Giuseppe Orsatti

Integrable Operators, dbar problems, KP and NLS hierarchy

In the theory of Integrable Operators , the main object of study is their Fredholm determinant, which in some scenarios represents a $\tau$-function of Integrable Systems. The works of Its, Izergin, Korepin and Slavnov (IIKS) unveiled an deep connection between the existence of the resolvent of integrable operators acting on curves $\Sigma \subset \mathbb{C}$ and the solution of a Riemann-Hilbert problem. In this talk we enlarge the IIKS theory also for Hilbert-Smith Integrable Operators acting on domains $\mathscr{D} \subset \mathbb{C}$. For this class of operators, the existence of the resolvent is given by the solution of a dbar problem. The existence of the resolvent assure the non vanishing of the Hilbert-Carleman determinant of the operator. Finally we show that when the operator has a particular dependence on parameters that we call times, the Hilbert-Carleman determinant represents a $\tau$-function of the Kadomtsev-Petviashvili equation or the nonlinear Schr\”odinger equation. In particular in this latter case the such kind of operators naturally arise as an infinite limit of a $N$-soliton solution. This talk is based on a joint work with Prof. Grava (University of Bristol) and Prof. Bertola (Concordia University)