FINITE RANK SINGULAR PERTURBATIONS AND DISTRIBUTIONS WITH DISCONTINUOUS TEST FUNCTIONS

Point interactions for the n-th derivative operator in one dimension a re investigated. Every such perturbed operator coincides with a selfad joint extension of the n-th derivative operator restricted to the set of functions vanishing in a neighborhood of the singular point. It is proven that the selfadjoint extensions can be described by the planes in the space of boundary values which are Lagrangian with respect to t he symplectic form determined by the adjoint operator. A distribution theory with discontinuous test functions is developed in order to dete rmine the selfadjoint operator corresponding to the formal expression L = (i d/dx)(n) + (n-1)Sigma(l,m=0) c(lm)delta((m))(.)delta((l)), c(lm ) = /c(ml), representing a finite rank perturbation of the n-th deriva tive operator with the support at the origin.