Completeness of the Dirac oscillator eigenfunctions

Abstract

Completeness of the Dirac oscillator eigenfunctions is proved in one and three spatial dimensions. Proofs are based on standard properties of the Hermite and the generalized Laguerre polynomials.

Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: The case of the relativistic harmonic oscillator

“…Therefore, the 1 + 1 dimensional Dirac equation allows us to explore the physical consequences of the negative-energy states in a mathematically simpler and more physically transparent way. In this spirit the two-dimensional version of the anomalous magnetic-like interaction linear in the space coordinate has also received attention [24][25][26][27][28][29]. Later this system was shown to be a particular case of a more general class of exactly solvable problems [30].…”

Spectroscopy of the Dirac oscillator perturbed by a surface delta potential

“…(5), in this case φ(R + ∆R) = (−1) n φ(R − ∆R) and the surface δ potential is actually transparent. In the limit λ → 0, the eigenvalues of the unperturbed system are recovered [21]. They correspond to the poles of G, namely the set of values…”

The Peremptory Influence of a Uniform Background for Trapping Neutral Fermions With an Inversely Linear Potential

“…Indeed, the two-dimensional version of the anomalous magnetic-like interaction linear in the radial coordinate, christened by Moshinsky and Szczepaniak [3] as Dirac oscillator, has also received attention. Nogami and Toyama [4], Toyama et al [5] and Toyama and Nogami [6] studied the behaviour of wave packets under the influence of that conserving-parity potential whereas Szmytkowski and Gruchowski [7] proved the completeness of the eigenfunctions. More recently Pacheco et al [8] studied some thermodynamics properties of the 1+1 dimensional Dirac oscillator, and a generalization of the Dirac oscillator for a negative coupling constant was presented in Ref.…”