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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
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