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

摘要

Let ${\cal S}^{lin}:=\{a(t)(-2\II \partial_t-\partial_r^2+V(t,r) | a\inC^{\infty}(\R/2\pi\Z), V\in C^{\infty}(\R/2\pi\Z\times\R)\}$ be the space of Schr\"odinger operatorsin $(1+1)$-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-Virasorogroup and containing the Virasoro group, is proved to be Hamiltonian for acertain Poisson structure on ${\cal S}^{lin}$. More precisely, theinfinitesimal action of $\sv$ appears to be part of a coadjoint action of a Liealgebra of pseudo-differential symbols, $\g$, of which $\sv$ is a quotient,while the Poisson structure is inherited from the correspondingKirillov-Kostant-Souriau form.