Two-body short-range systems in a time-periodic electric field
type: Item
instance of: scholarly article
title: Two-body short-range systems in a time-periodic electric field
zbMATH Open document ID: 1010.81083
DOI: 10.1215/S0012-7094-00-10517-0
author: Jacob Schach Møller
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Scattering problem for two \(\nu\)-dimensional particles interacting through a short-range potential and placed in an external time-periodic electric field may be described by the Hamiltonian \[ H(t)={p_1^2 \over {2m_1}} + {p_2^2 \over {2m_2}} -q_1 {\mathcal E}(t)\cdot x_1 -q_2 {\mathcal E}(t)\cdot x_2 + v(x_2-x_1)\;\text{ on} L^2({\mathbb R}^{2\nu}), \] where \(m_i\) and \(q_i, i=1,2\), are the masses and the charges of the two particles, respectively and \(x_1,x_2\) denote their positions. The electric field \(\mathcal E\) is periodic with some period \(T\), i.e. \({\mathcal E}(t+T) = {\mathcal E}(t)\) almost everywhere. The potential \(v\) is short-range with weak singularities, periodic in time with the same period as the field. Periodicity enables to compactify the time space variable. The resulting Hamiltonian \(\widehat H\) is called Floquet Hamiltonian. The author proves that the Floquet Hamiltonian has no bound states. Moreover, it is proved that wave operators exist and are unitary. Absence of a singular continuous spectrum for \(\widehat H\) is also proved.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 81U05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 1820757
zbMATH Keywords: \(\nu\)-dimensional particles
zbMATH Keywords: short-range potential
zbMATH Keywords: Floquet Hamiltonian
zbMATH Keywords: no bound states
zbMATH Keywords: absence of singular continuous spectrum
MaRDI profile type: Publication
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full work available at URL: s0012-7094-00-10517-0
OpenAlex ID: W1530838159
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description: scientific article; zbMATH DE number 1820757