Semiclassical limits, Lagrangian states and coboundary equations

Abstract

Assume that $f$ is a continuous transformation $f:S^1\to S^1$. We consider here the cases where $f$ is either the transformation $f(z)=z^2$ or $f$ is a smooth diffeomorphism of the circle $S^1$. Consider a fixed continuous potential $\tau :S^1=[0,1)\to \mathbb{R}$, $\nu \in \mathbb{R}$ and $\phi :S^1\to \mathbb{C}$ (a quantum state). The transformation ${\hat{F}}_{\nu }$ acting on $\phi :S^1\to \mathbb{C}$, ${\hat{F}}_{\nu }(\phi )=\varphi$, defined by $\varphi (z)={\hat{F}}_{\nu }(\phi (z))=\phi (f(z))e^{i\nu \tau (z)}$ describes a discrete time dynamical evolution of the quantum state $\phi$. Given $S:\mathbb{R}\to \mathbb{R}$ we define the Lagrangian state $${\phi }_x^S(z)=\sum _{k\in \mathbb{Z}}{e}^{\frac{iS(z-k)}{\hslash }}{e}^{-\frac{(z-k-x{)}^2}{4\hslash }}.$$ In this case ${\hat{F}}_{\nu }({\phi }_x^S(z))={\sum }_{k\in \mathbb{Z}}{e}^{\frac{iS(f(z)-k)}{\hslash }}{e}^{-\frac{(f(z)-k-x{)}^2}{4\hslash }}{e}^{i\nu \tau (z)}$. Under suitable conditions on $S$ the micro-support of ${\phi }_x^S(z)$, when $\hslash \to 0$, is $(x,S^{\prime }(x))$. One of meanings of the semiclassical limit in Quantum Mechanics is to take $\nu =\frac{1}{\hslash }$ and $\hslash \to 0$. We address the question of finding $S$ such that ${\phi }_x^S$ satisfies the property: $\forall x$, we have that ${\hat{F}}_{\nu }({\phi }_x^S)$ has micro-support on the graph of $y\to S^{\prime }(y)$ (which is the micro-support of ${\phi }_x^S$). In other words: which $S$ is such that ${\hat{F}}_{\nu }$ leaves the micro-support of ${\phi }_x^S$ invariant? This is related to a coboundary equation for $\tau$, twist conditions and the boundary of the fat attractor.