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

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.