Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

Abstract

Abstract. We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.

Inverse spectral theory for semiclassical Jaynes–Cummings systems

“…The problem treated in this paper belongs to a class of semiclassical inverse spectral questions which has attracted much attention in recent years, e.g. [21,24,25,15,29,34,39], which goes back to pioneer works of Bérard [1], Brüning-Heintze [3], Colin de Verdière [12,13], Duistermaat-Guillemin [19], and Guillemin-Sternberg [22], in the 1970s/1980s, and are closely related to inverse problems that are not directly semiclassical but do use similar microlocal techniques for some integrable systems, as in [40] (see also [41] and references therein).…”

Integrable systems, symmetries, and quantization

“…There are two natural ways of obtaining commuting operators for this system. One is to view the sphere S 2 as a symplectic reduction of C 2 and use invariant differential operators on R 2 , see [64]; another possibility is to perform Berezin-Toeplitz quantization of the S 2 , see [59]. Figure 3 shows the joint spectrum of the Jaynes-Cummings model which, as was the case with the spherical pendulum, nicely fits within the image of the classical moment map.…”

Open problems, questions and challenges in finite- dimensional integrable systems

“…A positive solution was given in [189,149], but the procedure was not constructive: one had to first let → 0 for a regular value c near the focus-focus value c 0 , and then take the limit c → c 0 , which doesn't help computing the Taylor series in an explicit way from the spectrum. There are various later refinements and extensions of Conjecture 7.5 [188,190,198], following other inverse spectral questions in semiclassical analysis (see for instance [234,115,118]) which concern integrable systems (or even collections of commuting operators) more general than semitoric, and even in higher dimensions. All of them essentially make the same general claim: "from the semiclassical joint spectrum of a quantum integrable system one can detect the principal symbols of the system".…”