  Our main theorem is in the generality of the axioms of Hilbert space, and the
theory of unbounded operators. Consider two Hilbert spaces such that their
intersection contains a fixed vector space D. It is of interest to make a
precise linking between such two Hilbert spaces when it is assumed that D is
dense in one of the two; but generally not in the other. No relative
boundedness is assumed. Nonetheless, under natural assumptions (motivated by
potential theory), we prove a theorem where a comparison between the two
Hilbert spaces is made via a specific selfadjoint semibounded operator.
Applications include physical Hamiltonians, both continuous and discrete
(infinite network models), and operator theory of reflection positivity.
