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

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.