A Hamiltonian action of the Schr\"odinger-Virasoro algebra on a space of periodic time-dependent Schr\"odinger operators in $(1+1)$-dimensions

TL;DR

This paper demonstrates that the Schr"odinger-Virasoro group acts in a Hamiltonian manner on a space of periodic, time-dependent Schr"odinger operators in (1+1) dimensions, revealing a deep geometric structure.

Contribution

It establishes a Hamiltonian action of the infinite-dimensional Schr"odinger-Virasoro group on Schr"odinger operators, connecting it to coadjoint orbits of pseudo-differential symbol Lie algebras.

Findings

The action is Hamiltonian with respect to a specific Poisson structure.

The infinitesimal action corresponds to a coadjoint action of a Lie algebra of pseudo-differential symbols.

The Poisson structure derives from the Kirillov-Kostant-Souriau form.

Abstract

Let ${\cal S}^{lin}:=\{a(t)(-2\II \partial_t-\partial_r^2+V(t,r) | a\in C^{\infty}(\R/2\pi\Z), V\in C^{\infty}(\R/2\pi\Z\times\R)\}$ be the space of Schr\"odinger operators in $(1+1)$-dimensions with periodic time-dependent potential. The action on ${\cal S}^{lin}$ of a large infinite-dimensional reparametrization group $SV$ with Lie algebra $\sv$ \cite{RogUnt06,Unt08}, called the Schr\"odinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on ${\cal S}^{lin}$. More precisely, the infinitesimal action of $\sv$ appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, $\g$, of which $\sv$ is a quotient, while the Poisson structure is inherited from the corresponding Kirillov-Kostant-Souriau form.