On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

Gilbert, Daphne

doi:10.1017/s0308210500018680

Cited by 80 publications

(87 citation statements)

References 19 publications

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“…The contents of this section clearly has its origin in the beautiful theory of Gilbert and Pearson [120] which is known as the principle of subordinacy. This theory has since then been simplified and extended by various people (see e.g., [119], [48], [145]). I follow ideas of [68], [139], [163], and [207].…”

Section: Notes On Literaturementioning

confidence: 99%

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Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

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%

“…The results stated in this appendix are collected from [16], [17], [68], [73], [119], [198], [199], [200], [202], [203], [206]. In this chapter an important role will be played by Borel measures ρ on R. Hence we fix some notation first.…”

Section: Herglotz Functionsmentioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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“…Then Σ ±,n,sc and Σ sc are minimal supports (cf., e.g., [4], [21], [22], [32], [33]) of dν ±,n,sc and dν tr sc , where dν tr = dν 1,1 + dν 2,2 abbreviates the trace measure of the 2×2 matrix-valued spectral measure dν p,q , 1 ≤ p, q ≤ 2 of H (derived from g(z, n 0 ), g(z, n 0 + 1), h(z, n 0 ), cf. (2.12)-(2.14)).…”

Section: Taking Into Account (326)-(329)mentioning

confidence: 99%

On isospectral sets of Jacobi operators

1996

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Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlet-type data.

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“…It is a natural extension of the Gilbert-Pearson theory of subordinacy [5,6], and it allows us to analyze the dimensional Hausdorff properties for a number of examples with the singular-continuous spectrum. Below we describe the main ideas of our study and some of the main results.…”

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confidence: 99%

“…The spectral measures of a line operator can be constructed from those of corresponding two half-line operators (a left and a right), and while the relations are not completely trivial, they do allow an extension of the subordinacy theory to this case. Gilbert [6] has shown that the absolutely continuous part of the spectral measures of a line operator is supported on the set of energies for which at least one of the half-line problems has no subordinate solution. The singular part is supported on the set of energies for which (5) has a solution which is subordinate both to the right and to the left.…”

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confidence: 99%

Dimensional Hausdorff Properties of Singular Continuous Spectra

Jitomirskaya

1996

Phys. Rev. Lett.

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We present an extension of the Gilbert-Pearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of one-dimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze these properties for several examples having the singular-continuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.PACS numbers: 02.30. Sa, 71.23.An, 72.15.Rn Singular continuous spectra have been extensively studied recently. Our interest here is in the classification and decomposition of such spectra with respect to dimensional Hausdorff measures. The measure-theoretical aspect of this point of view goes back to Rogers and Taylor [1], and it has been studied recently within spectral theory by Last [2] and by del Rio et al. [3] who have shown that the singular-continuous spectrum which is produced by localized rank-one perturbations of Anderson-model Hamiltonians in the localized regime [4] must be purely zero dimensional-in the sense that the associated spectral measures are supported on a set of zero Hausdorff dimension.The main purpose of this paper is to report a general method for spectral analysis of one-dimensional Schrödinger operators from this point of view. It is a natural extension of the Gilbert-Pearson theory of subordinacy [5,6], and it allows us to analyze the dimensional Hausdorff properties for a number of examples with the singular-continuous spectrum. Below we describe the main ideas of our study and some of the main results. Mathematically complete proofs of these results will be given elsewhere [7].Most of our discussion will be restricted to onedimensional discrete (tight-binding) Schrödinger operators of the form ͑Hc͒ ͑n͒ c͑n 1 1͒ 1 c͑n 2 1͒ 1 V͑n͒c͑n͒ .We shall consider two kinds of such operators: "line" operators acting on ᐉ 2 ‫͒ޚ͑‬ ͑2`, n ,`͒, and "half-line" operators acting on ᐉ 2 ‫ޚ͑‬ 1 ͒ ͑n . 0͒, which are considered with a phase boundary condition of the formwhere 2p͞2 , u , p͞2. Before formulating our main result, which would require some definitions, we would like to describe some of its applications. We stress at this point that the dimensional Hausdorff properties which we study are those which are associated with the spectral measures of the corresponding operators. The spectra themselves, as sets, are closed sets, and their dimensions may be larger than those which are associated with the spectral measures. A description of the precise spectral-theoretic scheme which underlies our study is given below.We start with a somewhat artificial example of half-line operators with sparse barrier potentials. More specifically, we consider potentials which vanish for all n's outside a sparse (fastly growing) sequence of points ͕L n ͖ǹ 1 where jV ͑L n ͒j !`as n !`. Simon and Spencer [8] have shown that the Schrödinger operators corresponding to such potentials have no absolutely continuous spectrum, and Gordon [9] has shown that if the jV ͑L n ͒j's grow suffi...

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On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

Cited by 80 publications

References 19 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

On isospectral sets of Jacobi operators

Dimensional Hausdorff Properties of Singular Continuous Spectra

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