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The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decaying potential. II: Internal gaps. (English. Russian original) Zbl 1070.47042
St. Petersbg. Math. J. 15, No. 2, 249-287 (2004); translation from Algebra Anal. 15, No. 2, 128-189 (2003).
The paper studies second-order operators with periodic coefficients in \(\mathbb{R}\), perturbed by a potential tending to zero toward the infinity and having an internal gap in the spectrum. The asymptotics (on the coupling constant tending to infinity) are analyzed for the number of eigenvalues (of the perturbed operator), which were “born” or “died” at the edges of the gap around eigenvalues of the unperturbed operator. The high-energy Weyl and threshold asymptotics of eigenvalues of the perturbed task are determined. The author assumes that competition may occur between the Weyl and the threshold contributions in the asymptotics of perturbation of eigenvalues on the right edge of the internal gap of the spectrum of the unperturbed operator.
The present work is the sequel of the article [M. Sh. Birman, A. Laptev and T. A. Suslina, St. Petersbg. Math. J. 12, No. 4, 535–567 (2001); translation from Algebra Anal. 12, No. 4, 36–78 (2001; Zbl 1070.47041), reviewed above], which was dedicated to the case of a semiinfinite gap.

MSC:
47F05 General theory of partial differential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47A55 Perturbation theory of linear operators
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