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

2023/04/14 by Fulsche, Robert, van Luijk, Lauritz
#47B25 #47G30 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2304.07153

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 L²(ℝ^d), Calderón-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.

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