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2015 | OriginalPaper | Chapter

6. A Short Introduction to the Asymptotic Theory of Rapidly Oscillating Integrals

Author : Franco Cardin

Published in: Elementary Symplectic Topology and Mechanics

Publisher: Springer International Publishing

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Abstract

In this chapter we develop some elements of the asymptotic theory of oscillating integrals; for the sake of simplicity, these elements will be modeled on Schrödinger equation and on its asymptotic solutions. The purpose is that of indicating, rapidly indeed, the profound connections among symplectic geometry, geometric solutions to H-J equations and solutions to Schrödinger equation in the so-called semi-classical limit: \(\hslash \rightarrow 0\) (ℏ is the Planck constant).

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previous chapter Calculus of Variations, Conjugate Points and Morse Index

next chapter Notes on Lusternik-Schnirelman and Morse Theories

I borrowed this quote from the book [64]

Fourier series in the compact case, Fourier transform in the non compact case.

The reader could verify that, at the end, we obtain a Nekhoroshev-like estimate (6.10), typical in perturbative theory of the Hamiltonian systems.

All these sets with the same dimension, this is possible thanks to some theorems about equivalence of generating functions, see Sect. 7.2.1.

Remember that \(\varepsilon = \hslash = h/2\pi\).

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Title: A Short Introduction to the Asymptotic Theory of Rapidly Oscillating Integrals
Author: Franco Cardin
Publisher: Springer International Publishing
Book: Elementary Symplectic Topology and Mechanics
Print ISBN: 978-3-319-11025-7

Electronic ISBN: 978-3-319-11026-4

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-11026-4_6

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