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Perturbation of the Hill operator by narrow potentials

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Abstract

We consider a perturbation of a periodic second order differential operator, defined on the real axis, which is a special case of the Hill operator. The perturbation is realized by a sum of two complex-valued potentials with compact supports. The potentials depend on two small parameters. 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. We obtain a sufficient condition, under fulfillment of which, the eigenvalues arise from the edges of non-degenerate lacunas of continuous spectrum, and construct their asymptotics. We also give a sufficient condition under which the eigenvalues do not arise.

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Authors and Affiliations

  1. Ufa Scientific Center of Russian Academy of Sciences, pr. Oktyabrya 71, Ufa, 450054, Russia

    A. R. Bikmetov

  2. M. Akmulla Bashkir State Pedagogical University, ul. Oktyabrskoi Revolutsii 3a, Ufa, 450000, Russia

    I. Kh. Khusnullin

Authors
  1. A. R. Bikmetov
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  2. I. Kh. Khusnullin
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Correspondence to A. R. Bikmetov.

Additional information

Original Russian Text © A.R. Bikmetov, I.Kh. Khusnullin, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 7, pp. 3–13.

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Bikmetov, A.R., Khusnullin, I.K. Perturbation of the Hill operator by narrow potentials. Russ Math. 61, 1–10 (2017). https://doi.org/10.3103/S1066369X17070015

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