Discrete spectrum in the gaps of a perturbed periodic Schrödinger operator. II: Nonregular perturbations.

*(English. Russian original)*Zbl 0911.35082
St. Petersbg. Math. J. 9, No. 6, 1073-1095 (1998); translation from Algebra Anal. 9, No. 6, 62-89 (1997).

Let \(A_\pm(\alpha)=A\mp\alpha V(x)\), where \(A\) is the \(\mathbb{Z}^d\)-periodic operator
\[
A=-\text{div} g(x)\text{grad}+p(x),\quad x\in\mathbb{R}^d,
\]
\(\alpha\) is a positive coupling constant and \(V\) is a nonnegative function which tends to zero in some sense, as \(| x|\to\infty\). Let \(\Lambda=(\lambda_-,\lambda_+)\) be a gap in the spectrum of \(A\), and denote by \(N(\lambda_\pm,\alpha)\) the number of eigenvalues of \(A_\pm(t)\) “born” at \(\lambda_\pm\) as \(t\) increase from 0 to \(\alpha\); as \(t\) increases, the eigenvalues of \(A_+(t)\) decrease and those of \(A_-(t)\) increase. The subject of the paper is the asymptotic behaviour of \(N(\lambda_\pm,\alpha)\) as \(\alpha\to\infty\). In a previous paper [M. Sh. Birman, in: Boundary value problems, Schrödinger operators, deformation quantization, Akademie-Verlag, Berlin, Math. Top. 8, 334-352 (1995; Zbl 0848.47032)] the author considered regular perturbations \(V\) which ensure that \(N(\lambda_+,\alpha)\) satisfies the Weyl-type asymptotic formula
\[
\lim_{\alpha\to\infty}\alpha^{-d/2}N(\lambda_+,\alpha)=(2\pi)^{-d}\omega_d \int_{\mathbb{R}^d}V^{d/2}(\text{det} g)^{-1/2}dx,
\]
where \(\omega_d\) is the volume of the unit ball in \(\mathbb{R}^d\): for \(d\geq 3\), \(V\) is regular if and only if \(V\in L_{d/2}(\mathbb{R}^d)\), while for \(d=2\), only some necessary and some (different) sufficiency conditions are known. In the present paper, the case of nonregular perturbations \(V\) is treated, and this when \(d\geq 3\). The conditions on \(V\) are formulated in terms of the “standard” operator \(-\Delta-\alpha V\), which is assumed to be such that \(N(0,\alpha)=O(\alpha^q)\) for some \(q>d/2\). The answers are given in terms of two model operators.

Reviewer: W.D.Evans (Cardiff)