Geodesic
Logarithmic corrections in a~quantization rule. The polaron spectrum
Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 1, pp. 78-93

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A nonlinear integrodifferential equation that arises in polaron theory is considered. The integral nonlinearity is given by a convolution with the Coulomb potential. Radially symmetric solutions are sought. In the semiclassical limit, an equation for the self-consistent potential is found and studied. The potential has a logarithmic singularity at the origin, and also a turning point at 1. The phase shifts at these points are determined. The quantization rule that takes into account the logarithmic corrections gives a simple asymptotic formula for the polaron spectrum. Global semiclassical solutions of the original nonlinear equation are constructed. 
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     author = {M. V. Karasev and A. V. Pereskokov},
     title = {Logarithmic corrections in a~quantization rule. {The} polaron spectrum},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1993_97_1_a5/}
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M. V. Karasev; A. V. Pereskokov. Logarithmic corrections in a~quantization rule. The polaron spectrum. Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 1, pp. 78-93. http://geodesic.mathdoc.fr/item/TMF_1993_97_1_a5/