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Quantum graph resonances by cut-off technique

Pavel Exner, Jiří Lipovský, Jan Pekař

TL;DR

The paper addresses identifying resonances in quantum graphs composed of a compact core and semi-infinite leads by using a cut-off technique. It establishes a general link between the spectra of the finite cut-off graph $ ext{Gamma}_L$ and the resonances of the full graph, replacing the exterior factor $ik$ with $- ext{cot}(kL)$ in the spectral condition and expressing the resulting determinant as $ ilde{F}(k)= sum_{j=0}^n(- ext{cot}(kL))^j c_j(k)$. Resonances are interpreted as poles of $F^{-1}$ and can be inferred from zeros of ${ m Re}igl(F(k)igr)$ and their crossings with $ ext{cot}(kL)$ or $ ext{tan}(kL)$ as $L$ varies, with the framework illustrated on cross-shaped and related graphs. The methodology provides a practical, rigorous route to detect and track resonances and their widths in quantum graphs, including convergence guarantees as the cut-off length grows.

Abstract

We demonstrate how resonances in a quantum graph consisting of a compact core and semi-infinite leads can be identified from the eigenvalue behavior of the cut-off system.

Quantum graph resonances by cut-off technique

TL;DR

The paper addresses identifying resonances in quantum graphs composed of a compact core and semi-infinite leads by using a cut-off technique. It establishes a general link between the spectra of the finite cut-off graph and the resonances of the full graph, replacing the exterior factor with in the spectral condition and expressing the resulting determinant as . Resonances are interpreted as poles of and can be inferred from zeros of and their crossings with or as varies, with the framework illustrated on cross-shaped and related graphs. The methodology provides a practical, rigorous route to detect and track resonances and their widths in quantum graphs, including convergence guarantees as the cut-off length grows.

Abstract

We demonstrate how resonances in a quantum graph consisting of a compact core and semi-infinite leads can be identified from the eigenvalue behavior of the cut-off system.
Paper Structure (3 sections, 1 theorem, 17 equations, 6 figures)

This paper contains 3 sections, 1 theorem, 17 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relating the cut-off graph spectra to resonances

Key Result

Proposition 3.1

Let $\Gamma$ consist of a compact part with $N$ edges and $M$ semi-infinite leads, and let $\Gamma_L$ be obtained from $\Gamma$ by cutting the leads to length $L$. The condition determining positive eigenvalues of the Laplacian on $\Gamma_L$ with coupling at $\mathcal{V}$ given by a matrix $U$ and D

Figures (6)

  • Figure 1: The loop graph with two external leads.
  • Figure 2: The cross-shaped graph.
  • Figure 3: The trajectory of the resonance pole in the lower complex half-plane starting at $k = 2\pi$ for the cross-shaped resonator with $\alpha = 1$. The color-coding (visible online) shows the dependence on $\lambda$ changing from blue ($\lambda = 1$) to red ($\lambda = 0$), the black points on the curve correspond to the values of $\lambda$ used in Figure \ref{['Resonances_lambdas_1_2']} and \ref{['Resonances_lambdas_2_2']}.
  • Figure 4: The numerical solutions of \ref{['Cross_cond_spec']} with $\alpha= 1$ for specific values of $\lambda$. The curves are colored (visible online) according to the value of derivative -- running between $-3$ and zero -- to highlight the resonances; points with derivative outside of the specified interval are colored black.
  • Figure 5: The numerical solutions of \ref{['Cross_cond_spec']} with $\alpha= 1$ for specific values of $\lambda$. The color coding is the same as in the previous figure.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 3.1
  • proof