Geodesic
Quantum mechanics of one-dimensional motion in a~field with the singularity
Teoretičeskaâ i matematičeskaâ fizika, Tome 74 (1988) no. 2, pp. 247-258

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The one-dimensional motion of a particle in a field with singularity $\lambda|x|^{-\nu}$, $0\nu2$ and $\nu=2$, $-1/4\lambda3/4$ is investigated quantum mechanically. A physically acceptable self-adjoint extension of the Hamiltonian is found. A perturbation theory is constructed for a confining even smooth potential. It is shown that in this case matrix elements of the perturbation and Rayleigh–Schrödinger coefficients exist only for $\nu3/2$. A way of calculating transmission coefficients for an asymptotically free potential is found. Examples of exact solutions $\nu=1$ and $\nu=2$ are given. 
@article{TMF_1988_74_2_a10,
     author = {V. B. Gostev and A. R. Frenkin},
     title = {Quantum mechanics of one-dimensional motion in a~field with the singularity},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {247--258},
     publisher = {mathdoc},
     volume = {74},
     number = {2},
     year = {1988},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1988_74_2_a10/}
}
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V. B. Gostev; A. R. Frenkin. Quantum mechanics of one-dimensional motion in a~field with the singularity. Teoretičeskaâ i matematičeskaâ fizika, Tome 74 (1988) no. 2, pp. 247-258. http://geodesic.mathdoc.fr/item/TMF_1988_74_2_a10/