Pseudo-differential Operators on Non-isotropic Heisenberg Groups with Multi-dimensional Centers

Abstract

We introduce non-isotropic Heisenberg groups with multi-dimensional centers and the corresponding Schrodinger representations. The Wigner and Weyl transforms are then defined. We prove the Stone-von Neumann theorem for the non-isotropic Heisenebrg group by means of Stone-von Neumann theorem for the ordinary Heisenebrg group. Using this theorem, the Fourier transform is defined in terms of these representations and the Fourier inversion formula is given. Pseudo-differential operators with operator-valued symbols are introduced and can be thought of as non-commutative quantization. We give necessary and sufficient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. We also give a characterization of trace class pseudo-differential operators and a trace formula for these trace class operators.