Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients

TL;DR

This thesis develops rigorous mathematical bounds for transmission, reflection, and Bogoliubov coefficients, which are crucial in quantum tunneling, black hole physics, and parametric oscillations, providing more precise understanding beyond approximation methods.

Contribution

It introduces new rigorous bounds for these coefficients across four distinct problems, advancing the mathematical tools available in quantum physics and black hole studies.

Findings

Established bounds on Bogoliubov coefficients

Derived bounds for Schwarzschild black hole greybody factors

Provided analytic bounds on transmission probabilities

Abstract

This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: 1) Bounding the Bogoliubov coefficients; 2) Bounding the greybody factors for Schwarzschild black holes; 3) Transformation probabilities and the Miller--Good transformation; 4) Analytic bounds on transmission probabilities.