Exact and semiclassical approach to a class of singular integral operators arising in fluid mechanics and quantum field theory

TL;DR

This paper analyzes a class of singular integral operators relevant in fluid mechanics and quantum field theory, providing exact solutions for special cases and semiclassical approximations for others, linking them to Riemannian symmetric spaces.

Contribution

It introduces a unified approach to analyze these operators, connecting exact solutions to geometric structures and developing semiclassical methods for general cases.

Findings

Exact eigenfunction expansions for three special parameter values

Semiclassical approximation derived for general cases

Connection established between operators and Riemannian symmetric spaces

Abstract

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated to Riemannian symmetric spaces of rank one with positive, negative or vanishing curvature. For all other cases an accurate semiclassical approximation is derived, based on the identification of the operators with a peculiar Schroedinger-like operator.