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


Author: Kh. K. Ishkin
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 28 (2016), nomer 1.
Journal: St. Petersburg Math. J. 28 (2017), 37-63
MSC (2010): Primary 34B24
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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Additional Information

Kh. K. Ishkin
Affiliation: Baskir State university, A. Z. Validi str. 32, 450074 Ufa, Russia
Email: Ishkin62@mail.ru

DOI: https://doi.org/10.1090/spmj/1438
Keywords: Nonselfadjoint differential operators, spectral instability, localization of the spectrum
Received by editor(s): February 6, 2014
Published electronically: November 30, 2016
Additional Notes: Supported by the Ministry of Education and Science of RF (grant no. 01201456408) and by RFBR (grant no. 15-01-01095)
Article copyright: © Copyright 2016 American Mathematical Society