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arXiv · 2304.07153

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

Robert Fulsche·Lauritz van Luijk
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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.

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Robert Fulsche, Lauritz van Luijk. 2023-04-14. A simple criterion for essential self-adjointness of Weyl pseudodifferential operators. https://doi.org/10.1007/s11868-025-00699-2

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