Transport and large deviations for schrodinger operators and mather measures

Abstract

In this mainly survey paper we consider the Lagrangian L(x, v) = 1 2 |v|2 − V (x), and a closed form w on the torus Tn. For the associated Hamiltonian we consider the the Schrodinger operator H β = − 1 Δ + V where β is large real 2β 2 parameter. Moreover, for the given form β w we consider the associated twist operator Hβw. We denote by(Hwβ )∗ the corresponding backward operator. We are interested in the positive eigenfunction ψβ associated to the the eigenvalue E β for the operator Hβw. We denote ψβ∗ the positive eigenfunction associated to the eigenvalueE β for the operator (Hβw)∗. Finally, we analyze the asymptotic limit of the probability νβ = ψβ ψβ∗ on the torus when β →∞. 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) = 12 |v|2 − V (x) − w x (v) and L + (x, v) = 12 |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.