M-functions for closed extensions of adjoint pairs of operators with   applications to elliptic boundary problems
  In this paper, we combine results on extensions of operators with recent results on the relation between the M-function and the spectrum, to examine the spectral behaviour of boundary value problems. M-functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that can not be detected by the M-function. 
B. M. Brown, G. Grubb and I. G. Wood