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Discrete spectrum of the periodic Schrödinger operator for non-negative perturbations. (English) Zbl 0828.34075
Demuth, Michael (ed.) et al., Mathematical results in quantum mechanics: International conference in Blossin (Germany), May 17-21, 1993. Basel: Birkhäuser Verlag. Oper. Theory, Adv. Appl. 70, 3-7 (1994).
The author considers a periodic operator \(A= -\text{div}(g(x)\text{ grad})+ p(x)\), \(g,p\in L_\infty(\mathbb{R}^d)\), \(g(x+ n)= g(x)\), \(p(x+ n)= p(x)\), while \(g\) is a positive definite matrix function. If \(g\) is constant then \(A\) is exactly the Schrödinger operator with lattice periodicity. A perturbed operator is given by \(A(\alpha)= A+ \alpha V\), \(V(x)\geq 0\), \(\alpha> 0\). Asymptotically \(V\in L_{\infty\text{ loc}}\) behaves like \(V(x)\sim x^{-2\sigma} f(\vartheta)\), \(\vartheta= {x\over |x|}\), \(\sigma> 0\), as \(|x|\to \infty\).
Let \((- \lambda, +\lambda)\) be a gap in the spectrum of \(A\), \(\lambda\) a fixed number in this gap. Let \(t\) be a constant, \(\alpha\geq t> 0\), let \(N(\alpha, \lambda)\) denote the number of eigenvalues of \(A(t)\) which cross the value \(\lambda\) as \(t\) increases from zero to \(\alpha\). The author wishes to determine the asymptotic behavior of \(N(\alpha, \lambda)\) as \(\alpha\to \infty\). Introducing \(W= \sqrt V\), the author defines the operator \(T(\lambda)= - W(A- \lambda I)^{- 1} W\), and shows that the asymptotic behavior of a related operator \(T_r\) (the first \(r\) summands in the expansion of \(T(\lambda)\)) determines upper asymptotic estimates for \(N(\alpha, \lambda)\). This in turn leads to upper asymptotic estimates expressed in terms of integrals of fairly arbitrary functions \(f\in L_q(S^{d- 1})\).
The author’s results extend those of A. V. Sobolev who basically considered the one-dimensional problem [Adv. Sov. Math. 7, 159-178 (1991; Zbl 0752.34046)] and [S. Alama, P. A. Deift and R. Hempel, Asymptotic Anal. 8, No. 4, 311-344 (1994; Zbl 0806.47042)].
For the entire collection see [Zbl 0791.00039].
Reviewer: V.Komkov (Roswell)

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47F05 General theory of partial differential operators