Elliptic Integrable Systems: solutions, deformations and integrability

Descrizione del progetto

Estensione del regno delle funzioni speciali con un focus sugli integrali ellittici

La matematica fornisce il linguaggio per descrivere le relazioni che governano le interazioni nel mondo che ci circonda. I sistemi di equazioni differenziali parziali sono fondamentali per queste descrizioni. L’ultimo decennio ha visto un’esplosione della ricerca sui modelli Calogero-Moser-Sutherland (CMS) rilevanti per le soluzioni esatte di importanti problemi di meccanica quantistica multiparticellare unidimensionale e le loro controparti classiche. Questi modelli sono fondamentali per numerosi campi non solo della matematica pura ma anche nella fisica teorica, compresa la teoria quantistica dei campi e la fisica della materia condensata. Il progetto ELLIS-SDI, finanziato dall’UE, estenderà questi modelli concentrandosi su funzioni e integrali ellittici e sulla corrispondenza tra i modelli CMS e le gerarchie di Painlevé, un’altra importante classe di equazioni differenziali.

Obiettivo

The proposal studies relativistic generalizations of quantum integrable models of Calogero-Moser-Sutherland (CMS) type and their deformations, and the correspondence between the CMS models and the Painlevé hierarchies. It is divided into several parts. The first part investigates the van Diejen model (which is the most complicated model in the CMS family) and its sophisticated limiting case proposed by Takemura. The aim is to construct exact eigenfunctions of these two models using the kernel function methods. These eigenfunctions belong to an emerging new class in the theory of special functions. The second part is devoted to the study of integrable deformations of the relativistic CMS model (Ruijsenaars model), going back to the works of Chalykh, Feigin, Veselov and Sergeev. In the trigonometric case, the deformed models are known to be integrable and the eigenfunctions of the principal Hamiltonian are given in terms of super-Macdonald polynomials. Using the kernel function identities, I will prove that all higher Hamiltonians of this model are diagonalized by the super-Macdonald polynomials. This will be used to establish orthogonality of the super-Macdonald polynomials. Extending this, I plan to establish integrability of the elliptic case. Furthermore, by using algebraic tools such as Cherednik operators and double affine Hecke algebras, and building upon a recent work of Chalykh, I will construct quantum Lax matrices for the deformed models in all cases. Lastly, we aim to find a conceptual link between elliptic Cherednik algebras and higher Painlevé systems. Namely, first we will obtain the classical Inozemtsev system from Cherednik algebra by a Hamiltonian reduction. By relating this to the recent results of Bertolo, Cafasso, and Rubtsov, we will then find an alternative and more natural interpretation of the higher Painlevé equations as isomonodromic deformations.