Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity

Damanik, David; Simon, Barry

doi:10.1155/imrn/2006/19396

Cited by 37 publications

(83 citation statements)

References 29 publications

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“…By the theorems found in the appendix to [6] (which codifies well-known results), when (3.63) holds, one has ([6, eqn. (A.27)])…”

Section: Clock Theorems For Bounded Variation Perturbations Of Free Amentioning

confidence: 67%

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Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Simon

2007

Comm Pure Appl Math

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Abstract. We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

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“…By the theorems found in the appendix to [6] (which codifies well-known results), when (3.63) holds, one has ([6, eqn. (A.27)])…”

Section: Clock Theorems For Bounded Variation Perturbations Of Free Amentioning

confidence: 67%

“…When (3.63) holds, the Jost function, u, can de defined on [−2, 2]; see, for example, the appendix to [6]. If u(2) = 0, we say there is a resonance at 2, and if u(2) = 0, we say that 2 is nonresonant.…”

Section: Clock Theorems For Bounded Variation Perturbations Of Free Amentioning

confidence: 99%

Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Simon

2007

Comm Pure Appl Math

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“…Statement ii) is one of several criteria (see Theorem 1 [16] or Theorem 1.10.1 [21]) which together are equivalent to A − z − J 0 being of Hilbert-Schmidt class. Statements iii) and iv) follow from Theorem A.6 [11] (see also [20]). The absolutely continuous part f (x)dx of the implied spectral measure is supported on [z − 2, z + 2], the positivity of which is demonstrated by a finite lower bound on a weighted Lebesgue integral of log |f (x)| (the so-called Quasi-Szegö condition); indeed, in case iii) an even stricter integral condition (the Szegö condition) on f (x) also holds (see [16,21]).…”

Section: Matrix Work and Spectramentioning

confidence: 99%

Continuum eigenmodes in some linear stellar models

Winfield

2016

Z. Angew. Math. Phys.

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Abstract. We apply parallel approaches in the study of continuous spectra to adiabatic stellar models. We seek continuum eigenmodes for the LAWE formulated as both finite difference and linear differential equations. In particular, we apply methods of Jacobi matrices and methods of subordinancy theory in these respective formulations. We find certain pressure-density conditions which admit positive-measured sets of continuous oscillation spectra under plausible conditions on density and pressure. We arrive at results of unbounded oscillations and computational or, perhaps, dynamic instability. IntroductionThe problem of radial, adiabatic (isentropic) stellar oscillations is well studied in cases where discrete eigenmodes are calculated in the frameworks of linearized differential equations and in parallel applications of operator theory (see, for instance, [2,3,9,22]).1 However, continuous eigenvalues, although previously suggested in the general literature, are typically absent, either excluded from particular physical models or explicitly avoided to assure dynamic or numerical stability [4,14,17,23] (see also the Appendix of [6]). In this article we investigate some cases where continuous spectra are present which have ramifications on the associated motion -and perhaps on the perturbation method itself. Our study involves linearized perturbations of the differential systemHere, m = M(r) is total mass measured from the origin to distance r from the center of a spherically symmetric star, and P and ρ are pressure and density, respectively, depending on r. Moreover, the differential mass dm is interpreted as that of a spherical mass shell (or mass element) with mean radius r about 1991 Mathematics Subject Classification. 47N50, 85-08, 85-A30. 1 In the general literature (cf. [9]), the problem is often referred to as the LAWE -the linear adiabatic wave equation.

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“…The relation between the Szegő asymptotics and the Jost function is given by (see, e.g., [2] and Theorem 13.9.2 in [17])…”

Section: Scattering Datamentioning

confidence: 99%

“…The goal of the present note is to obtain a complete characterization of the scattering data in Ryckman's class, and so demonstrate that the scattering theory goes beyond Guseinov's class (2). We analyze the scattering data and prove the uniqueness theorem in Section 2.…”

Section: Introductionmentioning

confidence: 96%

On the scattering problem in Ryckman's class of Jacobi matrices

Golinskiĭ¹,

Kheifets²,

Yuditskii³

2011

J. Spectr. Theory

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Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity

Abstract: Abstract. We present necessary and sufficient conditions on the Jost function for the corresponding Jacobi parameters a n − 1 and b n to have a given degree of exponential decay.

Cited by 37 publications

References 29 publications

Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Continuum eigenmodes in some linear stellar models

On the scattering problem in Ryckman's class of Jacobi matrices

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