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

Abstract

We address finding solutions y ∈ ʗ² (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 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.