Weyl-Titchmarsh M-Function Asymptotics for Matrix-Valued Schrödinger Operators
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line.
math.SP↗
arXiv subjects
Fritz Gesztesy
Publications and source records attributed to Fritz Gesztesy.
Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\bbR$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.
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A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure
We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-κ^2) =-κ- \int_0^a A(α) e^{-2ακ} dα+O(e^{-(2a -ε)κ}) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure ρ: A(α) = -2\int_{-\infty}^\infty λ^{-\frac12} \sin (2α\sqrtλ)\, dρ(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b<\infty. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.
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The Xi Operator and its Relation to Krein's Spectral Shift Function
We explore connections between Krein's spectral shift function $ξ(λ,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift operator $Ξ(J+K^*(H_0-λ-i0)^{-1}K)$ associated with the operator-valued Herglotz function $J+K^*(H_0-z)^{-1}K$, $\Im(z)>0$ in $\calH$, where $V=KJK^*$ and $J=\sgn(V)$. Our principal results include a new representation for $ξ(λ,H_0,H)$ in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections $(E_{J+A(λ)+tB(λ)}((-\infty,0)),E_J((-\infty,0)))$, $t\in\bbR$, where $A(λ)=\Re(K^*(H_0-λ-i0)^{-1}K)$, $B(λ)=\Im(K^*(H_0-λ-i0)^{-1}K)$ a.e. Moreover, introducing the new concept of a trindex for a pair of operators $(A,P)$ in $\calH$, where $A$ is bounded and $P$ is an orthogonal projection, we prove that $ξ(λ,H_0,H)$ coincides with the trindex associated with the pair $(Ξ(J+K^*(H_0-λ-i0)^{-1}K),Ξ(J))$. In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of $Ξ$-operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman-Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.
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Integrable Systems in the Infinite Genus Limit
We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.
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Borg-Type Theorems for Matrix-Valued Schrödinger Operators
A Borg-type uniqueness theorem for matrix-valued Schrödinger operators is proved. More precisely, assuming a reflectionless potential matrix and spectrum a half-line $[0,\infty)$, we derive triviality of the potential matrix. Our approach is based on trace formulas and matrix-valued Herglotz representation theorems. As a by-product of our techniques, we obtain an extension of Borg's classical result from the class of periodic scalar potentials to the class of reflectionless matrix-valued potentials.
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Some Applications of the Spectral Shift Operator
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators.
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The Spectral Shift Operator
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.
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Darboux-type transformations and hyperelliptic curves
We systematically study Darboux-type transformations for the KdV and AKNS hierarchies and provide a complete account of their effects on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies.
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The Cole-Hopf and Miura transformations revisited
An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail.
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The classical Boussinesq hierarchy revisited
We develop a systematic approach to the classical Boussinesq (cBsq) hierarchy based on an elementary polynomial recursion formalism. Moreover, the gauge equivalence between the cBsq and AKNS hierarchies is studied in detail and used to provide an effortless derivation of algebro-geometric solutions and their theta function representations of the cBsq hierarchy.
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Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies - An Analytic Approach
We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.
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Algebro-Geometric Solutions of the Boussinesq Hierarchy
We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function representation of all algebro-geometric quasi-periodic solutions and related quantities of the Bsq hierarchy.
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Some Applications of Operator-Valued Herglotz Functions
We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrödinger operators on a half-line and provide two illustrations of Livsic's result [44] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (1,1).
math.FA↗
On Matrix-Valued Herglotz Functions
We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schrödinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanlinna-Riesz-Herglotz representation. In particular, we study a class of linear fractional transformations M_A(z) of a given n \times n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuos part of the measure associated to M_A(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self-adjoint finite-rank perturbations of self-adjoint operators, self-adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.
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A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions
We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sGmKdV hierarchy, including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebro-geometric solutions. Although we mainly focus on sG-type equations, our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.
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An Addendum to Krein's Formula
We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A_1 and A_2 of a densely defined closed symmetric linear operator A with (possibly infinite) equal deficiency indices. In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M_1(z) and M_2(z) corresponding to A_1 and A_2.
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An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy
We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.
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