Calogero–Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators

Abstract

We construct certain eigenfunctions of the Calogero–Sutherland Hamiltonian for particles on a circle, with mixed boundary conditions. That is, the behaviour of the eigenfunction, as neighbouring particles collide, depend on the pair of colliding particles. This behaviour is generically a linear combination of two types of power laws, depending on the statistics of the particles involved. For fixed ratio of each type at each pair of neighbouring particles, there is an eigenfunction, the ground state, with lowest energy, and there is a discrete set of eigenstates and eigenvalues, the excited states and the energies above this ground state. We find the ground state and special excited states along with their energies in a certain class of mixed boundary conditions, interpreted as having pairs of neighbouring bosons and other particles being fermions. These particular eigenfunctions are characterized by the fact that they are in direct correspondence with correlation functions in boundary conformal field theory. We expect that they have applications to measures on certain configurations of curves in the statistical O(n) loop model. The derivation, although completely independent of results of conformal field theory, uses ideas from the ‘Coulomb gas’ formulation.