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.
Highlights
Введем новую переменнуюОтсюда следует уравнение 2-го порядка для функции f (x) (отмечаем, что c = ε / M > 1)
Исходная идея, лежащая в основе данного анализа, появилась много лет назад в работе Редже и Уиллера [1]
A unified theory based on singularities / S
Summary
Отсюда следует уравнение 2-го порядка для функции f (x) (отмечаем, что c = ε / M > 1). 0, уравнение для функции g(x) следует из (2) при формальных заменах:. Введем сокращающие запись обозначения (ε 2 − M 2 ) / 4 = E, ε 2 − M 2 / 2 = E′, для дальнейшего уравнения (4) достаточно будет представить в кратком виде. (5a)–(5e)), находим уравнение для функции F (x). Минимальный радиус сходимости Rconv = 1 покрывает всю физическую область изменения переменной x ∈ (0,1), r ∈ (1, ∞). Наибольший интерес представляют решения F+ и F−: F+ , cosh Σπe −i2Γr∗ < −− > A∗e −i2Гr∗ + B ∗e +i2Γr∗ , F− , sinh Σπe −i2Γr∗ < −− > A∗e −i2Γr∗ − B ∗e +i2Γr∗ ; они описывают ситуацию, когда частица падает справа на гравитационный барьер, при этом частично отражаясь и частично проходя сквозь него.
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