Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions

Polyakov, D.

doi:10.1090/spmj/1417

Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions
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by D. M. Polyakov
Translated by: S. Kislyakov

St. Petersburg Math. J. 27 (2016), 789-811

DOI: https://doi.org/10.1090/spmj/1417

Published electronically: July 26, 2016

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Abstract:

By the method of similar operators, the spectral properties of a fourth order differential operator are studied under periodic or semiperiodic boundary conditions. The spectrum asymptotics is obtained, together with some estimates for the spectral resolution for the operator in question. Also, the operator semigroup is constructed whose generator is equal to minus the operator under study.

References

S. G. Mikhlin, Variational methods in mathematical physics, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by T. Boddington; editorial introduction by L. I. G. Chambers. MR 0172493
Lothar Collatz, Eigenwertaufgaben mit technischen Anwendungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 19, Akademische Verlagsgesellschaft, Leipzig, 1949 (German). MR 0031337
V. A. Yakubovich and V. M. Starzhinskii, Linear differential equations with periodic coefficients. 1, 2, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1975. Translated from Russian by D. Louvish. MR 0364740
Evgeny Korotyaev and Igor Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Henri Poincaré 8 (2007), no. 6, 1151–1176. MR 2355344, DOI 10.1007/s00023-007-0331-y
Andrei Badanin and Evgeny Korotyaev, Spectral asymptotics for periodic fourth-order operators, Int. Math. Res. Not. 45 (2005), 2775–2814. MR 2182471, DOI 10.1155/IMRN.2005.2775
A. V. Badanin and E. L. Korotyaev, Spectral estimates for a fourth-order periodic operator, Algebra i Analiz 22 (2010), no. 5, 1–48 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 5, 703–736. MR 2828825, DOI 10.1090/S1061-0022-2011-01164-1
—, Eigenvalue asymptotics for fourth order operators on the unit interval, arXiv:1309. 3449.
Andrey Badanin and Evgeny L. Korotyaev, Even order periodic operators on the real line, Int. Math. Res. Not. IMRN 5 (2012), 1143–1194. MR 2899961, DOI 10.1093/imrn/rnr057
O. A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (2010), 195–207. MR 2653191, DOI 10.1007/s11856-010-0025-x
Volodymyr A. Mikhailets and Volodymyr M. Molyboga, Singular eigenvalue problems on the circle, Methods Funct. Anal. Topology 10 (2004), no. 3, 44–53. MR 2092532
Volodymyr A. Mikhailets and Volodymyr M. Molyboga, Uniform estimates for the semi-periodic eigenvalues of the singular differential operators, Methods Funct. Anal. Topology 10 (2004), no. 4, 30–57. MR 2109216
Volodymyr Molyboga, Estimates for periodic eigenvalues of the differential operator $(-1)^md^{2m}/dx^{2m}+V$ with $V$-distribution, Methods Funct. Anal. Topology 9 (2003), no. 2, 163–178. MR 1999778
M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967. MR 0216050
P. Dzhakov and B. S. Mityagin, Instability zones of one-dimensional periodic Schrödinger and Dirac operators, Uspekhi Mat. Nauk 61 (2006), no. 4(370), 77–182 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 4, 663–766. MR 2279044, DOI 10.1070/RM2006v061n04ABEH004343
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Douglas C. Schmidt and Gernot Metze, Modular replacement of combinational switching networks, IEEE Trans. Comput. C–24 (1975), no. 1, 29–48. MR 421888, DOI 10.1109/t-c.1975.224081
M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized method of eigenoscillations in diffraction theory, WILEY-VCH Verlag Berlin GmbH, Berlin, 1999. Translated from the Russian manuscript by Vladimir Nazaikinskii. MR 1713196
A. G. Baskakov, Methods of abstract harmonic analysis in the theory of perturbations of linear operators, Sibirsk. Mat. Zh. 24 (1983), no. 1, 21–39, 191 (Russian). MR 688589
A. G. Baskakov, The averaging method in the theory of perturbations of linear differential operators, Differentsial′nye Uravneniya 21 (1985), no. 4, 555–562, 732 (Russian). MR 791103
A. G. Baskakov, A theorem on splitting of an operator and some related problems in the analytic theory of perturbations, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 3, 435–457, 638 (Russian). MR 854591
A. G. Baskakov, Spectral analysis of perturbed non-quasi-analytic and spectral operators, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 4, 3–32 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 45 (1995), no. 1, 1–31. MR 1307054, DOI 10.1070/IM1995v045n01ABEH001621
A. G. Baskakov, A. V. Derbushev, and A. O. Shcherbakov, The method of similar operators in the spectral analysis of the nonselfadjoint Dirac operator with nonsmooth potential, Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 3, 3–28 (Russian, with Russian summary); English transl., Izv. Math. 75 (2011), no. 3, 445–469. MR 2847780, DOI 10.1070/IM2011v075n03ABEH002540
D. M. Polyakov, Spectral analysis of fourth-order nonselfadjoint operator with nonsmooth coefficients, Sibirsk. Mat. Zh. 56 (2015), no. 1, 165–184 (Russian, with Russian summary); English transl., Sib. Math. J. 56 (2015), no. 1, 138–154. MR 3407948, DOI 10.1134/s0037446615010140
—, On spectral properties of fourth-order differential operator, Vestnik VGU. Ser. fiz.-mat. 1 (2012), 179–181. (Russian)
D. M. Polyakov, On spectral properties of fourth order differential operator with periodic and semiperiodic boundary conditions, Russian Math. (Iz. VUZ) 59 (2015), no. 5, 64–68. Translation of Izv. Vyssh. Uchebn. Zaved. Mat. 2015, no. 5, 75–79. MR 3374274, DOI 10.3103/S1066369X15050096
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989