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Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

Abstract

We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δ Σ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σL p,loc(Σ), p > d − 1.

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Correspondence to I. V. Kachkovskiy.

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__________

Translated from Funktsional’ nyi Analiz i Ego Prilozheniya, Vol. 47, No. 2, pp. 27–37, 2013

Original Russian Text Copyright © by I. V. Kachkovskiy

The research was supported by King’s Annual Fund Studentship and King’s Overseas Research Studentship, King’s College, London.

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Kachkovskiy, I.V. Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder. Funct Anal Its Appl 47, 104–112 (2013). https://doi.org/10.1007/s10688-013-0015-y

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Key words

  • Schrödinger operator
  • periodic coefficients
  • absolutely continuous spectrum