Geodesic
Inversion transformation in the Schr\"{o}dinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 423-431

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The equivalence of Schrödinger equations with different types of potentials is established on the basis of an inversion transformation. It is shown that at zero energy the equation with a spherical potential is equivalent to the equation with an axisymmetric potential of special form. Exact solutions are obtained to problems of the motion of a particle with zero total energy in the field of two “fish-eye” potentials and also in the field of a potential with two Coulomb singularities. 
@article{TMF_1984_60_3_a8,
     author = {Yu. N. Demkov and N. V. Semenova},
     title = {Inversion transformation in the {Schr\"{o}dinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {423--431},
     publisher = {mathdoc},
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     number = {3},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1984_60_3_a8/}
}
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Yu. N. Demkov; N. V. Semenova. Inversion transformation in the Schr\"{o}dinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 423-431. http://geodesic.mathdoc.fr/item/TMF_1984_60_3_a8/