The tunneling effect through Schwarzschild barrier for spin 1/2 particle, analytical and numerical study

For Dirac particle, the general mathematical and numerical study of the tunneling process through the effective potential barrier generated by Schwarzschild black hole geometry is done. The main accent is given to analytical construction of the exact solutions for the problem. The study is based on the use of eight Frobenius solutions of the relevant second order radial differential equations with the complicated structure of the singular points. We construct such solutions in explicit form and prove that the power series involved in them are converged in the whole physical region of the variable: from Schwarzschild radius to infinity. Results for tunneling effect significantly differ for two situations: one when the particle falls on the barrier from inside of the black hole and another when the particle falls from outside. Mathematical structure of the derived asymptotic relations is exact, however their further study is based on numerical summing the convergent series. In calculations, the tools of the Mathematica system are used.