Discrete Dynamics in Nature and Society
Volume 4 (2000), Issue 4, Pages 297-308

Discretized representations of harmonic variables by bilateral Jacobi operators

Andreas Ruffing

Zentrum Mathematik, Technische Universität München, Arcisstrasse 21, München D-80333, Germany

Received 10 September 1999

Copyright © 2000 Andreas Ruffing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for a q-oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non-interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.