Inner products of resonance solutions in 1-D quantum barriers

TL;DR

This paper develops a regularization method for inner products of resonance, scattering, and bound states in 1D quantum barriers, revealing their orthogonality properties and the non-calculability of their norms.

Contribution

It introduces a Gaussian regularization for Gamow states' inner products, clarifying their orthogonality and divergence properties in 1D quantum barriers.

Findings

Most resonance states are orthogonal to bound states and Dirac kets.

Inner products of neighboring resonance states diverge.

Resonance states have properties between continuum and bound states.

Abstract

The properties of a prescription for the inner products of the resonance (Gamow states), scattering (Dirac kets), and bound states for 1-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, to the bound states, and to the Dirac kets, except when they are neighbors, 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 half way between the (continuum real) Dirac-delta orthogonality and the (discrete real) Kronecker-delta orthogonality of the bound states.