Semiclassical limits, Lagrangian states and coboundary equations

Abstract

Assume that [Formula: see text] is a continuous transformation [Formula: see text]. We consider here the cases where [Formula: see text] is either the transformation [Formula: see text] or [Formula: see text] is a smooth diffeomorphism of the circle [Formula: see text]. Consider a fixed continuous potential [Formula: see text], [Formula: see text] and [Formula: see text] (a quantum state). The transformation [Formula: see text] acting on [Formula: see text], [Formula: see text], defined by [Formula: see text] describes a discrete time dynamical evolution of the quantum state [Formula: see text]. Given [Formula: see text] we define the Lagrangian state [Formula: see text] In this case [Formula: see text]. Under suitable conditions on [Formula: see text] the micro-support of [Formula: see text], when [Formula: see text], is [Formula: see text]. One of the meanings of the semiclassical limit in Quantum Mechanics is to take [Formula: see text] and [Formula: see text]. We address the question of finding [Formula: see text] such that [Formula: see text] satisfies the property: [Formula: see text], we have that [Formula: see text] has micro-support on the graph of [Formula: see text] (which is the micro-support of [Formula: see text]). In other words: which [Formula: see text] is such that [Formula: see text] leaves the micro-support of [Formula: see text] invariant? This is related to a coboundary equation for [Formula: see text], twist conditions and the boundary of the fat attractor.