Threshold Virtual States of a Jacobi operator

TL;DR

This paper characterizes the parameter sets leading to virtual levels at the spectrum edge of Jacobi matrices with finite-rank perturbations, revealing their algebraic and hierarchical structure.

Contribution

It establishes that these parameter sets form algebraic varieties of codimension one and uncovers a hierarchical structure as the perturbation rank increases.

Findings

Parameter sets form algebraic varieties of codimension one.

The number of connected components depends on the perturbation support size.

A hierarchical structure underlies the critical varieties with increasing perturbation rank.

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.