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Gadylshin, R. R.; Khusnullin, I. Kh.

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.

Cited in 1 Document

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

Keywords:

periodic operator; perturbation; eigenvalue; asymptotic behavior
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\textit{R. R. Gadylshin} and \textit{I. Kh. Khusnullin}, Theor. Math. Phys. 173, No. 1, 1438--1444 (2012; Zbl 1280.81055); translation from Teor. Mat. Fiz. 173, No. 1, 127--134 (2012)
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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
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