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

Moritz Simon,Andreas Ruffing

doi:10.1155/ade.2005.109

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.

Highlights

This equation is obtained by standard separation methods—decomposing the wave function into a radial and an angular part—as they are taught in every first course on quantum mechanics, see for instance [5]
There is a quite general power series method for solving second-order differential equations of “Bessel-type” as discussed above; we refer for instance to [1]
Provided that the condition (3.10) is satisfied, one obtains in the special cases the following integrability situation: (i) y1 is square integrable with respect to a Gaussian weight if b ≥ 1/2; (ii) y1 is square integrable with respect to a Gaussian weight if b > 0; (iii) y1 is square integrable with respect to a Gaussian weight if b > −1

Summary

Motivation via quantum mechanics

In quantum mechanics, when considering the two-dimensional hydrogen atom in a strong magnetic field, one obtains the following radial Schrodinger equation for the “radial wave function” ψ of the electron: ψ (x) +. This equation is obtained by standard separation methods—decomposing the wave function into a radial and an angular part—as they are taught in every first course on quantum mechanics, see for instance [5]. Spectra of hydrogen in strong magnetic fields play a role, for example, in the strongly magnetic white dwarf stars In all of these physically interesting situations, the mathematical models behind are related to the differential equation (1.1). Motivated by (1.2), we want to see in which (weighted) sense the solutions of (1.7) are square integrable

Analytically solvable special cases

Power series and related difference equations

Square integrability with respect to Gaussian weights

Physical interpretation and perspectives