Advances in Difference Equations
Volume 2005 (2005), Issue 2, Pages 109-118
doi:10.1155/ADE.2005.109
Power series techniques for a special Schrödinger operator and related difference equations
Department of Mathematics, Centre for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, Garching 85747, Germany
Received 27 July 2004; Revised 16 February 2005
Copyright © 2005 Moritz Simon and 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.
Abstract
We address finding solutions y∈𝒞2(ℝ+) of the special (linear) ordinary differential equation
xy″(x)+(ax2+b)y′(x)+(cx+d)y(x)=0 for all x∈ℝ+, where a,b,c,d∈ℝ 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 e−x2 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.