We introduce a generalized index for certain meromorphic, unbounded,
operator-valued functions. The class of functions is chosen such that energy
parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic
partial differential operators, particularly, non-self-adjoint Schr\"odinger
operators, on bounded Lipschitz domains, and abstract operator-valued
Weyl-Titchmarsh M-functions and Donoghue-type M-functions corresponding to
closed extensions of symmetric operators belong to it.
The principal purpose of this paper is to prove index formulas that relate
the difference of the algebraic multiplicities of the discrete eigenvalues of
Robin realizations of non-self-adjoint Schr\"{o}dinger operators, and more
abstract pairs of closed operators in Hilbert spaces with the generalized index
of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract
Weyl-Titchmarsh M-functions, respectively.Comment: A natural continuation of our arXiv:1512.06962 submissio
