Transport and large deviations for Schrodinger operators and Mather measures

Abstract

In this mainly survey paper we consider the Lagrangian $L(x,v)=\frac{1}{2}|v{|}^2-V(x)$, and a closed form $w$ on the torus ${\mathbb{T}}^n$. For the associated Hamiltonian we consider the the Schrodinger operator ${\mathbf{H}}_{\beta }=-\frac{1}{2{\beta }^2}\Delta +V$ where $\beta$ is large real parameter. Moreover, for the given form $\beta w$ we consider the associated twist operator ${\mathbf{H}}_{\beta }^w$. We denote by $({\mathbf{H}}_{\beta }^w{)}^{\ast }$ the corresponding backward operator. We are interested in the positive eigenfunction ${\psi }_{\beta }$ associated to the the eigenvalue $E_{\beta }$ for the operator ${\mathbf{H}}_{\beta }^w$. We denote ${\psi }_{\beta }^{\ast }$ the positive eigenfunction associated to the the eigenvalue $E_{\beta }$ for the operator $({\mathbf{H}}_{\beta }^w{)}^{\ast }$. Finally, we analyze the asymptotic limit of the probability ${\nu }_{\beta }={\psi }_{\beta }{\psi }_{\beta }^{\ast }$ on the torus when $\beta \to \infty$. The limit probability is a Mather measure. We consider Large deviations properties and we derive a result on Transport Theory. We denote $L^{-}(x,v)=\frac{1}{2}|v{|}^2-V(x)-w_x(v)$ and $L^{+}(x,v)=\frac{1}{2}|v{|}^2-V(x)+w_x(v)$. We are interest in the transport problem from ${\mu }_{-}$ (the Mather measure for $L^{-}$) to ${\mu }_{+}$ (the Mather measure for $L^{+}$) for some natural cost function. In the case the maximizing probability is unique we use a Large Deviation Principle due to N. Anantharaman in order to show that the conjugated sub-solutions $u$ and $u^{\ast }$ define an admissible pair which is optimal for the dual Kantorovich problem.