For monotone Hamburger Hamiltonians with regularly varying parameters, the monodromy matrix grows like r times an inverse-function integral, yielding the exact Nevanlinna order 1/(2(β-1)) in the simply critical Jacobi case with 3/2<β<2.
Eigenvalue distribution of canonical systems: trace class and sparse spectrum
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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.
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Growth estimates for Nevanlinna matrices of order larger than one half
For monotone Hamburger Hamiltonians with regularly varying parameters, the monodromy matrix grows like r times an inverse-function integral, yielding the exact Nevanlinna order 1/(2(β-1)) in the simply critical Jacobi case with 3/2<β<2.