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