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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic completeness for classes of two, three, and four particle Schrödinger operators

Author: George A. Hagedorn
Journal: Trans. Amer. Math. Soc. 258 (1980), 1-75
MSC: Primary 81F10; Secondary 35P25
MathSciNet review: 554318
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Abstract: Formulas for the resolvent $ {(z\, - \,H)^{ - 1}}$ are derived, where $ H\, = \,{H_0}\, + \,{\Sigma _{i < j}}{\lambda _{ij}}{V_{ij}}$ is an N particle Schrödinger operator with the center of mass motion removed. For a large class of two-body potentials and generic couplings $ \{ {\lambda _{ij}}\} $, these formulas are used to prove asymptotic completeness in the $ N\, \leqslant \,4$ body problem when the space dimension is $ m\, \geqslant \,3$. The allowed potentials belong to a space of dilation analytic multiplication operators which fall off more rapidly than $ {r^{ - 2 - \varepsilon }}$ at $ \infty $. In particular, Yukawa potentials, generalized Yukawa potentials, and potentials of the form $ {(1\, + \,r)^{ - 2 - \varepsilon }}$ are allowed.

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Article copyright: © Copyright 1980 American Mathematical Society