FROBENIUS’ SOLUTIONS AND THE ANALYSIS OF THE TUNNELING EFFECT FOR SPIN 1/2 PARTICLE THROUGH THE SCHWARZSCHILD BARRIER
https://doi.org/10.29235/1561-8323-2018-62-3-274-280
Abstract
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.
About the Authors
E. M. Ovsiyuk
Mozyr State Pedagogical University named after I. P. Shamyakin.
Belarus
Belarus
Ovsiyuk Elena Mikhailovna – Ph. D. (Physics and Mathematics), Assistant Professor.
28, Studencheskaya Str., 247760, Mozyr.
Ya. A. Voynova
Minsk Suvorov Military School.
Belarus
Belarus
Voynova Yanina Aleksandrovna – Physics teacher.
29, Bogdanovich Str., 220029, Minsk.
V. M. Red’kov
B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.
Belarus
Belarus
Red’kov Viktor Mikhailovich – D. Sc. (Physics and Mathematics), Chief Researcher.
68-2, Nezavisimosti Ave., 220072, Minsk.
References
1. Regge T., Wheeler J. A. Stability of a Schwarzschild Singularity. Physical Review, 1957, vol. 108, no. 4, pp. 1063–1069. https://doi.org/10.1103/physrev.108.1063
2. Schwarzschild K. Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften. Sitzung vom 3. Februar 1916. pp. 189–196 (in German).
3. Chandrasekhar S. The Mathematical Theory of Black Holes. Oxford, Oxford University Press, 1983. 646 p.
4. Smoller J., Xie Chunjing. Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry. Annales Henri Poincare, 2012, vol. 13, no. 4, pp. 943–989. https://doi.org/10.1007/s00023-011-0145-9
5. Ovsiyuk E. M., Veko O. V., Rusak Yu. A., Chichurin A. V., Red’kov V. M. To Analysis of the Dirac and Majorana Particle Solutions in Schwarzschild Field. Nonlinear Phenomena in Complex System, 2017, vol. 20, no 1, pp. 56–72.
6. Red’kov V. M. Field particles in Riemannian space and the Lorentz group. Minsk, Belaruskaya Navuka Publ., 2009. 496 p. (in Russian).
7. Red’kov V. M. Tetrad formalism, spherical symmetry and Schrödinger basis. Minsk, Belaruskaya Navuka Publ., 2011. 496 p. (in Russian).
8. Ronveaux A. Heun’s Differential Equations. Oxford, Oxford University Press, 1995. 354 p.
9. Slavyanov S. Yu., Lay W. Special functions. A unified theory based on singularities. Oxford, Oxford University Press, 2000. 312 p.
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