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.
Categories: math.DS math-ph math.CA math.MP math.PR quant-ph