A modified local Weyl law and spectral comparison results for $\delta'$-coupling conditions

TL;DR

This paper develops a modified local Weyl law for Schrödinger operators on metric graphs with -coupling, enabling explicit spectral comparisons between different boundary conditions and confirming divergence phenomena.

Contribution

It introduces a novel modified local Weyl law for -coupling and provides explicit formulas for spectral distance comparisons between boundary conditions.

Findings

Explicit expression for eigenvalue distance between self-adjoint realizations.

Spectral distance diverges when comparing -coupling to anti-Kirchhoff conditions.

Confirms numerical divergence observed in prior work.

Abstract

We study Schr\"odinger operators on compact finite metric graphs subject to $\delta'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $\delta'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .