A HAMILTONIAN ACTION OF THE SCHRÖDINGER–VIRASORO ALGEBRA ON A SPACE OF PERIODIC TIME-DEPENDENT SCHRÖDINGER OPERATORS IN (1 + 1)-DIMENSIONS

Abstract

Let be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on of a large infinite-dimensional reparametrization group SV with Lie algebra [8, 10], called the Schrödinger–Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on . More precisely, the infinitesimal action of appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, , of which is a quotient, while the Poisson structure is inherited from the corresponding Kirillov–Kostant–Souriau form.

Keywords:

• Schrödinger–Virasoro Lie algebra
• time-dependent Schrödinger operators
• infinite-dimensional Lie algebras
• algebra of pseudo-differential symbols
• Poisson structure