  We show that non-round boundary points of the numerical range of an unbounded
operator (i.e. points where the boundary has infinite curvature) are contained
in the spectrum of the operator. Moreover, we show that non-round boundary
points, which are not corner points, lie in the essential spectrum. This
generalizes results of H\"ubner, Farid, Spitkovsky and Salinas and Velasco for
the case of bounded operators.
  We apply our results to non-selfadjoint Schr\"odinger operators, showing that
in this case the boundary of the numerical range can be non-round only at
points where it hits the essential spectrum.
