Note on an integral expression for the average lifetime of the bound state in 2D

TL;DR

This paper derives an exact expression for the average lifetime of a bound state in a 2D diffusion system by interpreting a divergent integral as a Stieltjes transform and calculating its finite part.

Contribution

It introduces a novel approach to evaluate divergent integrals in diffusion problems using Stieltjes transforms, providing exact results for bound state lifetimes.

Findings

Exact expression for the average lifetime derived

Divergent integral interpreted as a Stieltjes transform

Finite part of the integral calculated analytically

Abstract

Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was used to derive an exact expression for the average lifetime of the bound state. Here, we show that the corresponding divergent integral may be considered as the formal limit of a Stieltjes transform. Upon analytically calculating the Stieltjes transform one can obtain an exact expression for the finite part of the divergent integral and hence for the average lifetime.