Complex Fermi-curves of 2D Schrödinger operator and their applications

Modern algebraic-geometrical spectral theory of periodic operators is central for the construction of exact periodic solutions of a wide class of nonlinear models known as soliton systems. These solutions are expressed in terms of Riemann theta functions. The interaction between algebraic geometry and the soliton equations turns out to be beneficial for both sides. The famous Novikovs conjecture on the characterization of Jacobians of algebraic curves (more than 130 years old Riemann-Shottky problem) via solutions of the Kadomtsev-Petviashvili equation was proved by Shiota in 1986. Much stronger characterization was obtained by the author who proved Welters' trisecant conjecture.

In the talk we present these results and the solutions of another classical problem on characterization of Prym varieties via the theory of 2D Shrodinger operators integrable on one energy value. We then discuss another class of applications of the spectral theory that include a new approach to the construction of exact solutions to sigma-models. The latter is a joint work with A. Ilina and N. Nekrasov.