Victor Buchstaber (Steklov Mathematical Institute and University of Manchester)
buchstab@mi.ras.ru
We introduce a quasi-periodic function Ψ (t) = Ψ (t; v, w, g). Here the parameters v and w are points on the elliptic curve with parameters g=(g2, g3). For v = ± w, this function coincides with the Baker-Akhiezer function, important in soliton theory and the theory of Hirzebruch genera. The function Ψ (t) has many fundamental properties of the Baker-Akhiezer function, but it is not meromorphic because it can have branch points t=v or t= -v in the parallelogram of periods. We obtain the addition theorem for the function Ψ (t). As it follows from this theorem, the function Ψ (t) is an eigenfunction of differential operators L2 and L3 of orders 2 and 3 with doubly periodic coefficients. In the case v = ± w, the operator L2 is the classical Lamé operator and the pair L2 , L3 is the classical elliptic Lax pair.
The talk will be accessible for general mathematical audience. Main definitions will be introduced during the talk. New results presented in the talk were obtained in recent joint works with E.Yu. Bunkova.