Spectral function of Krein’s and Kotani’s string in the class \(\Gamma\). (English) Zbl 1284.47031

The author focuses on the asymptotic behavior of the spectral function associated with the generalized second-order Sturm-Liouville operator \[ \mathcal{L}=\frac{\mathrm{d}}{\mathrm{d}m(x)}\frac{\mathrm{d}}{\mathrm{d}x},\quad -\infty<x<\ell, \] where \(m\) is a string, i.e., a nondecreasing and right-continuous function \(m:(-\infty,\infty)\to[0,+\infty]\) satisfying \(m(-\infty+0)=0\), the Lebesgue-Stieltjes measure \(\mathrm{d}m\) describes the mass-distribution of the string \(m\), and \(\ell=\sup\{x: m(x)<\infty\}\). Note that the classical Sturm-Liouville operator \(a(x)\frac{\mathrm{d}^2}{\mathrm{d}x}+b(x)\frac{\mathrm{d}}{\mathrm{d}x}\) can be written as \(\mathcal{L}\). Necessary and sufficient conditions guaranteeing that the spectral function varies regularly with index \(1\) are established as main results. These results are closely related to the so-called de Haan theory. Two illustrative examples are also provided.


47E05 General theory of ordinary differential operators
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
60J60 Diffusion processes
60G51 Processes with independent increments; Lévy processes
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