Transport and large deviations for Schrodinger operators and Mather measures

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 ${\bf H}_\beta=\, -\,\frac{1}{2 \beta^2} \, \Delta +V$ where $\beta$ is large real parameter. Moreover, for the given form $\beta\, w$ we consider the associated twist operator ${\bf H}_\beta^w$. We denote by $({\bf H}_\beta^w)^*$ the corresponding backward operator. We are interested in the positive eigenfunction $\psi_\beta$ associated to the the eigenvalue $E_\beta$ for the operator ${\bf H}_\beta^{w}$. We denote $\psi_\beta^*$ the positive eigenfunction associated to the the eigenvalue $E_\beta$ for the operator $({\bf H}_\beta^{w})^*$. Finally, we analyze the asymptotic limit of the probability $\nu_\beta= \psi_\beta\, \psi_\beta^*$ 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 $u^*$ define an admissible pair which is optimal for the dual Kantorovich problem.