Schrödinger operator on the Wiener space

Ichiro Shigekawa (Kyoto University)

We consider a Schrödinger operator −L + V on an abstract Wiener space (B, H, μ). Here L is the Ornstein-Uhlenbeck operator and V is a scalar potential. We discuss the following problems:

1. Essential self-adjointness of a Schrödinger operator.

2. The domain of the Schrödinger operator.

3. Spectral gap of the Schrödinger operator.

To solve the first problem, we use the logarithmic Sobolev inequality and the theory of positive generalized functions. For the second problem, we use the intertwining property of operators of the form \sqrt{V} (−L+V) = A\sqrt{V}. Here A is a non-symmetric operator. For the third problem, we use the approximation method and show the norm convergence of associated resolvents. The uniform estimate of the Schrödinger operators plays an essential role.