Operators in divergence form and their Friedrichs and Kreĭn extensions

Abstract

For a densely defined nonnegative symmetric operator $\mathcal{A} = L_2^L_1$ in a Hilbert space, constructed from a pair $L_1 \subset L_2$ of closed operators, we give expressions for the Friedrichs and Kreĭn nonnegative selfadjoint extensions. Some conditions for the equality $(L_2^ L_1)^* = L_1^* L_2$ are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given