Eigenvalue distribution of canonical systems: trace class and sparse spectrum

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.