  The Evans function is a well known tool for locating spectra of differential
operators in one spatial dimension. In this paper we construct a
multidimensional analogue as the modified Fredholm determinant of a ratio of
Dirichlet-to-Robin operators on the boundary. This gives a tool for studying
the eigenvalue counting functions of second-order elliptic operators that need
not be self-adjoint. To do this we use local representation theory for
meromorphic operator-valued pencils, and relate the algebraic multiplicities of
eigenvalues of elliptic operators to those of the Robin-to-Robin and
Robin-to-Dirichlet operator pencils. In the self-adjoint case we relate our
construction to the Maslov index, another well known tool in the spectral
theory of differential operators. This gives new insight into the Maslov index
and allows us to obtain crucial monotonicity results by complex analytic
methods.
