Poisson groups and differential Galois theory of Schroedinger equation on the circle

TL;DR

This paper integrates projective geometry, differential Galois theory, and Poisson Lie groups to develop a Poisson structure on wave functions for Schroedinger equations on the circle, with applications to nonlinear equations and lattice operators.

Contribution

It introduces a novel Poisson structure on wave functions using a combined geometric and algebraic approach, extending to difference operators and lattice algebras.

Findings

Constructed a natural Poisson structure on wave functions at zero energy.

Applied the approach to KdV-like nonlinear equations.

Extended the lattice Poisson Virasoro algebra to difference operators.

Abstract

We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to second order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.