Spiraling attractors and quantum dynamics for a class of long-range magnetic fields
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: \[ \Omega_+^d=s-\lim_{t\to\infty}e^{itH}U_0(t): L^2((E_d,\infty)\times\mathbb{T})\to {\mathcal H}_{F_d}, \] \[ \Omega_+=s-\lim_{t\to\infty}U^*_0(t)e^{-itH}:{\mathcal H}_{E_d} \to L^2((E_d,\infty) \times\mathbb{T}). \] One can also give an approximate dynamics at negative times.
scientific article; zbMATH DE number 5167463
scientific article