Transport and large deviations for Schrodinger operators and Mather measures

TL;DR

This survey explores the asymptotic behavior of Schrödinger operators on tori, their connection to Mather measures, and applications to transport theory and large deviations.

Contribution

It provides a comprehensive analysis of the large deviations and transport properties of Schrödinger operators linked to Mather measures, including new insights into their asymptotic limits.

Findings

Limit probability measures are Mather measures.

Established large deviation principles for eigenfunctions.

Demonstrated the optimality of certain pairs in Kantorovich duality.

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 ${\bf H}_\beta=\, -\,\frac{1}{2 \beta^2} \, \Delta +V$ 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.