Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent 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.