  We obtain a complete asymptotic expansion for the eigenvalues of the
Dirichlet-to-Neumann maps associated with Schr\"odinger operators on compact
Riemannian surfaces with boundary. For the zero potential, we recover the
well-known spectral asymptotics for the Steklov problem. For nonzero
potentials, we obtain new geometric invariants determined by the spectrum of
what we call the parametric Steklov problem. In particular, for constant
potentials parametric Steklov problem, the total geodesic curvature on each
connected component of the boundary is a spectral invariant. Under the constant
curvature assumption, this allows us to obtain some interior information from
the spectrum of these boundary operators.
