Open AccessFROBENIUS’ SOLUTIONS AND THE ANALYSIS OF THE TUNNELING EFFECT FOR SPIN 1/2 PARTICLE THROUGH THE SCHWARZSCHILD BARRIER
Author(s) -
Е. M. Оvsiyuk,
Ya. A. Voynova,
В. М. Редьков
Publication year - 2018
Publication title -
doklady nacionalʹnoj akademii nauk belarusi
Language(s) - English
Resource type - Journals
eISSN - 2524-2431
pISSN - 1561-8323
DOI - 10.29235/1561-8323-2018-62-3-274-280
Subject(s) - quantum tunnelling , power series , schwarzschild metric , schwarzschild radius , frobenius solution to the hypergeometric equation , rectangular potential barrier , dirac equation , series (stratigraphy) , physics , asymptotic expansion , gravitational singularity , quantum mechanics , mathematics , mathematical physics , mathematical analysis , classical mechanics , generalized hypergeometric function , paleontology , hypergeometric function , biology , gravitation , hypergeometric function of a matrix argument , general relativity
For a Dirac particle, the general mathematical study of the particle tunneling process through an effective potential barrier generated by the Schwarzschild black hole background is done. The study is based on the use of 8 Frobenius’ solutions of the related second-order differential equation with 3 regular and 2 irregular singularities of the rank 2. Solutions of the radial equations are constructed in explicit form, and the convergence of the involved power series is proved in the physical range f the variable (1, ). r∈ +∞ Results for the tunneling effect are significantly different for two situations: one when the particle falls on the barrier from the inside and another when the particle falls from the outside. The mathematical structure of the derived asymptotic relations is exact, however the analytical expressions for the involved convergent powers series are unknown, and a further study of penetration and reflection coefficients should be based on the numerical summation of the power series.