$L^{\infty }$-Uniqueness of Generalized SCHRÖdinger Operators

Abstract

The main purpose of this paper is to show that the generalized Schrödinger operator ${\mathcal{A}}^{V}f=1/2\Delta f+b\nabla f-Vf$, $f\in C_0^{\infty }({\mathbb{R}}^d)$, is a pre-generator for which we can prove its $L^{\infty }({\mathbb{R}}^d,dx)$-uniqueness. Moreover, we prove the $L^1({\mathbb{R}}^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with this pre-generator.