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Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function

Robert Fulsche, Medet Nursultanov

Abstract

We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness of the spectrum and a criterion for the membership of the resolvents to Schatten classes. We give two side estimates for the lower bound of the essential spectrum. Our main tool in achieving this is Otelbaev's function.

Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function

Abstract

We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness of the spectrum and a criterion for the membership of the resolvents to Schatten classes. We give two side estimates for the lower bound of the essential spectrum. Our main tool in achieving this is Otelbaev's function.

Paper Structure

This paper contains 15 sections, 22 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Classes of potentials
  5. Formulation of the problem
  6. Auxiliary lemmas
  7. Otelbaev's function
  8. Two sided estimates of the spectral distribution function
  9. Upper bound for the spectral distribution function
  10. Lower bound for the spectral distribution function
  11. Applications
  12. Discreteness of the spectrum
  13. Schatten class membership of resolvents
  14. Estimates for the eigenvalues
  15. Examples

Key Result

Proposition \oldthetheorem

Let $\beta\geq 0$ and $\mu \in \mathfrak {M}_{\beta}$, then the form $a_\mu$ is lower semibounded with bound $-2\max\{\beta, \beta^2\}$, closed, symmetric and densely defined.

Figures (2)

  • Figure 1: The cumulative distribution function of the Cantor measure ("Devil's staircase")
  • Figure 2: The Otelbaev function of the Cantor measure

Theorems & Definitions (65)

  • Example \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem: Brinck
  • proof : Proof of Proposition \ref{['form']}
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: Glazman Lemma
  • Lemma \oldthetheorem
  • ...and 55 more