Introduction  
The construction of a proper model for a quantum relativistic oscillator is of general interest in physics, as a first approximation to periodic dynamics with positive energy for systems in which the velocity of constituents is not small with respect to c. However, it is difficult to generalize to the relativistic case any quantum equation for bound states. For a quantum oscillator, a straigthforward approach is to attach a covariant potential, to a free relativistic particle, but this procedure does not provide the required properties for a QRHO. 

Relativistic Hermite polynomia  
A new approach was presented in 1991 (this was my pH D thesis) which solves the previous problems in a different and untrivial way. Covariance is satisfied by taking a model system of Einstein’s general relativity; interaction is build up as a curvature of the spacetime. The relevant model is the de Sitter space with a constant curvature. As shown by S. Weinberg in the book “Gravitation and Cosmology”, the geodesics of this metric take a remarkably simple form. These geodesics can be seen as a demonstration of the fact that a free particle in deSitter space may describe a relativistic oscillator. The extension of this observation to the quantum mechanical terms was developed in the paper by V. Aldaya, J. Bisquert and J. NavarroSalas, “The quantum relativistic harmonic oscillator: Generalized Hermite polinomials” Phys. Lett. A 156, 381385 (1991). The quantization procedure makes use of the relatively simple symmetries of this curved space time; wave functions (generalized Hermite polinomia) are obtained, and an algebra of operators results that generalizes both the free relativistic particle and the normal quantum oscillator, in the proper limits. 


Papers on the QRHO 
V. Aldaya, J. Bisquert, J. Guerrero and J. NavarroSalas 
V. Aldaya, J. Bisquert, R. Loll and J. NavarroSalas 
V. Aldaya, J. Bisquert and J. NavarroSalas 
V. Aldaya, J.A. de Azcárraga, J. Bisquert and J.M. Cerveró 

Other papers on the QRHO 
P. Blasiak, A. Horzela, E. Kapuscik Alternative Hamiltonians and Wigner quantization Journal of Optics B: Quantum and Semiclassical Optics, 5, S245 (2003) 
Dattoli G, Lorenzutta S, Maino G, et al. The generating function method and properties of relativistic Hermite polynomials NUOVO CIMENTO B 113, 553 (1998) 
J. Tang Coherent states and squeezed states of massless and massive relativistic harmonic oscillators Physics Letters A, 219, 33 (1996) 
Beckers J, Ndimubandi J From nonrelativistic to relativistic properties of quantum harmonic oscillators PHYS SCRIPTA 54, 9 (1996) 
NAGEL B THE RELATIVISTIC HERMITE POLYNOMIAL IS A GEGENBAUER POLYNOMIAL J MATH PHYS 35, 1549 (1994) 
GAZEAU JP RELATIVISTIC HARMONICOSCILLATOR AND SPACE CURVATURE PHYS LETT A 179, 67 (1993 
ZARZO A, MARTINEZ A THE QUANTUM RELATIVISTIC HARMONICOSCILLATOR  SPECTRUM OF ZEROS OF ITS WAVEFUNCTIONS J MATH PHYS 34, 2926 (1993) 