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

Abstract

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of 〈∏l = 1 n|cos φ1 - cos θl||cos φ 2 - cos θl|〉, 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 〈∏l = 1 n|cos φ - cos θl|〉. By changing variables this average in turn is a special case of the function of t obtained by averaging ∏l=1n|t-xl|2q over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a dua..