Interfaces in spectral asymptotics and nodal sets

TL;DR

This survey reviews joint work on spectral interfaces in Schrödinger operators and Toeplitz Hamiltonians, focusing on the transition of asymptotics across hypersurfaces in physical and phase space, with new insights into Airy and Erf type behaviors.

Contribution

It provides a comprehensive overview of the detailed transition of spectral asymptotics across interfaces in various geometric and quantum settings, including new results in Bargmann-Fock space.

Findings

Asymptotics are of Airy type in Schrödinger setting.

Asymptotics are of Erf (Gaussian error function) type in Kähler setting.

Introduces interface asymptotics in Bargmann-Fock space.

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.