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Threshold Virtual States of a Jacobi operator

Saidakhmat N. Lakaev, Konstantin A. Makarov

Abstract

We prove that the set of parameters for which a virtual level appears at the edge of the continuous spectrum of a Jacobi matrix with a finite-rank diagonal perturbation constitutes an algebraic variety of codimension one. This variety partitions the parameter space into connected components, with their number determined by the size of the perturbation support. We also reveal a hierarchical structure underlying these critical varieties as the rank of the perturbation increases.

Threshold Virtual States of a Jacobi operator

Abstract

We prove that the set of parameters for which a virtual level appears at the edge of the continuous spectrum of a Jacobi matrix with a finite-rank diagonal perturbation constitutes an algebraic variety of codimension one. This variety partitions the parameter space into connected components, with their number determined by the size of the perturbation support. We also reveal a hierarchical structure underlying these critical varieties as the rank of the perturbation increases.

Paper Structure

This paper contains 10 sections, 15 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. The Jost function and perturbation determinant
  3. Critical Operators and the threshold virtual states
  4. Nodal hypersurfaces of the polynomials $Q_n$ and the Jost function $C_n$
  5. A simple perturbation result
  6. The geometry of the critical variety and the discrete spectrum
  7. The Jost function and perturbabtion determinant. Proofs
  8. Proof of Proposition \ref{['propjost']}
  9. Proof of Proposition \ref{['propdet']}
  10. A lemma from geometric perturbations theory

Key Result

Proposition 2.1

The Jost function $j_0^{(n)}(z)$ associated with the Jacoby operator $J_n$jac can explicitly be represented as where $\Theta=\Theta(z)$ is given by guk. Here $q_n$ are polynomials in $z$ and $v_1, \dots, v_n$ satisfying the recurrence relations with the initial data Moreover, the Jost function $j_{0}^{(0)}(z)$ associated with the unperturbed Jacobi matrix $J_0=J$ is given by $\blacktrianglelef

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 24 more