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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2017 Volume 208, Number 1, Pages 3–47 (Mi sm8637)

This article is cited in 48 papers

The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential

A. G. Baskakova, D. M. Polyakovb

a Voronezh State University
b Research Institute for Mathematics, Voronezh State University

Abstract: The paper is concerned with the spectral properties of second-order differential operators defined by periodic and quasi-periodic boundary conditions. We obtain asymptotic formulae for the eigenvalues, derive estimates of projections, give estimates for the equiconvergence of spectral decompositions, find sufficient conditions for operators to be spectral, and write down an asymptotic expansion for the semigroup of operators generated by the negative of the differential operator under consideration. Our estimates involve coefficients of the Fourier potential. The main results of the paper are obtained by using the method of similar operators.
Bibliography: 34 titles.

Keywords: spectrum of an operator, method of similar operators, Hill's differential operator, asymptotic behaviour of eigenvalues, semigroup of operators.

UDC: 517.927

MSC: 47A55, 47B40, 34L10, 34L20, 34L40

Received: 21.11.2015

DOI: 10.4213/sm8637


 English version:
Sbornik: Mathematics, 2017, 208:1, 1–43

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© Steklov Math. Inst. of RAS, 2025