P. Kurasov et J. Boman, FINITE RANK SINGULAR PERTURBATIONS AND DISTRIBUTIONS WITH DISCONTINUOUS TEST FUNCTIONS, Proceedings of the American Mathematical Society, 126(6), 1998, pp. 1673-1683
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.