Abstract:
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.
Citation:
M. V. Karasev, A. V. Pereskokov, “Logarithmic corrections in a quantization rule. The polaron spectrum”, TMF, 97:1 (1993), 78–93; Theoret. and Math. Phys., 97:1 (1993), 1160–1170