On the criteria of transversality and disjointness of nonnegative selfadjoint extensions of nonnegative symmetric operators

Abstract

We propose criteria of transversality and disjointness for the Friedrichs and Krein extensions of a nonnegative symmetric operator in terms of the vectors $\{ \varphi j , j \in J\}$ that form a Riesz basis of the defect subspace. The criterion is applied to the Friedrichs and Krein extensions of the minimal Schr¨odinger operator $\scr A$ d with point potentials. We also present a new proof of the fact that the Friedrichs extension of the operator $\scr A$ d is a free Hamiltonian.