On isospectral sets of Jacobi operators

Gesztesy, Fritz; Krishna, M.; Teschl, Gerald

doi:10.1007/bf02101290

Cited by 30 publications

(42 citation statements)

References 33 publications

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“…Reflectionless operators have attracted several mathematicians since they allow an explicit treatment. In the first two sections I follow closely ideas of [113] and [222]. However, I have added some additional material.…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…Reflectionless operators have attracted a considerable amount of interest recently in connection with inverse spectral theory [2], [22], [35], [36] and completely integrable lattices [7], [32]. In this section we show that the trace formulas of the previous section become particularly transparent in this case.…”

Section: Reflectionless Operatorsmentioning

confidence: 86%

“…Using this information to evaluate the exponential Herglotz representation of g(z, n) then implies ( [22], Lemma 1.1)…”

Section: Reflectionless Operatorsmentioning

confidence: 98%

“…λ ∈ σ ess (H). By [22], Lemma 3.3 the requirement (6.3) is equivalent to one of the following (i). For some n 0 ∈ Z, n 1 ∈ Z\{n 0 , n 0 + 1}, ξ(λ, n 0 ) = ξ(λ, n 0 + 1) = ξ(λ, n 1 ) = 1 2 for a.e λ ∈ σ ess (H).…”

Section: Reflectionless Operatorsmentioning

confidence: 99%

“…Those ingredients form the basis for the solution of the periodic initial value problem of the Toda equations (cf., e.g., [7], [10], [39]). Moreover, relation (1.3) was extended to certain reflectionless operators in [2] and successfully used in [2], [22] to solve inverse spectral problems for these operators.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Trace Formulas and Inverse Spectral Theory for Jacobi Operators

Teschl

1998

Communications in Mathematical Physics

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Abstract. Based on high energy expansions and Herglotz properties of Green and Weyl m-functions we develop a self-contained theory of trace formulas for Jacobi operators. In addition, we consider connections with inverse spectral theory, in particular uniqueness results. As an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit formulas for the coefficients in terms of minimal spectral data. Finally, trace formulas are applied to scattering theory with periodic backgrounds.

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On Matrix-Valued Herglotz Functions

2000

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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 MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(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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On isospectral sets of Jacobi operators

Cited by 30 publications

References 33 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Trace Formulas and Inverse Spectral Theory for Jacobi Operators

On Matrix-Valued Herglotz Functions

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