Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

TL;DR

This paper explores how the nodal deficiency of eigenfunctions relates to the spectral flow of Dirichlet-to-Neumann maps, providing explicit descriptions for Schrödinger operators on rectangles and mechanisms affecting eigenfunction contributions.

Contribution

It explicitly describes the spectral flow for Schrödinger operators with separable potentials on rectangles and identifies mechanisms influencing eigenfunction contributions to nodal deficiency.

Findings

Spectral flow encodes nodal deficiency information.

Explicit description of spectral flow for rectangular domains.

Mechanisms determine eigenfunction contribution to nodal deficiency.

Abstract

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schr\"odinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.