Trace Formulas and Inverse Spectral Theory for Jacobi Operators

Teschl, Gerald

doi:10.1007/s002200050419

Cited by 25 publications

(19 citation statements)

References 36 publications

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“…The section on inverse problems is taken from [222]. Additional results can be found in [88], [89], [92] and, in the case of finite Jacobi matrices, [104].…”

Section: Notes On Literaturementioning

confidence: 99%

“…In particular, recently Gesztesy and Simon showed [105] that all main results can be viewed as special cases of Krein's spectral shift theory [155]. Most parts of this section are taken from [222].…”

Section: Notes On Literaturementioning

confidence: 99%

“…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%

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Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

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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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“…The section on inverse problems is taken from [222]. Additional results can be found in [88], [89], [92] and, in the case of finite Jacobi matrices, [104].…”

Section: Notes On Literaturementioning

confidence: 99%

Section: Notes On Literaturementioning

confidence: 99%

See 1 more Smart Citation

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

Self Cite

323

349

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“…There is a second m-function that plays a role, 8) where H [1,n] is the n × n upper left corner of H. Section 2 relates these m-functions to solutions of the second-order difference equation and obtains relations between m ± (z, n) and m ± (z, n + 1) (of which (1.6) is a special case). Section 2 also contains some critical formulas expressing the diagonal Green's functions After a brief interlude in Section 5 obtaining the straightforward analog of Borg's twospectra theorem [11] (see also [12,54,55,57,59]) first considered in the Jacobi context by Hochstadt [46,47] (see also [10,27,40,41,45,48,69]), we turn in Section 6 to the question of determining H from a diagonal Green's function element (δ n , (H−z) −1 δ n ) when N < ∞. If n = 1 or N , the central inverse spectral theory result says G(z, n, n) uniquely determines H. For other n, there are always at least…”

Section: §1 Introductionmentioning

confidence: 99%

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

1997

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Abstract. We study inverse spectral analysis for finite and semi-infinite Jacobi matrices H. Our results include a new proof of the central result of the inverse theory (that the spectral measure determines H). We prove an extension of Hochstadt's theorem (who proved the result in the case n = N ) that n eigenvalues of an N × N Jacobi matrix, H, can replace the first n matrix elements in determining H uniquely. We completely solve the inverse problemThere is an enormous literature on inverse spectral problems for − [1,29,[56][57][58][59][60]64] and references therein), but considerably less for their discrete analog, the infinite and semi-infinite Jacobi matrices (see, e.g., [3, 4, 6-8, 13-22, 24, 26-28, 30, 32, 37, 38, 42-44, 50-52, 61-63, 66, 67, 69-71]) and even less for finite Jacobi matrices (where references include, e.g., [9,10,23,25,39,40,41,[45][46][47][48]). Our goal in this paper is to study the last two problems using one of the most powerful tools from the spectral theory of − d 2 dx 2 + V (x), the m-functions of Weyl. Explicitly, we will study finite N × N matrices of the form:

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“…However, note that special care has to be taken since the resolvents of H and H ∞ n0 live in different Hilbert spaces (cf. [6], [7] Appendix, or [13] …”

Section: Introductionmentioning

confidence: 99%

Spectral deformations of Jacobi operators.

Teschl

1997

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We extend recent work concerning isospectral deformations for one-dimensional Schrödinger operators to the case of Jacobi operators. We provide a complete spectral characterization of a new method that constructs isospectral deformations of a given Jacobi operator (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H ∞ n 0 on 2 (−∞, n 0 ) ⊕ 2 (n 0 , ∞) with a Dirichlet boundary condition at n 0 . The transformation moves a single eigenvalue of H ∞ n 0 and perhaps flips which side of n 0 the eigenvalue lives. On the remainder of the spectrum the transformation is realized by a unitary operator.

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Trace Formulas and Inverse Spectral Theory for Jacobi Operators

Cited by 25 publications

References 36 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

Spectral deformations of Jacobi operators.

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