First steps in symplectic and spectral theory of    integrable systems

Álvaro Pelayo; San Vű Ngọc

doi:10.3934/dcds.2012.32.3325

Article Contents

2012, Volume 32, Issue 10: 3325-3377. Doi: 10.3934/dcds.2012.32.3325

This issue Previous Article Next Article Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences

First steps in symplectic and spectral theory of integrable systems

Álvaro Pelayo^1, and
San Vű Ngọc^2,

1.
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
2.
Institut Universitaire de France, Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex

Received: April 2011

Revised: February 2012

Published: October 2012

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Abstract

The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities.
This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.

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Mathematics Subject Classification: Primary: 53D05, 37J35, 58J50, 81R12; Secondary: 58J53, 53D50, 35P05, 53D20.

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