The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

How close is the Dirichlet-to-Neumann (DtN) map to square root of corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot techniques involved can be traced back newly rediscovered manuscript Hörmander from 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates spectral asymptotics. In particular, we obtain results for DtN maps non-smooth boundaries Riemannian setting, operators Helmholtz equation differential forms.