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

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analysed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated with Riemannian symmetric spaces of rank 1 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.