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Jacobi matrices with lacunary spectrum

Ilya Losev

Abstract

We find asymptotics of entries of Jacobi matrices with lacunary spectral data under some additional growth conditions. We also prove the inverse results. In addition, we study connections between Jacobi matrices, canonical systems and de Branges spaces for lacunary spectral data

Jacobi matrices with lacunary spectrum

Abstract

We find asymptotics of entries of Jacobi matrices with lacunary spectral data under some additional growth conditions. We also prove the inverse results. In addition, we study connections between Jacobi matrices, canonical systems and de Branges spaces for lacunary spectral data

Paper Structure

This paper contains 21 sections, 14 theorems, 120 equations.

Table of Contents

  1. Introduction and main results
  2. Jacobi matrices and de Branges spaces
  3. Main results.
  4. Canonical Systems
  5. Proof of Theorem \ref{['ThrJacobi']}
  6. Proof of Lemmas \ref{['mainlemma']} and \ref{['lemmaPoleBound']}
  7. Root localization
  8. Invariance of the lacunary parameter
  9. Lower bound
  10. Upper bound
  11. Proof of Theorem \ref{['ThrRevearsed']}
  12. Root localization
  13. Lower bound
  14. Upper bound
  15. Remark about lacunarity of $\left\{\frac{\mu_k}{t_k}\right\}$
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\nu_k$ and $r_k$ be defined by FTHBmeasure. Suppose that $r_{k+1}>\lambda r_k$, $\nu_{k+1}>\varkappa \nu_k$ and $\frac{\nu_{k+1}}{r_{k+1}^2}<\theta \frac{\nu_k}{r_k^2},$ for some $\lambda>10^6,\, \varkappa>10^6,\, \theta<10^{-6}$. Then

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Lemma 2
  • ...and 20 more