A quasiclassical limit of the spectrum of a Schrödinger operator with complex periodic potential

D. V. Nekhaev   ; A. I. Shafarevich

Sbornik. Mathematics, Tome 208 (2017) no. 10, pp. 1535-1556

Résumé

The quasiclassical asymptotics of the spectrum of a one-dimensional Schrödinger operator with periodic complex potential that arises in the statistical mechanics of a Coulomb gas are described. The spectrum is shown to concentrate in a neighbourhood of a tree in the complex plane; the vertices of this tree are calculated explicitly, and the position of its edges can be investigated comprehensively. Equations are derived from which the asymptotic eigenvalues are found; these equations are conditions that certain special periods of a holomorphic form on the Riemann surface of constant classical energy are integers. Bibliography: 25 titles.

Détail

Keywords: quasiclassical asymptotics, nonselfadjoint operators, spectral graph, Stokes curves.

Bibliographie

[1] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques, Études mathematiques, Dunod, Paris; Gauthier-Villars, Paris, 1972, xvi+384 pp. | Zbl

[2] S. G. Reddy, P. J. Schmidt, D. S. Henningson, “Pseudospectra of the Orr–Sommerfeld operator”, SIAM J. Appl. Math., 53:1 (1993), 15–47 | DOI | MR | Zbl

[3] S. A. Stepin, “Non-self-conjugate singular perturbations: a model of transition from a discrete to a continuous spectrum”, Russian Math. Surveys, 50:6 (1995), 1311–1313 | DOI | MR | Zbl

[4] S. A. Stepin, “Model perekhoda ot diskretnogo spektra k nepreryvnomu v singulyarnoi teorii vozmuschenii”, Fundament. i prikl. matem., 3:4 (1997), 1199–1227 | MR | Zbl

[5] A. A. Shkalikov, “The limit behavior of the spectrum for large parameter values in a model problem”, Math. Notes, 62:6 (1997), 796–799 | DOI | DOI | MR | Zbl

[6] S. A. Stepin, A. A. Arzhanov, “Quasiclassical spectral asymptotics and the Stokes phenomenon for the Weber equation”, Dokl. Math., 63:3 (2001), 306–309 | MR | Zbl

[7] A. V. Dyachenko, A. A. Shkalikov, “On a model problem for the Orr–Sommerfeld equation with linear profile”, Funct. Anal. Appl., 36:3 (2002), 228–232 | DOI | DOI | MR | Zbl

[8] S. N. Tumanov, A. A. Shkalikov, “On the limit behaviour of the spectrum of a model problem for the Orr–Sommerfeld equation with Poiseuille profile”, Izv. Math., 66:4 (2002), 829–856 | DOI | DOI | MR | Zbl

[9] A. A. Shkalikov, “Spectral portraits of the Orr–Sommerfeld operator with large Reynolds numbers”, J. Math. Sci. (N. Y.), 124:6 (2004), 5417–5441 | DOI | MR | Zbl

[10] S. V. Gal'tsev, A. I. Shafarevich, “Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients”, Math. Notes, 80:3 (2006), 345–354 | DOI | DOI | MR | Zbl

[11] S. V. Galtsev, A. I. Shafarevich, “Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients”, Theoret. and Math. Phys., 148:2 (2006), 1049–1066 | DOI | DOI | MR | Zbl

[12] S. A. Stepin, V. A. Titov, “On the concentration of spectrum in the model problem of singular perturbation theory”, Dokl. Math., 75:2 (2007), 197–200 | DOI | MR | Zbl

[13] V. I. Pokotilo, A. A. Shkalikov, “Semiclassical approximation for a nonself-adjoint Sturm–Liouville problem with a parabolic potential”, Math. Notes, 86:3 (2009), 442–446 | DOI | DOI | MR | Zbl

[14] H. Roohian, A. I. Shafarevich, “Semiclassical asymptotics of the spectrum of a nonselfadjoint operator on the sphere”, Russ. J. Math. Phys., 16:2 (2009), 309–314 | DOI | MR | Zbl

[15] H. Roohian, A. I. Shafarevich, “Semiclassical asymptotic behavior of the spectrum of a nonselfadjoint elliptic operator on a two-dimensional surface of revolution”, Russ. J. Math. Phys., 17:3 (2010), 328–333 | DOI | MR | Zbl

[16] A. I. Esina, A. I. Shafarevich, “Quantization conditions on Riemannian surfaces and the semiclassical spectrum of the Schrödinger operator with complex potential”, Math. Notes, 88:2 (2010), 209–227 | DOI | DOI | MR | Zbl

[17] L. K. Kusainova, A. Zh. Monashova, A. A. Shkalikov, “Asymptotics of eigenvalues of a second-order non-self-adjoint differential operator on the axis”, Math. Notes, 93:4 (2013), 629–632 | DOI | DOI | MR | Zbl

[18] A. I. Esina, A. I. Shafarevich, “Analogs of Bohr–Sommerfeld–Maslov quantization conditions on Riemann surfaces and spectral series of nonself-adjoint operators”, Russ. J. Math. Phys., 20:2 (2013), 172–181 | DOI | MR | Zbl

[19] A. I. Esina, A. I. Shafarevich, “Asymptotics of the spectrum and eigenfunctions of the magnetic induction operator on a compact two-dimensional surface of revolution”, Math. Notes, 95:3 (2014), 374–387 | DOI | DOI | MR | Zbl

[20] A. I. Esina, A. I. Shafarevich, “Semiclassical asymptotics of eigenvalues for non-selfadjoint operators and quantization conditions on Riemann surfaces”, Acta Polytechnica, 54:2 (2014), 101–105 | DOI

[21] S. N. Tumanov, A. A. Shkalikov, “The limit spectral graph in semiclassical approximation for the Sturm–Liouville problem with complex polynomial potential”, Dokl. Math., 92:3 (2015), 773–777 | DOI | DOI | MR | Zbl

[22] T. Gulden, M. Janas, P. Koroteev, A. Kamenev, “Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces”, ZhETF, 144:3(9) (2013), 595–616

[23] C. M. Bender, D. C. Brody, H. F. Jones, B. K. Meister, “Faster than Hermitian quantum mechanics”, Phys. Rev. Lett., 98:4 (2007), 040403, 4 pp. | DOI | MR | Zbl

[24] M. A. Evgrafov, M. V. Fedoryuk, “Asymptotic behaviour as $\lambda\to\infty$ of the solution of the equation $w''(z)-p(z,\lambda)w(z)=0$ in the complex $z$-plane”, Russian Math. Surveys, 21:1 (1966), 1–48 | DOI | MR | Zbl

[25] M. V. Fedoryuk, Asymptotic analysis. Linear ordinary differential equations, Springer-Verlag, Berlin, 1993, viii+363 pp. | DOI | MR | MR | Zbl | Zbl