Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity

Details Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity

2005-01-06

arxiv.org/abs/cond-mat/0501110v2

Rev.Math.Phys.19:349-370,2007

Physics

Condensed Matter

Other Condensed Matter

20 pages, 5 figures, version accepted for publication: some typos corrected, references and comments added

Scientific paper

10.1142/S0129055X07002973

The one-dimensional problem of $N$ particles with contact interaction in the presence of a tunable transmitting and reflecting impurity is investigated along the lines of the coordinate Bethe ansatz. As a result, the system is shown to be exactly solvable by determining the eigenfunctions and the energy spectrum. The latter is given by the solutions of the Bethe ansatz equations which we establish for different boundary conditions in the presence of the impurity. These impurity Bethe equations contain as special cases well-known Bethe equations for systems on the half-line. We briefly study them on their own through the toy-examples of one and two particles. It turns out that the impurity can be tuned to lift degeneracies in the energies and can create bound states when it is sufficiently attractive. The example of an impurity sitting at the center of a box and breaking parity invariance shows that such an impurity can be used to confine asymmetrically a stationary state. This could have interesting applications in condensed matter physics.

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