Contact transformations for micro-differential operators

Abstract

Let X and Y be complex analytic manifolds of the same complex dimension, let p be a point in T * X\{0} and let q be a point in T * Y\{0}. Let ϵ (p) (resp. ϵ (q)) denote the graded ring of germs of micro-differential operators at p (resp. q). Supply ϵ (p) and ϵ (q) with a suitable and natural convergence structure. Let X: ϵ (p) ϵ → (q) be a continuous degree preserving ring isomorphism. We show that there exists a vector field ∂ tangent to X in a neighbourhood of л(q), a symplectic map ф defined on a conic open neighbourhood of q to a similar neighbourhood of p such that for an appropriate constant z and a suitable invertible element A. in ϵ (q) of degree 0. Uniqueness of this result is discussed and a global version is also given. Here denotes the duality between T(Y) and T *(Y). of do,(X) = 1, then this representation is canonical.