A pseudodifferential calculus on non-standard symplectic space

Abstract

We introduce a class of pseudodifferential operators acting on functions defined on an arbitrary symplectic space (ℝ²ⁿ , ω). These operators arise naturally when one considers the generalized commutation relations from non-commutative quantum mechanics. The connection with the usual Weyl operator with symbol a is made using a family of intertwiners W _g defined in terms of the cross-Wigner transform W(f, g). We show that if a belongs to some adequate Shubin symbol classes there is a simple relation between the eigenvalues of and those of .

Keywords:

• Weyl operators on phase space
• symplectic spaces
• non-commutative quantum mechanics

AMS Subject Classifications::

• Primary 35S99
• 35P05
• Secondary 35S05
• 45A75

Acknowledgements

This work has been financed by the Austrian Research Agency FWF (Project ‘Symplectic Geometry and Applications to TFA and QM’, Project number P20442-N13).