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Discrete spectrum of a periodic Schrödinger operator with variable metric perturbed by a nonnegative rapidly decaying potential


Author: V. A. Sloushch
Translated by: A. P. Kiselev
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 317-326
MSC (2010): Primary 35P20
DOI: https://doi.org/10.1090/spmj/1388
Published electronically: January 29, 2016
MathSciNet review: 3444465
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Abstract | References | Similar Articles | Additional Information

Abstract: The discrete spectrum is investigated that emerges in spectral gaps of the elliptic periodic operator $ A=-\mathrm {div} a(x)\mathrm {grad} +b(x)$, $ x\in \mathbb{R}^{d}$, perturbed by a nonnegative, ``rapidly'' decaying potential

$\displaystyle 0\le V(x)\sim v(x/\vert x\vert)\vert x\vert^{-\varrho }, \quad \vert x\vert\to +\infty ,\quad \varrho \ge d. $

The asymptotics of the number of eigenvalues for the perturbed operator $ B(t)=A+tV$, $ t>0$, that have crossed a fixed point of the gap, is established with respect to the large coupling constant $ t$.

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Additional Information

V. A. Sloushch
Affiliation: Department of Physics, Saint Petersburg State University, Ulyanovskaya st. 3, Petrodvorets, 198504 Saint Petersburg, Russia
Email: vsloushch@list.ru, v.slouzh@spbu.ru

DOI: https://doi.org/10.1090/spmj/1388
Keywords: Periodic Schr\"odinger operator, discrete spectrum, spectral gap, asymptotics with respect to a large coupling constant, Cwikel-type estimate.
Received by editor(s): September 9, 2014
Published electronically: January 29, 2016
Additional Notes: The work was supported by SPbSU grant 11.38.263.2014 and RFBR grant 14-01-00760
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2016 American Mathematical Society