The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r −β in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case β = 4 is elucidated. In general, whenever the parameter β is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.
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Dong, SH., Ma, ZQ. & Esposito, G. Exact Solutions of the Schrödinger Equation with Inverse-Power Potential. Found Phys Lett 12, 465–474 (1999). https://doi.org/10.1023/A:1021633411616
- quantum mechanics
- scattering states
- bound states