Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow

TL;DR

This paper investigates the limiting behavior of eigenfunctions of Sturm-Liouville operators with delta potentials as their strength varies, providing explicit formulas, characterizations, and numerical tools for analysis.

Contribution

It introduces explicit formulas and characterizations for eigenfunctions and eigenvalues across the spectral flow, extending previous spectral flow results.

Findings

Explicit formulas for limiting eigenfunctions

Characterization of eigenfunctions and eigenvalues for all spectral flow values

Development of spectrally accurate numerical tools

Abstract

We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.