Completeness of the Klein-Gordon oscillator eigenfunctions via Hermite and Laguerre polynomials

TL;DR

This paper proves the completeness of Klein-Gordon oscillator eigenfunctions in 1D and 3D using Hermite and Laguerre polynomials, simplifying the process compared to Dirac oscillators.

Contribution

It establishes the closure relations for Klein-Gordon eigenfunctions, leveraging properties of Hermite, Laguerre polynomials, and spherical harmonics.

Findings

Eigenfunctions form a complete basis in 1D and 3D.

Closure relations are verified using polynomial properties.

Scalar nature simplifies the proof compared to Dirac oscillators.

Abstract

Completeness of the Klein--Gordon oscillator eigenfunctions is proved in one and three spatial dimensions. The proofs establish the closure relations satisfied by the eigenfunctions and are based on standard properties of the Hermite and the generalized Laguerre polynomials, supplemented in three dimensions by the completeness of the spherical harmonics. The scalar nature of the Klein--Gordon field renders the argument strictly simpler than the analogous proof for the Dirac oscillator: no off-diagonal cancellation is required.