Interfaces in spectral asymptotics and nodal sets

Abstract

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.