First steps in symplectic and spectral theory of integrable systems

TL;DR

This paper explores initial steps towards a unified symplectic and spectral classification framework for finite-dimensional integrable Hamiltonian systems, emphasizing recent progress and potential pathways in the field.

Contribution

It proposes a preliminary approach to classify integrable systems using symplectic and spectral theory, focusing on non-hyperbolic, non-degenerate singularities and their quantum implications.

Findings

Identifies promising directions for classification

Highlights recent progress in non-degenerate singularities

Suggests strategies linking symplectic geometry and quantum spectroscopy

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.