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 $\varphi(H_V)$, where $\varphi$ is an arbitrary rapidly decreasing function on $\mathbb{R}$, and $\varphi(H_V)$ is defined via the spectral theorem.