Direct demonstration of the completeness of the eigenstates of the Schrodinger equation with local and non-local potentials bearing a Coulomb tail

TL;DR

This paper provides a rigorous and physically intuitive proof of the completeness of eigenstates for the Schrödinger equation with local and non-local potentials that include a Coulomb tail, addressing a longstanding challenge in quantum mechanics.

Contribution

It introduces a novel method to demonstrate the completeness of eigenstates for Schrödinger potentials with Coulomb tails, including non-local cases, avoiding complex measure theory.

Findings

Proof handles Coulomb tails with finite-range non-Coulomb parts

Method applies to non-local potentials

Demonstration is physically transparent and rigorous

Abstract

Demonstrating the completeness of wave functions solutions of the radial Schrodinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrodinger potentials. One can mention in particular unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided the non-Coulomb part of potentials vanishes after a finite radius. Non-locality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k = 0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.