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

Sloushch, V.

doi:10.1090/spmj/1388

Discrete spectrum of a periodic Schrödinger operator with variable metric perturbed by a nonnegative rapidly decaying potential
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by V. A. Sloushch
Translated by: A. P. Kiselev

St. Petersburg Math. J. 27 (2016), 317-326

DOI: https://doi.org/10.1090/spmj/1388

Published electronically: January 29, 2016

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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 \[ 0\le V(x)\sim v(x/|x|)|x|^{-\varrho }, \quad |x|\to +\infty ,\quad \varrho \ge d. \] 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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