A simple criterion for essential self-adjointness of Weyl pseudodifferential operators

TL;DR

This paper introduces a new criterion for the essential self-adjointness of Weyl pseudodifferential operators that avoids ellipticity assumptions, applicable to operators with sufficiently smooth symbols and operator-valued symbols in physics.

Contribution

It provides a novel self-adjointness criterion based on phase space calculus and extends applicability to infinite-dimensional operator-valued symbols.

Findings

Self-adjointness holds for symbols with bounded derivatives of order two and higher.

The criterion applies to hermitian operator-valued symbols in infinite-dimensional spaces.

The method combines phase space calculus, Calderón-Vaillancourt theorems, and Toeplitz operator results.

Abstract

We prove a new criterion for the essential self-adjointness of pseudodifferential operators that does not involve ellipticity-type assumptions. For example, we show that self-adjointness holds in case the symbol is $C^{2d+3}$ with derivatives of order two and higher being uniformly bounded. These results also apply to hermitian operator-valued symbols on infinite-dimensional Hilbert spaces, which are important to applications in physics. Our method relies on a phase space differential calculus for quadratic forms on $L^2(\mathbb{R}^d)$, Calder\'on-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.