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.