Quantized Contact Transformations and Pseudodifferential Operators ...
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Quantized Contact Transformations and Pseudodifferential Operators ...
In this article we establish a formula which gives quantized contact transformations of pseudodifferential operators in terms of symbols. As an application of the ...
Publ RIMS, Kyoto Univ. 26 (1990), 505-519
Quantized Contact Transformations and Pseudodifferential Operators of Infinite Order Dedicated to Professor Tosihusa Kimura on his 60th birthday By
Takashi AOKI*
Introduction
In this article we establish a formula which gives quantized contact transformations of pseudodifferential operators in terms of symbols. As an application of the formula, we define the characteristic sets by using symbols for pseudodifferential operators of infinite order and show that the sets are invariant under quantized contact transformations. The notion of quantization for contact transformations was introduced by Sato, Kawai and Kashiwara [14]. They had been inspired by Egorov [6], Hormander [7] and Maslov [13]. Roughly speaking, a quantization of a given contact transformation 0 is an extension of 0 to a ring isomorphism ^ between the rings of pseudodifferential operators. We denote by A the Lagrangean submanifold associated with 0. Let s be a non-degenerate section of a simple holonomic system with support A. Then the relation P(x, Dx)s = sQ(x', Dx.)
(= Q*(x', Dx,)s)
between two operators P, Q gives a quantization 0^ by setting Q = 4>x(P). Such a 0^ was first constructed in [14] for the sheaves $ and
