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Partial Differential Operators and Fourier Integral Operators in the Gevrey setting and applications

Benjamin Colmey, Richard Lascar

Abstract

We state and prove here semiclassical results about the construction of asymptotic solutions by the WKB method for pseudo-differential equations of real principal type. It is a Gevrey version; the smooth $C^\infty$ and the analytic ones may be found in Hörmander (Hormander1985) and Sjöstrand (Sjöstrand1982). We present here three versions depending on the degree of entrance in the complex domain.

Partial Differential Operators and Fourier Integral Operators in the Gevrey setting and applications

Abstract

We state and prove here semiclassical results about the construction of asymptotic solutions by the WKB method for pseudo-differential equations of real principal type. It is a Gevrey version; the smooth and the analytic ones may be found in Hörmander (Hormander1985) and Sjöstrand (Sjöstrand1982). We present here three versions depending on the degree of entrance in the complex domain.

Paper Structure

This paper contains 15 sections, 253 equations.

Table of Contents

  1. Introduction
  2. Notation 1.6.
  3. A convention more.
  4. A definition more.
  5. Stationary phase and symbol calculus in the Gevrey setting
  6. Proof.
  7. Proof.
  8. Proof.
  9. Proof.
  10. Proof of Theorem 2
  11. Proof of Theorem 1
  12. Corollary 3.5.
  13. Proof of Theorem 3
  14. Gevrey WF set of Lagrangian distributions
  15. Borel Lemma in Gevrey Symbol Classes