×

Perturbation of the Hill operator by narrow potentials. (English. Russian original) Zbl 1385.34060

Russ. Math. 61, No. 7, 1-10 (2017); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2017, No. 7, 3-13 (2017).
The operator, considered in the paper, is a special case of the Hill operator, a perturbation of a periodic second order differential operator, defined on the real axis, which describes the one-dimensional model of the Bloch electron in a crystal placed in an external electric field. The perturbation is realized by a sum of two complex-valued potentials with compact supports. One of them describes the lengths of the supports of the potentials and the reciprocal to the second one corresponds to the maximum values of the potentials. There are obtained sufficient conditions under which the eigenvalues arise and do not arise from the edges of non-degenerate lacunas of the continuous spectrum, their asymptotes are constructed. Such problems are important in quantum solid state physics and nanoelectronics.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
34D10 Perturbations of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kittel, Ch. Introduction to Solid State Physics (John Wiley and Sons, New York, 1953; Nauka, Moscow, 1978). · Zbl 0052.45506
[2] Rofe-Beketov, F. S. “Characteristic of A Test for the Finiteness of the Number of Discrete Levels Introduced Into the Gaps of a Continuous Spectrum by Perturbations of a Periodic Potential”, Sov. Math. Dokl. 5, 689-692 [in Russian]. · Zbl 0117.06004
[3] Zheludev, V. A. “On Eigenvalues of Perturbed Schrödinger Operator with Periodic Potential”, Probl.Math. Phys., No. 2, 108-123 (1967) [in Russian]. · Zbl 0167.44301
[4] Zheludev, V. A. “The Perturbation of the Spectrum of the Schrödinger Operator With a Complex-Valued Periodic Potential”, Probl.Mat. Fiz. 3, Spektral’. Teor., 31-48 (1968) [in Russian]. · Zbl 0169.48001
[5] Firsova, N. E. “Resonances of aHillOperator, Perturbed by an ExponentiallyDecreasingAdditive Potential”, Math. Notes of the Acad. Sci. USSR 36, No. 5, 854-861 (1984). · Zbl 0598.34017
[6] Firsova, N. E. “Levinson Formula for Perturbed Hill Operator”, Theor. and Math. Physics 62, No. 2, 130-140 (1985). · Zbl 0573.34022 · doi:10.1007/BF01033522
[7] Firsova, N. E. “An Inverse Scattering Problem for a Perturbed Hill’s Operator”, Math. Notes of the Acad. Sci. USSR 18, No. 6, 1085-1091 (1975). · Zbl 0328.34016
[8] Firsova, N. E. “Riemann Surface of Quasi-impulse and the Theory of Scattering forDisturbed HillOperator”, Zap. Nauchn. Sem. LOMI, No. 51, 183-196 (1975) [in Russian]. · Zbl 0349.34020
[9] Firsova, N. E. “Some Spectral Identities for the One-Dimensional Hill Operator”, Theor. and Math. Physics 37, No. 2, 1022-1027 (1978). · Zbl 0423.34038 · doi:10.1007/BF01036374
[10] Firsova, N. E. “The Direct and Inverse Scattering Problems for the One-Dimensional Perturbed Hill Operator”, Math. USSR, Sb. 58, 351-388 (1987). · Zbl 0627.34028 · doi:10.1070/SM1987v058n02ABEH003108
[11] Buslaev, V. S. “Adiabatic Perturbation of a Periodic Potential”, Theor. and Math. Physics 58, No. 2, 153-159 (1984). · Zbl 0557.34053 · doi:10.1007/BF01017921
[12] Buslaev, V. S., Dmitrieva, L. A. “Adiabatic Perturbation of Periodic Potential. II”, Theor. and Math. Physics 73, No. 3, 1320-1329 (1987). · Zbl 0643.34068 · doi:10.1007/BF01041915
[13] Gesztesy, F., Simon, B. “A Short Proof of Zheludev’s Theorem”, Trans. Amer.Math. Soc. 335, No. 1, 329-340 (1993). · Zbl 0770.34056
[14] Borisov, D. I., Gadyl’shin, R. R. “On the Spectrum of a Periodic Operator With a Small Localized Perturbation”, Izv.Math. 72, No. 4, 659-688. · Zbl 1169.34056
[15] Gadylshin, R. R., Khusnullin, I. Kh. “Perturbation of a Periodic Operator by a Narrow Potential”, Theor. and Math. Physics 173, No. 1, 1438-1444 (2012). · Zbl 1280.81055 · doi:10.1007/s11232-012-0124-4
[16] Gadylshin R.R., Khusnullin, I. Kh. “Perturbation of the Shrödinger Operator by aNarrowPotential”, Ufimsk. Mat. Zh. 3, No. 3, 55-66 (2011) [in Russian]. · Zbl 1249.35069
[17] Khusnullin, I. Kh. “Perturbed Boundary Eigenvalue Problem for the Schrödinger Operator on an Interval”, Comp.Math. and Math. Physics 50, No. 4, 646-664 (2010). · Zbl 1217.47038 · doi:10.1134/S096554251004007X
[18] Borisov, D. I., Karimov, R. Kh., Sharapov, T. F. “Initial Length Scale Estimate for Waveguides With Some Random Singular Potentials”, Ufimsk. Mat. Zh. 7, No. 2, 33-54 (2015) [in Russian]. · doi:10.13108/2015-7-2-33
[19] Eastham, M. S. P. The Spectral Theory of PeriodicDifferential Equations (Texts inMathematics, Scottish Academic Press, Edinburgh, 1973).
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.