Different factorizations of the relativistic finite-difference Schroedinger equation and q-oscillators

TL;DR

This paper explores various factorizations of the relativistic finite-difference Schrödinger equation, linking relativistic and nonrelativistic quantum mechanics through explicit transformations and generalizing the harmonic oscillator to a q-oscillator.

Contribution

It introduces new factorizations of the relativistic finite-difference Schrödinger equation and establishes a transformation connecting relativistic and nonrelativistic quantum mechanics.

Findings

Finite-difference generalizations of the harmonic oscillator are constructed.

Relativistic and nonrelativistic QM are shown to be different representations of the same theory.

An explicit transformation between these representations is derived.

Abstract

The concept of the one -- dimensional quantum mechanics in the relativistic configurational space (RQM) is reviewed briefly. The Relativistic Schroedinger equation (RSE) arising here is the finite-difference equation with the step equal to the Compton wave length of the particle. The different generalizations of the Dirac -- Infeld -- Hall factorizarion method for this case are constructed. This method enables us to find out all possible finite-difference generalizations of the most important nonrelativistic integrable case -- the harmonic oscillator. As it was shown (\cite{kmn},\cite{mir6}) in RQM the harmonic oscillator = $q$ -- oscillator. It is also shown that the relativistic and nonrelativistic QM's are different representations of the same theory. Thetransformation connecting these two representations is found in explicit form. It could be considered as the generalization of the Kontorovich -- Lebedev transformation.