×
Polyakov, D. M.

Spectral analysis of a fourth-order differential operator with periodic and antiperiodic boundary conditions. (English. Russian original) Zbl 1360.34171

St. Petersbg. Math. J. 27, No. 5, 789-811 (2016); translation from Algebra Anal. 27, No. 5, 117-152 (2015).
The author investigates the spectral properties of a fourth-order differential operator under periodic or semiperiodic boundary conditions, using the method of similar operators. Some patterns of harmonic analysis are used to develop this method of similar operators. Second approximations of the eigenvalues are obtained, along with spectral projection estimates for the operator considered. An operator semigroup, whose generator is equal to the negative of the operator under study, is also constructed. Practical applications of the resulting techniques are not mentioned, but the results are important and are elegantly presented in a number of theorems and lemmas.
Reviewer: Shlomo Carmi (Baltimore)

Cited in 15 Documents

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L05 General spectral theory of ordinary differential operators

Keywords:

similar operators; spectral properties; spectrum asymptotics; operator semigroup; eigenvalues
Cite Review PDF
Full Text: DOI

References:

[1] <Mihlin> S. G. Mikhlin, Variational methods in mathematical physics, 2nd ed., Nauka, 1970; English transl., Fist edition, Pergamon Press Book, New York, 1964. · Zbl 0119.19002
[2] <Kollatz> L. Collatz, Eigenwertaufgaten mit technischen Anweddungen, Math. Anwendungen Phys. Tech., Bd. 19, Akad. Verlag, Leipzig, 1949. · Zbl 0035.17504
[3] <Yakubovich> V. A. Yakubovich and V. M. Starzhinski\i , Linear differential equations with periodic coefficients and their applications, Nauka, Moscow, 1972; English transl., Halsted Press, New-York, 1975.
[4] <Korotyaev_Lobanov> E. Korotyaev and I. Lobanov, Schr\"odinger operators on zigzag nanotubes, Ann. Henri Poincare 8 (2007), no. 6, 1151-1176. · Zbl 1375.81098
[5] <Badanin_Korotyaev1> A. Badanin and E. Korotyaev, Spectral asymptotics for periodic fourth-order operators, Int. Math. Res. Not. IMRN 2005, no. 45, 2775-2814. · Zbl 1098.34020
[6] <Badanin_Korotyaev2> \bysame , Spectral estimates for a fourth-order periodic operator, Algebra i Analiz 22 (2010), no. 5, 1-48; English transl., St. Petersburg Math. J. 22 (2010), no. 5, 703-736. · Zbl 1230.34071
[7] <Bad_Kor_arXiv> \bysame , Eigenvalue asymptotics for fourth order operators on the unit interval, arXiv:1309. 3449.
[8] <Badanin_Korotyaev3> \bysame , Even order periodic operators on the real line, Int. Math. Res. Not. IMRN 2012, no. 5, 1143-1194. · Zbl 1239.34101
[9] Veliev O. A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (2010), 195-207. · Zbl 1204.34117
[10] <Molyboga_Mikhailets1> V. Mikhailets and V. Molyboga, Singular eigenvalue problems on the circle, Methods Funct. Anal. Topology 10 (2004), no. 3, 44-53. · Zbl 1077.47049
[11] <Molyboga_Mikhailets2> \bysame , Uniform estimates for the semi-periodic eigenvalues of the singular differential operators, Methods Funct. Anal. Topology 10 (2004), no. 4, 30-57. · Zbl 1072.34097
[12] <Molyboga> V. 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. · Zbl 1042.34108
[13] <Naimark> M. A. Naimark, Linear differential operators, 2nd ed., revised and augmented, Nauka, Moscow, 1969; English transl., Pt. I, II, Frederic Ungar Publ. Co., New York, 1967-1968. , · Zbl 0219.34001
[14] <Djakov_Mityagin> P. Dzhakov and B. S. Mityagin, Instability zones of one-dimensional periodic Schr\"odinger and Dirac operators, Uspekhi Mat. Nauk 61 (2006), no. 4, 77-182; English transl., Russian Math. Surveys 61 (2006), no. 4, 663-766. · Zbl 1128.47041
[15] <Kato> T. Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren Math. Wiss., Bd. 132, Springer-Verlag, Berlin, 1976. · Zbl 0342.47009
[16] <Danford> N. Dunford and J. T. Schwartz, Linear operators. Pt. III. Spectral operators, Pure Appl. Math., vol. 7, Intersci. Publ., New York, 1971. · Zbl 0243.47001
[17] <Agranovich> M. S. Agranovich, Spectral properties of diffraction problems, in: Generalized method of eigenoscillations in diffraction theory, Nauka, Moscow, 1974, pp. 289-416; English transl., Wilew, Berlin, 1999.
[18] <SMG_1983> 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; English transl., Sib. Math. J. 24 (1983), no. 1, 17-32. · Zbl 0546.46040
[19] <DU_1985> \bysame , The averaging method in the theory of perturbations of linear differential operators, Differ. Uravn. 21 (1985), no. 4, 555-562; English transl., Differ. Equ. 21 (1985), no. 4, 357-362. · Zbl 0611.34034
[20] <1986> \bysame , A theorem on spliting 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; English transl., Math. USSR-Izv. 28 (1987), no. 3, 421-444. · Zbl 0636.47019
[21] <1994> \bysame , Spectral analysis of perturbed non-quasi-analytic and spectral operators, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 4, 3-32; English transl., Izv. Math. 45 (1995), no. 1, 1-31. · Zbl 0851.47024
[22] <2011> A. G. Baskakov, A. V. Derbushev, and A. O. Sherbakov, 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; English transl., Izv. Math. 75 (2011), no. 3, 445-469. · Zbl 1219.47024
[23] <Polyakov1> D. M. Polyakov, Spectral analysis fourth-order nonselfajoint operator with nonsmooth coefficients, Sibirsk. Mat. Zh. 56 (2015), no. 1, 165-184; English transl., Sib. Math. J. 56 (2015), no. 1, 138-154. · Zbl 06455461
[24] <Polyakov2> \bysame , On spectral properties of fourth-order differential operator, Vestnik VGU. Ser. fiz.-mat. 1 (2012), 179-181. (Russian) · Zbl 1321.47101
[25] PolyakovIzvVuz \bysame , On spectral properties of fourth order differential operator with periodic and semiperiodic boundary conditions, Izv. Vyssh. Uchebn. Zaved. Mat. 2015, no. 5, 75-79; English transl., Russian Math. (Iz.VUZ) 59 (2015), no. 5, 64-68. · Zbl 1343.34196
[26] <Gohberg> I. C. Gohberg and M. G. Krein, Introduction to the theory of linear non-selfadjoint operators in Hilbert space, Nauka, Moscow, 1965; English transl., Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI, 1969. · Zbl 0138.07803
[27] <Zigmund> A. Zygmund, Trigonometric series. Vol. I, Cambridge Univ. Press, London, 1968. · Zbl 0157.38204
[28] <Henry> D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, 1981. · Zbl 0456.35001
[29] <Engel-Nagel> K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, of Grad. Texts in Math., vol. 194, Springer-Verlag, New York, 2000. · Zbl 0952.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.