Boundedness of spectral multipliers for Schrödinger operators on open sets

Abstract

Let $H_V$ be a self-adjoint extension of the Schrödinger operator $-\Delta +V(x)$ with the Dirichlet boundary condition on an arbitrary open set $\Omega$ of ${\mathbb{R}}^d$, where $d\ge 1$ and the negative part of potential $V$ belongs to the Kato class on $\Omega$. The purpose of this paper is to prove $L^p$-$L^q$-estimates and gradient estimates for an operator $\phi (H_V)$, where $\phi$ is an arbitrary rapidly decreasing function on $\mathbb{R}$, and $\phi (H_V)$ is defined via the spectral theorem.