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arXiv:2605.21791

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

Kevin Hernández

quant-phmath-phSubmitted May 20, 2026• Announced May 22, 2026
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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.

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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.
In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.
One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.