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Power series techniques for a special Schrödinger operator and related difference equations. (English) Zbl 1115.34002

The authors consider the ordinary differential equation \[ xy''(x)+(ax^2+b)y'(x)+(cx+d)y(x)=0, \quad x\in \mathbb R^+, \tag{1} \] where \(a,b,c,d\in \mathbb R\) are constant parameters. This ODE is a consequence of the radial Schrödinger equation for the radial wave function of the electron. The authors present two different approaches for solving the equation (1): separation method and power series method. They proved that the solutions are square integrable with respect to Gaussian weights. In addition the authors discuss the physical relevance of the obtained results.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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