Discrete basis representation of Ursell operators

The inverse Laplace transform of the two- and three-particle Ursell operators are shown to be related to scattering kernels. For a three-particle system the kernel is identical to Faddeev’s connected kernel. For well-behaved potentials, these kernels are compact with the consequence that they have a discrete spectrum and can thus be expressed in terms of discrete spectral representations. This leads to a method fur the direct computation of Ursell operators and the corresponding cluster integrals.