Weakly bound states in 2+\epsilon dimensions

Abstract

We study the critical behaviour near the threshold where a first bound state appears at some value of coupling constant in an attractive short-range potential in $2+\epsilon $ dimensions. We obtain general expression for the binding energy near the threshold and also demonstrate that the critical region is correctly described by an effective separable potential. The critical exponent of the radius of weakly bound state is shown to coincide with the correlation length exponent for the spin model in the large-N limit. In two dimensions, where the binding energy is exponentially small in coupling constant, we obtain a general analytic expression for the prefactor.