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Journal of Mathematical Physics

Bose Gas in One Dimension. II. Orthogonality of the Scattering States

M. Gaudin

Published Aug 1, 1971• 21 citations
Publisher

Abstract

A direct calculation gives the normalization integral for a basis of scattering wavefunctions of interacting bosons in one dimension.
DOI
10.1063/1.1665791

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arXiv:cond-mat/0404590•Apr 26, 2004•1 citation
Orthogonality catastrophe in fermionic systems is well known: in the thermodynamic limit, the overlap between the ground state wavefunctions with and without a single local scattering potential approaches zero algebraically as a function of the particle number NNN. Here we examine the analogous problem for bosonic systems. In the homogeneous case, we find that ideal bosons display an orthogonality stronger than algebraic: the wavefunction overlap behaves as exp[−λN1/3]{\rm exp}[-λN^{1/3}]exp[−λN1/3] in three dimensions and as exp[−λN/ln⁡2N]{\rm exp}[-λN/\ln ^2 N]exp[−λN/ln2N] in two dimensions. With interactions, the overlap becomes finite but is still (stretched-)exponentially small for weak interactions. We also consider the cases with a harmonic trap, reaching similar (though not identical) conclusions. Finally, we comment on the implications of our results for spectroscopic experiments and for (de)coherence phenomena.