Lattice two-body problem with arbitrary finite range interactions

Details Lattice two-body problem with arbitrary finite range interactions Lattice two-body problem with arbitrary finite range interactions

2010-01-21

arxiv.org/abs/1001.3805v2

Phys. Rev. A 81, 042102 (2010)

Physics

Condensed Matter

Other Condensed Matter

6 pages, 3 figures

Scientific paper

We study the exact solution of the two-body problem on a tight-binding one-dimensional lattice, with pairwise interaction potentials which have an arbitrary but finite range. We show how to obtain the full spectrum, the bound and scattering states and the "low-energy" solutions by very efficient and easy-to-implement numerical means. All bound states are proven to be characterized by roots of a polynomial whose degree depends linearly on the range of the potential, and we discuss the connections between the number of bound states and the scattering lengths. "Low-energy" resonances can be located with great precission with the methods we introduce. Further generalizations to include more exotic interactions are also discussed.

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