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

Claude Roger (1) , Jeremie Unterberger (2)

1 ICJ - Institut Camille Jordan
2 IECN - Institut Élie Cartan de Nancy

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

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Claude Roger, Jeremie Unterberger. A Hamiltonian action of the Schrödinger-Virasoro algebra on a space of periodic time-dependent Schrödinger operators in $(1+1)$-dimensions. 2008. ⟨hal-00358193⟩