We construct Fredholm equations for the resolvent of a three-body Hamiltonian with longrange interactions at energies which are negative and different from those of the two-body thresholds. Furthermore we show that the homogeneous equations have a non-trivial solution at the energy E, iff E is an eigenvalue of H. i: 1988 Academic Press, Inc.
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While qualitative features like asymptotic completeness are well understood in 3-body and even in n-body scattering theory (see [12,18,19]), the calculation of physical quantities like scattering and rearrangement amplitudes is still a difficult problem.
The problem of finding Fredholm equations for the resolvent of a three-body Hamiltonian was first solved by Faddeev [ 1 l] under suitable short-range conditions. However, the integral kernels of the Faddeev equations have no limit on the real axis in the case of Coulomb-interactions.
This case was considered on a formal level by Alt, Sandhas, and Ziegelmann [l] and by Vesselova [2] below the three-particle threshold (i.e., negative total energy). They have shown that the most singular terms of the integral kernels are two-body Lippman-Schwinger kernels. The aim of this paper is to construct equations for the three-body resolvent on suitable Banach spaces for negative energy and to prove their Fredholm property in a mathematically exact way. Since a simple squaring of the Faddeev equations as in [2] does not work and since a rigorous treatment of the equations in [ 1 ] would be more complicated, we introduce equations that differ from those in [ 1, 23. However, they are based on the Faddeev equations also. We consider potentials that decrease as /XI 'I2 -" at infinity, E > 0.
For negative energy the long-range problem can be reduced to a 2-body problem. The physical reason is that then the scattering channel with 3 free particles is closed. This does not remain valid for positive energy. In this situation our approach (essentially 2-body operators as input for the equations) breaks down. 54