On the Asymptotic Completeness for Particles in Constant Electromagnetic Fields

Skibsted, Erik

doi:10.1007/978-1-4612-0775-7_19

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…Hence e −itα·k total should be called a magnetic translation of the center of charge. The Avron-Herbst type formula for e −itH(E) like (3) was already obtained by Skibsted [2]. In fact, he introduced…”

Section: Introductionmentioning

confidence: 92%

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Remarks on the Avron-Herbst Type Formula for N-body Quantum Systems in Constant Electric and Magnetic Fields

Adachi¹,

Kiyose²

2019

NHMP

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In this paper, for N ≥ 2, we give a natural derivation of the Avron-Herbst type formula for the time evolution generated by an N -body Hamiltonian with constant electric and magnetic fields. By virtue of the formula, some scattering problems can be reduced to those in the case where the constant electric and magnetic fields are parallel to each other. As an application of the formula, we give the result of the asymptotic completeness for the systems which have the only charged particle and some neutral ones in crossed constant electric and magnetic fields.

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Section: Introductionmentioning

confidence: 92%

“…The case where d = 3 and E = 0 can be also dealt with quite similarly. The case where d = 3 and E = 0 can be treated by the results due to Skibsted [2]. We impose the following condition (V 1) 2,SR on V , which is stronger than (V 0) 2 : R), and satisfies the decaying condition…”

Section: Applicationmentioning

confidence: 99%

Remarks on the Avron-Herbst Type Formula for N-body Quantum Systems in Constant Electric and Magnetic Fields

Adachi¹,

Kiyose²

2019

NHMP

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“…Constant electric field can also be included. Skibsted [153] reduced this problem to scattering of non-interacting subsystems all having the same charge/mass ratio.…”

Section: Multiparticle Scattering In a Magnetic Fieldmentioning

confidence: 99%

Recent developments in quantum mechanics with magnetic fields

Erdős¹

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. AMS 2000 Subject Classification 81Q10, 81Q70Running title: Recent developments on magnetic fields * Partially supported by EU-IHP Network "Analysis and Quantum" HPRN-CT-2002-0027 and by Harvard University relativistic theory. Although the framework for QED has been clear since the 30's, the mathematical difficulties even to formulate the theory rigorously have not yet been resolved. In the low energy regime, however, massive quantum particles can be described non-relativistically. Electric and magnetic fields, with a good approximation, can be considered decoupled. Since typical magnetic fields in laboratory are relatively weak, as a first approximation one can completely neglect magnetic fields and concentrate only on quantum point particles interacting via electric potentials.The rigorous mathematical theory of Schrödinger operators has therefore started with studying the operator H = 1 2m p 2 + V (x) on L 2 (R d ) and its multi-particle analogues. Here x ∈ R d is the location of the particle in the d-dimensional configuration space, p = −i∇ x is the momentum operator and m is the mass, that can be set m = 1 2 with convenient units. The Laplace operator describes the kinetic energy of the particle and the real-valued function V (x) is the electric potential. Although both the kinetic and potential energy operators are very simple to understand separately, their sum exhibits a rich variety of complex phenomena which differ from their classical counterparts in many aspects. The mathematical theory of this operator is the most developed and most extensive in mathematical physics: the best recent review is by Simon [152].As a next approximation, classical magnetic fields are included in the theory, but spins are neglected. The kinetic energy operator is modified from p 2 to (p + A) 2 by the minimal substitution rule: p → p − eA and we set the charge to be e = −1. Here A : R d → R d is the magnetic vector potential that generates the magnetic field B according to classical electrodynamics. In d = 2 or d = 3 dimensions B = ∇ × A is a scalar or a vector field, respectively. In d = 1 dimension the vector potential can be removed by a unitary gauge transformation, e iϕ (p + A) 2 e −iϕ = p 2 , ϕ = A, therefore magnetic phenomena in R 1 are absent (they are present in the case of S 1 ).We will call the operator (p + A) 2 + V the magnetic Schrödinger operator. In general, even the kinetic energy part contains noncommuting operators, [(p + A) k , (p + A) ℓ ] = 0, and the theory of (p + A) 2 itself is more complicated than that of p 2 + V . The simplest case of constant magnetic field, B = const, is explicitly solvable. The resulting Landau-spectrum consists, in two dimensions, of infinitely degenerate eigenvalues at energies (2n+1)|B|, n = 0, 1, . . .. Notice that the magnetic spectrum is characteristically different from that ...

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“…Skibsted [23,24] studied the scattering theory for N-body quantum systems in combined constant electric and magnetic fields. His result needs the asymptotic completeness for the systems in a constant magnetic field only.…”

Section: Introductionmentioning

confidence: 99%

On the Mourre Estimates for Three Body Schroedinger Operators in a Constant Magnetic Field

Adachi

2007

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On the Asymptotic Completeness for Particles in Constant Electromagnetic Fields

Cited by 5 publications

References 9 publications

Remarks on the Avron-Herbst Type Formula for N-body Quantum Systems in Constant Electric and Magnetic Fields

Remarks on the Avron-Herbst Type Formula for N-body Quantum Systems in Constant Electric and Magnetic Fields

Recent developments in quantum mechanics with magnetic fields

On the Mourre Estimates for Three Body Schroedinger Operators in a Constant Magnetic Field

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