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POWER SERIES TECHNIQUES FOR A SPECIAL SCHRÖDINGER OPERATOR AND RELATED DIFFERENCE EQUATIONS MORITZ

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. | POWER SERIES TECHNIQUES FOR A SPECIAL SCHRODINGER 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 e 2 R of the special linear ordinary differential equation xy x ax2 b y x cx d y x 0 for all x e R where a b c d e 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-x 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 magnetic field one obtains the following radial Schrodinger equation for the radial wave function y of the electron y x y x e mX - mx2 y x 0 x e R . 1.1 x x x2 4 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 . For a more analytic approach to the subject see for instance 4 . We briefly describe the motivation physicists gave us cf. 2 3 to consider further analytic properties of the differential equation 1.1 . In quantum physics the problem of hydrogen atoms in strong magnetic fields has a particular meaning one motivation comes from experimental physics that is from atomic spectroscopy one would like to understand the spectra of highly excited hydrogen atoms in the so-called superstrong magnetic fields. But also in astrophysics 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 .