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

Hagedorn, George A.

doi:10.1090/S0002-9947-1980-0554318-4

Asymptotic completeness for classes of two, three, and four particle Schrödinger operators
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by George A. Hagedorn

Trans. Amer. Math. Soc. 258 (1980), 1-75

DOI: https://doi.org/10.1090/S0002-9947-1980-0554318-4

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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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