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 ${\bf H}_β=\, -\,\frac{1}{2 β^2} \, Δ+V$ where $β$ is large real parameter. Moreover, for the given form $β\, w$ we consider the associated twist operator ${\bf H}_β^w$. We denote by $({\bf H}_β^w)^*$ the corresponding backward operator. We are interested in the positive eigenfunction $ ψ_β$ associated to the the eigenvalue $ E_β$ for the operator ${\bf H}_β^{w} $. We denote $ ψ_β^*$ the positive eigenfunction associated to the the eigenvalue $ E_β$ for the operator $({\bf H}_β^{w})^* $. Finally, we analyze the asymptotic limit of the probability $ν_β= ψ_β\, ψ_β^*$ on the torus when $β\to \infty$. 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 $μ_{-}$ (the Mather measure for $L^{-}$) to $μ_{+}$ (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.