Open Access

Power series techniques for a special Schrödinger operator and related difference equations

Advances in Difference Equations20052005:517967

DOI: 10.1155/ADE.2005.109

Received: 27 July 2004

Published: 31 May 2005

Abstract

We address finding solutions y ʗ2 (+) of the special (linear) ordinary differential equation xy''(x) + (ax2 + b) y' (x) + (cx + d)y(x) = 0 for all x +, where a, b, c, d are constant parameters. This will be achieved in three special cases via separation and a power series method which is specified using difference equation techniques. Moreover, we will prove that our solutions are square integrable in a weighted sense—the weight function being similar to the Gaussian bell in the scenario of Hermite polynomials. Finally, we will discuss the physical relevance of our results, as the differential equation is also related to basic problems in quantum mechanics.

Authors’ Affiliations

(1)
Department of Mathematics, Centre for Mathematical Sciences, Munich University of Technology

Copyright

© Simon and Ruffing 2005