Semiclassical Limits of Distorted Plane Waves in Chaotic Scattering Without a Pressure Condition

In this paper, we study the semiclassical behavior of distorted plane waves, on manifolds that are Euclidean near infinity or hyperbolic near infinity, and of non-positive curvature. Assuming that there is a strip without resonances below the real axis, we show that distorted plane waves are bounded in $L^2_{loc}$ independently of $h$ and that they admit a unique semiclassical measure and we prove bounds on their $L^p_{loc}$ norms.

Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

Relationships between quantized algebras and their semiclassical limits

A Poisson ℂ-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a ℂ[[ħ]]-algebra Q = R[[ħ]] with star product * such that for any a,b Є R ⊆ Q, a*b = ab + B1(a,b)ħ + B2(a,b)ħ2 + … subject to {a,b}= ħ-1(a * b ‒ b * a)|ħ=0, … (**) where Bi : R ⨯ R → R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ħQ with Poisson bracket (**), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of ℂ then ħ - λ is a unit in Q and thus a so-called deformation of R, Q/(ħ - λ)Q, is trivial. Hence it seems that we need an appropriate 픽-subalgebra A of Q such that A contains all generators of Q, ħ є A and A is understandable at an algebraic point of view, where 픽 is a subring of C[[ħ]]. Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit. The results are illustrated by examples.

Semiclassical limits, Lagrangian states and coboundary equations

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.