Semiclassical analysis of low lying eigenvalues. II: Tunneling. (English) Zbl 0626.35070

Simon, Barry

Ann. Math. (2) 120, 89-118 (1984).

[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

Keywords:

tunneling; instanton; asymptotics; eigenvalue splitting; Schrödinger operator; quasiclassical limit; decay properties

Citations:

Zbl 0526.35027