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 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. Laboratoire de Physique Mathématique, Université Montpellier II, Pl. E. Bataillon, 34095 Montpellier, France, [email protected] 2On leave of absence from Landau Institute for Theoretical Physics, ul.Kosygina 2, 117940 Moscow, Russia. 3Dipartimento di Fisica, Università di Parma, and I.N.F.N., Gruppo Collegato di Parma, 43100 Parma, Italy, [name]@unipr.it