The X-ray problem revisited

TL;DR

This paper revisits the X-ray problem by analyzing the Green's function of a core-hole as a Fredholm determinant of Wiener-Hopf operators, revealing universal constants in the asymptotics using advanced operator theory.

Contribution

It introduces a rigorous operator-theoretic approach to compute the large T asymptotics of the core-hole Green's function, overcoming limitations of classical singular integral methods.

Findings

Derived universal constants in the asymptotics of the Fredholm determinant.

Established a connection between the X-ray problem and Wiener-Hopf operator theory.

Provided a rigorous framework for analyzing core-hole dynamics in the X-ray problem.

Abstract

In this letter we re-visit the X-ray problem. Assuming point interaction between the conduction electrons and the first instantaneously created core-hole, the latter's Green's function can be represented as a Fredholm determinant of certain Wiener-Hopf operators acting on L^2(0,T) with discontinuous symbols. Here the symbols are the local conduction electron Green's function in the frequency domain and T is the time the core-hole spends in the system before removal. In this situation, the classical theory of singular integral equations usually employed in the literature to compute the large T asymptotics of the Fredholm determinant ceased to be applicable. A rigorous theory first put forward in the context of operator theory comes into play and universal constants are found in the aymptotics.