Localization criterion for the spectrum of the Sturm–Liouville operator on a curve

Ishkin, Kh.

doi:10.1090/spmj/1438

Localization criterion for the spectrum of the Sturm–Liouville operator on a curve
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by Kh. K. Ishkin
Translated by: A. Plotkin

St. Petersburg Math. J. 28 (2017), 37-63

DOI: https://doi.org/10.1090/spmj/1438

Published electronically: November 30, 2016

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Abstract:

Two spectrum localization criteria are obtained for the Sturm–Liouville operator on a piecewise smooth curve. The first of them generalizes Marchenko’s well-known criterion. The second provides a necessary and sufficient condition on the potential under which the spectrum is asymptotically localized near a ray in the sense of a regularly distributed set relative to the order $\rho =1/2$, thus confirming Fedoryuk’s conjecture about the absence, in the general case, of an asymptotic formula for the spectrum of the problem $- v”= \mu \rho (x) v$, $0<x<1$, $v(0)=v(1)=0$.

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