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Semiclassical analysis of low lying eigenvalues. II: Tunneling. (English) Zbl 0626.35070
[For Part I see Ann. Inst. Henri Poincaré, Sect. A 38, 295-308 (1983; Zbl 0526.35027).]
The author discusses the asymptotics of the eigenvalue splitting of the Schrödinger operator \(-\Delta +\lambda^ 2V\) in the quasiclassical limit. The main result is that for the difference of the two first eigenvalues the following formula holds \[ \lim_{\lambda \to \infty}- \lambda^{-1} \ell n[E_ 1(\lambda)-E_ 0(\lambda)]=\rho (a,b), \] where \(\rho\) (a,b) is the Agmon metric between the zeros a,b of the otherwise nonnegative potential fulfilling certain additional properties; in particular V does not vanish at infinity. The result is obtained via certain decay properties of the corresponding eigenfunctions. Two alternative proofs are presented, one using Brownian motion, the other one using PDE techniques.
Reviewer: H.Siedentop

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81P20 Stochastic mechanics (including stochastic electrodynamics)
81S40 Path integrals in quantum mechanics
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