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Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

P. J. Forrester·N. E. Frankel·T. M. Garoni
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Abstract

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of $<\prod_{l=1}^n| \cosϕ_1-\cosθ_l| |\cosϕ_2-\cosθ_l|>$, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit log-gas considerations imply that the average factorizes into the product of averages of the form $<\prod_{l=1}^n|\cosϕ-\cosθ_l>$. By changing variables this average in turn is a special case of the function of $t$ obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large $n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to $\sqrt{N}$.

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BibTeXRIS

P. J. Forrester, N. E. Frankel, T. M. Garoni. 2003-05-22. Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions. https://doi.org/10.1063/1.1599954

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