Spectral triangles of non-selfadjoint Hill and Dirac operators

P. B. Djakov   ; B. S. Mityagin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 4, pp. 587-626

Résumé

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill–Schrödinger and Dirac operators. Let $L$ be a Hill operator or a one-dimensional Dirac operator on the interval $[0,\pi]$. If $L$ is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large $|n|$, close to $n^2$ in the Hill case or close to $n$ in the Dirac case ($n\in \mathbb{Z}$). There is one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-$ and $\lambda_n^+$ (counted with multiplicity). Asymptotic estimates are given for the spectral gaps $\gamma_n=\lambda_n^+-\lambda_n^-$ and the deviations $\delta_n=\mu_n-\lambda_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for $\gamma_n$ and $\delta_n$ are found for special potentials that are trigonometric polynomials. Bibliography: 45 titles.

Détail

Keywords: Hill operator, one-dimensional Dirac operator, periodic boundary conditions, antiperiodic boundary conditions, Dirichlet boundary conditions.

Bibliographie

[1] M. S. P. Eastham, The spectral theory of periodic differential equations, Texts Math. (Edinburgh), Scottish Acad. Press; Hafner Press, Edinburgh, 1973, viii+130 pp. | MR | Zbl

[2] W. Magnus, S. Winkler, Hill's equation, Interscience Tracts in Pure and Applied Mathematics, 20, Interscience Publishers John Wiley Sons, New York–London–Sydney, 1966, viii+127 pp. | MR | Zbl

[3] 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

[4] B. M. Brown, M. S. P. Eastham, K. M. Schmidt, Periodic differential operators, Oper. Theory Adv. Appl., 230, Birkhäuser/Springer Basel AG, Basel, 2013, viii+216 pp. | DOI | MR | Zbl

[5] 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

[6] P. Djakov, B. Mityagin, “Spectral gaps of Schrödinger operators with periodic singular potentials”, Dyn. Partial Differ. Equ., 6:2 (2009), 95–165 | DOI | MR | Zbl

[7] 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

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

[9] H. Hochstadt, “Estimates on the stability intervals for Hill's equation”, Proc. Amer. Math. Soc., 14:6 (1963), 930–932 | DOI | MR | Zbl

[10] H. Hochstadt, “On the determination of a Hill's equation from its spectrum”, Arch. Ration. Mech. Anal., 19 (1965), 353–362 | DOI | MR | Zbl

[11] V. A. Marčenko, I. V. Ostrovskii, “A characterization of the spectrum of Hill's operator”, Math. USSR-Sb., 26:4 (1975), 493–554 | DOI | MR | Zbl

[12] H. P. McKean, E. Trubowitz, “Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points”, Comm. Pure Appl. Math., 29:2 (1976), 143–226 | DOI | MR | Zbl

[13] E. Trubowitz, “The inverse problem for periodic potentials”, Comm. Pure Appl. Math., 30:3 (1977), 321–337 | DOI | MR | Zbl

[14] M. G. Gasymov, “Spectral analysis of a class of second-order nonselfadjoint differential operators”, Funct. Anal. Appl., 14:1 (1980), 11–15 | DOI | MR | Zbl

[15] V. A. Tkachenko, “Spectral analysis of a nonselfadjoint Hill operator”, Soviet Math. Dokl., 45:1 (1992), 78–82 | MR | Zbl

[16] V. A. Tkachenko, “Discriminants and generic spectra of nonselfadjoint Hill's operators”, Spectral operator theory and related topics, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994, 41–71 | MR | Zbl

[17] J.-J. Sansuc, V. Tkachenko, “Spectral properties of non-selfadjoint Hill's operators with smooth potentials”, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 371–385 | MR | Zbl

[18] T. Kappeler, B. Mityagin, “Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator”, SIAM J. Math. Anal., 33:1 (2001), 113–152 | DOI | MR | Zbl

[19] P. Djakov, B. Mityagin, “Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps”, J. Funct. Anal., 195:1 (2002), 89–128 | DOI | MR | Zbl

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

[21] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with distribution potentials”, Trans. Moscow Math. Soc., 2003 (2003), 143–192 | MR | Zbl

[22] R. O. Hryniv, Ya. V. Mykytyuk, “1-D Schrödinger operators with periodic singular potentials”, Methods Funct. Anal. Topology, 7:4 (2001), 31–42 | MR | Zbl

[23] P. Djakov, B. Mityagin, “Fourier method for one-dimensional Schrödinger operators with singular periodic potentials”, Topics in operator theory, v. 2, Oper. Theory Adv. Appl., 203, Systems and mathematical physics, Birkhäuser Verlag, Basel, 2010, 195–236 | DOI | MR | Zbl

[24] B. Grébert, T. Kappeler, B. Mityagin, “Gap estimates of the spectrum of the Zakharov–Shabat system”, Appl. Math. Lett., 11:4 (1998), 95–97 | DOI | MR | Zbl

[25] B. Grébert, T. Kappeler, “Estimates on periodic and {D}irichlet eigenvalues for the Zakharov–Shabat system”, Asymptot. Anal., 25:3-4 (2001), 201–237 ; “Erratum”, Asymptot. Anal., 29:2 (2002), 183 | MR | Zbl | MR

[26] P. Djakov, B. Mityagin, “Spectra of 1-D periodic Dirac operators and smoothness of potentials”, C. R. Math. Acad. Sci. Soc. R. Can., 25:4 (2003), 121–125 | MR | Zbl

[27] P. Djakov, B. Mityagin, “Instability zones of a periodic 1D Dirac operator and smoothness of its potential”, Comm. Math. Phys., 259:1 (2005), 139–183 | DOI | MR | Zbl

[28] P. Djakov, B. Mityagin, “Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials”, Math. Ann., 351:3 (2011), 509–540 | DOI | MR | Zbl

[29] P. B. Djakov, B. S. Mityagin, “Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials”, Dokl. Math., 83:1 (2011), 5–7 | DOI | MR | Zbl

[30] P. Djakov, B. Mityagin, “Riesz bases consisting of root functions of 1D Dirac operators”, Proc. Amer. Math. Soc., 141:4 (2013), 1361–1375 | DOI | MR | Zbl

[31] D. M. Levy, J. B. Keller, “Instability intervals of Hill's equation”, Comm. Pure Appl. Math., 16:4 (1963), 469–476 | DOI | MR | Zbl

[32] E. M. Harrell, II, “On the effect of the boundary conditions on the eigenvalues of ordinary differential equations”, Contributions to analysis and geometry (Baltimore, MD, 1980), Johns Hopkins Univ. Press, Baltimore, MD, 1981, 139–150 | MR | Zbl

[33] J. Avron, B. Simon, “The asymptotics of the gap in the Mathieu equation”, Ann. Phys., 134:1 (1981), 76–84 | DOI | MR | Zbl

[34] B. Anahtarci, P. Djakov, “Refined asymptotics of the spectral gap for the Mathieu operator”, J. Math. Anal. Appl., 396:1 (2012), 243–255 | DOI | MR | Zbl

[35] O. A. Veliev, “Isospectral Mathieu–Hill operators”, Lett. Math. Phys., 103:8 (2013), 919–925 | DOI | MR | Zbl

[36] E. A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., New York–Toronto–London, 1955, xii+429 pp. | MR | Zbl

[37] P. Djakov, B. Mityagin, “Divergence of spectral decompositions of Hill operators with two exponential term potentials”, J. Funct. Anal., 265:4 (2013), 660–685 | DOI | MR | Zbl

[38] P. Djakov, B. Mityagin, “Asymptotics of instability zones of the Hill operator with a two term potential”, J. Funct. Anal., 242:1 (2007), 157–194 | DOI | MR | Zbl

[39] P. Djakov, B. Mityagin, “The asymptotics of spectral gaps of a 1D Dirac operator with cosine potential”, Lett. Math. Phys., 65:2 (2003), 95–108 | DOI | MR | Zbl

[40] P. Djakov, B. Mityagin, “Multiplicities of the eigenvalues of periodic Dirac operators”, J. Differential Equations, 210:1 (2005), 178–216 | DOI | MR | Zbl

[41] 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

[42] 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

[43] 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

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

[45] A. Batal, “Characterization of potential smoothness and the Riesz basis property of the Hill–Schrödinger operator in terms of periodic, antiperiodic and Neumann spectra”, J. Math. Anal. Appl., 405:2 (2013), 453–465 | DOI | MR | Zbl