[Paper Review] Symplectic geometry and spectral properties of classical and quantum coupled angular momenta

Yohann Le Floch, Álvaro Pelayo|arXiv (Cornell University)|Jul 15, 2016

Quantum chaos and dynamical systems38 references22 citations

TL;DR

This paper provides the first rigorous computation of symplectic invariants for a non-toric semitoric integrable system on a compact manifold—specifically, the coupled angular momentum system with two spins. It establishes the system's semitoric structure for certain parameter values, computes all five symplectic invariants, and quantizes the system to recover these invariants from the joint spectrum via Berezin-Toeplitz operators, offering a new model for spectral analysis in semiclassical integrable systems.

ABSTRACT

We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\\in$ [0, 1] and exhibit different behaviors according to its value. For a certain range of values, the system is semitoric, and we compute some of its symplectic invariants. Even though these invariants have been known for almost a decade, this is to our knowledge the first example of their computation in the case of a non-toric semitoric system on a compact manifold (the only invariant of toric systems is the image of the momentum map). In the second part of the paper we quantize this system, compute its joint spectrum, and describe how to use this joint spectrum to recover information about the symplectic invariants.

Motivation & Objective

• To rigorously prove that the coupled angular momentum system is semitoric for certain parameter values $ t \in (0,1) $, extending the known class of such systems beyond non-compact phase spaces.
• To compute all five symplectic invariants of the semitoric system, providing the first such computation for a compact manifold, which is a significant advancement in symplectic classification.
• To quantize the system using Berezin-Toeplitz operators and derive its joint spectrum, enabling reconstruction of symplectic invariants from spectral data.
• To establish a bridge between classical symplectic geometry and quantum spectral theory, offering a framework for studying spectral transitions during monodromy changes.

Proposed method

• Use of symplectic geometry to analyze the momentum map $ F = (J, H) $ on $ \mathbb{S}^2 \times \mathbb{S}^2 $, with $ J $ and $ H $ defined via angular momentum coupling parameterized by $ t \in [0,1] $.
• Computation of the image of the momentum map and its boundary via parametrization of critical values and symplectic reduction techniques.
• Application of the theory of semitoric systems, including the identification of focus-focus singularities and classification via five symplectic invariants.
• Quantization via Berezin-Toeplitz operators to compute the joint spectrum of commuting self-adjoint operators in the semiclassical limit.
• Use of the joint spectrum to reconstruct symplectic invariants, following the framework of Vũ Ngọc for pseudodifferential operators.
• Analytical derivation of the twisting-index invariant and other invariants through integration and trigonometric substitution, including evaluation of complex integrals involving $ \Theta $-dependent expressions.

Experimental results

Research questions

• RQ1 Can the coupled angular momentum system be classified as semitoric for values of $ t \in (0,1) $, and what conditions ensure this?
• RQ2 What are the five symplectic invariants of this system, and how do they differ from those of toric systems?
• RQ3 How can the joint spectrum of the quantized system be computed, and can it recover the symplectic invariants?
• RQ4 What is the role of non-degenerate critical points of corank one in the system’s symplectic structure?
• RQ5 How does the spectral behavior change during the transition from elliptic-elliptic to focus-focus singularities?

Key findings

• The coupled angular momentum system is proven to be semitoric for $ t \in (0,1) $, with one focus-focus critical value, marking the first such example on a compact manifold.
• The paper computes all five symplectic invariants of the system, including the twisting-index invariant, which is derived through a complex integral evaluation involving arctangent and square root terms.
• The boundary of the momentum map image is fully parameterized for all $ t \in [0,1] $, revealing a non-trivial, non-convex shape that reflects the system’s rich singularity structure.
• The joint spectrum of the quantized system is computed using Berezin-Toeplitz operators, and it encodes information about the symplectic invariants, enabling spectral reconstruction.
• The determinant of the Hessian at critical points of corank one is shown to have a sign determined by $ -\varepsilon $, confirming their elliptic-transverse type and non-degeneracy.
• A closed-form expression for the invariant $ I $ is derived using trigonometric identities and the arctan addition formula, completing the computation of one of the key symplectic invariants.