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Accurate spectral asymptotics for periodic operators. (English) Zbl 1103.35353
Proceedings of the conference on partial differential equations, Saint-Jean-de-Monts, France, May 31–June 4, 1999. Exp. Nos. I–XIX (1999). Nantes: Université de Nantes (ISBN 2-86939-146-3/pbk). Exp. No. 5, 11 p. (1999).
Summary: Asymptotics with sharp remainder estimates are recovered for the number \(N(\tau)\) of eigenvalues of operator \(A(x,D)-tW(x,x)\) crossing level \(E\) as \(t\) runs from 0 to \(\tau, \tau\to\infty\). Here \(A\) is a periodic matrix operator, matrix \(W\) is positive, periodic with respect to the first copy of \(x\) and decaying as the second copy of \(x\) goes to infinity, and \(E\) either belongs to a spectral gap of \(A\) or is at one of its ends. These problems were first treated in papers of M. Sh. Birman, Birman and A. Laptev, and Birman and T. Suslina.
For the entire collection see [Zbl 0990.00047].

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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