Eigenvalue distribution of canonical systems: trace class and sparse spectrum

TL;DR

This paper analyzes the eigenvalue distribution of two-dimensional canonical systems with discrete spectra, providing formulas for eigenvalue densities, criteria for trace class resolvents, and algorithms for growth analysis of monodromy matrices.

Contribution

It introduces a formula for the eigenvalue counting function's Stieltjes transform, criteria for Schatten class resolvents, and an algorithm for monodromy matrix growth in canonical systems.

Findings

Derived a universal formula for eigenvalue density.

Established explicit criteria for trace class resolvent membership.

Developed an algorithm for monodromy matrix growth analysis.

Abstract

In this paper we consider two-dimensional canonical systems with discrete spectrum and study their eigenvalue densities. We develop a formula that determines the Stieltjes transform of the eigenvalue counting function up to universal multiplicative constants. An explicit criterion is given for the resolvents of the model operator to belong to a Schatten-von Neumann class with index 0<p<2, thus giving an answer to the long-standing question which canonical systems have trace class resolvents. For canonical systems with two limit circle endpoints we develop an algorithm for determining the growth of the monodromy matrix up to a small error. Moreover, we present examples to illustrate our results, show their sharpness and prove an inverse result giving explicit formulae.