A pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces

TL;DR

This paper develops a new pseudo-differential calculus based on arbitrary symplectic structures, extending Weyl calculus, and derives spectral and regularity results within modulation spaces.

Contribution

It introduces a pseudo-differential calculus on non-standard symplectic spaces and connects it to Weyl calculus through isometries, enabling new spectral and regularity analyses.

Findings

Operators act on functions/distributions on n, not n.

Spectral properties are characterized using Shubin's symbol classes.

Regularity results are established in modulation spaces.

Abstract

The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on $\mathbb{R}^{n}$ but rather on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of $L^{2}(\mathbb{R}^{n})\longrightarrow L^{2}(\mathbb{R}^{2n})$ \ indexed by $\mathcal{S}(\mathbb{R}^{n})$. This allows us obtain spectral and regularity results for our operators using Shubin's symbol classes and Feichtinger's modulation spaces.