Revista Matemática Iberoamericana


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Volume 34, Issue 3, 2018, pp. 1277–1322
DOI: 10.4171/RMI/1024

Published online: 2018-08-27

Boundedness of spectral multipliers for Schrödinger operators on open sets

Tsukasa Iwabuchi[1], Tokio Matsuyama[2] and Koichi Taniguchi[3]

(1) Tohoku University, Sendai, Japan
(2) Chuo University, Tokyo, Japan
(3) Chuo University, Tokyo, Japan

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.

Keywords: Schrödinger operators, functional calculus, Kato class

Iwabuchi Tsukasa, Matsuyama Tokio, Taniguchi Koichi: Boundedness of spectral multipliers for Schrödinger operators on open sets. Rev. Mat. Iberoamericana 34 (2018), 1277-1322. doi: 10.4171/RMI/1024