Asymptotic completeness for long-range many-particle systems with Stark effect. II

Abstract

We prove the existence and the asymptotic completeness of the Dollardtype modified wave operators for many-particle Stark Hamiltonians with long-range potentials.

Multidimensional Inverse Scattering in an Electric Field

“…In Herbst [11] and Simon [17] completeness in the two-body case was proven. In Adachi and Tamura [1] and in Herbst, Mo ller and Skibsted [12] completeness in the N-body case was proven. In the last two papers further references to contributions to twobody and multiparticle scattering are given.…”

“…Formula (1.13) shows that the proof that H 0 is essentially self-adjoint in the space of Schwartz reduces to the one in the twobody case. The Jacobi coordinates above are based in the pair of particles (1,2) in the sense that we have taken as the first coordinate ! 1 =x~2&x~1 the relative distance of the particles (1) and (2).…”

Schrödinger operators in the twentieth century

“…56 Once one has these estimates, they can be used to derive: ͑a͒ Sobolev estimates: As in the free case if V obeys the conditions of Theorem IV. 1…”

“…͑iii͒ At ␣ϭ 1 2 , (V L 2 ), we shift from there being a.c. spectrum for almost everywhere positive energy (␣Ͼ 1 2 ) to at least the possibility of very different spectrum.…”

“…249 Multiparticle completeness in electric fields has been studied by Herbst et al, 116 and Adachi and Tamura. 1 There is much literature on both constant and variable magnetic fields but an extensive review of it is beyond the scope of this paper. One can begin looking at the literature by consulting a series by Avron et al [19][20][21] and Chapter 6 of Cycon et al 53 and references therein.…”