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St. Petersburg Mathematical Journal

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On the spectrum of a limit-periodic Schrödinger operator


Authors: M. M. Skriganov and A. V. Sobolev
Translated by: the authors
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 815-833
MSC (2000): Primary 35P99.
Published electronically: July 27, 2006
MathSciNet review: 2241427
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Abstract: The spectrum of the perturbed polyharmonic operator $ H = (-\Delta)^l + V$ in $ \textsf{L}^2(\mathbb{R}^d)$ with a limit-periodic potential $ V$ is studied. It is shown that if $ V$ is periodic in one direction in $ \mathbb{R}^d$ and $ 8l > d+3$, $ d \not = 1 ({\mathrm{{mod}}}4)$, then the spectrum of $ H$ contains a semiaxis. The proof is based on the properties of periodic operators.


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Additional Information

M. M. Skriganov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: skrig@pdmi.ras.ru

A. V. Sobolev
Affiliation: Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom
Email: A.V.Sobolev@sussex.ac.uk

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00931-9
Keywords: Perturbed polyharmonic operator, limit-periodic potentials, Bethe--Sommerfeld conjecture.
Received by editor(s): April 6, 2005
Published electronically: July 27, 2006
Additional Notes: The first author acknowledges the support of RFBR (grant no. 05-01-00935).
Dedicated: In memory of O. A. Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society