Geodesic
The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential
Sbornik. Mathematics, Tome 208 (2017) no. 1, pp. 1-43 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. G. Baskakov; D. M. Polyakov. The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential. Sbornik. Mathematics, Tome 208 (2017) no. 1, pp. 1-43. http://geodesic.mathdoc.fr/item/SM_2017_208_1_a0/

[1] M. A. Naimark, Linear differential operators, v. I, II, Frederick Ungar Publishing Co., New York, 1967, 1968, xiii+144 pp., xv+352 pp. | MR | MR | MR | Zbl | Zbl

[2] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser Verlag, Basel, 1986, xii+367 pp. | DOI | MR | MR | Zbl | Zbl

[3] A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces”, Math. Notes, 80:6 (2006), 814–832 | DOI | DOI | MR | Zbl

[4] P. Djakov, B. S. Mityagin, “Instability zones of periodic 1-dimensional Schrödinger and Dirac operators”, Russian Math. Surveys, 61:4 (2006), 663–766 | DOI | DOI | MR | Zbl

[5] Ju. L. Daleckiĭ, M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Transl. Math. Monogr., 43, Amer. Math. Soc., Providence, RI, 1974, vi+386 pp. | MR | MR | Zbl | Zbl

[6] A. G. Baskakov, Zamena Krylova–Bogolyubova v teorii nelineinykh vozmuschenii lineinykh operatorov, Preprint No 80.19, In-t matem. AN USSR, Kiev, 1980, 44 pp. | MR

[7] A. G. Baskakov, “An abstract analog of the Krylov–Bogolyubov transformation in the perturbation theory of linear operators”, Funct. Anal. Appl., 33:2 (1999), 144–147 | DOI | DOI | MR | Zbl

[8] K. O. Friedrichs, Perturbation of spectra in Hilbert space, Lectures in Appl. Math., 3, Amer. Math. Soc., Providence, RI, 1965, xiii+178 pp. | MR | Zbl

[9] N. Dunford, J. T. Schwartz, Linear operators, v. III, Pure Appl. Math., 7, Spectral operators, Interscience Publishers [John Wiley Sons, Inc.], New York–London–Sydney, 1971 | MR | Zbl | Zbl

[10] A. G. Baskakov, “Methods of abstract harmonic analysis in the perturbation of linear operators”, Siberian Math. J., 24:1 (1983), 17–32 | DOI | MR | Zbl

[11] A. G. Baskakov, “A theorem on splitting an operator, and some related questions in the analytic theory of perturbations”, Math. USSR-Izv., 28:3 (1987), 421–444 | DOI | MR | Zbl

[12] A. G. Baskakov, T. K. Katsaran, “Spectral analysis of integrodifferential operators with nonlocal boundary conditions”, Differential Equations, 24:8 (1988), 934–941 | MR | Zbl

[13] A. G. Baskakov, “Spectral analysis of perturbed nonquasianalytic and spectral operators”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 1–31 | DOI | MR | Zbl

[14] A. G. Baskakov, “Rasscheplenie vozmuschennogo differentsialnogo operatora s neogranichennymi operatornymi koeffitsientami”, Fundament. i prikl. matem., 8:1 (2002), 1–16 | MR | Zbl

[15] A. G. Baskakov, A. V. Derbushev, A. O. Shcherbakov, “The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials”, Izv. Math., 75:3 (2011), 445–469 | DOI | DOI | MR | Zbl

[16] D. M. Polyakov, “Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients”, Siberian Math. J., 56:1 (2015), 138–154 | DOI | MR | Zbl

[17] D. M. Polyakov, “Method of similar operators in spectral analysis of a fourth-order nonself-adjoint operator”, Differ. Equ., 51:3 (2015), 421–425 | DOI | MR | Zbl

[18] D. M. Polyakov, “Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions”, St. Petersburg Math. J., 27:5 (2016), 789–811 | DOI | Zbl

[19] A. G. Baskakov, “Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations”, Sb. Math., 206:8 (2015), 1049–1086 | DOI | DOI | MR | Zbl

[20] F. Gesztesy, V. Tkachenko, “A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions”, J. Differ. Equations, 253:2 (2012), 400–437 | DOI | MR | Zbl

[21] A. A. Shkalikov, O. A. Veliev, “On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm–Liouville problems”, Math. Notes, 85:6 (2009), 647–660 | DOI | DOI | MR | Zbl

[22] A. A. Shkalikov, “On the basis problem of the eigenfunctions of an ordinary differential operator”, Russian Math. Surveys, 34:5 (1979), 249–250 | DOI | MR | Zbl

[23] N. B. Kerimov, Kh. R. Mamedov, “On the Riesz basis property of the root functions in certain regular boundary value problems”, Math. Notes, 64:4 (1998), 483–487 | DOI | DOI | MR | Zbl

[24] P. Djakov, B. Mityagin, “Spectral triangles of Schrödinger operators with complex potentials”, Selecta Math. (N.S.), 9:4 (2003), 495–528 | DOI | MR | Zbl

[25] P. Djakov, B. Mityagin, “Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators”, J. Funct. Anal., 263:8 (2012), 2300–2332 | DOI | MR | Zbl

[26] P. Djakov, B. S. Mityagin, “Riesz basis property of Hill operators with potentials in weighted spaces”, Tr. MMO, 75:2 (2014), 181–204, MTsNMO, M. | MR | Zbl

[27] A. S. Makin, “Convergence of expansions in the root functions of periodic boundary value problems”, Dokl. Math., 73:1 (2006), 71–76 | DOI | MR | Zbl

[28] Kh. R. Mamedov, “On the basis property in $L_p(0, 1)$ of the root functions of non-self adjoint Sturm–Lioville operators”, Eur. J. Pure Appl. Math., 3:5 (2010), 831–838 | MR | Zbl

[29] I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969, xv+378 pp. | MR | MR | Zbl | Zbl

[30] A. G. Baskakov, D. M. Polyakov, “Spectral properties of the Hill operator”, Math. Notes, 99:4 (2016), 598–602 | DOI | DOI | MR | Zbl

[31] A. G. Baskakov, A. A. Vorob'ev, M. Yu. Romanova, “Hyperbolic operator semigroups and Lyapunov's equation”, Math. Notes, 89:2 (2011), 194–205 | DOI | DOI | MR | Zbl

[32] A. Zygmund, Trigonometric series, v. I, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp. | MR | MR | Zbl | Zbl

[33] A. A. Kirillov, Elements of the theory of representations, Grundlehren Math. Wiss., 220, Springer-Verlag, Berlin–New York, 1976, xi+315 pp. | MR | MR | Zbl | Zbl

[34] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin–New York, 1981, iv+348 pp. | MR | MR | Zbl | Zbl