×

Perturbation of a periodic operator by a narrow potential. (English. Russian original) Zbl 1280.81055

Theor. Math. Phys. 173, No. 1, 1438-1444 (2012); translation from Teor. Mat. Fiz. 173, No. 1, 127-134 (2012).
Summary: We consider perturbations of a second-order periodic operator on the line; the Schrödinger operator with a periodic potential is a specific case of such an operator. The perturbation is realized by a potential depending on two small parameters, one of which describes the length of the potential support, and the inverse value of other corresponds to the value of the potential. We obtain sufficient conditions for the perturbing potential to have eigenvalues in the gaps of the continuous spectrum. We also construct their asymptotic expansions and present sufficient conditions for the eigenvalues of the perturbing potential to be absent.

MSC:

81Q15 Perturbation theories for operators and differential equations in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34D10 Perturbations of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatlit, Moscow (1963); English transl., Daniel Davey, New York (1966).
[2] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Acad. Press, Edinburgh (1973). · Zbl 0287.34016
[3] F. Gesztesy and B. Simon, Trans. Amer. Math. Soc., 335, 329–340 (1993).
[4] D. I. Borisov and R. R. Gadyl’shin, Izv. Math., 72, 659–688 (2008). · Zbl 1169.34056
[5] R. R. Gadyl’shin and I. Kh. Khusnullin, St. Petersburg Math. J., 22, 883–894 (2011). · Zbl 1232.35108
[6] I. Kh. Khusnullin, Comput. Math. Math. Phys., 50, 646–664 (2010). · Zbl 1217.47038
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.