On technically solving an effective QCD-Hamiltonian

Abstract

By their very nature, field-theoretical Hamiltonians are derived in momentum representation. To solve the corresponding integro-differential equations is more difficult than to solve the simpler differential equations in configuration space (`Schr\"odinger equation'). For the latter many different and very effective methods have been developed in the past. But rather than to Fourier-transform to configuration space - which is not always easy - the equations are solved here directly in momentum space, by using Gaussian quadratures. Special attention is given to the case where the potential in configuration space is linear and where the corresponding momentum-space kernel has an almost intractable 1/(\vec k - \vec k')^4 -singularity. Its regularization requires a certain technical effort, introducing suitable counter terms. The method is numerically reliable and fast, faster than other methods in the literature. It should be useful to and also applicable in other approaches, including phenomenological Schr\"odinger-type equations.