  This is largely a survey of results obtained jointly with Boris Hanin and
Peng Zhou on interfaces in spectral asymptotics, both for Schr\"odinger
operators on $L^2({\mathbb R}^d)$ and for Toeplitz Hamiltonians acting on
holomorphic sections of ample line bundles $L \to M$ over K\"ahler manifolds
$(M, \omega)$. By an interface is meant a hypersurface, either in physical
space ${\mathbb R}^d$ or in phase space, separating an allowed region where
spectral asymptotics are standard and a forbidden region where they are
non-standard. The main question is to give the detailed transition between the
two types of asymptotics across the hypersurface (i.e. interface). In the real
Schr\"odinger setting, the asymptotics are of Airy type; in the K\"ahler
setting they are of Erf (Gaussian error function) type. In addition, we
introduce the Bargmann-Fock space of a positive Hermitian line bundle and study
interface asymptotics in that setting.
