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Lecture Notes Nalini Anantharaman
 

Summary: Lecture Notes
Nalini Anantharaman
June 2007
Eigenfunctions of the laplacian on negatively curved manifolds : a
semiclassical approach.

Table of contents
1 An introduction to semiclassical analysis. 3
1.1 Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Three approaches to classical mechanics. . . . . . . . . 3
1.1.2 Quantum/wave mechanics. . . . . . . . . . . . . . . . 7
1.2 Weyl quantization. . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Born's probabilitic interpretation of the Schr˜odinger equation. 13
1.4 The semiclassical limit. . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Fourier transform. . . . . . . . . . . . . . . . . . . . . 14
1.4.2 The stationary phase method. . . . . . . . . . . . . . . 15
1.4.3 Pseudodi#erential operators. . . . . . . . . . . . . . . 16
1.5 Semiclassical measures, microlocal lifts. . . . . . . . . . . . . 21
2 Entropy and localization of eigenfunctions. 25
2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

  

Source: Anantharaman, Nalini - Centre de Mathématiques Laurent Schwartz, École Polytechnique

 

Collections: Mathematics