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

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2025年5月23日

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Robert Fulsche,
Lauritz van Luijk

摘要

We prove a new criterion for essential self-adjointness of pseudodifferential operators, which does not involve ellipticity-type assumptions. Essential self-adjointness is proved for symbols in $C^{2d+3}$ $C^{2 d + 3}$ 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 $L^2(\mathbb {R}^d)$ $L^{2} (R^{d})$ , Calderón-Vaillancourt type theorems and a recent self-adjointness result for Toeplitz operators.