Foundations of Physics Letters

, Volume 12, Issue 5, pp 465–474

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

  • Shi-Hai Dong
  • Zhong-Qi Ma
  • Giampiero Esposito

DOI: 10.1023/A:1021633411616

Cite this article as:
Dong, S., Ma, Z. & Esposito, G. Found Phys Lett (1999) 12: 465. doi:10.1023/A:1021633411616


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.

quantum mechanicsscattering statesbound states

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • Shi-Hai Dong
    • 1
  • Zhong-Qi Ma
    • 2
    • 3
  • Giampiero Esposito
    • 4
    • 5
  1. 1.Institute of High Energy PhysicsBeijingPeople's Republic of China
  2. 2.World LaboratoryChina Center for Advanced Science and TechnologyBeijing
  3. 3.Institute of High Energy PhysicsBeijingPeople's Republic of China
  4. 4.INFN, Sezione di NapoliMostra d'OltremareNapoliItaly
  5. 5.Dipartimento di Scienze FisicheComplesso Universitario di Monte S. AngeloNapoliItaly