Interfaces in Spectral Asymptotics and Nodal Sets

This is a survey of results obtained jointly with Boris Hanin and Peng Zhou on interfaces in spectral asymptotics, both for Schrödinger operators on L 2 ( ℝ d ) $$L^2(\mathbb {R}^d)$$ and for Toeplitz Hamiltonians acting on holomorphic sections of ample line bundles L → M over Kähler manifolds (M, ω). By an interface is meant a hypersurface, either in physical space ℝ d $$\mathbb {R}^d$$ 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 Schrödinger setting, the asymptotics are of Airy type; in the Kähler setting they are of Erf (Gaussian error function) type. A principal purpose of this survey is to compare the results in the two settings. Each is apparently universal in its setting. This is now established for Toeplitz operators, but in the Schrödinger setting it is only established for the simplest model operator, the isotropic harmonic oscillator. It is explained that the latter result is most comparable to the behavior of the canonical degree operator on the Bargmann-Fock space of a line bundle, a new construction introduced in these notes.