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

Patrizio Bifulco, Joachim Kerner

TL;DR

The paper investigates Schrödinger operators on finite metric graphs with δ'-coupling, deriving a modified local Weyl law and an explicit expression for the limiting mean eigenvalue distance between two δ'-realizations: $\lim_{N\to\infty} (1/N) \sum_{n=1}^N d_n(\beta,\beta') = (2/\mathcal{L}) \sum_{\mathsf v} \deg(\mathsf v)(1/\beta_{\mathsf v}-1/\beta'_{\mathsf v})$. It also provides a spectral comparison between δ'- and anti-Kirchhoff conditions, showing divergence in the mean spectral shift and thereby explaining prior numerical observations. The key methodological contribution is a modified local Weyl law tailored to δ'-couplings, combined with a Hellmann–Feynman/Hadamard-type perturbation analysis to obtain the asymptotic mean shifts, and extended results to include the potential term via $\frac{1}{\mathcal{L}} \int_{\mathcal{G}} q\,dx$. The results illuminate how graph topology, edge lengths, and vertex degrees govern eigenvalue statistics under singular couplings, with implications for spectral design on quantum graphs and related physics models.

Abstract

We study Schrödinger operators on compact finite metric graphs subject to $δ'$-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 $δ'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .

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

TL;DR

The paper investigates Schrödinger operators on finite metric graphs with δ'-coupling, deriving a modified local Weyl law and an explicit expression for the limiting mean eigenvalue distance between two δ'-realizations: . It also provides a spectral comparison between δ'- and anti-Kirchhoff conditions, showing divergence in the mean spectral shift and thereby explaining prior numerical observations. The key methodological contribution is a modified local Weyl law tailored to δ'-couplings, combined with a Hellmann–Feynman/Hadamard-type perturbation analysis to obtain the asymptotic mean shifts, and extended results to include the potential term via . The results illuminate how graph topology, edge lengths, and vertex degrees govern eigenvalue statistics under singular couplings, with implications for spectral design on quantum graphs and related physics models.

Abstract

We study Schrödinger operators on compact finite metric graphs subject to -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 -coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .
Paper Structure (9 sections, 5 theorems, 55 equations, 1 figure)

This paper contains 9 sections, 5 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The setting
  3. Main spectral comparison results
  4. A modified local Weyl law for $\delta'$-coupling conditions
  5. Proofs of the main spectral comparison results
  6. Proof of Theorem \ref{['MainResult1']}
  7. Proof of Theorem \ref{['MainResultII']}
  8. Some additional spectral comparison result
  9. Small-time asymptotics of the heat-kernel for bounded potentials

Key Result

Theorem 4

Let $\beta, \beta' \in (0,\infty)^{\vert \mathsf V \vert}$ and $H^q_\beta, H^q_{\beta'}$ be two corresponding self-adjoint Schrödinger operators subject to $\delta'$-coupling conditions on a finite compact, connected quantum graph $\mathcal{G}$ as described above. Then one has Furthermore, comparing the eigenvalues of $H^q_\beta, H^{q=0}_{\beta'}$ one obtains

Figures (1)

  • Figure 1: A bipartite graph with vertex set $\mathsf V = \mathsf V_1 \cup \mathsf V_1$ and $\beta(\mathcal{G}) = 1$.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 4: Comparison result $\delta^{\prime}$-conditions
  • Remark 5
  • Remark 6
  • Theorem 7: Comparison to anti-Kirchhoff conditions
  • Proposition 8: Local Weyl law for $\delta'$-coupling conditions
  • proof
  • Example 9: Spectral comparison on intervals
  • ...and 4 more