$L^\infty$-Uniqueness of Generalized SCHR\"Odinger Operators

TL;DR

This paper establishes the $L^ abla$-uniqueness of a generalized Schr"odinger operator and the $L^1$-uniqueness of solutions to the associated Fokker-Planck equation, contributing to the understanding of these operators' mathematical properties.

Contribution

It proves the $L^ abla$-uniqueness of a generalized Schr"odinger operator and the $L^1$-uniqueness of solutions to the related Fokker-Planck equation, extending existing theoretical frameworks.

Findings

Proves $L^ abla$-uniqueness of the generalized Schr"odinger operator.

Establishes $L^1$-uniqueness of weak solutions for the Fokker-Planck equation.

Provides conditions under which these uniqueness properties hold.

Abstract

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