthand for the Fock space F = F ((L2 (R × S 2 , du × d))),
(41)
du being the Lebesgue measure on R, and d the uniform measure on S 2 . F (X) denotes the bosonic Fock space over a (normed vector) space X: (SX ⊗n ), (42) F (X) := C ⊕ n1
where S is the projection onto the symmetric subspace of the tensor product. We adopt standard notation, e.g., is the vacuum vector, [ψ]n is the n-particle component of ψ ∈ F (X), d(A) is the second quantization of the operator A on X, N = d(1) is the number operator. The representation map πβ : A → B(H ) is the product β
πβ = πp ⊗ πf , where the ∗homomorphism πp : Ap → B(Hp ⊗ Hp ) is given by πp (A) = A ⊗ 1p . β
The representation map πf : Af → B(F ) is determined by the representation β β map of the Weyl algebra, πW : W → B(F ), according to (24). To describe πW , we point out that L2 (R+ × S 2 ) ⊕ L2 (R+ × S 2 ) is isometrically isomorphic to L2 (R × S 2 ) via the map uf (u, ), u > 0, (f, g) → h, h(u, ) = (43) ug(−u, ), u < 0. β
The representation map πW is given by β
πW = πFock ◦ Tβ ,
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JÜRG FRÖHLICH AND MARCO MERKLI
where the Bogoliubov transformation Tβ: W(L20 ) → W(L2 (R × S 2 )) acts as W (f ) → W (τβ f ), with τβ : L2 (R+ × S 2 ) → L2 (R × S 2 ) given by √ uf (u, ), u > 0, u √ (44) (τβ f )(u, ) = −βu ¯ 1−e − −uf (−u, ), u < 0. Remarks. (1) It is easily verified that Im τβ f, τβ g L2 (R×S 2 ) = Im f, g L2 (R+ ×S 2 ) , for all f, g ∈ L20 , so the CCR (19) are preserved under the map τβ . (2) In the limit β → ∞, the r.h.s. of (44) tends to uf (u, ), 0,
u > 0, u < 0,
which is identified via (43) with f ∈ L20 . Thus, Tβ reduces to the identity (an β imbedding), πW becomes the Fock representation of W(L20 ), as β → ∞, and we recover the zero temperature situation. It is useful to introduce the following notation. For f ∈ L2 (R × S 2 ), we define (f ), on the Hilbert space (40), by unitary operators, W (f ) = eiϕ(f ) , W
f ∈ L2 (R × S 2 ),
where ϕ(f ) is the selfadjoint operator on F given by a ∗ (f ) + a(f ) , (45) √ 2 and a ∗ (f ), a(f ) are the creation- and annihilation-operators on F , smeared out with f . One easily verifies that ϕ(f ) =
β (τβ f ). πW (W (f )) = W
The cyclic GNS vector is given by ρ0 = ρp0 ⊗ , where is the vacuum in F , and kn ϕn ⊗ Cp ϕn ∈ Hp ⊗ Hp . ρp0 =
(46)
n0
Here, {kn2 }∞ n=0 is the spectrum of ρ0 , {ϕn } is an orthogonal basis of eigenvectors of ρ0 , and Cp is an antilinear involution on Hp . The origin of Cp lies in the identification of l 2 (Hp ) (Hilbert–Schmidt operators on Hp ) with Hp ⊗ Hp , via |ϕ ψ| → ϕ ⊗ Cp ψ. We fix a convenient choice for Cp : it is the antilinear involution on Hp that has the effect of taking complex conjugates of components of vectors, in the basis in which the Hamiltonian Hp is diagonal, i.e., ψ(e) ∈ C, e = E, (Cp ψ)(e) = ψ(e) ∈ H, e ∈ [0, ∞).
253
THERMAL IONIZATION
By ψ(e) ∈ H for e ∈ [0, ∞), we understand the element in H obtained by complex conjugation of the components of ψ(e) ∈ H, in an arbitrary, but fixed, orthonormal basis of H. This Cp is also called the time reversal operator, and we have Cp H p Cp = H p . 2.1.4. The W ∗ -dynamical System (Mβ , σt,λ ) Let Mβ be the von Neumann algebra obtained by taking the weak closure (or equivalently, the double commutant) of πβ (A) in B(H ): Mβ = B(Hp ) ⊗ 1p ⊗ πf (Af ) ⊂ B(H ). β
Since ρ0 is strictly positive, ρp0 is cyclic and separating for the von Neumann β algebra πp (Ap ) = B(Hp )⊗1p . Similarly, is cyclic and separating for πf (Af ) , since it is the GNS vector of a KMS state (see, e.g., [BRII]). Consequently, ρ0 is cyclic and separating for Mβ . Let J be the modular conjugation operator associated to (Mβ , ρ0 ). It is given by J = Jp ⊗ Jf ,
(47)
where, for ϕ, ψ ∈ Hp , Jp (ϕ ⊗ Cp ψ) = ψ ⊗ Cp ϕ, and, for ψ = {[ψ]n }n0 ∈ F , [Jf ψ]n (u1 , . . . , un ) = [ψ]n (−u1 , . . . , −un ),
for n 1,
[Jf ψ]0 = [Jf ψ]0 ∈ C. Clearly, J ρ0 = ρ0 , and one verifies that Jp πp (A)Jp = 1p ⊗ Cp ACp , β (−e−βu/2 τβ (f )) = W (e−βu/2 τβ (f ))∗ , Jf π (W (f ))Jf = W W
(48) (49)
(f (−u, )). (f )Jf = W for f ∈ L20 . More generally, for f ∈ L2 (R × S 2 ), Jf W ( ) We now construct a unitary implementation of αt,λ w.r.t. πβ . Recall that πβ = β πp ⊗ πf , where πp : B(H) → B(Hp ⊗ Hp ) is continuous w.r.t. the strong topoloβ gies and πf : Af → B(F ) is continuous w.r.t. the norm topologies (because it is a ∗ homomorphism). We thus have, for A ∈ A, ( ) (A)) πβ (αt,λ
= πβ (αt,0 (A)) +
n1
t
dt1 · · ·
(iλ)n 0
tn−1
dtn [πβ (αtn ,0 (V ( ) )), [· · ·
0
· · · [πβ (αt1 ,0 (V ( ) )), πβ (αt,0 (A))] · · ·]].
(50)
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JÜRG FRÖHLICH AND MARCO MERKLI
Because πp (αt (A)) = eit Hp Ae−it Hp ⊗ 1p = eit (Hp ⊗1p −1p ⊗Hp ) πp (A)e−it (Hp ⊗1p −1p ⊗Hp ) , p
and (eiut τβ (f )) πW (αtW (W (f ))) = πW (W (eiωt f )) = W (τβ (f ))e−it d(u) = eit d(u)W β
β
= eit d(u)πW (W (f ))e−it d(u), β
so that πf (αt (a)) = eit d(u)πf (a)e−it d(u), β
f
β
a ∈ Af ,
we find that σt,0 (πβ (A)) := πβ (αt,0 (A)) = eit L0 πβ (A)e−it L0 , for all A ∈ A, where L0 is the selfadjoint operator on H , given by L0 = Hp ⊗ 1p − 1p ⊗ Hp + d(u),
(51)
commonly called the (noninteracting, standard) Liouvillian. One easily verifies that J eit L0 = eit L0 J.
(52)
Remark. There are other selfadjoint operators generating unitary implementations of σt,0 on H. Indeed, we may add to L0 any selfadjoint operator L0 affiliated with the commutant Mβ ; then L0 + L0 still generates a unitary implementation of σt,0 on H . However, the additional condition (52) fixes L0 uniquely, and the generator of this unitary group is called the standard Liouvillian for σt,0 . This terminology has been used before in [DJP]. The importance of considering the standard Liouvillian (as opposed to other generators of the dynamics) lies in the fact that its spectrum is related to the dynamical properties of the system; see Theorem 2.2. Notice that σt,0 is a group of ∗automorphisms of πβ (A), in particular, eit L0 πβ (A)e−it L0 = πβ (A), ∀t ∈ R. From Tomita–Takesaki theory, we know that J Mβ J = Mβ (the commutant), and since σt,0 (J πβ (V ( ) )J ) = J σt,0 (πβ (V ( ) ))J = J πβ (αt,0 (V ( ) ))J ∈ Mβ , we can write the multicommutator in (50) as [σtn ,0 (πβ (V ( ) ) − J πβ (V ( ) )J ), [· · · · · · [σt1 ,0 (πβ (V ( ) ) − J πβ (V ( ) )J ), σt,0(πβ (A))] · · ·]].
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THERMAL IONIZATION
( ) , It follows that the r.h.s. of (50) defines a ∗automorphism group of πβ (A), σt,λ which is implemented unitarily by ( )
( )
( ) ( ) (πβ (A)) = πβ (αt,λ (A)) = eit Lλ πβ (A)e−it Lλ , σt,λ
with ( ) ) − λJ πβ (V ( ) )J. L( ) λ = L0 + λπβ (V
It is not difficult to see (using Theorem 3.1) that the regularized Liouvillian L( ) λ is essentially selfadjoint on D = C0∞ ⊗ C0∞ ⊗ (F (C0∞ (R × S 2 )) ∩ F0 ) ⊂ H , ( )
( )
where F0 is the finite-particle subspace. Moreover, we have that J eit Lλ = eit Lλ J . We now explain how to remove the regularization ( → 0), obtaining a weak∗ continuous ∗automorphism group σt,λ of the von Neumann algebra Mβ . We recall that a ∗automorphism group τt on a von Neumann algebra M is called weak∗ continuous iff t → ω(τt (A)) is continuous, for all A ∈ M and for all normal states ω on M. From 1 (eiut τβ (gα )) − Gα ⊗ 1p ⊗ dt h (t){W πβ (V ( ) ) = 2i R α
J πβ (V ( ) )J =
1p ⊗ Cp Gα Cp ⊗
α
1 2i
R
(eiut τβ (gα ))∗ } −W (eiut e−βu/2 τβ (gα )) − dt h (t){W
(eiut e−βu/2 τβ (gα ))∗ }, −W where we recall that h (t) = (1/ )e−t / approximates the Dirac delta distribution concentrated at zero, one verifies that, in the strong sense on D, Gα ⊗ 1p ⊗ ϕ(τβ (gα )), lim πβ (V ( ) ) = 2
→0
α
lim J πβ (V
→0
( )
)J =
2
1p ⊗ Cp Gα Cp ⊗ ϕ(e−βu/2 τβ (gα )),
α
where the operator ϕ(f ) has been defined in (45). The symmetric operator Lλ , defined on D by Lλ = L0 + λI, with I=
α
Gα ⊗ 1p ⊗ ϕ(τβ (gα )) − 1p ⊗ Cp Gα Cp ⊗ ϕ(e−βu/2 τβ (gα )),
(53)
(54)
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JÜRG FRÖHLICH AND MARCO MERKLI
is essentially selfadjoint on D, for any real value of λ; (this will be shown to be a consequence of Theorem 3.1). Using Theorem 5.1 on invariance of domains, the Duhamel formula gives t ( ) ( ) eit Lλ = eit Lλ − iλ eisLλ (I − πβ (V ( ) ) + J πβ (V ( ) )J )e−i(s−t )Lλ 0 ( )
as operators defined on D, from which it follows that eit Lλ → eit Lλ , as → 0, in ( ) (A) → σt,λ (A), the strong sense on H. Consequently, for A ∈ πβ (A), we have σt,λ in the σ -weak topology of B(H). Notice that for A ∈ πβ (A), we have σt,λ (A) ∈ ( ) ( ) (A), σt,λ (A) ∈ πβ (A) ⊂ Mβ , and Mβ Mβ , because σt,λ (A) = w-lim →0 σt,λ is weakly closed. Clearly, σt,λ is a σ -weakly continuous ∗automorphism group of B(H). If A ∈ Mβ , there is a net {Aα } ⊂ πβ (A), s.t. Aα → A, in the weak operator topology. Thus, since σt,λ is weakly continuous, we conclude that σt,λ (A) = w-lim σt,λ(Aα ) ∈ Mβ . α
We summarize these considerations in a proposition. PROPOSITION 2.1. (Mβ , σt,λ ) is a W ∗ -dynamical system, i.e. σt,λ is a weak∗ continuous group of ∗automorphisms of the von Neumann algebra Mβ . Moreover, σt,λ is unitarily implemented by eit Lλ , where Lλ is given in (53), (54), and J eit Lλ = eit Lλ J,
for all t ∈ R.
2.1.5. Kernel of Lλ and Normal Invariant States Let P be the natural cone associated with (Mβ , ρ0 ), i.e., P is the norm closure of the set {AJ A ρ0 | A ∈ Mβ } ⊂ H . The data (Mβ , H , J, P ) is called the standard form of the von Neumann algebra Mβ . We have constructed J and P explicitly, starting from the cyclic and separating vector ρ0 . There is, however, a general theory of standard forms of von Neumann algebras; see [BRI, II, Ara, Con] for the case of σ -finite von Neumann algebras (as in our case), or [Haa] for the general case. Among the properties of standard forms, we mention here only the following: (P) For every normal state ω on Mβ , there exists a unique ξ ∈ P , s.t. ω(A) = ξ, Aξ , ∀A ∈ Mβ . Recall that a state ω on Mβ ⊂ B(H ) is called normal iff it is σ -weakly continuous, or, equivalently, iff it is given by a density matrix ρ ∈ l 1 (H ), as ω(A) = tr ρA, for all A ∈ Mβ . The uniqueness of the representing vector in the natural cone, according to (P), allows us to establish the following connection between the kernel of Lλ and the normal invariant states (see also, e.g., [DJP]).
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THERMAL IONIZATION
THEOREM 2.2. If Lλ does not have a zero eigenvalue, i.e., if ker Lλ = {0}, then there does not exist any σt,λ -invariant normal state on Mβ . Proof. We show below that, for all t ∈ R, eit Lλ P = P .
(55)
If ω is a normal state on Mβ , invariant under σt,λ , i.e., such that ω ◦ σt,λ = ω, for all t ∈ R, then, for a unique ξ ∈ P , ω(A) = ξ, Aξ = ω(σt,λ(A)) = e−it Lλ ξ, Ae−it Lλ ξ . Since (55) holds, and due to the uniqueness of the vector in P representing a given state, we conclude that eit Lλ ξ = ξ , for all t ∈ R, i.e. Lλ has a zero eigenvalue with eigenvector ξ . We now show (55). Notice that (55) is equivalent to eit Lλ P ⊆ P . Since P is a closed set, it is enough to show that for all A ∈ Mβ , eit Lλ AJ A ρ0 ∈ P . Since eit Lλ J = J eit Lλ , eit Lλ Ae−it Lλ ∈ Mβ , for all A ∈ Mβ , and BJ BJ P ⊂ P , for all B ∈ Mβ , we only need to prove that eit Lλ ρ0 ∈ P .
(56)
The Trotter product formula gives t
t
eit Lλ ρ0 = lim (ei n λI ei n L0 )n ρ0 , n→∞
and, since P is closed, (56) holds provided the general term under the limit is in P , for all n 1. We show that eisL0 P = P and eisλI P = P , for all s ∈ R. Remarking that eisL0 ρ0 = (eisHp ⊗ e−isHp ⊗ eis d(u)) ρ0 = (eisHp ⊗ 1p )J (eisHp ⊗ 1p ) ρ0 , where we use that Jp (eisHp ⊗ 1p )Jp = 1p ⊗ Cp eisHp Cp = 1p ⊗ e−isHp , recalling that eisL0 implements σt,0 , and arguing as above, we see that eit L0 P = P . The Trotter product formula gives N Gα ⊗ 1p ⊗ ϕ(τβ (gα )) − J Gα ⊗ 1p ⊗ ϕ(τβ (gα ))J ξ exp is α=1
= lim
n1 →∞
s τβ (gα ) × e 1 ⊗ 1p ⊗ W n1 i s G s τβ (gα ) J × × J e n1 1 ⊗ 1p ⊗ W n1 N s (Gα ⊗ 1p ⊗ ϕ(τβ (gα )) − × exp i n1 α=2 i ns G1
− J Gα ⊗ 1p ⊗ ϕ(τβ (gα ))J )
n1 ξ,
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JÜRG FRÖHLICH AND MARCO MERKLI
for all ξ ∈ P , and we may apply Trotter’s formula repeatedly to conclude that, since AJ AJ P ⊂ P , for A ∈ Mβ , and P is closed, we have that eisλI P = P , for all s ∈ R. 2 Remark. The proof of Theorem 2.2 uses property (P), which is satisfied in our case, because ρ0 is cyclic and separating for Mβ . This, in turn, is true because ρ0 has been chosen to be strictly positive. One may start with any reference state f of the form ωρp ⊗ ωβ , where ρ is any density matrix on Hp ; it may be of finite rank. The resulting von Neumann algebra (obtained as the weak closure of A f when represented on the GNS Hilbert space corresponding to (A, ωρp ⊗ ωβ )) is ∗isomorphic to Mβ . This is the reason we have not added to Mβ an index for the f density matrix ρ0 . More specifically, the GNS representation of (A, ωρp ⊗ ωβ ) is given by (H1 , π1 , 1 ), where H1 = Hp ⊗ K ρ ⊆ Hp ⊗ Hp , π1 (A ⊗ (W (f ))(h)) = A ⊗ 1p ⊗ 1 = ρp ⊗ .
R
(eiut τβ (f )), dt h(t)W
Here, Kρ is the closure of Ran ρ, ρp is given as in Equation (46). Consequently, β π1 (A) = B(Hp ) ⊗ 1p Kρ ⊗ πf (Af ) ∼ = Mβ .
In particular, π1 (A) and Mβ have the same set of normal states. Thus, our particular choice for the reference state is immaterial when examining properties of normal states. One may express this in the following way: (Mβ , H , J, P ) is a standard form for all the von Neumann algebras obtained from any reference state f (A, ωρp ⊗ ωβ ).
2.2. RESULT ON THERMAL IONIZATION Our main result in this paper is that the W ∗ -dynamical system (Mβ , σt,λ ) introduced above does not have any normal invariant states. THEOREM 2.3. Assume conditions (A1)–(A3) hold. For any inverse temperature 0 < β < ∞ there is a constant, λ0 (β) > 0, proportional to γ given in (39), such that the following holds. If 0 < |λ| < λ0 then the Liouvillian Lλ given in (53) and (54) does not have any eigenvalues. Remark. Since γ decays exponentially in β, for large β, Theorem 2.3 is a high temperature result (β has to be small for reasonable values of the coupling constant λ). From physics it is clear that thermal ionization takes place for arbitrary
259
THERMAL IONIZATION
positive temperatures (but not at zero temperature, where the coupled system has a ground state). Combining Theorems 2.3 and 2.2 yields our main result about thermal ionization. THEOREM 2.4 (Thermal ionization). Under the assumptions of Theorem 2.3, there do not exist any normal σt,λ-invariant states on Mβ . Remark. For λ = 0, the state ω0 , determined by the vector 0p ⊗ , where 0p = ϕ0 ⊗ϕ0 ∈ Hp ⊗Hp , and ϕ0 is the eigenvector of Hp , is a normal σt,0 -invariant state on Mβ . As we have explained in the introduction, the physical interpretation of Theorem 2.4 is that a single atom coupled to black-body radiation at a sufficiently high positive temperature will always end up being ionized. The proof of Theorem 2.3 is based on a novel virial theorem. 3. Virial Theorems and the Positive Commutator Method 3.1. TWO ABSTRACT VIRIAL THEOREMS Let H be a Hilbert space, D ⊂ H a core for a selfadjoint operator Y 1, and X a symmetric operator on D. We say the triple (X, Y, D) satisfies the GJN (Glimm– Jaffe–Nelson) condition, or that (X, Y, D) is a GJN-triple, if there is a constant k < ∞, s.t. for all ψ ∈ D: Xψ kY ψ, ±i{ Xψ, Y ψ − Y ψ, Xψ } k ψ, Y ψ .
(57) (58)
Notice that if (X1 , Y, D) and (X2 , Y, D) are GJN triples, then so is (X1 +X2 , Y, D). Since Y 1, inequality (57) is equivalent to Xψ k1 Y ψ + k2 ψ, for some k1 , k2 < ∞. THEOREM 3.1 (GJN commutator theorem). If (X, Y, D) satisfies the GJN condition, then X determines a selfadjoint operator (again denoted by X), s.t. D(X) ⊃ D(Y ). Moreover, X is essentially selfadjoint on any core for Y , and (57) is valid for all ψ ∈ D(Y ). Based on the GJN commutator theorem, we next describe the setting for a general virial theorem. Suppose one is given a selfadjoint operator 1 with core D ⊂ H, and operators L, A, N, D, Cn , n = 0, 1, 2, 3, all symmetric on D, and satisfying ϕ, Dψ = i{ Lϕ, Nψ − Nϕ, Lψ }, C0 = L, ϕ, Cn ψ = i{ Cn−1 ϕ, Aψ − Aϕ, Cn−1 ψ },
(59) n = 1, 2, 3,
(60)
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JÜRG FRÖHLICH AND MARCO MERKLI
where ϕ, ψ ∈ D. We assume that • (X, , D) satisfies the GJN condition, for X = L, N, D, Cn . Consequently, all these operators determine selfadjoint operators, which we denote by the same letters. • A is selfadjoint, D ⊂ D(A), and eit A leaves D() invariant. Remarks. (1) From the invariance condition eit A D() ⊂ D(), it follows that for some 0 k, k < ∞, and all ψ ∈ D(),
eit A ψ kek |t | ψ.
(61)
A proof of this can be found in [ABG], Propositions 3.2.2 and 3.2.5. (2) Condition (57) is phrased equivalently as ‘X kY , in the sense of Kato on D.’ (3) One can show that if (A, , D) satisfies conditions (57), (58), then the above assumption on A holds; see Theorem 5.1. THEOREM 3.2 (1st virial theorem). Assume that, in addition to (59), (60), we have, in the sense of Kato on D, D kN 1/2 , eit A C1 e−it A kek |t | N p , some 0 p < ∞, eit A C3 e−it A kek |t | N 1/2 ,
(62) (63) (64)
for some 0 k, k < ∞, and all t ∈ R. Let ψ be an eigenvector of L. Then there is a one-parameter family {ψα } ⊂ D(L) ∩ D(C1 ), s.t. ψα → ψ, α → 0, and lim ψα , C1 ψα = 0.
α→0
(65)
Remarks. (1) A sufficient condition for (63) to hold (with k = 0) is that N and eit A commute, for all t ∈ R, in the strong sense on D, and C1 kN p . This condition will always be satisfied in our applications. A similar remark applies to (64). (2) In a heuristic way, we understand C1 as the commutator i[L, A] = i(LA − AL), and (65) as ψ, i[L, A]ψ = 0, which is a standard way of stating the virial theorem, see, e.g., [ABG] and [GG] for a comparison (and correction) of virial theorems encountered in the literature. The result of the virial theorem is still valid if we add to the operator A a suitably small perturbation A0 : THEOREM 3.3 (2nd virial theorem). Suppose that we are in the situation of Theorem 3.2 and that A0 is a bounded operator on H s.t. Ran A0 ⊂ D(L) ∩ Ran P (N n0 ), for some n0 < ∞. Then i[L, A0 ] = i(LA0 − A0 L) is well defined
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in the strong sense on D(L), and we have, for the same family of approximating eigenvectors as in Theorem 3.2: lim ψα , (C1 + i[L, A0 ])ψα = 0.
α→0
(66)
In conjunction with a positive commutator estimate, the virial theorem implies a certain regularity of eigenfunctions. THEOREM 3.4 (Regularity of eigenfunctions). Suppose C is a symmetric operator on a domain D(C) s.t., in the sense of quadratic forms on D(C), we have that C P − B, where P 0 is a selfadjoint operator, and B is a bounded (everywhere defined) operator. Let ψα be a family of vectors in D(C), with ψα → ψ, as α → 0, and s.t. lim ψα , Cψα = 0.
(67)
α→0
Then ψ, Bψ 0, ψ ∈ D(P 1/2), and P 1/2 ψ ψ, Bψ 1/2 .
(68)
Remark. Theorem 3.4 can be viewed as a consequence of an abstract Fatou lemma, see [ABG], Proposition 2.1.1. We give a different, very short proof of (68) at the end of Section 6.
3.2. THE POSITIVE COMMUTATOR METHOD This method gives a conceptually very easy proof of absence of point spectrum. The subtlety of the method lies in the technical details, since one deals with unbounded operators. Suppose we are in the setting of the virial theorems described in Section 3.1, and that the operator C1 (or C1 + i[L, A0 ]) is strictly positive, i.e. C1 γ ,
(69)
for some γ > 0. Inequality (69) and the virial theorem immediately show that L cannot have any eigenvalues. Indeed, assuming ψ is an eigenfunction of L, we reach the contradiction 0 = lim ψα , C1 ψα γ lim ψα , ψα = γ ψ2 > 0. α→0
α→0
Although the global PC estimate (69) holds in our situation, often one manages to prove merely a localized version. Suppose g ∈ C ∞ (J ) is a smooth function with support in an interval J ⊆ R, g J1 = 1, for some J1 ⊂ J , s.t. g(L) leaves the form domain of C1 invariant. The same reasoning as above shows that if g(L)C1 g(L) γ g 2 (L),
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JÜRG FRÖHLICH AND MARCO MERKLI
for some γ > 0, then L has no eigenvalues in the interval J1 . The use of PC estimates for spectral analysis of Schrödinger operators has originated with Mourre [Mou], and had recent applications in [Ski, BFSS, DJ, Mer]. 4. Proof of Theorem 2.3 4.1. STRATEGY OF THE PROOF As in [JP, Mer], the starting point in the construction of a positive commutator is the adjoint operator Af = d(i∂u ), the second quantized generator of translation in the radial variable of the glued Fock space F , see (41). We formally have i[L0 , Af ] = d(1f ) = N 0. The kernel of this form is the infinite-dimensional space Hp ⊗ Hp ⊗ Ran P . Following [Mer], one is led to try to add a suitable operator A0 to Af , where A0 depends on the interaction λI , and is designed in such a way that i[L0 + λI, Af + A0 ] is strictly positive (has trivial kernel). This method is applicable if the (imaginary part) of the so-called level shift operator is strictly positive, or equivalently, if (39) is satisfied, but where the finite-dimensional projection p0 is replaced by the infinite-dimensional projection 1p . Such a positivity condition does not hold for reasonable operators Gα and functions gα . In order to be able to carry out our program, we add to Af a term Ap ⊗ 1p − 1p ⊗Ap that reduces the kernel of the commutator. A prime candidate for Ap would be the operator i∂e acting on Hp (we write simply i∂e instead of 0 ⊕ i∂e , c.f. (26)), since then i[L0 , Ap ⊗ 1p − 1p ⊗ Ap + Af ] = P+ (Hp ) ⊗ 1p + 1p ⊗ P+ (Hp ) + N, ⊕ where P+ (Hp ) = R+ de is the projection onto L2 (R+ , de; H ). The above form has now a one-dimensional kernel, Ran p0 ⊗ p0 ⊗ P . By adding a suitable operator A0 , as described above, one can obtain a lower bound on the commutator (and in particular, reduce its kernel to {0}), provided (39) is satisfied. However, the operator Ap chosen above has the inconvenience of not being selfadjoint, while our virial theorems require selfadjointness. We introduce a family of selfadjoint operators Aap , a > 0, that approximate i∂e in a certain sense (a → 0). The idea of approximating a nonselfadjoint A by a selfadjoint sequence was also used in [Ski]. We now define Aa and then explain, in the remainder of this subsection, how to prove Theorem 2.3. We define Aap as the generator of a unitary group on L2 (R+ , de; H ), which is induced by a flow on R+ . For the proof of the following proposition, and more information on unitary groups induced by flows, we refer to Section 7. PROPOSITION 4.1. Let ξ : R+ → R+ be a bounded, smooth vector field, s.t. ξ(0) = 0, ξ(e) → 1, as e → ∞, and (1 + e)ξ ∞ < ∞. Then ξ generates a
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THERMAL IONIZATION
global flow, and this flow induces a continuous unitary group on L2 (R+ , de; H). The generator Ap of this group is essentially selfadjoint on C0∞ , and it acts on C0∞ as Ap = i( 12 ξ (e) + ξ(e)∂e ),
(70)
where ξ (e) and ξ(e) are multiplication operators. Given a > 0, ξa (e) = ξ(e/a) is a vector field on R+ , and lima→0 ξa = 1, pointwise (except at zero). The generator Aap of the unitary group induced by ξa is given on its core, C0∞ , by e 11 e a Ap = i ξ +ξ ∂e . (71) 2a a a We define the selfadjoint operator Aa = Aap ⊗ 1p ⊗ 1f − 1p ⊗ Aap ⊗ 1f + Af , C1a
(72)
a
of iL with A (in the sense given in (60), see also and calculate the commutator Subsection 4.2): ⊕ ⊕ a C1 = ξa (e) de ⊗ 1p + 1p ⊗ ξa (e) de + N + λI1a , (73) R+
R+
where I1a is N 1/2 -bounded. In Section 4.3, we show that C1a + i[L, A0 ] Ma , where Ma is a bounded operator. We will see that s-lima→0+ Ma = M (see Proposition 4.6), where M is a bounded, strictly positive operator (see Proposition 4.8). Since Ma , M are bounded, we obtain from the virial theorem 0 = lim ψα , (C1a + i[L, A0 ])ψα ψ, (Ma − M)ψ + ψ, Mψ , α→0
(74)
for any eigenfunction ψ of L. Taking a → 0+ and using strict positivity of M (for small, but nonzero λ, see Proposition 4.8), gives a contradiction, and this will prove Theorem 2.3. 4.2. CONCRETE SETTING FOR THE VIRIAL THEOREMS The Hilbert space is the GNS representation space (40), and we set D = C0∞ ⊗ C0∞ ⊗ Df ,
(75)
where Df = F (C0∞ (R × S 2 )) ∩ F0 , and F0 denotes the finite-particle subspace of Fock space. The operator is given by = p ⊗ 1p + 1p ⊗ p + f , ⊕ e de + 1p = Hp P+ (Hp ) + 1p , p =
(77)
f = d(u2 + 1) + 1f .
(78)
(76)
R+
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JÜRG FRÖHLICH AND MARCO MERKLI
In (77), we have introduced P+ (Hp ), the projection onto the spectral interval R+ of Hp . It is clear that is essentially selfadjoint on D, and 1. The operator L is the interacting Liouvillian (53), and N = d(1)
(79)
is the particle number operator in F ≡ F (L2 (R × S 2 )). Clearly, X = L, N are symmetric operators on D, and the symmetric operator D on D (see (59)) is given by iλ {Gα ⊗ 1p ⊗ (−a ∗ (τβ (gα )) + a(τβ (gα ))) − D= √ 2 α − 1p ⊗ Cp Gα Cp ⊗ (−a ∗ (e−βu/2 τβ (gα )) + a(e−βu/2 τβ (gα )))}.
(80)
The operator A is given by Aa defined in (72). Notice that Aap leaves C0∞ invariant, Af leaves Df invariant, so Aa maps D into D(L). Furthermore, it is easy to see that L maps D into D(Aa ), hence the commutator of L with Aa is well defined in the strong sense on D. The same is true for the multiple commutators of L with Aa . Setting ξa (e) = ξ (e/a), ξa (e) = ξ (e/a), we obtain ⊕ ⊕ a ξa (e) de ⊗ 1p + 1p ⊗ ξa (e) de + N + λI1a , (81) C1 = R+
R+
1 1 ⊕ = ξ (e)ξa (e) de ⊗ 1p − 1p ⊗ ξ (e)ξa (e) de + λI2a , a R+ a a R+ a 1 ⊕ a (ξ (e)ξa (e)2 + ξa (e)2 ξa (e)) de ⊗ 1p + C3 = 2 a R+ a 1 ⊕ (ξ (e)ξa (e)2 + ξa (e)2 ξa (e)) de + λI3a , + 1p ⊗ 2 a R+ a C2a
⊕
(82)
(83)
where Ina = i n
n n j =0
k
(j )
(n−j )
{adAap (Gα ) ⊗ 1p ⊗ adAf
(ϕ(τβ (gα ))) +
α (j )
(n−j )
+ (−1) 1p ⊗ adAap (Cp Gα Cp ) ⊗ adAf j
(ϕ(eβu/2 τβ (gα )))},
(84)
for n = 1, 2, 3. We define the bounded selfadjoint operator A0 on H by A0 = iθλ(I R 2 − R 2 I ),
(85)
R 2 = (L20 + 2 )−1 .
(86)
with
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THERMAL IONIZATION
Here, θ and are positive parameters, and is the projection onto the zero eigenspace of L0 : = P0 ⊗ P , P0 = p0 ⊗ p0 , = 1 − ,
(87) (88) (89)
where p0 is the projection in B(Hp ) projecting onto the eigenspace corresponding to the eigenvalue E of Hp , i.e. p0 ψ = ψ(E) ∈ C, and P is the projection in B(F ) projecting onto C . We also introduce the notation R = R . Notice that the operator A0 satisfies the conditions given in Theorem 3.3 with n0 = 1. Moreover, [L, A0 ] = LA0 − A0 L extends to a bounded operator on the entire Hilbert space, and θλ θλ2 + 2 . (90) [L, A0 ] k This choice for the operator A0 was initially introduced in [BFSS] for the spectral analysis of Pauli–Fierz Hamiltonians (zero temperature systems), and was adopted in [Mer] to show return to equilibrium (positive temperature systems). The 2 key feature of A0 is that i[L, A0 ] = 2θλ2 I R I is a nonnegative operator. Assuming the Fermi Golden Rule condition (39), it is a strictly positive operator, as shows PROPOSITION 4.2. Assume condition (A3). For 0 < < 0 , we have 2
I R I
γ .
(91)
The proof is given in Section 8. We are now ready to verify that the virial theorems are applicable. a
PROPOSITION 4.3. The unitary group eit A leaves D() invariant (a > 0, t ∈ R), and, for ψ ∈ D(),
eit A ψ kek |t |/a ψ, a
(92)
where k, k < ∞ are independent of a. The proof is given in Section 8. Next, we verify the GJN conditions, and the bounds (62), (64), (63). The following result is useful.
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JÜRG FRÖHLICH AND MARCO MERKLI
PROPOSITION 4.4. Under conditions (35), (36), the multiple commutators of Gα with Aap are well defined in the strong sense on C0∞ , and, for any ψ ∈ C0∞ , we have that ad(n) Aap (Gα )ψ kψ,
(93)
for n = 1, 2, 3, and uniformly in a > 0. The proofs of this and the next proposition are given in Section 8. PROPOSITION 4.5. The virial theorems, Theorems 3.2 and 3.3, apply in the concrete situation described above, with the following identifications: the domain D of Section 3.1 is given in (75), the operators L, N, D, , A0 appearing in Theorems 3.2, 3.3 are chosen in (53), (79), (80), (76), (85), and the operator A is given by Aa in (72).
4.3. A LOWER BOUND ON C1a + i[L, A0 ] UNIFORM IN a In order to estimate C1a + i[L, A0 ] from below, we start with the following observation: in the sense of forms on D, ±λI1a
1 NP + kλ2 , 10
(94)
for some k independent of a > 0. This estimate follows in a standard way from the explicit expression for I1a , Equation (84), and the bound in (93). We conclude from (94), (81) that C1a + i[L, A0 ] Ma , where Ma =
⊕ R+
+
(95)
ξa (e) de ⊗ 1p + 1p ⊗
⊕ R+
ξa (e) de +
9 P − kλ2 + i[L, A0 ]. 10
(96)
The constant k on the r.h.s. is independent of a. Recalling that ξa → 1 a.e., we are led to define the bounded limiting operator M = P+ (Hp ) ⊗ 1p + 1p ⊗ P+ (Hp ) +
9 P − kλ2 + i[L, A0 ], 10
(97)
where k is the ⊕same constant as in (96). Using dominated convergence, one readily verifies that R+ ξa (e) de → P+ (Hp ), in the strong sense on Hp . PROPOSITION 4.6. lima→0+ Ma = M, strongly on H .
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THERMAL IONIZATION
Our next task is to show that M is strictly positive. 4.4. THE FESHBACH METHOD AND STRICT POSITIVITY OF M Recall that = P0 ⊗ P is the rank-one projection onto the zero eigenspace of L0 , see (87). We apply the Feshbach method to analyze the operator M, with the decomposition H = Ran ⊕ Ran . First, we note that M P 0 ⊗ P (P+ (Hp ) ⊗ 1p + 1p ⊗ P+ (Hp ) − kλ2 ) + 9 − kλ2 )P + i[L, A0 ]. + ( 10
(98)
Recalling the definitions of P0 and A0 , (88) and (85), one easily sees that P 0 (P+ (Hp ) ⊗ 1p + 1p ⊗ P+ (Hp )) P 0 , 2
2
i[L, A0 ] = −θλ2 (I I R + R I I ), in particular, i[L, A0 ] kθλ2 / 2 . Together with (98), this shows that there is a constant λ1 > 0 (independent of λ, θ, and of β β0 , for any β0 > 0 fixed), s.t. M := M Ran > 12 ,
(99)
provided |λ|,
θλ2 < λ1 . 2
(100)
It follows from Equation (99) that the resolvent set of M, ρ(M), contains the interval (−∞, 1/2), and for m < 1/2: (M − m)−1 < ( 12 − m)−1 .
(101)
For m ∈ ρ(M), we define the Feshbach map F,m applied to M by F,m (M) = (M − M(M − m)−1 M).
(102)
The operator F,m (M) acts on the space Ran . In our specific case, Ran ∼ = C, hence F,m (M) is a number. (If Ran had dimension n, then F,m (M) would be represented by an n × n matrix.) The following crucial property is called the isospectrality of the Feshbach map (see, e.g., [BFS, DJ]): m ∈ ρ(M) ∩ σ (M) ⇐⇒ m ∈ ρ(M) ∩ σ (F,m (M)),
(103)
where σ (·) denotes the spectrum. Hence by examining the spectrum of the operator F,m (M), one obtains information about the spectrum of M. The idea is, of course, that it is easier to examine the former operator, since it acts on a smaller space.
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JÜRG FRÖHLICH AND MARCO MERKLI
PROPOSITION 4.7. Assume condition (A3) and let 0 < < 0 . Then |λ| 2 θλ2 F,m (M) 2 γ 1 − kθ 1 + , −k γθ
(104)
uniformly in m < 1/4. Proof. Recall the structure of F,m (M), given in (102). We show that −M(M − m)−1 M is small, as compared to M, and that the latter is strictly positive. Estimate (101) gives −M(M − m)−1 M −4MM,
(105)
for m < 1/4. An easy calculation shows that 2
2
2
M = i[L, A0 ] = θλLR I = θλ(L0 R I + λI R I ), and using that L0 R 1, R 1/ , we obtain the bound θλ2 R I ψ, Mψ θ|λ| + k
(106)
2
for any ψ ∈ H , where we have used that Ran R I ⊂ Ran P (N 1), and I P (N 1) k. Combining (106) with (105) yields 2
−M(M − m)−1 M −kθ 2 λ2 (1 + |λ|/ )2 I R I . Furthermore, we have that 2
M = i[L, A0 ] − kλ2 = 2θλ2 I R I − kλ2 . These observations and the definition of the Feshbach map, (102), show that |λ| 2 2 2 I R I − kλ2 , F,m (M) 2θλ 1 − kθ 1 + which, by Proposition 4.2, yields (104).
2
Estimate (104) tells us that there is a λ2 > 0 s.t. F,m (M)
θλ2 γ ,
provided conditions (100) hold, and |λ| 2 < λ2 , 0 < < 0 . + θ 1+ γθ
(107)
(108)
Notice that all these estimates are independent of m < 1/4. Using the isospectrality property of the Feshbach map, (103), we conclude that if the bounds (100)
THERMAL IONIZATION
269
and (108) are imposed on the parameters, and if m < 1/4 and m ∈ σ (M), then 2 m > θλ γ . Consequently, θλ2 1 θλ2 γ = γ. M min , 4 Fix a θ < λ2 /4 and an < min{ 0 , γ θλ2 }. Then, defining √ λ1 λ0 = min λ1 , √ , , θ (100) and (108) are satisfied for |λ| < λ0 . PROPOSITION 4.8. There is a choice of the parameters θ and , and of λ0 > 0 (depending on θ, , β) s.t. if |λ| < λ0 then M>
θλ2 γ.
(109)
We have λ0 kγ for some k independent of β β0 (for any β0 > 0 fixed), i.e., λ0 ∼ eβE is exponentially small in β, as β → ∞ (see Remark (1) after (39)). Proposition 4.8 completes the proof of Theorem 2.3, according to the argument given in (74). 5. Some Functional Analysis The following two theorems are useful in our analysis. Their proofs can be found in [Frö]. THEOREM 5.1 (Invariance of domain, [Frö]). Suppose (X, Y, D) satisfies the GJN condition, (57), (58). Then the unitary group, eit X , generated by the selfadjoint operator X leaves D(Y ) invariant, and Y eit X ψ ek|t | Y ψ,
(110)
for some k 0, and all ψ ∈ D(Y ). THEOREM 5.2 (Commutator expansion, [Frö]). Suppose D is a core for the selfadjoint operator Y 1. Let X, Z, ad(n) X (Z) be symmetric operators on D, where ad(0) X (Z) = Z, (n−1) (Z)ψ, Xψ − Xψ, ad(n−1) (Z)ψ }, ψ, ad(n) X (Z)ψ = i{ adX X
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JÜRG FRÖHLICH AND MARCO MERKLI
for all ψ ∈ D, n = 1, . . . , M. We suppose that the triples (ad(n) X (Z), Y, D), n = 0, 1, . . . , M, satisfy the GJN condition (57), (58), and that X is selfadjoint, with D ⊂ D(X), eit X leaves D(Y ) invariant, and (110) holds. Then M−1 n
t ad(n) X (Z) − n! n=1 tM−1 t −itM X dt1 · · · dtM eitM X ad(M) , − X (Z)e
eit X Ze−it X = Z −
0
(111)
0
as operators on D(Y ). Remark. This theorem is proved in [Frö], under the assumption that (X, Y, D) satisfies (57), (58). However, [Frö]’s proof only requires the properties of the group eit X indicated in our Theorem 5.2. An easy, but useful result follows from (110). PROPOSITION 5.3. Suppose that the unitary group eit X leaves D(Y ) invariant, for some operator Y , and that estimate (110) holds. a function χ on R with For isX ˆ ds. If χˆ has compact Fourier transform χˆ ∈ L1 (R), we define χ(X) = R χ(s)e support, then χ(X) leaves D(Y ) invariant, and, for ψ ∈ D(Y ), Y χ(X)ψ ekR χˆ L1 (R) Y ψ,
(112)
for any R s.t. suppχˆ ⊂ [−R, R]. The proof is obvious. Proposition 5.4 states a similar result, but for a function whose Fourier transform is not necessarily of compact support. PROPOSITION 5.4. Suppose (X, Y, D) satisfies the GJN condition, and so do the triples (ad(n) X (Y ), Y, D), for n = 1, . . . , M, and for some M 1. Moreover, assume that, in the sense of Kato on D(Y ), ±ad(M) X (Y ) kX, for somek 0. For ∞ isX ˆ , χ ∈ C0 (R), a smooth function with compact support, define χ(X) = χ(s)e where χˆ is the Fourier transform ofχ. Then χ(X) leaves D(Y ) invariant. R isX ˆ , then χR (X) → χ(X) in operator Proof. For R > 0, set χR (X) = −R χ(s)e norm, as R → ∞. From the invariance of domain theorem, we see that χR (X) leaves D(Y ) invariant. Let ψ ∈ D(Y ), then using the commutator expansion theorem above, we have R χˆ (s)eisX (e−isX Y eisX − Y )ψ Y χR (X)ψ = χR (X)Y ψ + −R M−1 R (−s)n (n) isX adX (Y ) + χˆ (s)e = χR (X)Y ψ − n! −R n=1 sM−1 s M −isM X (M) isM X ds1 · · · dsM e adX (Y )e ψ. (113) + (−1) 0
0
271
THERMAL IONIZATION
The integrand of the s-integration in (113) is bounded in norm by k(|s|M + 1)(Y ψ + Xψ) k(|s|M + 1)Y ψ, isM X ψ XeisM X ψ Xψ. Since χˆ is of where we have used that ad(M) X (Y )e rapid decrease, it can be integrated against any power of |s|, and we conclude that the r.h.s. of (113) has a limit as R → ∞. Since Y is a closed operator, it follows that χ(X)ψ ∈ D(Y ). 2
PROPOSITION 5.5. Let χ ∈ C0∞ (R), χ = F 2 0. Suppose (X, Y, D) satisfies the GJN condition. Suppose F (X) leaves D(Y ) invariant. Let Z be a symmetric operator on D s.t., for some M 1, and n = 0, 1, . . . , M, the triples (ad(n) X (Z), Y, D) satisfy the GJN condition. Moreover, we assume that the multiple commutators, for n = 1, . . . , M, are relatively X 2p -bounded in the sense of Kato on D, for some p 0. In other words, there is some k < ∞, s.t. ∀ψ ∈ D, 2p ad(n) X (Z)ψ k(ψ + X ψ),
n = 1, . . . , M.
Then the commutator [χ(X), Z] = χ(X)Z − Zχ(X) is well defined on D and extends to a bounded operator. Proof. We write F, χ instead of F (X), χ(X). Since F leaves D(Y ) invariant, we have that [χ, Z] = F [F, Z] + [F, Z]F, as operators on D(Y ). We expand the commutator (s)eisX (Z − e−isX ZeisX ) [F, Z] = F M−1 s n (n) isX (s)e adX (Z) + = F n! n=1 sM−1 s (M) ds1 · · · dsM e−isM X adX (Z)eisM X . +
(114)
0
0
Multiplying this equation from the right with F (and noticing that F commutes with eisM X ), we see immediately that [F, Z]F is bounded, and hence F [F, Z] = 2 −([F, Z]F )∗ is bounded, too. PROPOSITION 5.6. Suppose (X, Y, D) is a GJN triple. Then the resolvent (X − z)−1 leaves D(Y ) invariant, for all z ∈ {C | |Im z| > k}, for some k > 0. Proof. Suppose Im z < 0 (the case Im z > 0 is dealt with similarly). We write the resolvent as ∞ −1 dt ei(X−z)t , (X − z) = i 0
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JÜRG FRÖHLICH AND MARCO MERKLI
and it follows from Theorem 5.1 that for ψ ∈ D(Y ), ∞ −1 dt e(Im z+k)t < ∞, Y (X − z) ψ Y ψ 0
provided Im z < −k.
2
6. Proof of the Virial Theorems and the Regularity Theorem Proof of Theorem 3.2. We start by introducing some cutoff operators, and the regularized (cutoff, approximate) eigenfunction. Let g1 ∈ C0∞ ((−1, 1)) be a real valued function, s.t. g1 (0) = 1, and set g = g12 ∈ C0∞ ((−1, 1)), g(0) = 1. Pick a real valued function f on R with the properties that f (0) = 1 and fˆ ∈ C0∞ (R) (Fourier transform). We set x f 2 (y) dy, f1 (x) = −∞
−1/2 ˆ ˆ f ∗ fˆ(s), it follows that so that f1 (x) = f 2 (x). Since f 1 (s) = is f1 (s) = (2π ) fˆ1 has compact support, and is smooth except at s = 0, where it behaves like s −1 . We have f1(n) = (is)n fˆ1 ∈ C0∞ , for n 1. Let α, ν > 0 be two parameters and define the cutoff-operators gˆ1 (s)eisνN ds, g1,ν = g1 (νN) = R
gν =
2 g1,ν ,
fα = f (αA) =
fˆ(s)eisαA ds.
R
For η > 0, define 1 η η ds fˆ1 (s)eisαA = (f1,α )∗ . f1,α = α R\(−η,η) η
η
f1,α leaves D() invariant, and f1,α k/α, where k is a constant independent of η; this can be seen by noticing that f1 ∞ < ∞. Suppose that ψ is an eigenfunction of L with eigenvalue e: Lψ = eψ. Since ψ ∈ D(L), then ψ = (L + i)−1 ϕ, for some ϕ ∈ H . Let {ϕn } ⊂ D() be a sequence of vectors converging to ϕ. Then ψn := (L + i)−1 ϕn −→ ψ,
n −→ ∞,
(115)
and moreover, ψn ∈ D(). The latter follows because the resolvent (L + i)−1 leaves D() invariant, see Proposition 5.6; without loss of generality, we assume
273
THERMAL IONIZATION
that k = 1. Moreover, by Proposition 5.3, we know that fα leaves D() invariant (see also (61)), and gν leaves D() invariant ( commutes with N in the strong sense on D). Hence, the regularized eigenfunction ψα,ν,n = fα gν ψn satisfies ψα,ν,n ∈ D(), ψα,ν,n → ψ, as α, ν → 0, n → ∞. Notice that in the definition of ψn , we introduced the resolvent of L, so that we have (L − e)ψn → 0, as n → ∞, which we write as (L − e)ψn = o(n).
(116)
We now prove the estimate 1 √ η (117) | if1,α (L − e) gν ψn | k ( ν + o(n)), α where k is some constant independent of η, α, ν, n. This estimate follows from the bound √ (118) (L − e)gν ψn k( ν + o(n)), which is proven as follows. We have that (L − e)gν ψn = gν (L − e)ψn + + g1,ν [L, g1,ν ]ψn + + [L, g1,ν ]g1,ν ψn ,
(119) (120) (121)
and the r.h.s. of (119) is√o(n), by (116). Let us show that both (120) and (121) are bounded above by k ν, uniformly in n. The commutator expansion of Theorem 5.2 (see also (114)) yields s isνN ds1 e−is1 νN g1,ν Deis1 νN , (122) g1,ν [L, g1,ν ] = ν ds gˆ1 (s)e R
0
as operators on D(), where D is given in (80). We use that g1,ν commutes with eisνN . From (62), we see that for any φ ∈ D(), g1,ν Deis1 νN φ =
| ϕ, g1,ν Deis1 νN φ | Dg1,ϕ φ sup ϕ ϕ ϕ∈D,ϕ =0 ϕ∈D,ϕ =0
k and consequently,
sup
1 N 1/2 g1,ν ϕ φ k √ φ, ϕ ν ϕ∈D,ϕ =0 sup
s ds|gˆ1 (s)| ds1 g1,ν Deis1 νN φ 0 R √ k ν ds|s gˆ1 (s)| φ.
g1,ν [L, g1,ν ]φ ν
R
(123)
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JÜRG FRÖHLICH AND MARCO MERKLI
Thus, the desired bound for (120) is proven, and the same bound is established for (121) by proceeding in a similar way. This proves (118). η η Next, since f1,α leaves D() invariant, the commutator [f1,α , L] is defined in the strong sense on D(), and Theorem 5.2 yields η
[f1,α , L] =
s2 ds fˆ1 (s)eisαA sC1 + α C2 + 2 R\(−η,η) s s1 2 isαA ds fˆ1 (s)e ds1 ds2 +α R\(−η,η)
0
0
ds3 e−is3 αA C3 eis3 αA .
0
For n 1, we have ds(is)n fˆ1 (s)eisαA = f1(n) (αA) = R
s2
R\(−η,η)
(124)
ds(is)n fˆ1 (s)eisαA − Rη,n ,
where the remainder term η ds(is)n fˆ1 (s)eisαA Rη,n = − −η
satisfies Rη,n = (Rη,n )∗ , and Rη,n kn η, with a constant kn that does not depend on α, η. We obtain from (124) η , L] [f1,α
α = −i(f1 (αA) + Rη,1 )C1 − (f1 (αA) + Rη,2 )C2 + 2 s1 s2 s 2 isαA ˆ ds f1 (s)e ds1 ds2 ds3 e−is3 αA C3 eis3 αA . +α R\(−η,η)
0
0
0
(125)
Recalling that f1 (αA) = f 2 (αA) = fα2 , we write −ifα2 C1 = −ifα C1 fα − s isαA ds1 αsC2 + α 2 − ifα ds fˆ(s)e R
= −ifα C1 fα − αfα fα C2 − s 2 isαA ˆ ds1 − iα fα ds f (s)e R
0
0 s1
s1
−is2 αA
ds2 e
is2 αA
C3 e
0
ds2 e−is2 αA C3 eis2 αA ,
(126)
0
where fα = f (αA). Remarking that fα fα = 12 (f 2 ) (αA) = 12 f1 (αA), we obtain
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THERMAL IONIZATION
from (125), (126): η
[f1,α , L] α = −ifα C1 fα − αf1 (αA)C2 − iRη,1 C1 − Rη,2 C2 + s s1 2 s2 2 isαA ds fˆ1 (s)e ds1 ds2 ds3 e−is3 αA C3 eis3 αA − +α 0 R\(−η,η) 0 0 s s1 − iα 2 fα ds fˆ(s)eisαA ds1 ds2 e−is2 αA C3 eis2 αA . (127) R
0
0
Consequently, taking into account estimate (64), we obtain that η , L] gν ψn = C1 ψα,ν,n − Re iα f (αA)C2 gν ψn + Re Rη,1 C1 gν ψn − i[f1,α 2 α α (128) − Re i Rη,2 C2 gν ψn + O √ , 2 ν
as we show next. We have taken the real part on the right side, since the left side is a real number. To estimate the remainder term, we use condition (64) to obtain 1 e−is3 αA C3 eis3 αA gν ψn k √ eαk |s3 | , ν uniformly in n, so the middle line in (127) is estimated from above by α 2 αk K α2 3 αk |s| ˆ ds|f1 (s)| |s| e k√ e ds|fˆ1 (s)| |s|3 , k√ ν R ν R where K < ∞ is such that suppfˆ1 ⊂ [−K, K]. The √ exponential is bounded 2 uniformly in 0 α < 1, hence the r.h.s. is k(α / ν). The last line in (127) is analyzed in the same way and (128) follows. Finally, we observe that − Re iαf (αA)C2 gν ψn α = − i[f (αA), C2 ] gν ψn 2 2 s α2 α (s)eisαA =− ds f ds1 e−is1 αA C3 eis1 αA =O √ , 2 R ν 0 gν ψn where we use (64) again, as above. A similar estimate yields 2 α α η α Re i Rη,2 C2 gν ψn = −i [Rη,2 C2 ] gν ψn = O √ , 2 4 ν and using the bound (63), we have that η Rη,1 C1 gν ψn = O p . ν
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JÜRG FRÖHLICH AND MARCO MERKLI
Combining this with (128) and (117) shows that √ ν + o(n) η α2 | C1 ψα,ν,n | k +√ + p . α ν ν Notice that C1 ψα,ν,n =
(129)
ds fˆ(s)C1 eisαA gν ψn −→ C1 ψα,ν ,
as n → ∞, where ψα,ν := fα gν ψ. This follows from the boundedness condition (63) and from ψn → ψ, n → ∞, see (115). Consequently we obtain by taking the limit n → ∞ in (129) √ η α2 ν +√ + p . | C1 ψα,ν | k α ν ν Choose, for instance, ν = α 3 , η = α 3p+δ , for any δ > 0, then lim C1 ψα,α3 = 0.
α→0
This concludes the proof of the theorem.
2
Proof of Theorem 3.3. We adopt the definitions and notation introduced in the proof of Theorem 3.2. It suffices to prove lim ψα , i[L, A0 ]ψα = 0,
α→0
where we set ψα = ψα,ν |ν=α 3 ; see in the proof of Theorem 3.2. The scalar product can be estimated by | ψα , i[L, A0 ]ψα | 2| (L − e)ψα , A0 ψα | 2P (N n0 )(L − e)ψα A0 ψα . We have P (N n0 )(L − e)ψα,ν = lim P (N n0 )[L, fα ]gν ψn + n→∞
+ lim P (N n0 )fα [L, gν ]ψn . n→∞
(130) (131)
Using condition (63), we easily find (expanding the commutator [L, fα ]) that P (N n0 )[L, fα ]g√ ν ψn kn0 α. Similarly, using (62), we find that P (N 2 n0 )fα [L, gν ]ψn k ν. It follows that P (N n0 )(L − e)ψα kn0 α. Proof of Theorem 3.4. The inequality C P − B, the continuity of B, and (67) imply that for any > 0, there is an α0 ( ), s.t. if α < α0 ( ) then ψα , P ψα ψ, Bψ + .
(132)
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THERMAL IONIZATION
Let µφ be the spectral measure of P corresponding to some φ ∈ H . Then ψα , P ψα = p dµψα (p) = lim pχ(p R) dµψα (p), R+
R→∞
R+
where χ(p R) is the indicator of [0, R]. We obtain from (132) lim ψα , χ(P R)P ψα = lim χ(P R)P 1/2ψα 2
R→∞
R→∞
ψ, Bψ + ≡ k.
√ We have χ(P R)P 1/2ψ√ R 1/2ψ − ψα + k, and taking α → 0 R gives χ(P R)P 1/2ψ k, uniformly in R, so limR→∞ 0 p dµψ (p) exand is finite, by the monotone convergence theorem. Since D(P 1/2) = {ψ | ists ∞ 1/2 ), and P 1/2 ψ ψ, Bψ . 2 0 p dµψ (p) < ∞}, we have that ψ ∈ D(P
7. Flows and Induced Unitary Groups Let R ⊆ Rn be a Borel set of Rn (with nonempty interior), let X be a vector field on Rn , and consider the initial value problem for x ∈ R: d xt = X(xt ), xt |t =0 = x. (133) dt We assume that X has the property that, for any initial condition x ∈ R, there is a unique, global (for all t ∈ R) solution xt ∈ R to (133). Let t denote the corresponding flow and assume t is a diffeomorphism of R into R, for all t ∈ R. The following properties of the flow will be needed: s+t = s ◦ t , −1 t = −t , 0 = 1. The Jacobian determinant of t (x) is given by Jt (x) = |det t (x)|,
(134)
t )i (x)). where t (x) is the matrix ( ∂( ∂xj Let µ: R → R+ be a continuous function which is C 1 on the interior of R and which is strictly positive except possibly on a set of measure zero. We write dµ for the absolutely continuous measure µ(x) dx, where dx denotes Lebesgue measure on Rn . Given a Hilbert space H, consider L2 (R, dµ; H), the space of square integrable functions ψ: R → H, equipped with the scalar product ψ, φ = ψ(x), φ(x) H dµ(x).
R
On the Hilbert space L2 (R, dµ; H), the flow t induces a strongly continuous unitary group, Ut , defined by µ( t (x)) ψ( t (x)), (135) (Ut ψ)(x) = Jt (x) µ(x)
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JÜRG FRÖHLICH AND MARCO MERKLI
for ψ ∈ L2 (R, dµ; H). To check that Ut preserves the norm, we make the change of variables y = t (x) to arrive at 2 |(Ut ψ)(x)| dµ(x) = Jt (x)|ψ( t (x))|2 µ( t (x)) dx R R −1 2 Jt ( −1 = t (y))|det( t ) (y)| |ψ(y)| µ(y) dy. R −1 We observe that Jt ( −1 t (y))|det( t ) (y)| = |det 1| = 1, hence Ut ψ = ψ. Next, using that t +s = t ◦ s , one easily shows that Jt +s (x) = Jt ( s (x))Js (x), and that
µ( t ( s (x))) µ( s (x)) µ( t +s (x)) = , µ(x) µ( s (x)) µ(x) hence t → Ut is a unitary group. In order to see that the unitary group is strongly continuous and to calculate its generator, we impose some additional assumptions on µ and X. (1) X is C ∞ and bounded, (2) for any compact set M ⊂ R, there is a k < ∞ s.t. ∂t |t =0 Jt (x) k, uniformly in x ∈ M, k, uniformly (3) for any compact set M ⊂ R, there is a k < ∞ s.t. X (x)∇µ(x) µ(x) in x ∈ M, (4) t → {Jt (x)µ( t (x))}1/2 is C 1 in a neighbourhood (−t0 , t0 ) of zero, and for any compact set M ⊂ R, there is a k < ∞ s.t. we have the estimate |{Jt (x)µ( t (x))}1/2 | < f (x), for |t| < t0 , where f ∈ L2loc (R, dx). If X is C ∞ then so is t (x) (jointly in (t, x)), and using that t X( s (x)) ds, t (x) = x + 0 t X ( s (x)) s (x) ds, t (x) = 1 +
(136)
0
it follows immediately that t (x) x + |t| X∞ ,
(137)
where the subscript ∞ denotes the supremum norm over x ∈ R. In order to obtain an estimate on t (x) (the operator norm on B(Rn ), i.e. the matrix norm, for x fixed), we recall Gronwall’s lemma. If µ: R → R+ is continuous, and ν: R → R+ is locally integrable, then the inequality t ν(s)µ(s) ds, µ(t) c + t0
279
THERMAL IONIZATION
where c 0, and t t0 , implies that t
µ(t) ce
t0
ν(s) ds
(138)
.
Equation (136) implies t (x)
t
1 + X ∞ 0
s (x) ds,
so Gronwall’s lemma yields the estimate t (x) exp(X ∞ t),
∀t > 0.
A similar bound holds for t < 0, and hence t (x) exp(X ∞ |t|),
t ∈ R,
(139)
from which it follows that Jt (x) exp(nX ∞ |t|).
(140)
For ψ ∈ C0∞ , 1 − ∂t |t =0 (Ut ψ)(x) i 1 ∇µ(x) · X(x) 1 1 ∂t |t =0 Jt (x) + + X(x) · ∇ ψ(x) =− i 2 2 µ(x) = (Aψ)(x),
(141)
which defines an operator A on C0∞ . Notice that due to conditions (1–3), A maps C0∞ into L2 (R, dµ; H). PROPOSITION 7.1. Assume conditions (1–4) hold. Then for any ψ ∈ C0∞ , in the strong sense on L2 (R, dµ; H), −
1 Ut − 1 ψ −→ Aψ, i t
t −→ 0.
(142)
Consequently, C0∞ is in the domain of definition of the selfadjoint generator of the unitary group Ut , and on C0∞ , this generator can be identified with the operator A of Equation (141). Proof. Invoking the dominated convergence theorem, it is enough to verify that 2 11 − (Ut − 1)ψ(x) − (Aψ)(x) i t
H
(143)
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JÜRG FRÖHLICH AND MARCO MERKLI
is bounded above by a dµ-integrable function which is independent of t, for small t. We write 2 1 1 ψ( t (x))2 + (143) ( J (x)µ( (x)) − µ(x)) (144) t t H µ(x) t 1 (145) + 2 ψ( t (x)) − ψ(x)2H + t (146) + (Aψ)(x)2H . Clearly, (146) is integrable, and, using the continuity properties of ψ and and the bound (139), one sees that (145) is bounded above by a t-independent function that is dµ-integrable (use the mean value theorem). Next, if ψ has support in a ball of radius ρ in R ⊂ Rn , then ψ ◦ t has support in the ball of radius ρ + |t|X∞ ρ + X∞ , for |t| 1. This follows from (137). Let χ(x) denote the indicator function on the ball of radius ρ + X∞ , then we have for |t| < t0 with t0 as in condition (4), 2 1 1 ( Jt (x)µ( t (x)) − µ(x)) (144) kχ(x) µ(x) t 1 |f (x)|2 , kχ(x) µ(x) where we have used the mean value theorem and condition (4). The latter function is dµ-integrable. 2 Proof of Proposition 4.1. Since ξ is globally Lipshitz (with Lipshitz constant ξ ∞ ), we have existence and uniqueness of global solutions to the initial value problem (133). Due to uniqueness and the fact that R t → et = 0 is a solution (since ξ(0) = 0), we see that t (e) ∈ (0, ∞), for all t ∈ R, e ∈ (0, ∞), so t is a diffeomorphism in R+ . It is not difficult to verify that conditions (1–4) above are satisfied. Consequently, it follows from Proposition 7.1 that C0∞ ⊂ D(A), and that A is given by (70) on C0∞ . Since ξ is infinitely many times differentiable, A leaves 2 C0∞ invariant. Hence C0∞ is a core for A.
8. Proofs of Some Propositions Proof of Proposition 4.2. Since I = 0 and I R 2 (p¯ 0 ⊗ p¯0 )I = 0, we have 2
I R I = I R 2I = I R 2(p¯ 0 ⊗ p0 + p0 ⊗ p¯ 0 )I + I R 2(p0 ⊗ p0 )I .
(147)
It is not difficult to see that I R 2(p0 ⊗ p0 )I → 0, as → 0, so the last term in (147) does not contribute effectively to a lower bound in the limit → 0.
281
THERMAL IONIZATION
Let J be the modular conjugation operator introduced in (47). Using the relations J 2 = J , Jp0 ⊗ p¯ 0 = p¯0 ⊗ p0 J , J R 2 = R 2 J , J I = −I J , and the invariance of ϕ0 ⊗ ϕ0 ⊗ under J , one verifies easily that I R 2 (p0 ⊗ p¯0 )I = I R 2 (p¯ 0 ⊗ p0 )I (Gα ⊗ 1p ⊗ a(τβ (gα ))) = α,α
p¯0 ⊗ p0 × (Hp ⊗ 1p − E + Lf )2 + 2
× (Gα ⊗ 1p ⊗ a ∗ (τβ (gα ))), where Lf = d(u) and where τβ has been defined in (44). We pull the annihilation operator through the resolvent, using the pull through-formula (for f ∈ L2 (R × S 2 )) f¯(u, )(Lf + u)a(u, ), a(f )Lf = R×S 2
and then contract it with the creation operator. This gives the bound 2
I R I E u2 × du d −βu e −1 S2 −∞ p¯ 0 ∗ F (−u, ) p × p0 F (−u, ) 0 ⊗ p0 ⊗ P , (Hp − E + u)2 + 2 where we restricted the domain of integration over u to (−∞, E) ⊂ R− (as → 0, tends to the Dirac distribution δ(Hp − E + u), hence u = −Hp + E ∈ (Hp −E+u)2 + 2 (−∞, E)), and where we used (44). The desired result (91) now follows by making the change of variable u → −u in the integral, and by remembering the definition of γ , (39). 2 a
Proof of Proposition 4.3. First, we prove a bound on p eit Ap ψ, for ψ ∈ C0∞ . Let at denote the flow generated by the vector field ξa . Then, for each e ∈ [0, ∞), a ((p − 1p )eit Ap ψ)(e) = eψ( at (e)), and it Aap 2 e2 ψ( at (e))2 de (p − 1p )e ψ = R + ( a−t (y))2 ψ(y)2 ( a−t ) (y) dy, (148) = R+
we make the change of variables y = at (e). Recall that at (y) = y + where t a 0 ξ( s (y)/a) ds, so | at (y)| |y| + |t| ξ ∞ .
(149)
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JÜRG FRÖHLICH AND MARCO MERKLI
Next ( at ) (y) = 1 + |( at ) (y)|
t
1 a a 0 a ξ ( s (y)/a)( s ) (y) ds
t
1+ 0
yields
1 ξ ∞ |( at ) (y)| ds, a
(150)
and Gronwall’s estimate, (138), implies that
|( at ) (y)| eξ ∞ |t |/a .
(151)
Using (151) and (149) in (148) yields
(p − 1p )eit Ap ψ2 eξ ∞ |t |/a a
ξ ∞ |t |/a
2e
R+
(y + ξ ∞ |t|)2 ψ(y)2 dy
(1 + ξ ∞ |t|)2 ((p − 1p )ψ + ψ)2 ,
from which it follows that √ a p eit Ap ψ 4 2(1 + ξ ∞ |t|)eξ ∞ |t |/a p ψ √ 4 2e(ξ ∞ +ξ ∞ )|t |/a p ψ.
(152)
Estimate (152) holds for all ψ ∈ C0∞ , which is a core for p . Next, let ψ ∈ D(p ), and let {ψn } ⊂ C0∞ be a sequence, s.t. ψn → ψ, p ψn → p ψ, as n → ∞. If χR denotes the cutoff function χ(p R), for R > 0, we have a
a
χR p eit Ap ψ χR p eit Ap ψn + Rψ − ψn √ 4 2e(ξ ∞ +ξ ∞ )|t |/a p ψn + Rψ − ψn . Taking n → ∞ yields
√ a χR p eit Ap ψ 4 2e(ξ ∞ +ξ ∞ )|t |/a p ψ, a
uniformly in the cutoff parameter R. This shows that eit Ap ψ ∈ D(p ), and (152) is valid for all ψ ∈ D(p ). We complete the proof of the proposition by examining f eit Af ψ. Let ψ ∈ Df . Then one finds the following bound for the n-particle component: n 2 it Af 2 2 [(f − 1f )e ψ]n = (uj + 1)ψn (u1 − t, . . . , un − t) j =1 n 2 = ((uj + t)2 + 1)ψn (u1 , . . . , un ) j =1 2 n (2(1 + t 2 ))2 (u2j + 1)ψn (u1 , . . . , un ) . j =1
283
THERMAL IONIZATION
It follows that (f − 1f )eit Af ψ 2(1 + t 2 )f ψ, for all ψ ∈ Df , so that f eit Af ψ 2(1 + t 2 )f ψ + ψ 3et f ψ, for all ψ ∈ Df . A similar argument as above shows that this estimate extends to 2 all ψ ∈ D(f ). Proof of Proposition 4.4. We denote the fiber of Aap by Aap (e), i.e. e 11 e a ξ +ξ ∂e , Ap (e) = i 2a a a
(153)
see also (71). For ψ ∈ C0∞ , we have (Aap Gα ψ)(e) = Aap (e)(Gα ψ)(e) =
Aap (e)Gα (e, E)ψ(E)
+
Aap (e)
R+
Gα (e, e )ψ(e ) de .
Due to the regularity property (36), we can take the operator Aap (e) inside the integral (dominated convergence theorem), and obtain the estimate a 2 2 Aap (e)Gα (e, E)2H de + (154) Ap Gα ψ |ψ(E)|
+
R+
R+
R+
Aap (e)Gα (e, e )ψ(e )H
de
2 de.
(155)
Using (153) and the bound |a −1 ξ (e/a)| e−1 supe0 eξ (e) ke−1 , it is easily seen that the integrand of (154) is bounded above by k(e−1 Gα (e, E)2H + ∂1 Gα (e, E)2H ), which is integrable, due to (35). We estimate the integrand in (155) by Aap (e)Gα (e, e )ψ(e )H k(e−1 Gα (e, e )B(H) + ∂1 Gα (e, e )B(H) )ψ(e )H , and using Hölder’s inequality, we arrive at ψ(e)2 de × (155) k R + {e−1 Gα (e, e )2B(H) + ∂1 Gα (e, e )2B(H) } de de . × R+
R+
By condition (36), the double integral is finite. We conclude that Aap Gα ψ kψ.
(156)
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JÜRG FRÖHLICH AND MARCO MERKLI
One also finds that Gα Aap ψ kψ, e.g., by noticing that Gα Aap ψ =
sup φ−1 | φ, Gα Aap ψ | =
0 =φ∈C0∞
sup φ−1 | Aap Gα φ, ψ |
0 =φ∈C0∞
and using (156). Consequently, we have shown (93) for n = 1. The proof for n = 2, 3 follows the above lines. For instance, in order to show boundedness of the third multi-commutator, a typical term to estimate is Aap Aap Gα Aap ψ, for ψ ∈ C0∞ . We shall sketch the proof that this term is bounded, all other ones being treated similarly. We have Aap Aap Gα Aap ψ =
sup φ−1 | φ, Aap Aap Gα Aap ψ |,
(157)
0 =φ∈C0∞
and the scalar product equals φ(e), Aap (e)2 Gα (e, e )Aap (e )ψ(e ) H de de . R+
(158)
R+
Recalling (153), one can calculate the operator A2p (e)2 Gα (e, e ). It can be written as a sum of terms, involving multiplications by functions with argument e, and derivatives ∂1 Gα (e, e ), ∂12 Gα (e, e ). Using the formulas for the adjoints of derivatives of ∂11,2 Gα (e, e ), see (32), we obtain [Aap (e)2 Gα (e, e )]∗ , and (158) becomes Aap (e )[Aap (e)2 Gα (e, e )]∗ φ(e), ψ(e ) H de de , (159) R+
R+
due to selfadjointness of Aap (e ) on H, and the fact that for all e ∈ R+ , [Aap (e)2 Gα (e, e )]∗ φ(e) ∈ D(Aap (e )), which follows from condition (36). Moreover, the same condition allows us to estimate |(159)| R+
R+
Aap (e )[|Aap (e)2 Gα (e, e )]∗ B(H) φ(e)H ψ(e )H de de
φψ kφ ψ,
R+
R+
Aap (e )[Aap (e)2 Gα (e, e )]∗ 2B(H)
de de
1/2
where we have used Hölder’s inequality. This shows that (157) kψ.
2
Proof of Proposition 4.5. We have mentioned before (90) that A0 satisfies the conditions of Theorem 3.3, so it suffices to verify the conditions of Theorem 3.2. We need to check that (X, , D) is a GJN triple, for X = L, N, D, Cna , n = 1, 2, 3, and that (61), (62), (64), (63) are satisfied. Proposition 4.3 shows that (61)
285
THERMAL IONIZATION
holds. The operator D, given in (80), is clearly N 1/2 -bounded in the sense of Kato on D, since Gα are bounded operators, and g˜α , e−βu/2 g˜α are square-integrable. Hence (62) holds. Recalling Remark (1) after Theorem 3.2, and noticing that N a commutes with eit A , in the strong sense on D, and that C3a kN 1/2 , in the sense of Kato on D (see (83)), we see that (64) is verified. Similarly, C1a kN in the sense of Kato on D, see (81), so (63) holds. It remains to show that the above mentioned triples satisfy the GJN properties. We first look at (L, , D). Clearly, Lψ kψ, for ψ ∈ D. Moreover, L0 commutes with in the strong sense on D, so we need only consider the interaction term in the verification of (58). Due to condition (37), we have for all ψ ∈ C0∞ : p Gα ψ kψ, Gα p ψ kψ. Consequently, for ψ ∈ D: | Gα ⊗ 1p ⊗ ϕ(g˜α )ψ, ψ − ψ, Gα ⊗ 1p ⊗ ϕ(g˜α )ψ | kψ ϕ(g˜ α )ψ + + | Gα ⊗ 1p ⊗ ϕ(g˜α )ψ, f ψ − f ψ, Gα ⊗ 1p ⊗ ϕ(g˜α )ψ | kψ ϕ(g˜ α )ψ + kψ ϕ((u2 + 1)g˜α )ψ kψ 1/2 ψ k(ψ2 + 1/2 ψ2 ) k ψ, ( + 1)ψ
2k ψ, ψ . (160) 1/2
We used in the third step that ϕ(g˜α ) and ϕ((u2 + 1)g˜α ) are relatively f bounded, in the sense of Kato on D. This follows since (u2 + 1)g˜α ∈ L2 (R × S 2 , du × d), due to conditions (33) and (34). The same estimates hold for 1p ⊗ Cp Gα Cp ⊗ ϕ(e−βu/2 g˜a ), hence we have shown that (L, , D) is a GJN triple. It is clear that N in the sense of Kato on D, and since N commutes with in the strong sense on D, we see immediately that (N, , D) is a GJN triple. Next, consider (D, , D). Since D has the same structure as I , c.f. (54) and (80), the proof that (D, , D) is a GJN triple goes as the one for (L, , D). We examine (Cna , , D), n = 1, 2, 3, a > 0. Recall that the Cna are given in (81)–(83). Each Cna has a term that acts purely on the particle space. This term is a bounded multiplication operator that commutes with , in the strong sense on D. a , , D) are GJN triples. Therefore, we need only show that (N + λI1a , , D), (I2,3 Since we have shown it for (N, , D), it suffices to treat (Ina , , D), n = 1, 2, 3, a > 0. We take the general term in the sum of (84): (j )
(n−j )
X := adAap (Gα ) ⊗ 1p ⊗ adAf
ϕ(g˜α )).
(j )
Since adAap (Gα ) is bounded, j = 1, 2, 3 (see Proposition 4.4), and (n−j )
adAf
(ϕ(g˜α )) = ϕ((i∂u )n−j g˜α ) 1/2
is relatively f -bounded, in the sense of Kato on D (this follows from ∂uk g˜α ∈ L2 (R × S 2 ), k = 1, 2, 3, due to (33), (34)), then it is clear that Xψ kψ,
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ψ ∈ D. Next, we verify condition (58) as above in (160): | Xψ, ψ − ψ, Xψ | kψ ϕ((u2 + 1)(i∂)n−j g˜α )ψ kψ 1/2 ψ, since (u2 + 1)(∂u )k g˜α ∈ L2 (R × S 2 ), for k = 1, 2, 3, due to (33) and (34).
2
Acknowledgements We thank I. M. Sigal for numerous stimulating discussions on the subject matter of this paper. M.M. is grateful to him for some ideas leading to an early version of Theorem 3.2.
References [ABG] [Ara]
[AW] [BFS] [BFSS]
[BRI, II] [Con] [DJ] [DJP] [Frö] [FGS] [GG] [Haa] [JP]
Amrein, W., Boutet de Monvel, A., Georgescu, V.: C0 -Groups, Commutator Methods and Spectral Theory of N-body Hamiltonians, Birkhäuser, Basel, 1996. Araki, H.: Some properties of modular conjugation operator of von Neumann algebras and a noncommutative Radon–Nicodym theorem with a chain rule, Pacific J. Math. 50(2) (1974), 309–354. Araki, H. and Woods, E.: Representations of the canonical commutation relations describing a nonrelativistic infinite free bose gas, J. Math. Phys. 4 (1963), 637–662. Bach, V., Fröhlich, J. and Sigal, I. M.: Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137(2) (1995), 299–395. Bach, V., Fröhlich, J., Sigal, I. M. and Soffer, A.: Positive commutators and the spectrum of Pauli–Fierz Hamiltonians of atoms and molecules, Comm. Math. Phys. 207(3) (1999), 557–587. Bratteli, O. and Robinson, D.: Operator Algebras and Quantum Statistical Mechanics I, II, 2nd edn, Texts Monogr. Phys., Springer, Berlin, 1987. Connes, A.: Caractérisation des algèbres de von Neumann comme espaces vectroriels ordonnés, Ann. Inst. Fourier (Grenoble) 24 (1974), 121–155. Derezi´nski, J. and Jakši´c, V.: Spectral theory of Pauli–Fierz operators, J. Funct. Anal. 180 (2001), 243–327. Derezi´nski, J., Jakši´c, V. and Pillet, C.-A.: Perturbation theory for W ∗ -dynamics, Liouvilleans and KMS-states, Preprint. Fröhlich, J.: Application of commutator theorems to the integration of representations of Lie algebras and commutation relations, Comm. Math. Phys. 54 (1977), 135–150. Fröhlich, J., Griesemer, M. and Schlein, B.: Asymptotic completeness for Rayleigh scattering, Ann. Inst. H. Poincaré 3 (2002), 107–170. Georgescu, V. and Gérard, C.: On the virial theorem in quantum mechanics, Comm. Math. Phys. 208 (1999), 275–281. Haagerup, U.: The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283. Jakši´c, V. and Pillet, C.-A.: On a model for quantum friction III. Ergodic properties of the spin-boson system, Comm. Math. Phys. 178 (1996), 627–651.
THERMAL IONIZATION
[Mer] [Mou] [Ski]
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Merkli, M.: Positive commutators in nonequilibrium quantum statistical mechanics, Comm. Math. Phys. 223 (2001), 327–362. Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 91 (1981), 391–408. Skibsted, E.: Spectral analysis of N-body systems coupled to a bosonic field, Rev. Math. Phys. 10(7) (1998), 989–1026.
Mathematical Physics, Analysis and Geometry 7: 289–308, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
289
A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions ZHIJUN QIAO1,2 and SHENGTAI LI2
1 Department of Mathematics, University of Texas–Pan American, Edinburg, TX 78539, USA 2 Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Received: 18 March 2003; in final form: 29 August 2003) Abstract. In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives: ut = ∂x5 u−2/3 , (u−1/3 )xx − 2(u−1/6 )2x , u uxxt + 3uxx ux + uxxx u = 0.
ut = ∂x5
The first two are the positive members of the hierarchy, and the first equation was a reduction of an integrable (2 + 1)-dimensional system (see B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A 102 (1984), 15–17). The third one is the first negative member. All nonlinear equations in the hierarchy are shown to have 3×3 Lax pairs through solving a key 3×3 matrix equation, and therefore they are integrable. Under a constraint between the potential function and eigenfunctions, the 3 × 3 Lax pair and its adjoint representation are nonlinearized to be two Liouville-integrable Hamiltonian systems. On the basis of the integrability of 6N-dimensional systems we give the parametric solution of all positive members in the hierarchy. In particular, we obtain the parametric solution of the equation ut = ∂x5 u−2/3 . Finally, we present the traveling wave solutions (TWSs) of the above three representative equations. The TWSs of the first two equations have singularities, but the TWS of the 3rd one is continuous. The parametric solution of the 5th-order equation ut = ∂x5 u−2/3 can not contain its singular TWS. We also analyse Gaussian initial solutions for the equations ut = ∂x5 u−2/3 , and uxxt + 3uxx ux + uxxx u = 0. Both of them are stable. Mathematics Subject Classifications (2000): 37K10, 58F07, 35Q35. Key words: Hamiltonian system, matrix equation, zero curvature representation, parametric solution, traveling wave solution.
1. Introduction The inverse scattering transformation (IST) method plays a very important role in solving integrable nonlinear evolution equations (NLEEs) [17]. These NLEEs include the well-known KdV equation [22] which is related to a 2nd order operator (i.e. Hill operator) problem [23, 25], the remarkable Ablowitz–Kaup–Newell– Segur (AKNS) equations [1, 2] which are associated with the Zakharov–Shabat (ZS) spectral problem [33], and other higher-dimensional integrable equations.
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In the theory of integrable system, it is significant for us to find new integrable evolution equations. Kaup [19] studied the inverse scattering problem for cubic eigenvalue equations of the form ψxxx + 6Qψx + 6Rψ = λψ, and showed a 5thorder partial differential equation (PDE) Qt + Qxxxxx + 30(Qxxx Q + (5/2)Qxx Qx ) + 180Qx Q2 = 0 (called the KK equation) integrable. Afterwards, Kupershmidt [21] constructed a super-KdV equation and presented the integrability of the equation through giving bi-Hamiltonian property and Lax form. Recently, Degasperis and Procesi [12] proposed a new integrable equation: mt + umx + 3mux = 0, m = u − uxx , called the DP equation, which has the peaked soliton solution. The DP equation is actually a member with b = 3 in the family mt + umx + bmux = 0, m = u − uxx , b = constant. It has been already proven that only b = 2, 3 are integrable cases [26]. With b = 2, it works out the equation mt + umx + 2mux = 0, which was first derived in Camassa and Holm [8] (1993) by using asymptotic expansions for Euler’s equations governing inviscid incompressible flow in the shallow water regime. It was thereby shown to be bi-Hamiltonian and integrable and to have the peaked soliton solution. Its billiard solutions, piecewise smooth solutions and algebro-geoemtric solutions were successively treated in Alber et al. [3–6] (1994, 1995, 1999, 2001), Constantin and McKean [10] (1999) and in Qiao [29] (2003). Before Camassa and Holm [8] (1993), families of integrable equations similar to shallow water equation were known to be derivable in the general context of hereditary symmetries in Fokas and Fuchssteiner [16] (1981). However, this equation was not written explicitly, nor was it derived physically as a water wave equation and its solution properties were not studied before Camassa and Holm [8] (1993). See Fuchssteiner [15] (1996) for an insightful history of how the shallow water equation is associated with the hereditary symmetries and symplectic structures. The DP equation (i.e. the equation with b = 3) was proven integrable, associated with a 3rd-order spectral problem [11]: ψxxx = ψx − λmψ, and related to the canonical Hamiltonian system under a new nonlinear Poisson bracket (called Peakon bracket) [18]. In 2002, we extended the DP equation to an integrable hierarchy and dealt with its parametric solution and stationary solutions [28]. In this paper, we propose a new integrable hierarchy. In particular, the following three representatives in the hierarchy ut = ∂x5 u−2/3 , (u−1/3 )xx − 2(u−1/6 )2x , ut = ∂x5 u uxxt + 3uxx ux + uxxx u = 0,
(1) (2) (3)
are shown to have bi-Hamiltonian operator structure and to be integrable. The first two are the positive members of the hierarchy. But the third one is the first negative member of the hierarchy. Konopelchenko and Dubrovsky [20] pointed out that Equation (1) is a reduction of a (2 + 1)-dimensional equation. Here we will deal
A NEW INTEGRABLE HIERARCHY
291
with its spectral problem and parametric representation of solution from the point of constraint view. All nonlinear equations in the hierarchy are shown to have 3 × 3 Lax pairs through solving a key 3 × 3 matrix equation, and therefore they are integrable. After being imposed on a constraint between the potential function and eigenfunctions, the 3×3 Lax pair and its adjoint representation are nonlinearized to be two Liouville-integrable Hamiltonian systems. On the basis of the integrability of 6N-dimensional systems we give the parametric solution of all positive members in the hierarchy. In particular, we obtain the parametric solution of the equation ut = ∂x5 u−2/3 . Furthermore, we obtain the traveling wave solutions (TWSs) for Equations (1), (2), and (3). The first two look like a class of cusp soliton solutions (but not cusp soliton [32]). The TWSs of Equations (1) and (2) have singularities, but the TWS of Equation (3) is continuous. Additionally, the parametric solution of the 5th-order Equation (1) can not include its singular TWS. Equation (3) has a compacton-like and a parabolic cylinder solution. We also analyse the initial Gaussian solutions for equations ut = ∂x5 u−2/3 and uxxt + 3uxx ux + uxxx u = 0. Both of them are stable (see Figures 1 and 2). The whole paper is organized as follows. In the next section we describe how to connect the above three equations to a spectral problem and how to cast them into a new hierarchy of NLEEs, and also give the bi-Hamiltonian operators for the whole hierarchy. In Section 3, we construct the zero curvature representations for the new hierarchy through solving a key 3 × 3 matrix equation. In particular, we obtain the Lax pair of Equations (1), (2), (3), and therefore they are integrable. In Section 4, we show that the 3rd order spectral problem and its adjoint representation related to the above three equations are nonlinearized as a completely integrable Hamiltonian system under a constraint in R6N . In Section 5 we present the parametric solution
Figure 1. Stable solution for the equation uxxt + 3uxx ux + uxxx u = 0 under the Gaussian initial condition. A shock is developed during the time integration. This figure is very like the Burgers case uxxt + 3uxx ux + uxxx u + uxxxx = 0 through adding small viscosity term uxxxx to the equation. For instance, when = −0.01, the equation uxxt + 3uxx ux + uxxx u + uxxxx = 0 has Figure 2.
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Figure 2. Stable solution for the equation uxxt + 3uxx ux + uxxx u + uxxxx = 0, = −0.01 under the Gaussian initial condition. This figure is almost same as Figure 1.
for the positive hierarchy of NLEEs. We particularly get the parametric solution of Equation (1). Moreover, in section 6 we obtain the traveling wave solutions for Equations (1), (2), and (3), and also analyse the initial Gaussian solutions for the equations ut = ∂x5 u−2/3 , and uxxt + 3uxx ux + uxxx u = 0. Finally, in Section 7 we give some conclusions. 2. Spectral Problem, Hamiltonian Operators, and a New Hierarchy Let us consider the following spectral problem ψxxx = −λuψ
(4)
and its adjoint representation ∗ = λuψ ∗ . ψxxx
(5)
Then, we have their functional gradient δλ/δu with respect to the potential u λψψ ∗ ∇λ δλ = ≡ , δu E E
(6)
where ∇λ = λψψ ∗ , uψψ ∗ dx = constant, E=
(7)
and = (−∞, ∞) or = (0, T ). In this procedure, we need the boundary condition of u decaying at infinities or of u being periodic with period T . Usually, we compute the functional gradient δλ/δu of the eigenvalue λ with respect to the potential u by using the method in [13, 9, 31]. Fokas and Anderson constructed
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A NEW INTEGRABLE HIERARCHY
hereditary symmetries and Hamiltonian systems by using the isospectral eigenvalue problems [13]. Later, Cao developed the functional gradient procedure to the nonlinearization method [9], which closely connects finite-dimensional integrable systems to nonlinear integrable partial differential equations (also see details in [27]). Taking derivatives five times on both sides of Equation (7), we find (∇λ)xxxxx = −3λ2 (2u∂ + ∂u)(ψψx∗ − ψ ∗ ψx ), (ψψx∗ − ψ ∗ ψx )xxx = (u∂ + 2∂u)∇λ, which directly lead to K∇λ = λ2 J ∇λ,
(8)
where K = ∂ 5, J = −3(2u∂ + ∂u)∂ −3 (u∂ + 2∂u).
(9) (10)
Obviously, K, J are antisymmetric, and both of them are Hamiltonian operators because they satisfy the Jacobi identity. Now, according to this pair of Hamiltonian operators, we define the hierarchy of nonlinear evolution equations associated with the spectral problems (4) and (5). Let G0 ∈ Ker J = {G ∈ C ∞ (R) | J G = 0} and G−1 ∈ Ker K = {G ∈ C ∞ (R) | KG = 0}. We define the Lenard sequence Lj G0 , j 0, j ∈ Z, (11) Gj = Lj +1 G−1 , j < 0, where L = J −1 K is called the recursion operator. Therefore we obtain a new hierarchy of nonlinear evolution equations: utk = J Gk ,
∀k ∈ Z.
(12)
Apparently, this hierarchy includes the positive members (k 0) and the negative members (k < 0), and possesses the bi-Hamiltonian structure because of the Hamiltonian properties of K, J . Let us now give specific equations in the hierarchy (12). • Choosing G−1 = 1/6 ∈ Ker K yields the first negative member of the hierarchy: ut + vux + 3vx u = 0,
u = vxx .
(13)
This equation is actually: vxxt + 3vxx vx + vxxx v = 0 which is equivalent to ∂ 2 (vt + vvx ) = 0. It has the compacton-like solution [30]. Obviously, v = c1 x + c0 (c1 , c0 are two constants) is a special solution of this equation.
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• Choosing G0 = u−2/3 ∈ Ker J leads to the second positive member of the hierarchy: ut = ∂x5 u−2/3 .
(14)
Konopelchenko and Dubrovsky pointed out that this equation is integrable and is a reduction of a (2 + 1)-dimensional equation [20]. But they did not study solutions of the equation. In the following, we study the relation between the equation and finite-dimensional integrable system and will find that it has parametric solution as well as the traveling wave solution which looks like a cusp. • Choosing another element G0 = ((u−1/3 )xx −2(u−1/6 )2x )/u in the kernel Ker J gives the following positive member of the hierarchy: (u−1/3 )xx − 2(u−1/6 )2x . (15) u This equation also has a cusp-like trveling wave solution. See this in Section 6. ut = ∂x5
Of course, we may generate further nonlinear equations by selecting other elements from the kernel elements of J , K. In the following, we will see that all equations in the hierarchy (12) are integrable. Particularly, the above three Equations (13), (14), (15) are integrable. 3. Zero Curvature Representations Letting ψ = ψ1 , we change Equation (4) to the following 3 × 3 matrix spectral problem x = U (u, λ), 0 1 0 0 1 , U (u, λ) = 0 −λu 0 0
ψ1 = ψ2 . ψ3
(16) (17)
Apparently, the Gateaux derivative matrix U∗ (ξ ) of the spectral matrix U in the direction ξ ∈ C ∞ (R) at point u is 0 0 0 d 0 0 (18) U∗ (ξ ) U (u + ξ ) = 0 d =0 −λξ 0 0 which is obviously an injective homomorphism, i.e. U∗ (ξ ) = 0 ⇔ ξ = 0. For any given C ∞ -function G, we construct the following 3 × 3 matrix equation with respect to V = V (G) Vx − [U, V ] = U∗ (KG − λ2 J G).
(19)
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THEOREM 1. For spectral problem (16) and an arbitrary C ∞ -function G, the matrix equation (19) has the following solution −G − 3λ∂ −2 ϒG 3(G + λ∂ −3 ϒG) −6G 2G 3(−G + λ∂ −3 ϒG) , V = λ −G − 3λ∂ −1 uG 2 −3 −1 −G + 3λ∂ −2 ϒG −G − 3λ u∂ ϒG G − 3λ∂ uG (20) where ∂ = ∂x = ∂/∂x, ϒ = u∂ + 2∂u, and the superscript ‘’ means the derivative in x. Therefore, J = −3ϒ ∗ ∂ −3 ϒ (ϒ ∗ is the conjugate of ϒ). Proof. Let us set V11 V12 V13 V = V21 V22 V23 , V31 V32 V33 and subsitute it into Equation (19), which is an overdetermined equation. Using calculation techniques in [27], we obtain the following results: V11 V12 V13 V21 V22 V23 V31 V32 V33
= −λG − 3λ2 ∂ −2 ϒG, = 3(λG + λ2 ∂ −3 ϒG), = −6λG, = −λG − 3λ2 ∂ −1 uG , = 2λG , = 3λ(−G + λ∂ −3 ϒG), = −λG − 3λ3 u∂ −3 ϒG, = λG − 3λ2 ∂ −1 uG , = −λG + 3λ2 ∂ −2 ϒG, 2
which completes the proof.
THEOREM 2. Let G0 ∈ Ker J , G−1 ∈ Ker K, and let each Gj be given through Equation (11). Then, 1. each new vector field Xk = J Gk , k ∈ Z satisfies the following commutator representation Vk,x − [U, Vk ] = U∗ (Xk ),
∀k ∈ Z;
(21)
2. the new hierarchy (12), i.e. utk = Xk = J Gk ,
∀k ∈ Z,
(22)
possesses the zero curvature representation Utk − Vk,x + [U, Vk ] = 0,
∀k ∈ Z,
(23)
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ZHIJUN QIAO AND SHENGTAI LI
where Vk =
V (Gj )λ2(k−j −1) ,
k−1 k > 0, j =0 , k = 0, = 0, −1 − j =k , k < 0,
(24)
and V (Gj ) is given by Equation (20) with G = Gj . Proof. 1. For k = 0, it is obvious. For k < 0, we have Vk,x − [U, Vk ] = −
−1 (Vx (Gj ) − [U, V (Gj )])λ2(k−j −1) j =k
= −
−1
U∗ (KGj − λ2 KGj −1 )λ2(k−j −1)
j =k
= U∗
−1
KGj −1 λ
2(k−j )
− KGj λ
2(k−j −1)
j =k
= U∗ (KGk−1 − KG−1 λ2k ) = U∗ (KGk−1 ) = U∗ (Xk ). For the case of k > 0, it is similar to prove. 2. Noticing Utk = U∗ (utk ), we obtain Utk − Vk,x + [U, Vk ] = U∗ (utk − Xk ). The injectiveness of U∗ implies the second result holds.
2
From Theorem 2, we immediately obtain the following corollary. COROLLARY 1. The new hierarchy (12) has Lax pair: ψxxx = −λuψ, λ2(k−j )−1 [−6Gj ψxx + 3(Gj + λ∂ −3 ϒGj )ψx ψt k = − (Gj + 3λ∂ −2 ϒGj )ψ],
(25)
(26)
where all symbols are the same as in Thereom 2 and Thereom 1. So, all equations in the hierarchy (12) have Lax pairs and are therefore integrable. In particular, we have the following specific examples. • When we choose G−1 = 1/6, Equation (13) has the following Lax pair: (27) x = U (u, λ), (28) t = V (u, λ),
A NEW INTEGRABLE HIERARCHY
297
where u = uxx , U (u, λ) is defined by Equation (17), and V (u, λ) is given by vx −v λ−1 0 −v . V (u, λ) = 0 (29) λvu 0 −vx Apparently, Lax pair (27) and (28) is equivalent to ψxxx = −λuψ, ψt = λ−1 ψxx − vψx + vx ψ.
(30) (31)
• In a similar way, choosing G0 = u−2/3 gives the Lax pair of Equation (14), i.e. ut = (u−2/3 )xxxxx , ψxxx = −λuψ, ψt = −6λu−2/3 ψxx + 3λ(u−2/3 )x ψx − λ(u−2/3 )xx ψ.
(32) (33)
This Lax pair is different from/inequivalent to the result in [20]. • Furthermore, through choosing G0 = ((u−1/3 )xx − 2(u−1/6 )2x )/u, we find that the new equation (2) has the Lax pair: ψxxx = −λuψ, ψt = −6λG0 ψxx + 3λ(G0 + 3λu−1/3 )ψx − λ(G0 + 9λ(u−1/3 )xx )ψ.
(34) (35)
4. 6N-dimensional Integrable System To discuss solutions of the hierarchy (12), we want to use the constrained method [9, 27] which leads finite-dimensional integrable systems to nonlinear integrable partial differential equations. Because Equation (4)/(16) is a 3rd order eigenvalue problem, we have to investigate itself together with its adjoint problem when we adopt the nonlinearized procedure. Ma and Strampp [24] already studied the AKNS and its adjoint problem, a 2 × 2 case, by using the so-called symmetry constraint method. Now, we are dealing with 3 × 3 spectral problem (16) related to the hierarchy (12). Let us go back to spectral problem (16) and consider its adjoint problem ∗ 0 0 uλ ψ1 0 ∗ , ∗ = ψ2∗ , (36) x∗ = −1 0 ψ3∗ 0 −1 0 where ψ ∗ = ψ3∗ . Let λj (j = l, . . . , N) be N distinct spectral values of (16) and (36), and q1j , q2j , q3j and p1j , p2j , p3j be the corresponding spectral functions, respectively. Then we have
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q1x = q2 , q2x = q3 , q3x = −u q1 ,
(37)
p1x = u p3 , p2x = −p1 , p3x = −p2 ,
(38)
and
where = diag(λ1 , . . . , λN ), qk = (qk1 , qk2 , . . . , qkN )T , pk = (pk1 , pk2 , . . . , pkN )T , k = 1, 2, 3. Let us consider the above two systems in the symplectic space (R6N , dp ∧ dq), and introduce the following constraint: −2/3
u
=
N
∇λj ,
(39)
j =1
where ∇λj = λj q1j p3j is the functional gradient of λj for spectral problems (16) and (36). Then Equation (39) reads u =
q1 , p3 −3/2 .
(40)
Under this constraint, Equation (37) and its adjoint problem (38) are cast in a canonical Hamiltonian form in R6N : qx = {q, H + }, px = {p, H + },
(41)
with the Hamiltonian H + = q2 , p1 + q3 , p2 + √
2 ,
q1 , p3
(42)
where p = (p1 , p2 , p3 )T , q = (q1 , q2 , q3 )T ∈ R6N , · , · stands for the standard inner product in RN , and {· , ·} represents the Poisson bracket of two functions F1 , F2 defined by: 3 ∂F1 ∂F2 ∂F1 ∂F2 , , − {F1 , F2 } = ∂qi ∂pi ∂pi ∂qi i=1
(43)
which is antisymmetric and bilinear and satifies the Jacobi identity. To see the integrability of system (41), we take into account the time part t = Vk and its adjoint problem t∗ = −VkT ∗ , where Vk is defined by Vk = k−1 2(k−j −1) , and V (Gj ) is given by Equation (20) with G = Gj . j =0 V (Gj )λ
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A NEW INTEGRABLE HIERARCHY
Let us first look at V1 case. Then the corresponding time part is: −(u−2/3 )xx 3(u−2/3 )x −6u−2/3 2(u−2/3 )xx −3(u−2/3 )x , t = λ −(u−2/3 )xxx + 6λu1/3 −2/3 −2/3 1/3 −(u )xxxx (u )xxx + 6λu −(u−2/3 )xx and its adjoint problem is: −2/3 (u )xx (u−2/3 )xxx + 6λu1/3 −2(u−2/3 )xx t∗ = λ−3(u−2/3 )x −2/3 6u 3(u−2/3 )x
(44)
−(u−2/3 )xxxx −(u−2/3 )xxx − 6λu−1/3 ∗ . (45) (u−2/3 )xx
Noticing the following relations u1/3 =
q1 , p3 −1/2 , (u−2/3 )x =
q2 , p3 −
q1 , p2 , (u−2/3 )xx =
q3 , p3 +
q1 , p1 − 2 λq2 , p2 , (u−2/3 )xxx = 3(
q2 , p1 −
q3 , p2 ), (u−2/3 )xxxx = 6
q3 , p1 + 3
q1 , p3 −3/2 (
2 q1 , p2 +
2 q2 , p3 ), we obtain nonlinearized systems of the time parts (44) and (45), and furthermore cast them into the following canonical Hamiltonian system in R6N : qt1 = {q, F1+ }, pt1 = {p, F1+ },
(46)
with the Hamiltonian 1 F1+ = − (
q1 , p1 +
q3 , p3 )2 2 + 2
q2 , p2 (
q1 , p1 +
q3 , p3 −
q2 , p2 ) + 3(
q2 , p3 −
q1 , p2 )(
q2 , p1 −
q3 , p2 ) − 6
q1 , p3
q3 , p1 6 (
2 q1 , p2 +
2 q2 , p3 ). +√
q1 , p3
(47)
A direct computation leads to the following theorem. THEOREM 3. {H + , F1+ } = 0,
(48)
that is, two Hamiltonian flows commute in R6N . For general case Vk , k > 0, k ∈ Z, we consider the following Hamiltonian functions
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1 (
2j +1 q1 , p1 +
2j +1 q3 , p3 )(
2(k−j )−1 q1 , p1 2 j =0 k−1
Fk+ = −
+
2(k−j )−1 q3 , p3 ) +2
k−1
2j +1 q2 , p2 (
2(k−j )−1 q1 , p1 +
2(k−j )−1 q3 , p3
j =0
−
2(k−j )−1 q2 , p2 ) +3
k−1
(
2j +1 q2 , p3 −
2j +1 q1 , p2 )(
2(k−j )−1 q2 , p1
j =0
−
2(k−j )−1 q3 , p2 ) −6
k−1
2j +1 q1 , p3
2(k−j )−1 q3 , p1
j =0
3 (
2j q1 , p1 −
2j q3 , p3 )(
2(k−j ) q1 , p1 2 j =0 k
−
−
2(k−j ) q3 , p3 ) −3
k
(
2j q2 , p3 −
2j q1 , p2 )(
2(k−j ) q2 , p1
j =0
+
2(k−j ) q3 , p2 ) + 3H + (
2k q1 , p2 +
2k q2 , p3 ).
(49)
Through a lengthy calculation, we find {H + , Fk+ } = 0,
{Fl+ , Fk+ } = 0,
k, l = 1, 2, . . . .
(50)
That is, THEOREM 4. All canonical Hamiltonian flows (Fk+ ) commute with the Hamiltonian system (41). In particular, the Hamiltonian systems (41) and (46) are compatible and therefore integrable in the Liouville sense. Remark 1. In the proof procedure of this theorem, we have used the following two facts: q1 , p2 + q2 , p3 = c1 , and q1 , p1 − q3 , p3 = c2 . They always hold along x-flow in R6N . Here c1 , c2 are two constants. Remark 2. In fact, the involutive functions Fk+ are generated from nonlinearization of the time part t = Vk and its adjoint problem t∗ = −VkT ∗ 2(k−j −1) under the constraint (39), where Vk is defined by Vk = k−1 , and j =0 V (Gj )λ V (Gj ) is given by Equation (20) with G = Gj . In this calculation, we have used the following equalities:
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Gj = −
2j +1 q1 , p3 , j = 0, 1, 2 . . . , Gj =
2j +1 q2 , p3 −
2j +1 q1 , p2 , GJ =
2j +1 q3 , p3 +
2j +1 q1 , p1 − 2
2j +1 q2 , p2 , 2j +1 q2 , p1 −
2j +1 q3 , p2 ), G j = 3(
2j +1 q3 , p1 + 3
q1 , p3 −3/2 (
2j +2 q1 , p2 +
2j +2 q2 , p3 ), G j = 6
∂ −1 mGj =
2j q3 , p2 +
2j q2 , p1 , ∂ −2 ϒGj =
2j q1 , p1 −
2j q3 , p3 , ∂ −3 ϒGj = −(
2j q1 , p2 +
2j q2 , p3 ).
5. Parametric Solution Since Hamiltonian flows (H + ) and (Fk+ ) are completely integrable in R6N and their Poisson brackets {H + , Fk+ } = 0 (k = 1, 2, . . .), their phase flows gHx + , gFtk + k commute [7]. Thus, we can define their compatible solution as follows:
0 0 , t ) q(x q(x, tk ) k t , k = 1, 2, . . . , (51) = gHx + gFk + k p(x, tk ) p(x 0 , tk0 ) where x 0 , tk0 are the initial values of phase flows gHx + , gFtk + . k
THEOREM 5. Let q(x, tk ) = (q1 , q2 , q3 ) , p(x, tk ) = (p1 , p2 , p3 )T be a solution of the compatible Hamiltonian systems (H+ ) and (Fk+ ) in R6N . Then T
1 u=
q1 (x, tk ), p3 (x, tk ) 3
(52)
satisfies the positive equation of the hierarchy utk = J Lk · u−2/3 ,
k = 1, 2, . . . ,
(53)
where the operators L = J −1 K, J , K are given by Equations (10) and (9), respectively. Proof. Direct computation completes the proof. 2 In particular, we have the following result. THEOREM 6. Let p(x, t), q(x, t) (p(x, t) = (p1 , p2 , p3 )T , q(x, t) = (q1 , q2 , q3 )T ) be a common solution of the two integrable compatible flows (41) and (46), then 1 u=
q1 (x, t), p3 (x, t) 3
(54)
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satisfies the equation: ut = ∂x5 u−2/3 .
(55)
Proof. Taking derivatives in x five times on both sides of Equation (54), we obtain ∂x5 u−2/3 = 9u(
2 q3 , p3 −
2 q1 , p1 ) + 3ux (
2 q1 , p2 +
2 q2 , p3 ),
(56)
where 3 (
2 q1 , p2 +
2 q2 , p3 )(
q2 , p3 −
q1 , p2 ) . ux = − u 2
q1 , p3 On the other hand, taking derivative in t on both sides of Equation (54) yields 3
p3 , q˙1 +
q1 , p˙3 ut = − u 2
q1 , p3 ∂F +
1 3
p3 , ∂p1 −
q1 , =− u 2
q1 , p3
∂F1+ ∂q3
.
Substituting expression of F1+ into the above equality and calculating, we find that final result is the same as the right-hand side of Equation (56), which completes the proof. 2
6. Traveling Wave Solutions First, let us compute the traveling wave solution of Equation (3). Set u = f (ξ ), ξ = x − ct (c is some constant speed), then after substituting this setting into Equation (3) we obtain −cf + 3f f + f f = 0, i.e. (f 2 − 2cf ) = 0. Therefore, (f − c)2 = Aξ 2 + Bξ + C,
∀A, B, C ∈ R.
(57)
So, the equation uxxt + 3uxx ux + uxxx u = 0 has the following traveling wave solution (58) u(x, t) = c ± A(x − ct)2 + B(x − ct) + C. Let us discuss specific cases as follows:
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303
• When c = 0, we get stationary solution (59) u(x) = ± Ax 2 + Bx + C, ∀A, B, C ∈ R, which may be a straight line, circle, ellipse, parabola, and hyperbola according to different choices of constants A, B, C. • When c = 0 and A = 0, then we have B 2 4AC − B 2 + , ∀A, B, C ∈ R. (60) u(x, t) = c ± A x − ct + 2A 4A Therefore with 4AC − B 2 = 0 this solution becomes √ B u(x, t) = c ± A x − ct + , ∀A > 0, B ∈ R. 2A
(61)
Setting c = 1, A = 1, B = 0 yields u(x, t) = 1 − |x − t|,
(62)
and u(x, t) = 1 + |x − t|. (63) The former looks like a compacton solution [30, 14]. The latter is a “V”-type solution. • When c = 0 and A = 0, then we have (64) u(x, t) = c ± B(x − ct) + C, ∀B, C ∈ R, which is a parabolic traveling wave solution if B = 0 and becomes a constant solution if B = 0. In particular, √ (65) u(x, t) = 1 + x − t, x − t 0, and √ (66) u(x, t) = 1 − x − t, x − t 0 are two specific solutions. So, the 3rd-order equation uxxt + 3uxx ux + uxxx u = 0 has the continuous traveling wave solution (58). In addition, we also have the Gaussian initial solution of this 3rd-order equation, which is stable (see Figure 1). Second, we give the traveling wave solution of the 5th-order equation (1). Set u = ξ −γ , ξ = x −ct (c is a constant speed to be determined), then after substituting this setting into Equation (1) we obtain γ =
12 , 5
c = − 336 . 625
(67)
So, the 5th-order equation (1) has the following traveling wave solution u = (x +
336 −12/5 t) . 625
(68)
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Figure 3. This is the stable solution for the 5th-order equation ut = ∂x5 u−2/3 under the Gaussian initial condition.
Figure 4. This is the stable solution for the Harry–Dym equation ut = ∂x3 u−1/2 under the Gaussian initial condition.
Although at each time solution (68) has singular point at x = −(336/625)t, this 5th-order equation has the smooth and stable traveling wave solution under the Gaussian initial condition (see Figure 3). So, Figure 3 of the equation ut = ∂x5 u−2/3 has a slight difference from Figure 4 of the Harry–Dym equation ut = ∂x3 u−1/2 .
A NEW INTEGRABLE HIERARCHY
305
Figure 5. Solution near singular point.
Third, we give the traveling wave solution for the new integrable 7th-order equation (2). Set u = ξ −γ , ξ = x − ct (c is a constant speed to be determined), then we have γ =
18 , 7
c=
31680 . 117649
(69)
So, the 7th-order equation (1) has the following traveling wave solution (see Figure 5) u = (x −
31680 −18/7 t) . 117649
(70)
Furthermore, we propose the following new equations: ut = ∂xl u−m/n ,
l 1, n = 0, m, n ∈ Z.
This equation has the following traveling wave solution l−1 m m(l − 1) −n(l−1)/(m+n) −k . , c= u(x, t) = (x − ct) n k=1 m + n
(71)
(72)
Apparently, if mn + n2 > 0 this solution has singularity at point x = ct at each time, and if mn + n2 < 0 this solution is a polynomial traveling wave solution which is smooth. Remark 3. Here are the cusp-like traveling wave solutions with singularities u(x, t) = (x − 29 t)−4/3
(73)
and u(x, t) = (x +
336 −12/5 t) 626
(74)
for the Harry–Dym equation ut = ∂ 3 (u−1/2 ) and the 5th-order equation ut = ∂ 5 (u−2/3 ).
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7. Conclusions In Section 5, we obtain the parametric solution (54) of the 5th-order equation (1). This parametric solution does not include its traveling wave solution u = (x + (336/625)t)−12/5 because the parametric solution is smooth everywhere, but the traveling wave solution has singularity. The traveling wave solutions u = (x + (336/625)t)−12/5 for the equation ut = 5 −2/3 ∂ u and u = (x−(31680/117649)t)−18/7 for the equation ut = ∂x5 (((u−1/3 )xx − 2(u−1/6 )2x )/u) are singular at each time. That is, the singularity property travels with the time t (see Figure 5). Actually, when n(m + n) > 0 the traveling wave solution (72) for general Equation (71) is also matching this property. A natural question arises here: is Equation (71) integrable for all l 1, m, n ∈ Z or for what kind of l 1, m, n ∈ Z is it integrable? We will discuss this elsewhere. The Harry–Dym equation has the cusp-like traveling wave solution u(x, t) = (x − (2/9)t)−4/3 , but this is not cusp soliton which Wadati described in [32], because the current traveling wave solution is singular, but the cusp is continuous. If we consider other constraints between the potential and eigenfunctions, then we can still get parametric solutions for the other two equations (u−1/3 )xx − 2(u−1/6 )2x , u + 3uxx ux + uxxx u = 0.
ut = ∂x5 uxxt
Acknowledgments The first author is much indebted to Dr. Darryl Holm for his invitation and providing an opportunity to join his research project. He would like to express his sincere thank to Prof. Konopelchenko for showing his paper [20] and Prof. Magri for his fruitful discussion during their visit at Los Alamos National Laboratory. He is grateful to the referee for reminding literatures [13, 16]. This work was supported by the Foundation for the Author of National Excellent Doctoral Dissertation (FANEDD) of PR China, and also the Doctoral Programme Foundation of the Insitution of High Education of China. References 1. 2. 3. 4.
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Rosenau, P. and Hyman, J. M.: Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70 (1993), 564–567. Tu, G. Z.: An extension of a theorem on gradients of conserved densities of integrable systems, Northeast. Math. J. 6 (1990), 26–32. Wadati, M., Ichikawa, Y. H. and Shimizu, T.: Cusp soliton of a new integrable nonlinear evolution equation, Progr. Theoret. Phys. 64 (1980), 1959–1967. Zakharov, V. E. and Shabat, A. B.: Exact theory of two dimensional self focusing and one dimensional self modulation of waves in nonlinear media, Soviet Phys. JETP 34 (1972), 62–69.
Mathematical Physics, Analysis and Geometry 7: 309–331, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Generalized Value Distribution for Herglotz Functions and Spectral Theory Y. T. CHRISTODOULIDES and D. B. PEARSON Department of Mathematics, University of Hull, Cottingham Rd., Hull HU6 7RX, UK. e-mail:
[email protected];
[email protected] (Received: 17 April 2003; in final form: 8 June 2004) Abstract. We generalize the theory of value distribution for a class of functions defined as boundary values of Herglotz functions, by considering other measures than Lebesgue measure. The link with compositions of Herglotz functions is presented, and precise relations for the associated measures are obtained. We also consider uniformly convergent sequences of Herglotz functions on compact subsets of the upper half-plane, and prove that the corresponding sequence of Herglotz measures and the generalized value distribution of these functions also converge. Mathematics Subject Classification (2000): 34L05. Key words: generalized value distribution, Herglotz functions, spectral theory.
1. Introduction Let F (z) be a Herglotz function, that is analytic in the upper half-plane and with positive imaginary part. Then F admits the integral representation ([1, 7, 10]) t 1 − 2 dρ(t), (1) F (z) = aF + bF z + t +1 R t −z where aF , bF are real constants and the function ρ(t) is nondecreasing, rightcontinuous, and unique up to an additive constant, for given F . In particular, aF and bF are given by aF = Re F (i),
bF = lim
s→∞
1 Im F (is), s
and ρ(t) gives rise to a Borel measure µ through the relation ρ(b) − ρ(a) = µ((a, b]) for intervals (a, b]. Thus µ is the Herglotz measure corresponding to F , and it satisfies the integral condition 1 dρ(t) < +∞, (2) 2 R 1+t which is a sufficient condition to ensure convergence of the representation of F (z) in (1).
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We denote by F+ (λ) the boundary value of F at the point λ ∈ R, defined by F+ (λ) = limε→0+ F (λ + iε). Thus, F+ (λ) is the limiting value of F (z) as z approaches the point λ on the real line, vertically from the upper half-plane. The measure µ corresponding to F may be decomposed into its absolutely continuous and singular parts, µ = µa.c. + µs ([3, 12, 15]). In the special case that F has real boundary values almost everywhere, µ is purely singular. More generally, µa.c. is concentrated on the set of λ ∈ R for which Im F+ (λ) > 0, and the density function f of µa.c. is given by f (λ) = π1 Im F+ (λ), λ ∈ R. For an analysis of the supports of the measures µa.c. and µs see [14]. In [4, 5, 13], a theory of value distribution for boundary values of Herglotz functions has been developed, with applications to spectral analysis and to the Weyl–Titchmarsh theory of the m-function [8] for Sturm–Liouville equations. The starting point of this theory is a study of the properties of the value distribution function M, given for Borel subsets of the real line in the case that µ is purely singular by M(A, S) = A ∩ F+−1 (S), where | · | denotes Lebesgue measure. The main purpose of the current paper is to extend the existing theory to allow a description of value distribution involving measures other than Lebesgue. To outline in broad terms the direction that this extension [6] might take, consider the following simple example, again in the special case that µ is purely singular. For given Herglotz function F , define a measure ν on Borel subsets of R by ν(A) = α A ∩ F+−1 (R− ) + β A ∩ F+−1 (R+ ), where α, β are positive constants. If α = β = 1, the measure ν reduces to Lebesgue measure. More generally, ν is an absolutely continuous measure weighted according to the sign of F+ (λ). Moreover, we have ∞ 0 µy (A) dy + β µy (A) dy, ν(A) = α −∞
0
where {µy } (y ∈ R) is a one-parameter family of measures defined by the Herglotz function F (for precise definitions of µy see Section 2 below, in particular Equations (3)–(6)). Defining an absolutely continuous measure dσ , having density function α on R− and β on R+ , we can write ∞ µy (A) dσ (y). ν(A) = −∞
Again, with α = β = 1 the measure dσ reduces to Lebesgue measure. More generally, we may apply the theory of boundary values for Herglotz functions by noting that dσ is the Herglotz measure associated with the function φ(z) = iπβ +
GENERALIZED VALUE DISTRIBUTION FOR HERGLOTZ FUNCTIONS
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(α − β)log z, and that ν is the Herglotz measure associated with the composed function φ ◦ F , given by (φ ◦ F )(z) = iπβ + (α − β)log F (z). If νS = ν|F+−1 (S) denotes the restriction of ν to F+−1 (S) for an arbitrary Borel set S, we may describe the generalized value distribution of F+ (that is, the value distribution with ν replacing Lebesgue measure) by means of the value distribution function Mν given by Mν (A, S) = ν A ∩ F+−1 (S) = νS (A). In the following sections, we shall extend these ideas to cover the case in which σ is a general absolutely continuous measure and ν is a general measure weighted according to the values of F+ (λ). Moreover, the general features of the theory, involving measures σ and ν, generalized value distribution function Mν , and a composed Herglotz function φ ◦ F with associated measure ν, apply quite generally to Herglotz functions F , whether or not the measure µ is purely singular. Within this context of generalized value distribution we prove continuity of Mν with respect to the Herglotz function F , in the sense that Mν (A, S; Fn ) converges to Mν (A, S; F ) whenever Fn → F uniformly for z in compact subsets of C+ . The main results of the paper are summarized below in Lemma 2.6, and Theorems 3.4 and 3.8. The paper is organized as follows. In Section 2 we consider the Herglotz measure dσ and associated Herglotz function φ(z), and define corresponding generalized measures ν, νS for the Herglotz function F , where S is an arbitrary Borel set. We verify that the measure dνS is absolutely continuous provided dσ is absolutely continuous, and prove that dνS is also a Herglotz measure. In Lemma 2.6 we show that, for any Borel set B, we have νS (B) = µ(φS ◦F ) (B) − bφ µ(B), where µ(φS ◦F ) is the measure corresponding to the composed Herglotz function (φS ◦ F ), and φS (z) is the Herglotz function having the same representation as φ(z), except that now integration with respect to σ takes place over the set S. Here, µ is the measure corresponding to the Herglotz function F , and bφ is the constant appearing in the representation of F in (1). Hence, in the case that bφ = 0, then νS is precisely the measure corresponding to the composed Herglotz function φS ◦ F . In Section 3 we consider a sequence of Herglotz functions Fn with corresponding measures µn , converging uniformly as n → ∞ to the Herglotz function F (z), on compact subsets of the upper half-plane. We prove in Theorem 3.4 that, in that case, we have µn ((a, b]) → µ((a, b]), provided the points a and b are not discrete points of any of the measures µn or µ, the measure corresponding to F . We make use of the standard result regarding Herglotz measures of intervals whose endpoints are not discrete points of the measure. Also, in order to control limits close to the real axis, we use complex contour integration in the upper half-plane [11], where the functions Fn , by assumption, satisfy convergence conditions. Under the further assumption that the measure dσ is absolutely continuous, we will prove in Theorem 3.8 that the generalized value distribution of Fn converges, as n → ∞, to the generalized value distribution of F .
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2. Generalized Value Distribution for Herglotz Functions Given a Herglotz function F , we may construct a one-parameter family of Herglotz functions Fy defined by Fy (z) =
1 , y − F (z)
y ∈ R,
with corresponding measures µy and representations 1 t − 2 dµy (t). Fy (z) = ay + by z + t +1 R t −z The integral µy (A) dy,
(3)
(4)
(5)
S
where A, S are arbitrary Borel sets and µy are the measures corresponding to the Herglotz functions Fy , defines the value distribution mapping for the Herglotz function F , as the following result shows. THEOREM 2.1. Let F be a Herglotz function with associated measure µ. Suppose that F has real boundary values almost everywhere with respect to Lebesgue measure, so that µ is purely singular. Then, for any Borel sets A, S ⊆ R we have µy (A) dy = A ∩ F+−1 (S), (6) S
where F+ is the boundary value of F as z approaches the real axis, µy are the measures corresponding to the Herglotz functions Fy defined in (3), and |·| denotes Lebesgue measure. In particular, we have µy (A) dy = |A|. (7) R
Proof. See [13, 14].
2
Hence S µy (A) dy is the Lebesgue measure of the points in A for which the boundary value of F is in S. Equation (6) also holds in the case when F has real boundary values almost everywhere on A, and (7) holds for arbitrary Herglotz functions F , that is without the assumption that F has real boundary values almost everywhere. The integral in (5) may be regarded as a spectral average of a family of measures over the set S. There is an extensive literature on spectral averaging with applications to spectral analysis. See [9] for a recent review of this theory. A unifying feature of the treatment in [9] is to consider the average of a one parameter family of
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measures, where each measure is obtained from the Herglotz measure corresponding to the composition of a given Herglotz function with a one-parameter group of automorphisms of the upper half plane. Results such as (7) then follow in a natural way from this underlying theory. In the present paper, we will extend the theory by considering spectral averaging with respect to a wider class of measures, for which we shall establish a connection with ideas of value distribution for boundary values of Herglotz functions. In this more general context, we will need to consider compositions with arbitrary Herglotz functions rather than Möbius transformations as in [9], and this theme will be continued in a subsequent publication. It will be important for later calculations to know how the constants by in (4) depend on y. The following lemma provides an answer to this question. LEMMA 2.2. For a given Herglotz function F (z), the nonnegative constants by appearing in (4) are zero, except possibly for a single value of y. Moreover, by may be strictly positive for some value y = y0 ; for any given y0 ∈ R, the condition by0 > 0 is equivalent to the condition that the point t = 0 is a discrete point of the measure dg, where dg is the measure corresponding to the Herglotz function G(z) defined by G(z) =
1 . y0 − F (− 1z )
(8)
Proof. The constants by are given by
1 1 1 . by = lim Im Fy (is) = lim Im s→∞ s s→∞ s y − F (is)
Suppose that F (is) = α(s) + iβ(s), with α, β real. Then, β(s) 1 . s→∞ s [y − α(s)]2 + [β(s)]2
by = lim
If by > 0, then α(s) → y and β(s) → 0 as s → ∞, so that F (is) → y as s → ∞. That is, F (is) → y as s → ∞ is a necessary condition for by > 0, where y is any real number. So there can only be at most one value of y such that by > 0, since F (is) can not tend to two different limits as s → ∞. Consider now the Herglotz function G(z) defined in (8), and in particular, the limit
= lim+ w Im G(iw), w→0
w ∈ R.
Writing w = 1/s, we have 1 1 .
= lim Im s→∞ s y0 − F (is) Therefore, by0 = and thus by0 > 0 is equivalent to > 0.
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Suppose that G admits the representation t 1 G(z) = aG + bG z + − 2 dg(t). t +1 R t −z Then,
= lim+ w Im G(iw) = lim+ w→0
w→0
R
w2 dg(t). t 2 + w2
(9)
The integrand in (9) has 0 as its pointwise limit, as w → 0+ , except in the case when t = 0, in which case the limit is 1. Note also that 1 w2 , 2 t 2 + w2 t +1
0 < w 1,
where the function 1/(1 + t 2 ) is integrable with respect to the Herglotz measure dg. Hence, we can apply the Lebesgue dominated convergence theorem in (9), to deduce that w2 dg(t) = g {0} . lim+ 2 2 w→0 R t +w Hence by0 > 0 is equivalent to t = 0 being a discrete point of the dg measure, and moreover in that case we have by0 = g({0}). We now extend the idea of value distribution associated with a Herglotz function. Given a Herglotz function F (z), and a Borel subset S of R, we define the integral-measures dν and dνS by µy (A) dσ (y), (10) ν(A) = R
and
Mν (A, S) = νS (A) =
µy (A) dσ (y)
(11)
S
respectively, for any Borel subset A of R. Here, µy are the measures corresponding to the Herglotz functions Fy which were defined in (3), and dσ is a Herglotz measure. 2 LEMMA 2.3. Suppose that the measure dσ is absolutely continuous. Then, the measures dν and dνS are absolutely continuous. Proof. Let A be a Borel set having zero Lebesgue measure. Then, from (7) we have µy (A) = 0 almost everywhere, and since dσ is absolutely continuous it follows that ν(A) = νS (A) = 0. This proves the absolute continuity of dν 2 and dνS .
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The following lemma shows that, in the special case µa.c. = 0, νS (A) is the νS -measure (and also the ν-measure) of the points in A for which the boundary value of F is in S. LEMMA 2.4. Suppose that F has real boundary values almost everywhere, i.e. the measure µ is purely singular. Then, we have (12) νS (A) = νS A ∩ F+−1 (S) = ν A ∩ F+−1 (S) . Proof. Suppose µ is purely singular. Note that the functions Fy will also have real boundary values almost everywhere, and hence the measures µy will also be purely singular. The support of µy will be the set {λ ∈ R : F+ (λ) = y}. So, for y ∈ S, the support of µy will be a subset of F+−1 (S), and we can write νS (A) = µy (A) dσ (y) S = µy A ∩ F+−1 (S) dσ (y) = νS A ∩ F+−1 (S) . S
On the other hand, if y ∈ / S, the support of µy will be a subset of R\F+−1 (S), and so we also have µy A ∩ F+−1 (S) dσ (y) = ν A ∩ F+−1 (S) , νS (A) = R
2
and the lemma is proved.
Thus dνS agrees with dν on F+−1 (S), though in general these measures are different, and we have νS = ν|F+−1 (S) in this case. We refer to dνS (and Mν ) defined in (11), with dσ absolutely continuous (so that by Lemma 2.3 dνS will also be absolutely continuous), as the generalized value distribution function for the Herglotz function F . LEMMA 2.5. The measure νS is a Herglotz measure. Proof. By considering characteristic functions of measurable sets, simple functions, and finally measurable functions h for which the integrals are absolutely convergent, one may easily verify the identity h(t) dνS (t) = h(t) dµy (t) dσ (y). (13) R
S
R
From the representation of the Herglotz functions Fy in (4), we obtain 1 dµy (t). Im Fy (i) = by + 1 + t2 R
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Therefore, we have
1 dµy (t) dσ (y) = Im Fy (i) − by dσ (y). 2 S R 1+t S
(14)
Suppose F (i) = A + iB, with B > 0. Then Im Fy (i) = B/[(y − A)2 + B 2 ], and it can easily be shown that Im Fy (i) const./(1 + y 2 ), which is integrable with respect to dσ . Since by are nonnegative, the integrals in (14) are finite, and hence we have from (13) 1 1 dνS (t) = dµy (t) dσ (y) < +∞, 2 2 R 1+t S R 1+t which is a sufficient condition for dνS to be a Herglotz measure, for arbitrary Borel set S. 2 Furthermore, as we shall prove next, dνS may be expressed in terms of compositions of Herglotz functions. Let HS (z) and φ(z) be Herglotz functions corresponding to the Herglotz measures dνS and dσ , respectively, with the following representations: t 1 − 2 dνS (t), (15) HS (z) = aH + bH z + t +1 R t −z t 1 − 2 dσ (t). (16) φ(z) = aφ + bφ z + t +1 R t −z Let also φS (z) be the Herglotz function having the same representation as that of φ(z) in (16), except that integration takes place over the set S instead of R, that is t 1 − 2 dσ (t). (17) φS (z) = aφ + bφ z + t +1 S t −z Moreover, let the composed Herglotz function (φS ◦ F )(z) have the following representation: t 1 − 2 dµ(φS ◦F ) (t). (18) (φS ◦ F )(z) = a(φS ◦F ) + b(φS ◦F ) z + t +1 R t −z LEMMA 2.6. For any Borel subset B of R, we have νS (B) = µ(φS ◦F ) (B) − bφ µ(B),
(19)
where µ(φS ◦F ) is the measure corresponding to the composed Herglotz function (φS ◦ F ), µ is the measure corresponding to the Herglotz function F , and the constant bφ appears in the representation of the Herglotz function φS in (17) (and also in (16)). Note that here we do not assume absolute continuity of the measure dσ .
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Proof. From the representation of HS (z) in (15) we have 1 Im HS (z) = bH Im z + Im dνS (t). t −z R The function Im[1/(t − z)] is continuous and hence measurable, and thus we have from Equation (13) 1 1 dνS (t) = dµy (t) dσ (y). Im Im t −z t −z R S R The representation of the functions Fy in (4) leads to 1 dµy (t). Im Fy (z) = by Im z + Im t −z R Combining these equations we obtain 1 dσ (y) − Im z by dσ (y). Im HS (z) = bH Im z + Im y − F (z) S S
(20)
Note that by Lemma 2.2 we have by ∗ , y = y ∗ , by = 0, y = y ∗ , where y ∗ is the point for which by > 0, if this point exists. Also, from (17) we have 1 dσ (t). Im φS (z) = bφ Im z + Im t −z S Therefore, we obtain from (20)
Im HS (z) = bH Im z + Im φS F (z) −
− bφ Im F (z) − by ∗ (Im z)σ {y ∗ ∩ S} .
(21)
Suppose that the points a and b are not discrete points of any of the measures νS , µ(φS ◦F ) or µ. By using the standard result regarding Herglotz measures of intervals (a, b] whose endpoint are not discrete points of the measures [14], we have 1 b Im HS (λ + iε) dλ νS (a, b] = lim+ ε→0 π a 1 b Im φS F (λ + iε) dλ − = lim+ ε→0 π a 1 b Im F (λ + iε) dλ − bφ lim ε→0+ π a (22) = µ(φS ◦F ) (a, b] − bφ µ (a, b] ,
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since 1 lim+ ε→0 π
b
Im(λ + iε) dλ = 0. a
Now take arbitrary points a and b. We shall show that (22) still holds. Since the set of discrete points of the measures νS , µ(φS ◦F ) , and µ is countable, given any ε > 0, there are points in the intervals (a − ε, a) and (b, b + ε) respectively, which are not discrete points of either of the measures νS , µ(φS ◦F ) , or µ. Hence, we can construct two sequences of such points, {ci }i∈N and {di }i∈N , with ci → a− and di → b+ . We then have, on taking the limit of νS ((ci , di ]) νS [a, b] = µ(φS ◦F ) [a, b] − bφ µ [a, b] . By the same argument, we have νS {x} = µ(φS ◦F ) {x} − bφ µ {x} ,
∀x ∈ R.
Combining these two equations, we obtain νS (a, b] = µ(φS ◦F ) (a, b] − bφ µ (a, b] ,
(23)
for all points a and b of R. Equation (23) implies that νS and (µ(φS ◦F ) − bφ µ) are measures defined on the algebra of countable unions of intervals of the form (a, b]. The fact that νS is a Herglotz measure, implies that νS ((−N, N]) is finite for any integer N
1 2 dνS (t) < +∞ . νS (−N, N] (1 + N ) 2 R 1+t Since R = N∈N (−N, N], it follows that νS , and also (µ(φS ◦F ) −bφ µ), are σ -finite measures. Hence, there is a unique extension of these measures to the collection of Lebesgue measurable sets, called the corresponding Lebesgue–Stieltjes measure, and restricting this measure to the Borel sets, we have a unique extension to all Borel sets. Therefore, with this extended measure we have shown that νS (B) = µ(φS ◦F ) (B) − bφ µ(B), for all Borel sets B, and Lemma 2.6 is proved.
2
Note that a special case of the result is the case where φ is a Möbius transformation (see [9]). In that case, unless φ is a linear transformation, we have bφ = 0.
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3. Herglotz Functions and Uniform Convergence Let Fn be a sequence of Herglotz functions with associated measures µn , and integral representations t 1 − 2 dµn (t). (24) Fn (z) = an + bn z + t +1 R t −z Throughout this section we assume that the functions Fn (z) converge uniformly in the limit n → ∞ to the Herglotz function F (z), on compact subsets of the upper half-plane. We consider the µn measure of an interval (a, b] where a, b are not discrete points of the measure µ. The following lemma allows us to control the µn measure of neighbourhoods of a and b. LEMMA 3.1. Let Fn (z) be a sequence of Herglotz functions, given by Equation (24), converging uniformly as n → ∞ to the Herglotz function F (z), given by Equation (1), on compact subsets of the upper half-plane. Suppose that a and b (a < b) are not discrete points of the measure µ. Let ε > 0 be given. Then, there exists δ0 with 0 < δ0 < (b − a)/2, and N0 ∈ N depending on δ0 and ε, such that if n > N0 then µn (J0 ) < ε, where J0 = [a − δ0 , a + δ0 ] ∪ [b − δ0 , b + δ0 ]. Proof. From the representations (24), (1) of the functions Fn and F respectively we have δ2 dµn (t), δ Im Fn (a + iδ) = bn δ 2 + 2 2 R (t − a) + δ and δ Im F (a + iδ) = bF δ 2 +
R
δ2 dµ(t). (t − a)2 + δ 2
Similar expressions hold for δ Im Fn (b + iδ) and δ Im F (b + iδ). It is easy to verify that since a and b are not discrete points of the measure µ, we have lim δ Im F (a + iδ) = lim+ δ Im F (b + iδ) = 0.
δ→0+
(25)
δ→0
(Note that 1 δ2 , 2 2 (t − a) + δ (t − a)2 + 1
∀t ∈ R,
for 0 < δ < 1, which implies that we can use the Lebesgue dominated convergence theorem.) Let ε > 0 be given. In view of (25), we can choose δ0 > 0 such that ε δ0 Im F (a + iδ0 ) + F (b + iδ0 ) < . 8
(26)
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Since the functions Fn (z) converge to F (z) at the points z = a+iδ0 and z = b+iδ0 , there is an N0 ∈ N such that, if n > N0 then
ε δ0 Im Fn (a + iδ0 ) − F (a + iδ0 ) + Fn (b + iδ0 ) − F (b + iδ0 ) < . (27) 8 It follows from (26) and (27) that for n > N0 we have δ0 Im Fn (a + iδ0 ) + Fn (b + iδ0 ) δ02 δ02 ε 2 + dµn (t) < . = 2bn δ0 + 2 2 2 2 4 (t − b) + δ0 R (t − a) + δ0 Since bn δ02 0, (28) implies δ02 ε dµ (t) < n 2 2 4 R (t − a) + δ0
and R
(28)
δ02 ε . dµ (t) < n 2 4 (t − b)2 + δ0
For t ∈ [a − δ0 , a + δ0 ] we have (t − a)2 δ02 , so that δ02 /((t − a)2 + δ02 ) 1/2, and hence a+δ0 a+δ0 ε 1 δ02 dµn (t) . dµ (t) < n 2 2 4 a−δ0 2 a−δ0 (t − a) + δ0 Therefore, for n > N0 we have ε µn [a − δ0 , a + δ0 ] < . 2 Similarly, for n > N0 we have ε µn [b − δ0 , b + δ0 ] < . 2 Hence µn (J ) < ε, provided n > N0 , as stated in the lemma.
2
Remark 3.2. The proof of Lemma 3.1 does not require uniform convergence of Fn (z). It is sufficient that Fn (z) converge to F (z) at the points z = a + iδ0 and z = b + iδ0 . LEMMA 3.3. With the same notation as in the statement of Lemma 3.1, suppose that a and b (a < b) are not discrete points of the measure µ, and let ε > 0 be given. Then δ > 0 can be chosen such that c 1 t − b t − a < ε, − dµ(t) ds (29) π 2 2 (t − b)2 + s 2 0 J (t − a) + s where c > 0 is arbitrary, J = (a − δ, a + δ) ∪ (b − δ, b + δ), and δ is taken to lie in the interval 0 < δ < (b − a)/2.
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Proof. Let c > 0 be a constant, and define a function K(c, t) by c |t − b| |t − a| + ds, K(c, t) = (t − a)2 + s 2 (t − b)2 + s 2 0 so that K(c, t) π . Since µ({a}) = µ({b}) = 0, it follows that there is a δ, with 0 < δ < (b − a)/2, such that µ(J ) < ε, where J = [a − δ, a + δ] ∪ [b − δ, b + δ]. Since the double integral in (29) is absolutely convergent we can change the order of integration [2] to obtain c 1 t −b t −a − dµ(t) ds π 2 2 (t − b)2 + s 2 0 J (t − a) + s 1 K(c, t) dµ(t) µ(J ) < ε, π J by our choice of δ in the definition of J , and (29) is proved.
2
THEOREM 3.4. Let Fn (z) be a sequence of Herglotz functions with corresponding measures µn . Suppose that Fn (z) converge uniformly, as n → ∞, to the Herglotz function F (z), on compact subsets of the upper half-plane, and that the points a and b (a < b) are not discrete points of any of the measures µn (n ∈ N) or µ, the measure corresponding to F (z). Then, we have (30) µn (a, b] → µ (a, b] . Proof. By the standard result regarding Herglotz measures of intervals (a, b] whose endpoints are not discrete points of the measure [14], we have µn (a, b] − µ (a, b] 1 b 1 b Im Fn (λ + iw) dλ − lim+ Im F (λ + iw) dλ. (31) = lim+ w→0 π w→0 π a
a
The problem is to control the behaviour of the functions Fn (z) and F (z) close to the real axis. We also need to control the behaviour of integrals near the endpoints a and b of the interval (a, b], on a contour perpendicular to the real axis. Given any ε > 0, set ε0 =
επ , 6M(b − a)
where the constant M is defined to be M=
2 Im F (is0 ) + 1, s0
for any s0 > 0. The role of ε0 and the constant M will become clear shortly.
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Let z = λ+iw, for some fixed w, 0 < w < ε0 , and let A = a +iw, B = b+iw. Then µn (a, b] − µ (a, b] B B 1 1 Fn (z) dz − lim+ Im F (z) dz. (32) = lim+ Im w→0 π w→0 π A
A
Now let C = b + iε0 , D = a + iε0 , and consider the contour ABCD. It follows by Cauchy’s theorem that Fn (z) dz = F (z) dz = 0, ABCD
so that
ABCD
B A
D
Fn (z) dz = A
C
Fn (z) dz +
B
Fn (z) dz + D
Fn (z) dz. C
A similar expression holds for F (z). On the contour AD let z = a+is, w s ε0 , on DC let z = s + iε0 , a s b, and on the contour CB let z = b + is, w s ε0 . Then, we have B Fn (z) dz Im A b ε0 Fn (s + iε0 ) ds. Fn (a + is) − Fn (b + is) ds + Im = Re a
w
From the representations of Fn in (24) we have Re Fn (a + is) − Fn (b + is) t −b t −a − dµn (t). = (a − b)bn + 2 2 (t − b)2 + s 2 R (t − a) + s Thus, we obtain B Fn (z) dz Im A b ε0 (a − b)bn ds + Im Fn (s + iε0 ) ds + = a w ε0 t −b t −a − (t) ds. dµ + n 2 2 (t − b)2 + s 2 w R (t − a) + s A similar expression holds for Im the function P (s, t) be defined by P (s, t) =
B A
(33)
(34)
F (z) dz. For the remainder of the proof let
t −b t −a − . 2 2 (t − a) + s (t − b)2 + s 2
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Then
µn (a, b] − µ (a, b] 1 ε0 lim+ P (s, t) dµn (t) ds − w→0 π w R ε0 1 P (s, t) dµ(t) ds + − lim+ w→0 π w R ε0 1 (a − b)(bn − bF ) ds + + lim+ w→0 π w b 1 Im Fn (s + iε0 ) − F (s + iε0 ) ds . (35) + π a From the representations of Fn (z) in (24) we have 1 1 Im Fn (is) = bn + dµn (t), 2 2 s R t +s and thus bn < (1/s) Im Fn (is), ∀s ∈ R. Similarly, bF < (1/s) Im F (is), ∀s ∈ R. In particular, bF < (1/s0 ) Im F (is0 ), and bn < (1/s0 ) Im Fn (is0 ). Since Fn → F , as n → ∞, at z = is0 , there is an N1 ∈ N such that if n > N1 , then (1/s0 )|Im[Fn (is0 ) − F (is0 )]| < 1/2. Hence bn < (1/s0 ) Im F (is0 ) + (1/2), so that 2 1 |bn − bF | bn + bF < Im F (is0 ) + < M. s0 2 With an application of the Lebesgue dominated convergence theorem, it follows that ε0 1 ε0 ε lim 1 (36) (a − b)(bn − bF ) ds < (b − a)M ds = . w→0+ π π 0 6 w Since, by assumption, Fn (z) → F (z) uniformly, as n → ∞, on the horizontal contour joining the points a+iε0 and b+iε0 , there is an N2 ∈ N such that if n > N2 then |Fn (s + iε0 ) − F (s + iε0 )| < επ/6(b − a), a s b. Hence, we also have b 1 ε (37) Im Fn (s + iε0 ) − F (s + iε0 ) ds < . π 6 a
By Lemma 3.1, there is a δ1 > 0, a corresponding set J1 = [a − δ1 , a + δ1 ] ∪ [b − δ1 , b + δ1 ], and an N3 ∈ N such that, if n > N3 , then µn (J1 ) < ε/18. By changing the order of integration and applying the Lebesgue dominated convergence theorem we obtain ε0 lim 1 P (s, t) dµn (t) ds w→0+ π w J ε0 1 1 P (s, t) ds dµn (t) = π J1 0 ε 1 (38) K(ε0 , t) dµn (t) µn (J1 ) < , π J1 18
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provided that n > N3 . The function K(c, t), t ∈ R and any c > 0 fixed, is defined in the proof of Lemma 3.3. Also, by Lemma 3.3, there is a δ2 > 0 and a corresponding set J2 = [a − δ2 , a + δ2 ] ∪ [b − δ2 , b + δ2 ] such that ε0 1 ε P (s, t) dµ(t) ds < . π 18 0 J2 Hence ε0 1 lim P (s, t) dµ(t) ds w→0+ π w ε0J2 1 ε = P (s, t) ds dµ(t) < . π J2 18 0
(39)
With δ = min{δ1 , δ2 }, and J = [a − δ, a + δ] ∪ [b − δ, b + δ], (38) and (39) hold with J1 and J2 replaced by J . It remains to estimate the integrand for t ∈ R\J , and it is not difficult to verify that on this set we have P (s, t) cJ 1 , 1 + t2 for some constant cJ . The fact that µn and µ are Herglotz measures now implies that the integrals P (s, t) dµn (t) c1 = R\J
and
c2 =
P (s, t) dµ(t) R\J
converge absolutely. Note that 1 dµn (t) = cJ Im Fn (i) − bn cJ Im Fn (i), |c1 | cJ 2 R 1+t
(40)
and thus c1 is bounded uniformly in n, since Fn (i) → F (i). Hence, it follows by the Lebesgue dominated convergence theorem that 1 ε0 P (s, t) dµn (t) ds lim w→0+ π w R\J 1 ε0 P (s, t) dµn (t) ds, (41) = π 0 R\J and similarly for the second double integral. This shows that the integration with respect to Lebesgue measure ds must be performed on the interval [0, ε0 ].
GENERALIZED VALUE DISTRIBUTION FOR HERGLOTZ FUNCTIONS
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Suppose that |c1 | K, for all n ∈ N, for some constant K > 0. We now let c = max{K, |c2 |}, ε1 = επ/9c , and set ε = (1/2) min{ε1 , ε2 }. We then have ε ε 1 . (42) P (s, t) dµ (t) − P (s, t) dµ(t) ds n 9 π 0 R\J R\J
So far, from (36)–(39), (41), and (42) we have µn (a, b] − µ (a, b] 5ε ε0 + P (s, t) dµn (t) − P (s, t) dµ(t) ds . < 9 ε R\J R\J
(43)
But our previous arguments show that ε0 1 < ε , P (s, t) dµ (t) ds n 18 π ε J provided n > N3 , and ε0 1 ε P (s, t) dµ(t) ds < . π 18 ε J Therefore, from (43) we find µn (a, b] − µ (a, b] 2ε 1 ε0 + P (s, t) dµn (t) − P (s, t) dµ(t) ds . < 3 π ε R R
(44)
By using Equation (33) and a similar expression for F (z), and substituting in (44), we obtain µn (a, b] − µ (a, b] 1 ε0 2ε + Fn (a + is) − F (a + is) ds + 3 π ε 1 ε0 Fn (b + is) − F (b + is) ds + + π ε 1 ε0 (b − a)|bF − bn | ds. (45) + π ε Since Fn (z) → F (z) uniformly, as n → ∞, on compact subsets of the upper half-plane, we can find an N4 ∈ N such that, if n > N4 , then ε 1 ε0 Fn (a + is) − F (a + is) + Fn (b + is) − F (b + is) ds < . π ε 6 Also, from (37) we have ε 1 ε0 (b − a)|bF − bn | ds < . π ε 6
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Let N = max N1 , N2 , N3 , N4 . Then, if n > N, we have from (45) µn (a, b] − µ (a, b] < ε, 2
so that Theorem 3.4 is proved.
Remark 3.5. Again, rather than uniform convergence of Fn (z), it is sufficient that Fn (z) converge to F (z) at a point z = is0 , as n → ∞, for any s0 1, and uniformly on the -shaped contour consisting of the following parts: the horizontal contour joining the points a +iε0 and b+iε0 , and the vertical contours {z = a +is : ε s ε0 }, {z = b + is : ε s ε0 }, where the positive constants ε0 , ε were defined in the proof of Theorem 3.4. LEMMA 3.6. Suppose that the measure dσ corresponding to the Herglotz function φ(z) is absolutely continuous, and let hσ (y) be the density function of dσ . For n ∈ N, let a family of sets Xn be defined by Xn = {y ∈ R : hσ (y) > n}. Then 1 lim dσ (y) = 0. (46) n→∞ X 1 + y 2 n Proof. The result follows from the Lebesgue dominated convergence theorem, using the bound 1 χn (y) , 2 1+y 1 + y2
y ∈ R, n ∈ N,
where χn is the characteristic function of Xn .
2
Define now a sequence of Herglotz functions Fyn (z) (y ∈ R) by Fyn (z) =
1 , y − Fn (z)
y ∈ R, n ∈ N,
having measures µny and integral representations t 1 n n n − 2 dµny (t). Fy (z) = ay + by z + t +1 R t −z
(47)
(48)
The measures µny satisfy the following bounds. LEMMA 3.7. Let Fn be a sequence of Herglotz functions, given by (24), such that Fn (i) lie in a compact subset of C+ . Then, for any fixed N and any y ∈ R there exists a constant c > 0 independent of y and n such that µny [−N, N]
c , 1 + y2
n ∈ N.
GENERALIZED VALUE DISTRIBUTION FOR HERGLOTZ FUNCTIONS
327
Proof. Let D be a compact subset of C+ such that Fn (i) ∈ D for all n. Let KD be a constant such that |z| KD for all z ∈ D, and δD = infz∈D Im z > 0. For any y ∈ R and n ∈ N we have N 1 n 2 dµny (t) µy [−N, N] (1 + N ) 2 1 + t −N 1 2 n 2 n n dµ (t) = (1 + N ) Im F (i) − b (1 + N ) y y y 2 R 1+t 2 n (1 + N )Im Fy (i), since byn 0 for all y ∈ R and n ∈ N. Let Fn (i) = An + iBn , and note that Bn 1 Im Fyn (i) = Im = , y − Fn (i) (y − An )2 + Bn2 where 0 < δD Bn KD and |An | KD . It is straightforward to show that Im Fyn (i) k/(1 + y 2 ), where k is a constant, and the lemma follows with c = 2 (1 + N 2 )k. THEOREM 3.8. Let Fn (z) be a family of Herglotz functions with corresponding measures µn , such that Fn (z) → F (z) uniformly, as n → ∞, on compact subsets of the upper half-plane. Suppose that the measure dσ is absolutely continuous with respect to Lebesgue measure. Then, for any Borel set S and any bounded Borel set B, we have n µy (B) dσ (y) = µy (B) dσ (y), (49) lim n→∞
S
S
where the measures µny appear in (48), and the measures µy in (4). Proof. Suppose B ⊆ [−N, N], and consider the family A of all Lebesgue measurable subsets A of [−N, N] which satisfy Equation (49). A is nonempty; to see this, first note that by Lemma 3.7 there exists a constant K1 > 0 such that µny [−N − 1, N + 1] K1
1 , 1 + y2
(50)
which is integrable with respect to dσ . Thus, for any subinterval (a, b] of [−N, N], an application of the Lebesgue dominated convergence theorem gives n µy (a, b] dσ (y) = lim µny (a, b] dσ (y). lim n→∞
S
S n→∞
Since Fn (z) → F (z) uniformly, as n → ∞, on compact subsets of the upper halfplane, it can easily be verified that also Fyn (z) → Fy (z) uniformly, as n → ∞, on compact subsets of the upper half-plane. Therefore, it follows from Theorem 3.4 that µny ((a, b]) → µy ((a, b]) as n → ∞, provided that the endpoints a and b are
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not discrete points of any of the measures µny or µy . Hence, such an interval (a, b] satisfies (49). Next we show that A is closed under countable unions of disjoint sets. There also exists a constant K2 > 0 such that µy [−N − 1, N + 1] K2
1 , 1 + y2
(51)
for all y ∈ R and n ∈ N (this follows in a similar way as the result in Lemma 3.7). Let the sets S0 and S1 be defined by S0 = {y ∈ S : hσ (y) C},
S1 = {y ∈ S : hσ (y) > C},
(52)
where hσ is the density function of dσ , S is an arbitrary Borel set, and C > 0 is a constant. Let ε > 0 be given. Then, (50), (51) and Lemma 3.6 enable us to choose the constant C such that we have both ε (53) µny [−N − 1, N + 1] dσ (y) < , 6 S1 and
ε µy [−N − 1, N + 1] dσ (y) < . 6 S1
Let {Ak } be a disjoint sequence of sets in A. Then ε n µy Ak dσ (y) µny [−N − 1, N + 1] dσ (y) < , 6 S1 S1 k and
µy S1
(54)
(55)
Ak dσ (y)
k
ε µy [−N − 1, N + 1] dσ (y) < . 6 S1
(56)
Also, for each k we have n µy (Ak ) dσ (y) C µny (Ak ) dy S0 S 0 C µny (Ak ) dy = C|Ak |, R
by Equation (7) in the statement of Theorem 2.1, where | · | stands for Lebesgue measure. Since C|Ak | C [−N, N] = 2NC < +∞, k
GENERALIZED VALUE DISTRIBUTION FOR HERGLOTZ FUNCTIONS
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it follows by a discrete version of the Lebesgue dominated convergence theorem that n µy (Ak ) dσ (y) = µy (Ak ) dσ (y). lim n→∞
k
S0
S0
k
Therefore, we have n µy Ak dσ (y) = lim µny (Ak ) dσ (y) lim n→∞
S0
n→∞
k
=
S0
µy (Ak ) dσ (y)
S0
k
=
k
µy S0
Ak dσ (y).
k
Hence, there is an N1 ∈ N such that, if n > N1 then 2ε n < . µ A µ A dσ (y) − dσ (y) k y k y 3 S0 S 0 k k
(57)
Combining (55), (56), and (57), we can see that for n > N1 we have µn Ak dσ (y) − µy Ak dσ (y) < ε, y S
k
S
k
which shows that A is closed under countable unions of disjoint sets. Now take any measurable subset B of [−N, N]. There is an open measurable set G ⊂ [−N − 1, N + 1] such that G ⊃ B and |G| < |B| + ε/6C. Suppose G = B ∪ B0 where B and B0 are disjoint, so that B0 = G\B and thus B0 is measurable. Then, |G| = |B| + |B0 |, which implies that |B0 | ε/6C. We have µny (G) = µny (B) + µny (B0 ), and from Lemma 3.7 µny (B0 ) is bounded so that we can subtract µny (B0 ) from both sides of the equation to obtain µny (B) = µny (G) − µny (B0 ). Similarly, we have µy (B) = µy (G) − µy (B0 ). Therefore, µn (B) dσ (y) − µy (B) dσ (y) y S S n µ (G) − µy (G) + µn (B0 ) + µy (B0 ) dσ (y). (58) y y S
As before, we split integration over the set S to integration over the disjoint sets S0 and S1 . From (53) we have ε µny (B0 ) dσ (y) < . 6 S1
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Also,
µny (B0 ) dσ (y)
C
S0
Hence, µny (B0 ) dσ (y) < S
µny (B0 ) dy
S0
C
ε µny (B0 ) dy = C|B0 | . 6 R
ε . 3
Similarly, we have ε µy (B0 ) dσ (y) < . 3 S
(59)
(60)
Since it is an open set, G is the union of a countable collection of disjoint subintervals of [−N − 1, N + 1], and we may assume that the endpoints of these intervals are not discrete points of any of the measures µny or µ. Thus, by the first part of this proof (where now we consider disjoint measurable subsets of [−N − 1, N + 1], rather than [−N, N], satisfying (49)), it follows that there is an N2 ∈ N such that if n > N2 then µn (G) dσ (y) − µy (G) dσ (y) < ε . (61) y 3 S S Combining (59), (60), and (61), we see from (58) that, if n > N2 , then µn (B) dσ (y) − µy (B) dσ (y) < ε, y S
S
which completes the proof of Theorem 3.8.
2
References 1. 2. 3. 4. 5. 6. 7. 8.
Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space I, Pitman, London, 1981. Apostol, T. M.: Mathematical Analysis, Addison Wesley, Massachusetts, 1960. Bartle, G. R.: The elements of Integration, Wiley, New York, 1966. Breimesser, S. V.: Asymptotic value distribution for solutions of the Schrödinger equation. PhD Thesis, University of Hull, 2001. Breimesser, S. V. and Pearson, D. B.: Asymptotic value distribution for solutions of the Schrödinger equation, Math. Phys., Anal. and Geometry 3 (2000), 385–403. Christodoulides, Y. T.: Spectral theory of Herglotz functions and their compositions, and the Schrödinger equation. PhD Thesis, University of Hull, 2001. Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Eastham, M. S. P. and Kalf, H.: Schrödinger-type Operators with Continuous Spectra, Pitman, London, 1982.
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9.
10. 11. 12. 13. 14. 15.
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Gesztesy, F. and Makarov, K. A.: SL2 (R), exponential Herglotz representations, and spectral averaging, St. Petersburg Math. J. (to appear). (Available on Los Alamos archive under math.SP/0203142.) Herglotz, G.: Über Potenzreihen mit Positivem, reelem Teil in Einheitskreis, Sächs Acad. Wiss. Leipzig 63 (1911), 501–511. Markushevich, A. I.: Theory of Functions of a Complex Variable I, Prentice-Hall, New Jersey, 1965. Munroe, M. E.: Measure and Integration, Addison-Wesley, Massachusetts, 1971. Pearson, D. B.: Value distribution and spectral theory, Proc. Lond. Math. Soc. 68(3) (1994), 127–144. Pearson, D. B.: Quantum Scattering and Spectral Theory, Academic Press, London, 1988. Wheeden, L. R. and Zygmund, A.: Measure and Integral, Marcel Dekker, New York, 1977.
Mathematical Physics, Analysis and Geometry 7: 333–345, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Spectral Theory of Herglotz Functions and Their Compositions Y. T. CHRISTODOULIDES and D. B. PEARSON Department of Mathematics, University of Hull, Cottingham Rd., Hull HU6 7RX, UK. e-mail:
[email protected];
[email protected] (Received: 17 April 2003; in final form: 8 June 2004) Abstract. Recent developments in the theory of value distribution for boundary values of Herglotz functions [5], with applications to the spectral analysis of Herglotz measures and differential operators [2, 3] lead in a natural way to the investigation of measures which relate (through the Herglotz representation theorem) to the composition of a pair of Herglotz functions F, G. The present paper provides results on the boundary values of composed Herglotz functions and on the terms of their Herglotz representation which are dominant at large |z|. Mathematics Subject Classifications (2000): 30D40, 30D05. Key words: boundary values, Herglotz functions, spectral theory.
1. Introduction Let G(z) be a Herglotz function, that is analytic in the upper half-plane, with positive imaginary part. Then G admits the integral representation [1, 7, 10] t 1 − 2 dρ(t), (1) G(z) = aG + bG z + t +1 R t −z where aG , bG are real constants and the function ρ(t) is nondecreasing, rightcontinuous, and unique up to an additive constant, for given G. In particular, aG and bG are given by aG = Re G(i),
1 Im G(is), s→+∞ s
bG = lim
and ρ(t) defines a Borel measure, with ρ{(a, b]} = ρ(b)−ρ(a) for intervals (a, b]. Thus dρ is the Herglotz measure corresponding to G, and satisfies the integral condition 1 dρ(t) < ∞. (2) 2 R 1+t For given G, we can define a one-parameter family {Gy } of Herglotz functions (y ∈ R) by Gy (z) = 1/(y −G(z)). Let dρy be the Herglotz measure corresponding
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to Gy . Then if dσ is a Herglotz measure (that is, a Borel measure satisfying the integrability condition R (1/(1 + t 2 )) dσ (t) < ∞), we can define the averaged measure νS over a Borel set S by νS (A) = S ρy (A) dσ (y), for any Borel set such that the integral is finite. If G has real boundary value almost everywhere, with boundary value function G+ (λ) = limε→0+ G(λ + iε)(λ ∈ R), the measure νS , with ν = νR , defines the value distribution of G+ (λ), through the formula νS (A) = ν A ∩ G−1 + (S) . See [2, 3, 11] for the theory of value distribution of boundary values of Herglotz functions, with applications to the spectral theory of Sturm–Liouville equations and the Weyl Titchmarsh m-function [8]. In the special case that the measure dσ is Lebesgue measure | · |, it follows that dν is also Lebesgue measure, and we have the resulting formula in that case ρy (A) dy = A ∩ G−1 + (S) . S
More generally [5] one may verify that if dσ is purely absolutely continuous, then so are νS and ν. Moreover, if we define a Herglotz function corresponding to measure dσ by t 1 − 2 dσ (t), φS (z) = aφ + bφ z + t +1 S t −z then νS is given for Borel sets B by νS (B) = ρ(φS ◦G) (B) − bφ ρ(B). In particular we can take bφ = 0, in which case νS is precisely the Herglotz measure corresponding to the composition φS ◦ G of the two Herglotz functions φS and G. The analysis of value distribution theory for Herglotz functions, in term of Herglotz measures more general than Lebesgue measure, is treated in [5], with applications to spectral theory. As a key element of that analysis, the present paper deals with the Herglotz representation for composed functions. In particular, we shall consider the two related questions: given two Herglotz functions F , G, (i) How does the coefficient bF ◦G of the term linear in z of the Herglotz representation for F ◦G relate to the corresponding coefficients bF , bG of the functions F , G? We provide a complete description of this coefficient. (ii) What can be deduced regarding the boundary values (F ◦ G)+ (λ) of the composed Herglotz function F ◦ G? This is an important question in view of the role of the composed function in determining generalized value distribution. One might expect that the boundary value (F ◦ G)+ (λ) is given quite simply by F+ (G+ (λ)). But the validity of such a result depends on a careful analysis of the mode of convergence of Herglotz functions to their boundary values,
SPECTRAL THEORY OF HERGLOTZ FUNCTIONS
335
raising such questions as whether or not G(z) approaches G+ (λ) as a nontangential limit. We resolve some of these questions and show that the boundary value of F ◦ G is indeed F+ (G+ (λ)) on a set characterized by the notion of approximate monotonicity, and for which the complement is of Lebesgue measure zero. This paper may be regarded as a sequel to [6]. See [6] for introductory material on spectral averaging and its connection with the composition of Herglotz functions, as well as references on these and related areas. The main results of the paper are summarized below in Theorems 2.2 and 3.2. The paper is organized as follows. In Section 2, we consider the coefficient bF ◦G of the z term in the Herglotz representation for the composed function. Theorem 2.2 presents the results of this analysis, which include various special cases. In particular, if the measure dµ corresponding to the Herglotz function F has no discrete component, then we have bF ◦G = bF bG . In the case that dµ does have a discrete point, the canonical result bF ◦G = bF bG may be violated, depending on the measure dρ for the second Herglotz function G, and we provide also a complete description of this case. The identities and estimates required to obtain these results would appear to be of interest in their own right. Section 3 presents an analysis of boundary behaviour for the composed Herglotz function F ◦ G. The boundary values of F ◦ G are important both to determine the spectral density for the corresponding measure in the case of a.c. measure and, in the case of real boundary values, to determine the value distribution of the composed function on the real line. The main problem of this analysis is to obtain some control of the angle at which G(λ + iε) approaches the real line in the limit as ε → 0+ , if G+ (λ) is real. We make use of general results on the boundary behaviour of analytic functions, as well as local real analysis for a class of functions which are measurable but which may exhibit no local continuity and need not be locally bounded. The main result of this analysis is presented in Theorem 3.2, and depends on an investigation of the consequences of approximate monotonicity, constancy, and oscillation. 2. Coefficient b of Composed Herglotz Functions Let G be an arbitrary Herglotz function with representation given by Equation (1), and let F be a Herglotz function t 1 − 2 dµ(t). (3) F (z) = aF + bF z + t +1 R t −z Throughout this section, we define the function c(s) (s ∈ R+ ) in terms of the spectral measure dρ for G by 1 dρ(t). (4) c(s) = 2 + t2 s R
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The following lemma provides a simple criterion for a Herglotz measure to be finite. LEMMA 2.1. With c(s) given by Equation (4), the measure dρ is finite if and only if a := lims→∞ c(s)s 2 < ∞, and infinite if and only if a = ∞. Proof. Suppose first that c(s)s 2 → a < ∞ as s → ∞. Then if dρ is an infinite measure there exists N0 ∈ N such that ρ((−N0 , N0 ]) > a. Hence N0 s2 s2 dρ(t) lim dρ(t) = ρ (−N , N ] > a, a = lim 0 0 s→∞ R s 2 + t 2 s→∞ −N s 2 + t 2 0 which is a contradiction. So dρ is a finite measure in this case. Conversely, if dρ is finite then s2 dρ(t) = ρ(R) < ∞, lim c(s)s 2 = lim s→∞ s→∞ R s 2 + t 2 where we have used the Lebesgue dominated convergence theorem. The condition for dρ to be infinite follows easily. 2 Note that, in terms of the Herglotz function G, the condition for finite measure dρ can be written lim s Im G(is) − bG s 2 := a < ∞. s→∞
The following theorem provides a complete characterization of the term bz in the representation for F ◦ G. THEOREM 2.2. Let bF ◦G z be the term linear in z in the Herglotz representation for the composed function F ◦ G. Then if either bG = 0 or if the measure dρ is infinite, we have bF ◦G = bF bG . If bG = 0 and dρ is a finite measure, then bF ◦G = a1 µ{t0 }, where a = ρ(R) and t dρ(t). t0 = aG − 2 R 1+t (Observe that if t0 is not a discrete point of the measure dµ then bF ◦G = bF bG . Also note that if dρ is the zero measure then bG = 0.) Proof. The constant bF ◦G is given by 1 bF ◦G = lim Im F G(is) . s→∞ s From the representations (1), (3) of the Herglotz functions G, F , we have 1 Im F G(is) = bF bG + bF c(s) + s 1 dµ(t) + + bG 2 2 2 R [t − A(s)] + s [bG + c(s)] c(s) dµ(t), (5) + 2 + s 2 [b + c(s)]2 [t − A(s)] G R
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SPECTRAL THEORY OF HERGLOTZ FUNCTIONS
where the function A(s) is defined by t (1 − s 2 ) dρ(t) A(s) = aG + 2 2 2 R (s + t )(1 + t )
(6)
and c(s) is given in Equation (4). Note that c(s) → 0 as s → ∞, since we can apply the Lebesgue dominated convergence theorem with s2
1 1 2 +t 1 + t2
for s 1,
and this bound is integrable with respect to the measure dρ. Hence Equation (5) implies that 1 dµ(t) + bF ◦G = bF bG + lim bG 2 2 2 s→∞ R [t − A(s)] + s [bG + c(s)] c(s) dµ(t), (7) + lim 2 s→∞ R [t − A(s)] + s 2 [bG + c(s)]2 provided both these limits separately exist. In the special case that dρ is the zero measure (so that bG = 0) we have A(s) = aG , c(s) = 0, and 1 dµ(t). bF ◦G = bF bG + lim bG 2 2 2 s→∞ R (t − aG ) + s bG It is straightforward to verify that const. 1 2 1 + t2 (t − aG )2 + s 2 bG
(t ∈ R, s 1)
which is integrable with respect to dµ, and hence the Lebesgue dominated convergence theorem gives bF ◦G = bF bG in this case. We now consider the case in which bG = 0 and dρ is finite. Then, from (7) we obtain 1 dµ(t). (8) bF ◦G = lim 1 s→∞ R [t − A(s)]2 + s 2 c(s) c(s) Note that |t|s 2 t (1 − s 2 ) (s 2 + t 2 )(1 + t 2 ) (s 2 + t 2 )(1 + t 2 ) 1,
s 1,
so that with finite measure dρ an application of the Lebesgue dominated convergence theorem gives t dρ(t) := t0 . (9) lim A(s) = aG − s→∞ 1 + t2 R
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By Lemma 2.1 we also have lims→∞ c(s)s 2 = ρ(R) = a = 0, from which we may deduce that t2 + 1 const., 1 [t − A(s)]2 + c(s)s 2 c(s) for all t ∈ R and for s sufficiently large. Since 1/(1 + t 2 ) is integrable with respect to dµ, we can apply the Lebesgue dominated convergence theorem to the limit on the right-hand side of Equation (8). Noting that c(s) → 0 and A(s) → t0 , we deduce that if t0 is not a discrete point of dµ, then bF ◦G = 0. If t0 is a discrete point of dµ, then 1 . bF ◦G = µ {t0 } lim 1 s→∞ [t − A(s)]2 + c(s)s 2 c(s) 0 From (6) and (9), applying the Schwarz inequality we have the estimate
t0 − A(s)
2
2 t dρ(t) 2 2 R s +t t2 c(s) dρ(t), 2 2 R s +t =
implying that 2 1
t0 − A(s) = 0. s→∞ c(s) lim
Hence 1 1 = µ {t0 } , 2 s→∞ c(s)s a
bF ◦G = µ {t0 } lim
which completes the case bG = 0 with dρ finite. Now consider the case bG = 0 with dρ an infinite measure. Here we shall make use of an inequality for A(s), which may be verified by the Schwarz inequality. For s 1, we have 1/2 1/2 1 s2t 2 A(s) − aG s dρ(t) dρ(t) 2 2 2 2 2 2 R s +t R (s + t )(1 + t ) const. s c(s). (10) We now verify the bound, for all s sufficiently large, t2 + 1 const., 1 [t − A(s)]2 + c(s)s 2 c(s)
(t ∈ R).
(11)
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To do this, note first of all that (t 2 + 1)/((t − aG )2 + 1) is bounded, so that to verify (11) it is sufficient to show that (t − aG )2 + 1 const., 1 [t − A(s)]2 + c(s)s 2 c(s)
(t ∈ R).
Here it is equivalent to replace t ∈ R by t + aG , so that it remains to verify that t2 + 1 const. 1 [t − (A(s) − aG )]2 + c(s)s 2 c(s)
(12)
Given a positive constant K1 , we consider the discriminant D of the quadratic expression in t given by 2 1
t − A(s) − aG + c(s)s 2 − K1 t 2 . c(s) Since D(s) = 4K1 s 2 c(s) +
4K1 (A(s) − aG )2 − 4s 2 , c(s)
where c(s) → 0 as s → ∞, we can use (10) to show that D(s) < 0 for all s sufficiently large and for any (fixed) value of K1 sufficiently small. Hence the quadratic is strictly positive for all s sufficiently large, for this value of K1 . Since also c(s)s 2 → ∞ in this case, it follows that (12) holds and (11) is verified. We can now apply the Lebesgue dominated convergence theorem to Equation (8) to deduce that bF ◦G = 0 in the case bG = 0 with dρ infinite. In the remaining case, with bG > 0, similar arguments may be used to confirm the bounds that allow the Lebesgue dominated convergence theorem to be applied to the two integrals on the right-hand side of Equation (7). In this case, both integrals are easily found to converge to zero, and the result bF ◦G = bF bG in this case completes the proof of Theorem 2.2. 2
3. Boundary Values of Composed Herglotz Functions Given a Herglotz function F , the boundary value F+ (λ) is defined for (Lebesgue) almost all λ ∈ R by F+ (λ) = limε→0+ F (λ + iε), and is the limit of F (z) as z approaches the real axis in a direction at right angles to R. Such limits may not be appropriate while considering the boundary behaviour of a composed Herglotz function F ◦ G, since if λ ∈ R is a point such that G+ (λ) exists and is real then, in the limit ε → 0+ , G(λ + iε) will not approach the point G+ (λ) at right angles to R in general, and it is not clear how the limit of F (G(λ + iε)) should be evaluated. To circumvent this problem we need boundary limits in more general regions.
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By a wedge-shaped region with vertex λ ∈ R, we shall mean a set of the form {z ∈ C+ , α < arg(z − λ) < β}, with α < β < π . By the wedgy (nontangential) limit of F at the point λ ∈ R we shall mean the limit as z approaches λ along a simple curve ending at λ, and lying entirely in a wedge-shaped region with vertex at λ. From the theory of analytic functions [4] it is known that if the limit of F at λ exists along a simple curve, then the limit also exists along any other simple curve ending at λ and contained in a wedge-shaped area with vertex λ. Moreover these various limits agree, for given λ, and exist for almost all λ ∈ R. We will denote by Fw (λ) the wedgy limit of F at λ ∈ R; thus, Fw (λ) = F+ (λ) whenever either limit exists. Boundary behaviour of Herglotz functions can be highly irregular. For example, F+ (λ) may exhibit such discontinuities that F+ assumes every real value in every subinterval of R [12]. To describe such possible boundary behaviour, we shall need a number of ideas and results drawn from local real analysis. The following definitions will be found useful. DEFINITION 3.1. Let f be a Lebesgue measurable real valued function, finite almost everywhere on R, and let If = {x ∈ R; f (x) is finite}. Then f is said to be approximately right monotonic increasing at x ∈ If provided that lim+ t ∈ [x, x + h] ∩ If : f (t) > f (x) / h = 1, h→0
where | · | stands for Lebesgue measure. Approximately left monotonic increasing is defined similarly, and a function which is both right and left approximately monotonic increasing at x is said to be approximately monotonic increasing at x. Approximately monotonic decreasing is defined in a similar way. If for some x, with f (x) = y, x is a point of density for the set f −1 ({y}), then f is said to be approximately constant at x. In that case, the set f −1 ({y}) must have strictly positive measure. The function f is said to be approximately oscillatory (to the right) at a point x ∈ If , if there exist sequences {hn }, {hn } of positive numbers, approaching zero, such that t ∈ [x, x + hn ] ∩ If ; f (t) < f (x) / hn and t ∈ [x, x + h ] ∩ If ; f (t) > f (x) / h n n both approach 1 in the limit n → ∞. A similar definition applies to the left of the point x, and a function which is approximately oscillatory to both the right and left will be described as approximately oscillatory. Measurable functions can be categorized in terms of the above definitions, as follows.
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SPECTRAL THEORY OF HERGLOTZ FUNCTIONS
THEOREM 3.1. Let f be real-valued, measurable and finite almost everywhere. Then, at almost all x ∈ R, f is either (i) (ii) (iii) (iv)
approximately monotonic increasing, approximately monotonic decreasing, approximately constant, or approximately oscillatory. 2
Proof. See [9]. We can use Theorem 3.1 to prove the following result for Herglotz functions.
THEOREM 3.2. Let F , G be arbitrary Herglotz functions, and denote by IG , I˜G the sets IG = {λ ∈ R : G+ (λ) exist and G+ (λ) ∈ R} and I˜G = {λ ∈ R : G+ (λ) exists and Im G+ (λ) > 0}, where G+ (λ) = limε→0+ G(λ + iε). Then at almost all λ ∈ R we have lim (F ◦ G)(λ + iε) = (F ◦ G)+ (λ) =
ε→0+
F+ G+ (λ) , λ ∈ IG , F G+ (λ) , λ ∈ I˜G .
(13)
Proof. The conclusion of Theorem 3.2 is easily verified in the case λ ∈ I˜G , and follows from the continuity of F at G+ (λ). It remains to consider the case λ ∈ IG . Let A be the set A = {λ ∈ IG ; Fw (G+ (λ)) exists}. Thus F+ (G+ (λ)) exists for λ ∈ A. If B ⊂ R is the set of points at which Fw fails to exist, then IG \A = G−1 + (B). −1 But |B| = 0, implying that |IG \A| = |G+ (B)| = 0, since the inverse image of any set of measure zero will have measure zero. We first verify (13) for λ ∈ IG with the simplifying assumption that G has real boundary value almost everywhere; this will be so if and only if the measure dρ is purely singular. Then |Ac | = 0 and we can define a function g, real almost everywhere, by g(λ) = G+ (λ). From Theorem 3.1, at almost all λ ∈ A, g is approximately monotonic increasing or decreasing, approximately constant, or approximately oscillatory. The case of approximate constancy may be ruled out. If λ0 ∈ A is a point of approximate constancy of g then |λ ∈ IG : G+ (λ) = g(λ0 )| > 0, which contradicts the fact that the inverse image of any singleton set must have zero Lebesgue measure. Now suppose that λ is a point of A at which g is approximately monotonic increasing or decreasing. By considering the function log(G(z) − g(λ)), one may verify the result (see [9])
Arg G(λ + iε) − g(λ) =
R
εξ(t) dt , (t − λ)2 + ε 2
(14)
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Y. T. CHRISTODOULIDES AND D. B. PEARSON
where ξ(t) is almost everywhere the characteristic function of the set {t ∈ IG : g(t) < g(λ)}. On the right-hand side of (14), the limit as ε → 0+ may be identified (see [11]) with λ+h ξ(t) dt, lim+ π h→0 λ−h 2h in the sense that if either limit exists then both limits exist and are equal. Hence lim+ Arg G(λ + iε) − g(λ) ε→0 π lim+ t ∈ [λ − h, λ + h] ∩ IG : g(t) < g(λ) / h, = 2 h→0
(15)
so that the limit on the left-hand side is π/2. Hence, in the limit as ε → 0+ , G(λ + iε) approaches g(λ) along a path contained in a wedge-shaped region with vertex at g(λ). Since λ ∈ A, we know that Fw (G+ (λ)) exists, and it follows that (F ◦ G)+ (λ) = F+ G+ (λ) . If, on the other hand, π Arg G(λ + ih1 ) − g(λ) + ε 2 and π Arg G(λ + ih2 ) − g(λ) − ε, 2 then by continuity in h of Arg(G(λ+ih )−g(λ)) we can find h between h1 and h2 such that (19) holds, with again 0 < h < h. Since h can be taken arbitrarily small, we can construct a decreasing sequence {hn } of positive numbers, converging to zero, and such that G(λ + ihn ) lies in the wedge shaped region with vertex g(λ) and angle 2ε, defined by the inequalities π π − ε < Arg G(λ + iεn ) − g(λ) < + ε. 2 2 Since λ ∈ A and Fw (G+ (λ)) exists, it follows again that (F ◦ G)+ (λ) = F+ G+ (λ) . Finally, consider the general case in which G takes boundary values having strictly positive imaginary part on a set of positive Lebesgue measure. As noted earlier, ˜ real almost for λ ∈ I˜G we have (F ◦ G)+ (λ) = F (G+ (λ)). Define a function g, everywhere, by G+ (λ), λ ∈ IG , g(λ) ˜ = 0, λ ∈ I˜G .
SPECTRAL THEORY OF HERGLOTZ FUNCTIONS
343
Again by Theorem 3.1, at almost all λ, g˜ is approximately monotonic increasing or decreasing, approximately constant, or approximately oscillatory. At λ ∈ IG , g˜ can be approximately constant only if G+ (λ) = 0. However, G+ (λ) = 0 only on a set of λ values having Lebesgue measure zero, so that for almost all λ ∈ IG we can rule out the possibility that g˜ is approximately constant. Denote by Id the set of λ ∈ A such that λ is a point of density of A. Then IG \Id has Lebesgue measure zero, and for λ ∈ Id we have εξ(t) dt , Arg G(λ + iε) − g(λ) ˜ = 2 2 R (t − λ) + ε where now ξ(t) =
1 Im log G+ (t) − g(λ) ˜ , π
for almost all t. Next suppose λ ∈ A is a point at which g is approximately oscillatory. Given any ε with 0 < ε < π/2, we first fix δ, β, with 0 < δ < β, such that β π 1 dt > − ε. 2 2 δ 1+t Next, note that the definition of approximately oscillatory implies that, for any h > 0, we can find h1 satisfying 0 < h1 < h, such that t ∈ [λ, λ + βh1 ] ∩ IG : g(t) < g(λ) > (β − δ)h1 . (16) Proceeding as before from Equation (14), we have λ+βh1 h1 ξ(t) dt Arg G(λ + ih1 ) − g(λ) (t − λ)2 + h21 λ λ+βh1 β π h1 dt 1 = dt > − ε, 2 2 2 2 δ 1+t λ+δh1 (t − λ) + h1
(17)
since on the interval λ < t < λ + βh1 we have ξ(t) = 1 for a set of points having measure at least (β − δ)h1 and h1 /((t − λ)2 + h21 ) is a decreasing function of t, so that the minimum value is attained when integration is carried out on the interval λ + δh1 < t < λ + βh1 . Similarly, we can find h2 satisfying 0 < h2 < h, such that t ∈ [λ, λ + βh2 ] ∩ IG : g(t) > g(λ) > (β − δ)h2 . In that case, with χ0 the characteristic function of {t ∈ IG : ξ(t) = 0}, we have h2 ξ(t) dt Arg G(λ + ih2 ) − g(λ) = 2 2 R (t − λ) + h2 h2 dt π− 2 2 ξ(t )=0 (t − λ) + h2
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Y. T. CHRISTODOULIDES AND D. B. PEARSON
π−
λ+βh2
λ
h2 χ0 (t) dt (t − λ)2 + h22
λ+βh2
h2 dt (t − λ)2 + h22 λ+δh 2 π π − ε = + ε. < π− 2 2 π−
(18)
Having found h1 , h2 to satisfy (17) and (18) respectively, we now define h by π h = h1 if Arg G(λ + ih1 ) − g(λ) < + ε, 2 and h = h2
π if Arg G(λ + ih2 ) − g(λ) > − ε. 2
Then in either case we have π π − ε < Arg G(λ + ih ) − g(λ) < + ε. 2 2
(19)
Note that ξ(t) satisfies ˜ < g(λ), ˜ 1, t ∈ IG , g(t) ξ(t) = ˜ > g(λ), ˜ 0, t ∈ IG , g(t) and 0 < ξ(t) < 1, t ∈ I˜G . If λ is a point at which g˜ is approximately monotonic increasing or decreasing, then Equation (15) holds with g replaced by g, ˜ and as before we may deduce that (F ◦ G)+ (λ) = F+ (G+ (λ)). On the other hand, if g˜ is approximately oscillatory at λ, we can follow the same argument as before to construct a decreasing sequence {hn } of positive numbers, converging to zero, such that π π − ε < Arg G(λ + ihn ) − g(λ) ˜ < + ε, 2 2 where ε in the interval 0 < ε < π/2 is arbitrary. It follows again in this case that (F ◦ G)+ (λ) = F+ G+ (λ) , and this completes the proof of Theorem 3.2.
2
References 1.
Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space I, Pitman, London, 1981.
SPECTRAL THEORY OF HERGLOTZ FUNCTIONS
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
345
Breimesser, S. V.: Asymptotic value distribution for solutions of the Schrödinger equation, PhD Thesis, University of Hull, 2001. Breimesser, S. V. and Pearson, D. B.: Asymptotic value distribution for solutions of the Schrödinger equation, Math. Phys., Anal. and Geometry 3 (2000), 385–403. Caratheodory, C.: Theory of Functions of a Complex Variable II, Chelsea, New York, 1954. Christodoulides, Y. T.: Spectral theory of Herglotz functions and their compositions, and the Schrödinger equation, PhD Thesis, University of Hull, 2001. Christodoulides, Y. T. and Pearson, D. B.: Generalised value distribution for Herglotz functions, and spectral theory, Math. Phys., Anal. and Geometry (2003) (to appear). Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Eastham, M. S. P. and Kalf, H.: Schrödinger-type Operators with Continuous Spectra, Pitman, London, 1982. Elsken, T., Pearson, D. B. and Robinson, P. M.: Approximate monotonicity: theory and applications, J. London Math. Soc. 53(2) (1996), 489–502. Herglotz, G.: Über Potenzreihen mit positivem, reelem Teil in Einheitskreis, Sächs. Acad. Wiss. Leipzig 63 (1911), 501–511. Pearson, D. B.: Value distribution and spectral theory, Proc. London Math. Soc. 68(3) (1994), 127–144. Pearson, D. B.: Quantum Scattering and Spectral Theory, Academic Press, London, 1988.
Mathematical Physics, Analysis and Geometry 7: 347–349, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
347
Errata: “Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential” ALEXANDRE STOJANOVIC Institut Mathématique de Jussieu, Physique mathématique et Géométrie, Université Paris 7, case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France. e-mail:
[email protected] Abstract. The errata concern mainly the last computations for the universality of the local statistics of eigenvalues at the edge of the spectrum in parts (iii) of Theorems 2.3 and 2.4.
1. Corrections • Page 343, (1.20): there is a factor 1/2 in front of the right-hand side. • Page 346, (2.10): It can be simplified in gj k = aj k −
2n−1
2n−1
sj m tm ak .
=k−d m=2n−d
• Page 347, before Theorem 2.3: The definition of b(x, y) must be +∞ 1 1, Ai(t) dt , with cβ = bβ (x, y) = Ai(x) cβ − 0, 2 y
if β = 1, if β = 4.
The precedent definition corresponds to cβ = 1/2, what is wrong. • Page 347, Theorem 2.3, part (ii): −s (x − y) must be replaced by s (x − y) in the expression of the matrix kernel τ1 (x, y). • Page 348, Theorem 2.3, part (iii): w v , Zj + Zj − cj n2/3 cj n2/3 must be replaced by wj vj , Z + , Zj − j cj n2/3 cj n2/3 with w2 = w, v2 = v, w1 = −v, and v1 = −w, because sign(cj ) = (−1)j . θ1 (x, y) is now only defined by the first expression, in which we replace b(x, y) by b1 (x, y), the limit is now taken independently of the parity of n and the result holds for j = 1 and j = 2. Math. Phys. Anal. Geom. 3(4) (2000), 339–373.
348
• •
• •
ALEXANDRE STOJANOVIC
Moreover, we have to multiply by (−1)j the matrix kernel θ1 (xp , xq ) in the expression of the limiting scaled correlation functions. Page 349, Theorem 2.4, part (ii): −s (2(x − y)) must be replaced by s (2(x − y)) in the expression of the matrix kernel τ4 (x, y). Page 349, Theorem 2.4, part (iii): We modify the expression of the interval as in the case β = 1. Replace b(x, y) by b4 (x, y) in the expression of θ4 (x, y) and multiply it by 1/2. Page 350, Theorem 2.4, part (iii): We have to multiply by (−1)j the matrix kernel θ4 (xp , xq ) in the expression of the limiting scaled correlation functions. Page 363: In fact, we have x y 1 K + , Z + Z = (−1)j a(x, y). lim n j j n→∞ cj n2/3 cj n2/3 cj n2/3
• Page 367, last line: We have (−1)σ0 +j at the numerator of the fraction. • Page 369, last line: replace (−1)n+p by (−1)2n+p . • Page 371, bottom: In fact c has the same value as c if n is odd. 2. Explanations We give explanations for the mistakes about the end of the computations for the universality of the local statistics of eigenvalues at the edge of the spectrum. The mistakes in the proof of the computations of the terms depending on Hn and Gn have a common origin: if · ∞ := · L∞ ((Zj −δ,Zj +δ)2 ) , then we have said that ψn+p ⊗ (ε ψn+q ) − ψn+q ⊗ (ε ψn+p )∞ = O(n−1/2), (ε ψn+p ) ⊗ (ε ψn+q ) − (ε ψn+q ) ⊗ (ε ψn+p )∞ = O(n−7/6), which is wrong. There are some contributions at the level of constants of integration, which make wrong the arguments based on the antisymmetry. Thus, the estimates, pages 363–364, for εµ Gn , ελ εµ Gn and for the part called βn in the expression of Hn are false. Since now, there are contributions of εµ Gn , ελ εµ Gn , εµ βn and ελ εµ βn and the method of computations is completely different at this level. Now, we need equivalents of the coefficients gj k and not only estimates, what requires more computations. The details are given in the revision www.physik.uni-bielefeld.de/bibos/preprints, 02-07-098, BiBoS, Bielefeld, May 2002 of the preprint [13]. The method consists in doing what we have done on pages 367–368, for the computations of the contribution of the term αn (λ). First, we have to give explicitly the values of the coefficients gj k in function of the coefficients aj k , aj k and cj k . Secondly, we have to give the equivalents of these coefficients gj k ; these coefficients are rational functions in terms of the coefficients aj k , aj k and cj k . The denominators are factors from the expression of det D; equivalents are given on pages 370–372. We just have to compute the
349
ERRATA
equivalents of the numerators to get the results. We give the results for the new versions of Lemmas 4.3 and 4.4: • For β = 1, n even, we have √ √ n −t √ ( 1 + u − 1 − u) + O(n2/3), gn−3,n−2 = √4 n −t √ 1 − u + O(n2/3). gn−1,n−2 = 2 • For β = 1, n odd, we have √ √ n√ −t(− 1 + u + 1 − u) + O(n2/3), gn−2,n−3 = 4 gn−1,n−2 = O(n2/3). • For β = 4, we have g2n,2n+1
g2n+1,2n+2
√ √ n√ = −2t 1 + 2u + 1 − 2u × 4 √ 1 + 2u − (−1)n (1 − 2u2 )1/4 + O(n2/3), × √ 1 + 2u + (−1)n (1 − 2u2 )1/4
√ √ n√ = −2t 1 + 2u + 1 − 2u − 4 √ √ − 1 + 2u − 1 − 2u × √ 1 + 2u − (−1)n (1 − 2u2 )1/4 + × √ 1 + 2u + (−1)n (1 − 2u2 )1/4 + O(n2/3).
Mathematical Physics, Analysis and Geometry 7: 351–352, 2004.
351
Contents of Volume 7 (2004)
Volume 7 No. 1
2004
CRAIG ROBERTS / Relating Thomas–Whitehead Projective Connections by a Gauge Transformation
1–8
IVAN G. AVRAMIDI / Heat Kernel Asymptotics of Zaremba Boundary Value Problem
9–46
A. KOKOTOV and D. KOROTKIN / Tau-functions on Hurwitz Spaces
47–96
Volume 7 No. 2
2004
DIMITRI J. FRANTZESKAKIS, IOANNIS G. STRATIS and ATHANASIOS N. YANNACOPOULOS / On Equilibria of the Twofluid Model in Magnetohydrodynamics
97–117
ROSTYSLAV O. HRYNIV and YAROSLAV V. MYKYTYUK / Transformation Operators for Sturm–Liouville Operators with Singular Potentials 119–149 ABEL KLEIN and ANDREW KOINES / A General Framework for Localization of Classical Waves: II. Random Media 151–185 P. COULTON and G. GALPERIN / Forces along Equidistant Particle Paths 187–192
Volume 7 No. 3
2004
INÉS PACHARONI and JUAN A. TIRAO / Three Term Recursion Relation for Spherical Functions Associated to the Complex Projective Plane 193–221 RICARDO URIBE-VARGAS / Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities 223–237 JÜRG FRÖHLICH and MARCO MERKLI / Thermal Ionization
239–287
352 Volume 7 No. 4
CONTENTS OF VOLUME 7
2004
ZHIJUN QIAO and SHENGTAI LI / A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions 289–308 Y. T. CHRISTODOULIDES and D. B. PEARSON / Generalized Value Distribution for Herglotz Functions and Spectral Theory 309–331 Y. T. CHRISTODOULIDES and D. B. PEARSON / Spectral Theory of Herglotz Functions and Their Compositions 333–345 ALEXANDRE STOJANOVIC / Errata: “Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential” 347–349
