Quantum semitoric systems form a large class of quantum Hamiltonian integrable systems with circular symmetry which has received great attention in the past decade. They include systems of high interest to physicists and mathematicians such as the Jaynes-Cummings model (1963), which describes a two-level atom interacting with a quantized mode of an optical cavity, and more generally the so-called systems of Jaynes-Cummings type. In this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of Jaynes-Cummings type. We prove, assuming the Bohr-Sommerfeld rules hold, that if the joint spectrum of two of these systems coincide up to O( 2 ), then the systems are isomorphic. arXiv:1407.5159v2 [math.SP] 3 Aug 2014 YOHANN LE FLOCHÁLVARO PELAYO SAN VŨ NGO . C 1 The notion of isomorphism for semitoric systems is recalled in Definition 2.1.

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.

No abstract

No abstract