Integrable systems, symmetries, and quantization

Abstract

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system? ForewordThese notes, after a general introduction, are split into four parts:• Integrable systems and action-angle coordinates (Section 3), where the basic notions about Liouville integrable systems are recalled.• Almost-toric singular fibers (Section 4), where emphasis is laid on a Morselike theory of integrable systems.• Semi-toric systems (Section 5), where we introduce the recent semi-toric systems and their classification.

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".…”

Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z)

“…The semi-global 3 structure of a focus-focus singular point is determined by a formal power series in two variables up to the suitable notion of isomorphism (see [43,45]). The Taylor series invariant is one element of R[[X, Y ]] 0 for each of the m f focus-focus points.…”

“…The Taylor series invariant is one element of R[[X, Y ]] 0 for each of the m f focus-focus points. Thanks to [43] we understand why the isomorphism need not be taken into account when there is a global S 1 -action, that is, as in the case of semitoric systems [37], provided one assumes everywhere that the Eliasson isomorphisms preserve the global S 1 -action and the R 2orientation, in which case the Taylor series is unique (for the general case uniqueness is up to a (Z 2 × Z 2 )-action, see [43]).…”

“…The marked interior points are in fact the images of the focus-focus points of the integrable system and the formal power series in two variables determines the semiglobal model around the focus-focus fiber up to a suitable notion of isomorphism (cf. [43,45]). The collection of integers labeling the focus-focus points for each polygon is the twisting index invariant.…”

Correction to: Inverse spectral theory for semiclassical Jaynes–Cummings systems

“…In other words, these two invariants are not independent; there indeed exists a representative of the polygonal invariant for which the corresponding twisting integer is zero, but one cannot start from any weighted polygon and assume that the associated twisting integer vanishes. A good way to define the twisting index jointly with the polygon invariant is explained in [2].…”