Inner products of resonance solutions in 1D quantum barriers

The properties of a prescription for the inner products of resonance (Gamow states), scattering (Dirac kets) and bound states for one-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other and to the Dirac kets, except when they are neighbours, in which case the inner product is divergent. Therefore, as it happens for the continuum scattering states, the norm of the resonant ones remains non-calculable. Thus, they exhibit properties halfway between the (continuum real) Dirac-δ orthogonality and the (discrete real) Kronecker-δ orthogonality of the bound states.