Spiraling attractors and quantum dynamics for a class of long-range magnetic fields
type: Item
instance of: scholarly article
title: Spiraling attractors and quantum dynamics for a class of long-range magnetic fields
zbMATH Open document ID: 1125.35069
author: Ira Herbst
author: Horia D. Cornean
author: E. Skibsted
published in: Journal of Functional Analysis
publication date: 2007-06-26T00:00:00Z
review text: In this paper one studies the classical and quantum dynamics of a two-dimensional particle in a magnetic field of the form \[ \mathbb{B}(x)=B(r, \theta)dx_1 \wedge dx_2,\;x=(r\cos\theta,r\sin\theta)\in\mathbb{R}^2, \] where \(B\) is smooth, \(B(r,\theta)=b(\theta)/r\) for \(r\geq 1\), \(b(\theta)\) nonnegative. the associated transverse magnetic vector potential is \(a(x)=(-\sin\theta,\cos\theta)b (\theta)\), \(r\geq 1\). The corresponding classical Hamiltonian is (in polar coordinates) \(h(r,\theta,\rho,\ell):= \rho^2/2+(\ell/r-b(\theta))^2/2\), where \(\rho=\frac{x}{|x|}\cdot(\xi-a)\) is the radial velocity and \(\ell=x_1p_2-x_2p_1\) is the canonical angular momentum. Define the extended configuration space \({\mathcal A}:= \{(t,r,\theta);\;t>0,r>0,\theta\in\mathbb{T}\}\) and introduce a symplectic form on \(T^*{\mathcal A}\) by \(d\theta\wedge d\ell+dr\wedge d\rho+dtrd\tau\). One can construct a solution \(S\) to the Hamilton-Jacobi equation \(\partial_tS+h(r, \theta;\partial_rS,\partial_\theta S)=0\) such that the associated Lagrangian manifold \({\mathcal L}\subset T^*{\mathcal A}\) is attractive for all energies above a certain threshold \[ E_d\leq E_b:= \max_{\theta\in[0,2\pi]}b^2(\theta)|_2. \] The main result of this paper is the construction, for that energy regime, of a simple approximate evolution based on this attractor, that completely describes the quantum dynamics of the system. More exactly, let \(H=\tfrac 12(p-a)^2\) be the corresponding quantum Hamiltonian. One can define a family of isometries \(L^2((E_d,\infty)\times\mathbb{T})<f\mapsto U_0(t)f\in L^2(\mathbb{R}_+\times \mathbb{T})\), \(t\geq 1\), depending on \(S\), such that if \({\mathcal H}_{E_d}k= {\mathbf 1}_{(E_d, \infty)\setminus\sigma_{pp}(H)}(H)L^2(\mathbb{R}_+\times \mathbb{T})\), then the following limits exist and define unitary operators which are mutually inverse: \[ \Omega_+^d=s-\lim_{t\to\infty}e^{itH}U_0(t): L^2((E_d,\infty)\times\mathbb{T})\to {\mathcal H}_{F_d}, \] \[ \Omega_…
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 81U05
zbMATH DE Number: 5167463
zbMATH Keywords: classical and quantum dynamics
zbMATH Keywords: long range magnetic fields
zbMATH Keywords: asymptotic completeness
zbMATH Keywords: attracting periodic orbit
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.04.032
OpenAlex ID: W2048751916
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DOI: 10.1016/J.JFA.2006.04.032
description: scientific article; zbMATH DE number 5167463