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On a theorem of Deift and Hempel. (English) Zbl 0647.35063
The paper presents in strikingly simple proof of a previous result in P. A. Deift and R. Hempel [Commun. Math. Phys. 103, 461-490 (1986; Zbl 0594.34022)] on eigenvalues of a Schrödinger operator \(H_ 0+\lambda W\) in spectral gaps of \(H_ 0\). More precisely, the following theorem has been proven:
Theorem. Let \(V,W\in L^{\infty}({\mathbb{R}}^{\nu})\) be real-valued, \(\nu\in {\mathbb{N}}\), supp(W) compact, \(W_-(x)\geq 1\) for all x in some open ball of radius \(\epsilon >0\). (Here \(W_{\mp}:=(| W|_{\mp}W)/2)\). Define \(H_ 0=-\Delta +V\), \(H=H_ 0+\lambda W\), \(\lambda\geq 0\) on \(H^ 2({\mathbb{R}}^{\nu})\). Let \((a,b)\subset {\mathbb{R}}\setminus \sigma (H_ 0)\) be a spectral gap of \(H_ 0\) and assume \(E_ 0\in (a,b)\). Then there exists a sequence of positive numbers \(\lambda_ n\uparrow \infty\) such that \(E_ 0\in \sigma_ p(H_{\lambda_ n})\), \(n\in {\mathbb{N}}.\)
The proof (in contrast to that of Deift-Hempel (loc. cit.)) is based on the use of a self-adjoint Birman-Schwinger kernel. While the condition supp(W) compact has not been used there, the present theorem avoids the notion of “exceptional levels” introduced in that paper. In the meantime generalizations of the results of Deift-Hempel and the above theorem appeared in a preprint by S. Alama, P. Deift and R. Hempel [Eigenvalue branches of the Schrödinger operator \(H-\lambda W\) in a gap of \(\sigma(H)\), New York Univ. (1988)].
Reviewer: F.Gesztesy

35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
Full Text: DOI
[1] Deift, P.A., Hempel, R.: On the existence of eigenvalues of the Schrödinger operatorH??W in a gap of ?(H). Commun. Math. Phys.103, 461-490 (1986) · Zbl 0594.34022 · doi:10.1007/BF01211761
[2] Simon, B.: Brownian motion,L p properties of Schrödinger operators and the localization of binding. J. Funct. Anal.35, 215-229 (1980) · Zbl 0446.47041 · doi:10.1016/0022-1236(80)90006-3
[3] Kato, T.: Monotonicity theorems in scattering theory. Hadronic J.1, 134-154 (1978) · Zbl 0426.47004
[4] Deift, P.A.: Applications of a commutation formula. Duke Math. J.45, 267-310 (1978) · Zbl 0392.47013 · doi:10.1215/S0012-7094-78-04516-7
[5] Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001
[6] Hempel, R.: A left-indefinite generalized eigenvalue problem for Schrödinger operators. Habilitationsschrift, University of Munich, FRG, 1987 · Zbl 0754.35026
[7] Alama, S., Deift, P., Hempel, R.: Eigenvalue problems arising in the theory of the color of crystals (to appear) · Zbl 0676.47032
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