Finite rank singular perturbations and distributions with discontinuous test functions

Abstract:

Point interactions for the $n$-th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint 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 the symplectic form determined by the adjoint operator. A distribution theory with discontinuous test functions is developed in order to determine the selfadjoint operator corresponding to the formal expression \[ L=\left (i\frac d{dx}\right )^n+\sum ^{n-1}_{l,m=0}c_{lm}\delta ^{(m)}(\cdot ) \delta ^{(l)},\qquad c_{lm}=\overline {c_{ml}},\] representing a finite rank perturbation of the $n$-th derivative operator with the support at the origin.