## Semiclassical theory of shape resonances in quantum mechanics.(English)Zbl 0704.35115

Mem. Am. Math. Soc. 399, 123 p. (1989).
The authors study the resonances of the Schrödinger operator $$H(\lambda)=-\Delta +\lambda^ 2V+U$$ in the semiclassical limit $$\lambda \to +\infty$$. Here V is assumed to be a non negative potential in $$C^ 3({\mathbb{R}}^ n)$$, which can be written at infinity as a sum of functions homogeneous of negative order. U is in some $$L^ p$$ space and may have a finite number of singularities. Using a method of geometric perturbation theory, they prove the existence of resonances for H($$\lambda$$) near energy levels for which V admits a compact well. (These energy may also correspond to thresholds for H($$\lambda$$).) They also obtain an estimate on the width of these resonances, involving the Agmon distance between the classical turning surfaces.
Reviewer: A.Martinez

### MSC:

 35P99 Spectral theory and eigenvalue problems for partial differential equations 35J10 Schrödinger operator, Schrödinger equation 47A55 Perturbation theory of linear operators 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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