Hamiltonian dynamics and spectral theory for spin-oscillators

TL;DR

This paper investigates the symplectic dynamics and spectral properties of spin-oscillators, revealing their rich singularity structures and applying semiclassical analysis and differential geometry to understand their quantization.

Contribution

It provides a detailed analysis of the singularities in spin-oscillators and develops a quantization framework combining semiclassical and geometric methods.

Findings

Identification of multiple types of singularities in spin-oscillators

Analysis of dynamics around the focus-focus singularity

Spectral characterization of quantized spin-oscillator systems

Abstract

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.