Excitons in quantum wires

This paper deals with excitons in quantum wires. We first study these excitons as the limit of excitons in D dimensions when $D \rightarrow 1$. In order to do it, we have had to find a new resolution of the hydrogen atom Schrödinger equation: besides the fact that the usual resolution found in textbooks is not valid for D exactly equal to 1, it is, surprisingly enough, inconsistent since it relies on two hypergeometric functions which are not independent for the parameters of physical interest! In a second part, we write down the exact potential felt by the exciton relative motion along the wire in terms of the wire confinement. This allows a quite precise determination of the effective Coulomb potential for this 1D motion, which is of crucial importance to obtain a meaningfull finite value for the exciton ground state energy. In a last part, we study the dependence of the exciton energies on the wire area and anisotropy. While the quantitative results are here given for cylindrical and rectangular wires with infinite barriers, we show how they can easily be extended to any particular wire shape and barrier height