Wannier-Mott exciton formed by electron and hole separated in parallel quantum wires

We analyze a Wannier-Mott exciton in which the electron and hole are constrained to move in two separated and parallel quantum wires. We expand the electron-hole interaction potential in terms of multipoles by assuming that both the electron and hole experience transverse harmonic confinements in the $x$ and $y$ directions, both being in their respective transverse ground states. For the resulting $z$-dependent Schrödinger equation, we calculate in detail eigenenergies and eigenfunctions for the exciton ground and first excited states as functions of the transverse dimension of the wires and their separation distance. Also, we calculate the higher eigenenergies and eigenfunctions approximately by using a WKB formalism.