# Boundedness of spectral multipliers for Schrödinger operators on open sets

### Tsukasa Iwabuchi

Tohoku University, Sendai, Japan### Tokio Matsuyama

Chuo University, Tokyo, Japan### Koichi Taniguchi

Chuo University, Tokyo, Japan

## 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.

## Cite this article

Tsukasa Iwabuchi, Tokio Matsuyama, Koichi Taniguchi, Boundedness of spectral multipliers for Schrödinger operators on open sets. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 1277–1322

DOI 10.4171/RMI/1024