Complex Lagrangian manifolds and splitting of the eigenvalues for Beltrami - Laplace operators on 2D sufraces with integrable geodesic flow
by
Andrei I. Shafarevich
Moscow State University

It is well known that asymptotics of eigenvalues of Laplace and Schroedinger operators on a certain Riemannian manifold are connected with geometrical and topological characteristics of (real) Lagrangian submanifolds of the contangent bundle of the manifold. The aim of the talk is to deminstrate the analogous connection between exponentially small distance between eigenvalues and geometrical characteristics of complex Lagrangian manifolds. This connection is realized in the form of a formula, which expresses splitting of the eigenvalues for Laplace operator on a 2D surface with quadratically integrable geodesic flow in terms of special cycles and cocycles of the corresponding complex Lagrangian manifold.

Date received: June 2, 1999