Shape resonances in quantum mechanics

Abstract

We prove the existence of shape resonances for Schrödinger operators of the form H(λ)=−Δ+λ²V+U, λ=n⁻¹, in the semiclassical limit in any number of dimensions. The potential V is non-negative, vanishing at infinity as 0(|x|−α),α>0, and forms a barrier about a compact region in which V has finitely many zeros. U∈L ²_loc is any real function which is bounded above and continuous except at a finite number of points. In addition, V and U are assumed to be dilation analytic in a neighborhood of infinity. The resonances shown to exist correspond as λ→∞ to the eigenvalues of a particle confined to the region containing the zeros of V. The width of a resonance near one of these eigenvalues λE is proved to be bounded above by c exp(−2β(λ)(ρ_E−ε)), for any ε>0 and where c>0 is a constant. β(λ) depends upon α and is given by λ for α>2, λ ln λ for α=2, and λ^1/α+1/2 for 0<α<2. The factor ρ_E satisfies $\mathop {\lim }\limits_{\lambda \to \infty } \rho _E (\lambda ) < \infty$, and β(λ)ρ_E is the leading asymptotic to the geodesic distance in the Agmon metric ds²=(λ²V+U−λE)₊dx² between the turning surfaces given by {x|λ²V+U=λE}.

supported in part by US NSF Grant No. DMS-8507040.

For the Proceedings of the International Conference on Differential Equations and Mathematical Physics, University of Alabama, Birmingham, March, 1986.