Scalar products of elementary distributions

The Schroedinger equation with point interaction in one dimension is revisited in a simple framework where the ''singular'' potential is defined as a symmetric operator in a natural way. The main tool is a scalar product of the elementary distributions constructed after a commutative (and well-ordered) field extension of R followed by complexification. A contact with the hyper-real numbers that arise in nonstandard analysis is possible but not essential, our extensions of R and C being obtained by a quite elementary method