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Mat. Sb., 2017, Volume 208, Number 1, Pages 3–47 (Mi msb8637)  

This article is cited in 20 scientific papers (total in 20 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.

Funding Agency Grant Number
Russian Science Foundation 14-21-00066
This work was supported by the Russian Science Foundation (grant no. 14-21-00066).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8637

Full text: PDF file (767 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.927
MSC: 47A55, 47B40, 34L10, 34L20, 34L40
Received: 21.11.2015

Citation: A. G. Baskakov, D. M. Polyakov, “The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential”, Mat. Sb., 208:1 (2017), 3–47; Sb. Math., 208:1 (2017), 1–43

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. G. Baskakov, T. K. Katsaran, T. I. Smagina, “Linear differential second-order equations in Banach space and splitting of operators”, Russian Math. (Iz. VUZ), 61:10 (2017), 32–43  mathnet  crossref  isi
    2. G. V. Garkavenko, N. B. Uskova, “Asimptotika sobstvennykh znachenii raznostnogo operatora s rastuschim potentsialom i polugruppy operatorov”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:4 (2017), 6–17  mathnet  crossref
    3. A. G. Baskakov, L. Yu. Kabantsova, T. I. Smagina, “Invertibility conditions for second-order differential operators in the space of continuous bounded functions”, Differ. Equ., 54:3 (2018), 285–294  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. G. Baskakov, N. B. Uskova, “A generalized Fourier method for the system of first-order differential equations with an involution and a group of operators”, Differ. Equ., 54:2 (2018), 277–281  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    5. I. N. Braeutigam, D. M. Polyakov, “On the asymptotics of eigenvalues of a fourth-order differential operator with matrix coefficients”, Differ. Equ., 54:4 (2018), 450–467  crossref  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    6. D. M. Polyakov, “A one-dimensional Schrödinger operator with square-integrable potential”, Siberian Math. J., 59:3 (2018), 470–485  mathnet  crossref  crossref  isi  elib
    7. A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Linear differential operator with an involution as a generator of an operator group”, Oper. Matrices, 12:3 (2018), 723–756  crossref  mathscinet  zmath  isi  scopus
    8. A. G. Baskakov, N. B. Uskova, “Spectral analysis of differential operators with involution and operator groups”, Differ. Equ., 54:9 (2018), 1261–1265  crossref  crossref  zmath  isi  elib  scopus
    9. S. Kozlukov, “The method of similar operators in the study of the spectra of the adjacency matrices of graphs”, International Conference Applied Mathematics, Computational Science and Mechanics: Current Problems, Journal of Physics Conference Series, 973, IOP Publ. Ltd, 2018, 012036  crossref  isi  scopus
    10. A. G. Baskakov, N. B. Uskova, “Fourier method for first order differential equations with involution and groups of operators”, Ufa Math. J., 10:3 (2018), 11–34  mathnet  crossref  isi
    11. D. M. Polyakov, “On the Spectral Characteristics of Non-Self-Adjoint Fourth-Order Operators with Matrix Coefficients”, Math. Notes, 105:4 (2019), 630–635  mathnet  crossref  crossref  isi  elib
    12. N. B. Uskova, “Matrichnyi analiz spektralnykh proektorov vozmuschennykh samosopryazhennykh operatorov”, Sib. elektron. matem. izv., 16 (2019), 369–405  mathnet  crossref  mathscinet
    13. A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Similarity techniques in the spectral analysis of perturbed operator matrices”, J. Math. Anal. Appl., 477:2 (2019), 930–960  crossref  mathscinet  zmath  isi  scopus
    14. I. A. Krishtal, N. B. Uskova, “Spektralnye svoistva differentsialnykh operatorov pervogo poryadka s involyutsiei i gruppy operatorov”, Sib. elektron. matem. izv., 16 (2019), 1091–1132  mathnet  crossref
    15. I. N. Broitigam, D. M. Polyakov, “Ob asimptotike sobstvennykh znachenii differentsialnogo operatora tretego poryadka”, Algebra i analiz, 31:4 (2019), 16–47  mathnet
    16. I. N. Braeutigam, D. M. Polyakov, “Asymptotics of eigenvalues of infinite block matrices”, Ufa Math. J., 11:3 (2019), 11–28  mathnet  crossref  isi
    17. A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Metod podobnykh operatorov v issledovanii spektralnykh svoistv vozmuschennykh differentsialnykh operatorov pervogo poryadka”, Materialy Voronezhskoi zimneimatematicheskoi shkolySovremennye metody teorii funktsiii smezhnye problemy.28 yanvarya–2 fevralya 2019 g.Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 3–18  mathnet  crossref
    18. O. Cabri, “On the Riesz basis property of the root functions of a discontinuous boundary problem”, Math. Methods Appl. Sci., 42:18 (2019), 6733–6740  crossref  mathscinet  zmath  isi  scopus
    19. A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “On the spectral analysis of a differential operator with an involution and general boundary conditions”, Eurasian Math. J., 11:2 (2020), 30–39  mathnet  crossref
    20. S. A. Nazarov, “Porogovye rezonansy i virtualnye urovni v spektre tsilindricheskikh i periodicheskikh volnovodov”, Izv. RAN. Ser. matem., 84:6 (2020), 73–130  mathnet  crossref
  •  Sbornik: Mathematics (from 1967)
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