Bosons in the form of ultra cold alkali atoms can be confined to a one dimensional (1d) domain by the use of harmonic traps. This motivates the study of the ground state occupations λi of effective single particle states φi, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The λi and φi are the eigenvalues and eigenfunctions respectively of the one body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations λi are asymptotically proportional to √ N in both the circular and harmonically trapped cases.

No abstract

Okamoto has obtained a sequence of τ-functions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

With · denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study isfor I = (0, s) and I = (s, ∞), where χ (l) I = 1 for λ l ∈ I and χ (l) I = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown thatẼ N (I ; a, µ) is a τ -function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second-order second-degree differential equation, or in terms of the solution of certain difference equations. The cases µ = 0 and µ = 2 are of particular interest, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞),Ẽ N (I ; a, µ) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group, and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard-edge and softedge scaled limits ofẼ N (I ; a, µ). In particular, in the hard-edge scaled limit it is shown that the limiting quantityẼ hard ((0, s); a, µ) can be evaluated as a τ -function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.

Abstract:The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

The probabilities for gaps in the eigenvalue spectrum of finite N × N random unitary ensembles on the unit circle with a singular weight, and the related Hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second-order second-degree ordinary differential equations (ODEs) which are related to certain Painlevé-VI transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to Painlevé-V transcendents. Using transformations relating the second-degree ODE and transcendent we prove an identity for the scaled bulk limit which leads to a simple expression for the spacing probability density function. We also relate all the variables appearing in the Fredholm determinant formalism to particular Painlevé transcendents, in a simple and transparent way, and exhibit their scaling behaviour.

Abstract. The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle w(z) = m j=1 (z − z j (t)) ρ j , consisting of m ∈ Z >0 finite singularities, difference equations with respect to the bi-orthogonal polynomial degree n (Laguerre-Freud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables z j (t) (Schlesinger equations) are derived completely characterising the system.

The loop equation formalism is used to compute the 1/N expansion of the resolvent for the Gaussian β ensemble up to and including the term at O(N −6 ). This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at O(N −6 ). The latter contain non-integrable singularities at the endpoints of the support -we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings β = 1, 2 and 4 there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at β = 1 and 4, which complements the known third order linear differential equations for these quantities at β = 2.2010 Mathematics Subject Classification. 15B52; 60K35; 62E13; 33C45. 1 1 2 [δ(λ − 1) + δ(λ + 1)] −