Document Type

Theses, Ph.D

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This item is available under a Creative Commons License for non-commercial use only

Disciplines

1.1 MATHMATICS

Publication Details

Successfully submitted for the award of Doctor of Philosophy (Ph.D) to the Dublin Institute of Technology, October, 2011.

Abstract

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), [17], but it is useful to include this case as it clarifies the more general case. In particular, the more algebraic approach used for the n-particle case is illustrated in this simple example. In Chapter 3 the case n = 3 is examined. This is useful for illustrative purposes as the scattering states can still be written explicitly term by term and it is not yet necessary to introduce the complicated notation used in the general case. On the other hand, this case shows up certain technical difficulties to do with the non-commutativity of the permutation group (S3) which do not occur in the 2-particle case. Finally, Chapter 4 contains the proof of the completeness relation in the general n-particle case. The method used is the same as in the 3-particle case, but the algebra is much more complicated. In particular a number of interesting lemmas and one theorem is proved. The first lemma for 3-particle case and its generalisation - theorem for n-particle case essentially concerns the Yang-Baxter relation for this model, as first written by Yang. Indeed, Yang proposed his version of these relations as a consistency condition for the Bethe Ansatz solution of the model but never actually gave a complete proof of the consistency given these relations. Here a complete inductive proof is given. Some algebraic manipulation reduces the left-hand side of the completeness relation to a simpler form. Another lemma, which seems to be new, then shows that this expression does not contain divergent terms and consists of a sum of integrals similar to those encountered in the 2- and 3-particle cases. Evaluation of these integrals then leads to the required _-relation.

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