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

Caudrelier, V. & Crampe, N. (2007). Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity. Reviews in Mathematical Physics (rmp), 19(04), pp. 349-370. doi: 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 a stationary state asymmetrically. This could have interesting applications in condensed matter physics.

Publication Type: Article
Additional Information: Electronic version of an article published as V. CAUDRELIER and N. CRAMPÉ, Rev. Math. Phys. 19, 349 (2007). DOI: 10.1142/S0129055X07002973 © copyright World Scientific Publishing Company, Reviews in Mathematical Physics.
Publisher Keywords: Coordinate Bethe ansatz, one-dimensional bosons, delta interactions, impurity, Bethe equations
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Departments: School of Science & Technology > Mathematics
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