On a Certain Semiclassical Problem on Wiener Spaces

Abstract

We study asymptotic behavior of the spectrum of a Schrödinger type operator $L_V^{\lambda }=L-{\lambda }^{2}V$ on the Wiener space as $\lambda \to \infty$. Here $L$ denotes the Ornstein–Uhlenbeck operator and $V$ is a nonnegative potential function which has finitely many zero points. For some classes of potential functions, we determine the divergence order of the lowest eigenvalue. Also tunneling effect is studied.