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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)

##### MSC:
 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