Symplectic Geometry and Spectral Properties of Classical and Quantum Coupled Angular Momenta

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 ∈ [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.

A family of compact semitoric systems with two focus-focus singularities

“…This system was originally introduced in Sadovskií & Zĥilinskií [31] and studied in detail in Le Floch & Pelayo [18], where it is shown that there exist two fixed values t − , t + ∈ (0, 1) with t − < t + which depend on R 1 , R 2 such that 1) if t − < t < t + then (J R , H t ) is a semitoric system with exactly one focusfocus point, 2) if t > t + or t < t − the (J R , H t ) is a semitoric system with exactly zero focus-focus points (these are known as systems of toric type, see Section 2 of Vũ Ngo . c [35]), 3) if t = t − or t = t + then (J R , H t ) has a degenerate singular point, and thus is not a semitoric system.…”

“…By Equations (18) and (17) we see that X J = ∂ z1 +∂ z2 so the flow of J rotates θ 1 and θ 2 by a common angle. Thus, the S 1 -action produced by the flow of X J preserves the angle difference θ 1 − θ 2 .…”

“…These systems each have exactly one focus-focus singularity. In the present work we describe semitoric systems on M = S 2 ×S 2 which have two focus-focus singular points, generalizing the system from [18]. More precisely, the main result of this paper is the following.…”

“…The image of the momentum map for the system in Theorem 1.2 is plotted in Figure 1 for various choices of (s 1 , s 2 ) and γ is plotted in Figure 2. The coupled angular momenta system from Le Floch & Pelayo [18] is exactly the one parameter family of systems obtained from the system in Theorem 1.2 by taking s 2 = 0, so the momentum map image of the coupled angular momentum system is the bottom row of images in Figure 1.…”

“…c classification predicts many systems and gives certain properties of those systems, one thing that has thus far been lacking are explicit examples of semitoric systems giving the symplectic manifold (M, ω) and the momentum map F . Le Floch & Pelayo [18] explicitly describe the coupled angular momenta system (originally described in [31], see Example 2.12) and details of the coupled spin oscillator (see Example 2.13) are spread over several papers. These systems each have exactly one focus-focus singularity.…”