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

Robert Fulsche, Lauritz van Luijk

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 $L^2(\mathbb{R}^d)$, Calderón-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.

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

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 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 , Calderón-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.
Paper Structure (1 section, 9 theorems, 15 equations)

This paper contains 1 section, 9 theorems, 15 equations.

Table of Contents

  1. Corrected proof of Theorem 20 in [1]

Key Result

Theorem 1

Let $f\in C^{2d+3}({\mathbb R}^{2d},{\mathbb R})$ with derivatives of order $2$ to $2d+3$ being uniformly bounded, i.e., $\|\partial^\gamma f\|_\infty<\infty$ for multi-indices $\gamma\in\mathbb N_0^{2d}$ with $2\le|\gamma|\le2d+3$. Then $\mathrm{op}^{w}(f)$ is an essentially self-adjoint operator o

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2: ESA, see \ref{['appendix']}
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5
  • Remark
  • Proposition 6: ESA
  • Lemma 7: ESA
  • Lemma 8
  • ...and 4 more