Trace Formula for the Perturbation of Partial Differential Operator and Cyclic Cocycle on a Generalized Heisenberg Group

Abstract

A trace formula related to the operator $P\left( {i\frac{\partial }{{\partial {{x}_{1}}}}, \ldots ,i\frac{\partial }{{\partial {{x}_{n}}}}} \right) + Q$ +Q is established, where P(·) is a homogeneous real polynomial of degree m and Q is a bounded self-adjoint operator satisfying e ^itx Qe ^-itx — Q ∈ L ¹ and e ^iR Qe ^-iR- Q ∈L ¹ for every R ∈ D _m where D _m is the set of all partial differential operator R of order ≤ m with real constant coefficients. The form of the related cyclic one-cocycle on the group of all the elements of e ^iR e ^ixt e ^tθ for R ∈D _m ,t ∈ Rⁿ,θ ∈ R¹ is determined.