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Eigenvalue bounds for Schrödinger operators on quantum graphs with $δ$-coupling conditions

Duc Hoang Cao

TL;DR

This work addresses how to bound eigenvalues of Schrödinger operators on finite metric graphs with $\delta$-coupling at vertices. It develops a variational framework using linear combinations of Dirichlet- and Neumann-eigenfunctions to control the potential term and graph-induced geometry, encapsulated by the quantities $M(\Gamma)$ and $M_k(\Gamma)$. The authors prove sharp upper bounds for the principal eigenvalue and for higher eigenvalues, in regimes of large and small coupling strengths $\alpha$, and establish universal bounds as $\alpha\to\infty$ that weaken topology dependence. The results generalize and interpolate between known Neumann/Kirchhoff and Dirichlet bounds, are sharp in relevant limits, and provide quantitative tools for spectral analysis on quantum graphs.

Abstract

We prove sharp upper bounds for eigenvalues of Schrödinger operators on quantum graphs with $δ$-coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the strength of the couplings, and as the coupling strengths grow, the dependence on the topology gets weaker, answering a question of Rohleder and Seifert. We obtain those bounds via the variational characterisation, comparing with appropriate linear combinations of eigenfunctions with Dirichlet and Neumann vertex conditions.

Eigenvalue bounds for Schrödinger operators on quantum graphs with $δ$-coupling conditions

TL;DR

This work addresses how to bound eigenvalues of Schrödinger operators on finite metric graphs with -coupling at vertices. It develops a variational framework using linear combinations of Dirichlet- and Neumann-eigenfunctions to control the potential term and graph-induced geometry, encapsulated by the quantities and . The authors prove sharp upper bounds for the principal eigenvalue and for higher eigenvalues, in regimes of large and small coupling strengths , and establish universal bounds as that weaken topology dependence. The results generalize and interpolate between known Neumann/Kirchhoff and Dirichlet bounds, are sharp in relevant limits, and provide quantitative tools for spectral analysis on quantum graphs.

Abstract

We prove sharp upper bounds for eigenvalues of Schrödinger operators on quantum graphs with -coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the strength of the couplings, and as the coupling strengths grow, the dependence on the topology gets weaker, answering a question of Rohleder and Seifert. We obtain those bounds via the variational characterisation, comparing with appropriate linear combinations of eigenfunctions with Dirichlet and Neumann vertex conditions.

Paper Structure

This paper contains 6 sections, 9 theorems, 71 equations.

Table of Contents

  1. Introduction and Main Results
  2. Acknowledgements
  3. Notation and definitions
  4. Upper bounds for eigenvalues
  5. Upper bounds for the principal eigenvalue
  6. Upper bounds for higher eigenvalues

Key Result

Theorem 1.1

Let $\Gamma$ be a connected metric graph with Betti number $\beta$, i.e. $\beta:=|E|-|V|+1$, the length of the longest edge $\ell_{\max}$, and $P$ pendants, i.e. vertices of degree one. Suppose further that $\Gamma$ is not a cycle. We define Let $p\in [1,\infty]$ and $H_q$ be the Schrödinger operator on $\Gamma$ with potential $q\in L^p(\Gamma)$ and coupling strengths $\alpha_v\ge 0$ for all $v\i

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 6 more