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

TL;DR

This paper investigates symplectic covariance properties of Shubin and Born-Jordan pseudo-differential operators, revealing weaker forms of covariance and extending understanding beyond the Weyl operators' well-known properties.

Contribution

It demonstrates that Shubin's { au}-dependent operators have a weaker symplectic covariance with invertible non-unitary intertwiners, and shows Born-Jordan operators maintain covariance under a subgroup of the metaplectic group.

Findings

Weaker symplectic covariance for { au}-operators with non-unitary intertwiners.

Born-Jordan operators exhibit metaplectic covariance under a subgroup.

Extension of symplectic covariance properties beyond Weyl operators.

Abstract

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