  Starting with an adjoint pair of operators, under suitable abstract versions
of standard PDE hypotheses, we consider the Weyl M-function of extensions of
the operators. The extensions are determined by abstract boundary conditions
and we establish results on the relationship between the M-function as an
analytic function of a spectral parameter and the spectrum of the extension. We
also give an example where the M-function does not contain the whole spectral
information of the resolvent, and show that the results can be applied to
elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.
