1209.1849 (Nuno Costa Dias et al.)

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

Nuno Costa Dias, Maurice de Gosson, Franz Luef, João Nuno Prata

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.

View original: http://arxiv.org/abs/1209.1849