General inner products for energy eigenstates

TL;DR

This paper develops a unified framework for inner products of energy eigenstates in 1D quantum systems, using a Gaussian regularization to encompass various boundary conditions and solution types.

Contribution

It introduces a Master Solution and Master Inner Product that generalize all types of energy eigenstate inner products in 1D quantum potentials.

Findings

Inner products are fully determined by boundary conditions.

Outgoing and Incoming conditions specify momentum locations in complex plane.

Unified framework applies to bound, anti-bound, and scattering states.

Abstract

The features of the inner products between all the types of real and complex-energy solutions of the Schr\"odinger equation for 1-dimensional cut-off quantum potentials are worked out using a Gaussian regularization. A general Master Solution is introduced which describes any of the above solutions as particular cases. From it, a Master Inner Product is obtained which yields all the particular products. We show that the Outgoing and the Incoming Boundary Conditions fully determine the location of the momenta respectively in the lower and upper half complex plane even for purely imaginary momenta (anti-bound and bound solutions).