[Paper Review] FIRST STEPS IN SYMPLECTIC AND SPECTRAL THEORY OF INTEGRABLE SYSTEMS

Álvaro Pelayo, San Vũ Ngọc|arXiv (Cornell University)|Aug 17, 2016

Nonlinear Waves and Solitons60 references20 citations

TL;DR

This paper proposes a unified framework for classifying finite-dimensional integrable Hamiltonian systems using symplectic invariants and spectral data, leveraging recent advances in symplectic geometry and microlocal analysis of Toeplitz operators. It focuses on non-hyperbolic, non-degenerate singularities and outlines a path toward solving inverse spectral problems in quantum integrable systems, particularly in molecular 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.

Motivation & Objective

• To develop a unified symplectic and spectral theory for finite-dimensional integrable Hamiltonian systems.
• To address inverse spectral problems in quantum mechanics by reconstructing classical systems from their joint spectra.
• To identify a complete set of invariants that classify integrable systems up to symplectic isomorphism.
• To bridge classical symplectic geometry with quantum spectral theory using microlocal techniques.
• To provide a systematic approach for classifying systems like the coupled spin-oscillator, spherical pendulum, and Kowalevski top.

Proposed method

• Utilize symplectic linearization theorems for non-degenerate singularities as a foundational tool.
• Apply singular affine structures to analyze the geometry of singular fibers in integrable systems.
• Employ microlocal analysis of Toeplitz operators to study semiclassical joint spectra in compact phase spaces.
• Focus on semitoric systems (two degrees of freedom with one periodic, proper component) as a test case.
• Use the framework of action-angle variables and semiclassical quantization to relate classical invariants to quantum spectra.
• Leverage results from Duistermaat, Colin de Verdière, and V\'u Ngo\.c on spectral asymptotics and symplectic invariants.

Experimental results

Research questions

• RQ1 Can the semiclassical joint spectrum of a quantum toric integrable system determine the underlying classical symplectic manifold and Poisson-commuting functions up to symplectic isomorphism?
• RQ2 What invariants characterize integrable systems with non-hyperbolic, non-degenerate singularities, particularly focus-focus type?
• RQ3 How can microlocal analysis of Toeplitz operators be used to solve inverse spectral problems in non-cotangent bundle phase spaces?
• RQ4 To what extent do spectral data encode global geometric and topological features of integrable systems?
• RQ5 What is the role of singular affine structures in classifying the symplectic geometry of integrable systems?

Key findings

• The paper establishes that symplectic linearization theorems for non-degenerate singularities are essential tools for analyzing the local and global structure of integrable systems.
• It identifies the microlocal analysis of Toeplitz operators as a key method for studying quantum integrable systems on compact symplectic manifolds, such as the sphere.
• The authors demonstrate that the semiclassical joint spectrum of a quantum integrable system can, in principle, determine the classical system up to symplectic isomorphism, under suitable conditions.
• The framework provides a path to classify well-known systems like the coupled spin-oscillator, spherical pendulum, and Kowalevski top via their symplectic and spectral invariants.
• The program is designed to reconcile the vast literature on specific integrable systems with modern symplectic and spectral theory through a unified invariant-based classification.
• The authors highlight that degenerate and hyperbolic singularities remain largely unexplored, posing major challenges for future research.