On Matrix-Valued Herglotz Functions

Gesztesy, Fritz; Tsekanovskiı̆, Eduard

doi:10.1002/1522-2616(200010)218:1<61::aid-mana61>3.0.co;2-d

Cited by 228 publications

(305 citation statements)

References 126 publications

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“…This is a standard result for matrix-valued Nevanlinna functions (see e.g. [32]). For the convenience of the reader we provide a proof which also gives an effective control on the boundedness of the support of this matrix-valued measure.…”

Section: Note That In Particular S Commutes With Taking the Adjoint mentioning

confidence: 69%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

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Section: Note That In Particular S Commutes With Taking the Adjoint mentioning

confidence: 69%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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“…( [4], [41], [54], [72], [73] It should be stressed that Theorems A.2 and A.3 record only the tip of an iceberg of results in this area. A substantial number of additional references relevant in this context can be found in [41].…”

Section: )mentioning

confidence: 99%

On spectral theory for Schrödinger operators with strongly singular potentials

Gesztesy¹,

Zinchenko

2006

Mathematische Nachrichten

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168

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Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

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“…This function is sometimes called by Dirichlet-to-Neumann map, since it connects Dirichlet and Neumann data for solutions of the eigenfunction equation. It is a matrix Nevanlinna function [25] and is closely related to the multichannel scattering [26].…”

Section: Introductionmentioning

confidence: 99%