Boundedness of spectral multipliers for Schrödinger operators on open sets (Q1989547)

Summary: 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 \(d\geq1\) 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.