‘Probabilistic’ approach to Richardson equations

Abstract

It is known that solutions of Richardson equations can be represented as stationary points of the ‘energy’ of classical free charges on the plane. We suggest considering the ‘probabilities’ of the system of charges occupying certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of ‘probability’ has similarities with the square of the Laughlin wavefunction. Next, we introduce the ‘partition function’, from which the ground state energy of the initial quantum-mechanical system can be determined. The ‘partition function’ is given by a multidimensional integral, which is similar to the Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval where potential acts. In this case, the ‘partition function’ is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields a quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insight into the physics of Cooper-paired states.