A quantitative formula for the imaginary part of a Weyl coefficient
Jakob Reiffenstein
University of Vienna, Austria
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Abstract
We investigate two-dimensional canonical systems on an interval, with positive semi-definite Hamiltonian . Let be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of along the imaginary axis up to multiplicative constants, which are independent of . We also provide versions of this result for Sturm–Liouville operators and Krein strings.
Using classical Abelian–Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function with respect to the spectral measure , and boundedness of the distribution function of relative to a given comparison function.
We study in depth Hamiltonians for which approaches or (at least on a subsequence). It turns out that this behavior of imposes a substantial restriction on the growth of . Our results in this context are interesting also from a function theoretic point of view.
Cite this article
Jakob Reiffenstein, A quantitative formula for the imaginary part of a Weyl coefficient. J. Spectr. Theory 13 (2023), no. 2, pp. 555–591
DOI 10.4171/JST/457