  We present a way of defining the Dirichlet-to-Neumann operator on general
Hilbert spaces using a pair of operators for which each one's adjoint is
formally the negative of the other. In particular, we define an abstract
analogue of trace spaces and are able to give meaning to the
Dirichlet-to-Neumann operator of divergence form operators perturbed by a
bounded potential in cases where the boundary of the underlying domain does not
allow for a well-defined trace. Moreover, a representation of the
Dirichlet-to-Neumann operator as a first-order system of partial differential
operators is provided. Using this representation, we address convergence of the
Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of
the leading coefficients converge in the weak operator topology. We also
provide some extensions to the case where the bounded potential is not coercive
and consider resolvent convergence.
