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).
Often, people in some unjustified fear of physics say you can’t write an equation for life. Well, perhaps we can. As a matter of fact, we very possibly already have the equation to a sufficient approximation when we write the equation of quantum mechanics: \(\displaystyle{i\hslash \frac{\partial \psi } {\partial t} = H\psi.}\)
Richard Feynman, Chapter 41 of his Lectures on Physics.
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Notes
- 1.
I borrowed this quote from the book [64]
- 2.
Fourier series in the compact case, Fourier transform in the non compact case.
- 3.
The reader could verify that, at the end, we obtain a Nekhoroshev-like estimate (6.10), typical in perturbative theory of the Hamiltonian systems.
- 4.
All these sets with the same dimension, this is possible thanks to some theorems about equivalence of generating functions, see Sect. 7.2.1.
- 5.
Remember that \(\varepsilon = \hslash = h/2\pi\).
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Cardin, F. (2015). A Short Introduction to the Asymptotic Theory of Rapidly Oscillating Integrals. In: Elementary Symplectic Topology and Mechanics. Lecture Notes of the Unione Matematica Italiana, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-11026-4_6
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