Asymptotic completeness in classical $N$-body systems

Abstract We consider a Schrödinger particle placed in an external electromagnetic field of the form typical for scattering settings in the field theory: $$F=F^\mathrm {ret}+F^\mathrm {in}=F^\mathrm {adv}+F^\mathrm {out}$$ F = F ret + F in = F adv + F out , where the current producing $$F^{\mathrm {ret}/\mathrm {adv}}$$ F ret / adv has the past and future asymptotes homogeneous of degree $$-3$$ - 3 , and the free fields $$F^{\mathrm {in}/\mathrm {out}}$$ F in / out are radiation fields produced by currents with similar asymptotic behavior. We show that with appropriate choice of electromagnetic gauge the particle has ‘in’ and ‘out’ states reached with no further modification of the asymptotic dynamics. We use a special quantum mechanical evolution ‘picture’ in which the free evolution operator has well-defined limits for $$t\rightarrow \pm \infty $$ t → ± ∞ , and thus the scattering wave operators do not need the free evolution counteraction. The existence of wave operators in this setting is established, but the proof of asymptotic completeness is not complete: more precise characterization of the asymptotic behavior of the particle for $$|\mathbf {x}|=|t|$$ | x | = | t | would be needed.

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Asymptotic completeness for $N \leq 4$ particle systems with the Coulomb-type interactions

Duke Mathematical Journal ◽

10.1215/s0012-7094-93-07110-4 ◽

1993 ◽

Vol 71 (1) ◽

pp. 243-298 ◽

Cited By ~ 5

Author(s):

I. M. Sigal ◽

A. Soffer

Keyword(s):

Asymptotic Completeness ◽

Particle Systems ◽

Coulomb Type

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Breakdown of Regularity of Scattering for Mass-Subcritical NLS

International Mathematics Research Notices ◽

10.1093/imrn/rnaa072 ◽

2020 ◽

Author(s):

Gyu Eun Lee

Keyword(s):

Initial Data ◽

Wave Operator ◽

Schrodinger Equation ◽

Scattering Problem ◽

Asymptotic Completeness ◽

Nonlinear Schrodinger Equation ◽

Mild Form ◽

Scattering State ◽

Subcritical Nonlinearity ◽

Class C

Abstract We study the scattering problem for the nonlinear Schrödinger equation $i\partial _t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $\Sigma$ holds and the wave operator is well defined on $\Sigma$. We show that there exists $0<\beta <p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta }$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.

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