$L^\infty$-uniqueness of Schr\"odinger operators restricted in an open domain

TL;DR

This paper investigates the conditions under which Schr"odinger operators on open domains are unique in the $L^$ space, linking this to the uniqueness of solutions to the heat equation.

Contribution

It establishes the $L^$-uniqueness of Schr"odinger operators on open domains, providing a key equivalence with the uniqueness of associated heat equation solutions.

Findings

Proves $L^$-uniqueness for Schr"odinger operators.

Links operator uniqueness to heat equation solution uniqueness.

Provides criteria for operator uniqueness in open domains.

Abstract

Consider the Schr\"odinger operator ${\cal A}=-\frac{\Delta}{2}+V$ acting on space $C_0^\infty(D)$, where $D$ is an open domain in $\R^d$. The main purpose of this paper is to present the $L^\infty(D,dx)$-uniqueness for Schr\"odinger operators which is equivalent to the $L^1(D,dx)$-uniqueness of weak solutions of the heat diffusion equation associated to the operator $\cal A$.