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

Moritz Simon and Andreas Ruffing

 Department of Mathematics, Centre for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, Garching 85747, Germany
Received 27 July 2004; Revised 16 February 2005

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.