Báo Cáo Báo cáo hóa học POWER SERIES TECHNIQUES FOR A SPECIAL SCHRÖDINGER OPERATOR AND RELATED DIFFERENCE EQ

POWER SERIES TECHNIQUES FOR A SPECIAL SCHRÖDINGER OPERATOR AND RELATED DIFFERENCE EQUATIONS MORITZ SIMON AND ANDREAS RUFFING Received 27 July 2004 and in revised form 16 February 2005

We address finding solutions y ∈ 2 (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 2 function being similar to the Gaussian bell eưx 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.

1. Motivation via quantum mechanics In quantum mechanics, when considering the two-dimensional hydrogen atom in a strong