Symplectic Covariance Properties for Shubin and Born-Jordan\n Pseudo-Differential Operators

2011/04/27 by Maurice A. de Gosson, de Gosson, Maurice A. · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #NA #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1104.5198

openalex publication_date 2011/04/27 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Among all classes of pseudo-differential operators only the Weyl operators\nenjoy the property of symplectic covariance with respect to conjugation by\nelements of the metaplectic group. In this paper we show that there is,\nhowever, a weaker form of symplectic covariance for Shubin's \τ-dependent\noperators, in which the intertwiners no longer are metaplectic, but still are\ninvertible non-unitary operators. We also study the case of Born--Jordan\noperators, which are obtained by averaging the \τ-operators over the\ninterval [0,1] (such operators have recently been studied by Boggiatto and his\ncollaborators). We show that metaplectic covariance still hold for these\noperators, with respect top a subgroup of the metaplectic group.\n

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• Resolvent trace asymptotics for operators in the Shubin class

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