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Semiclassical asymptotics for the spectral function of long-range Schrödinger operators. (English) Zbl 0692.35069
The authors study the spectral function for the semiclassical Schrödinger operator, \(P=-h^ 2\Delta +V(x)\) on \({\mathbb{R}}^ n\), when V is a long range potential, satisfying some general hypothesis so that complex scaling method may be applied. After fixing the general hypothesis (on the nature of V), the authors study the case of an energy level which does not contain trapped trajectories and obtain expansions, in h, of the spectral function of P, locally uniformly. Next, authors study a case with trapped trajectories, the “potential well inside an island”, which gives rise to the well-known shape resonances. Finally, the authors obtain pointwise expansions which roughly speaking are the sum of a non-trapping term corresponding to the potential obtained by filling up the well, and a term of Breit-Wigner type corresponding to the resonances.
Reviewer: N.D.Sengupta

MSC:
35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
81Q15 Perturbation theories for operators and differential equations in quantum theory
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