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Splitting Quantum Graphs

Nathaniel Smith, Alim Sukhtayev

Abstract

We derive a counting formula for the eigenvalues of Schrödinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue problems into smaller subgraph eigenvalue problems. These methods provide a simple way to calculate the spectra of operators with localized potentials.

Splitting Quantum Graphs

Abstract

We derive a counting formula for the eigenvalues of Schrödinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue problems into smaller subgraph eigenvalue problems. These methods provide a simple way to calculate the spectra of operators with localized potentials.
Paper Structure (21 sections, 12 theorems, 195 equations, 7 figures)

This paper contains 21 sections, 12 theorems, 195 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notations
  3. Summary of Paper
  4. General Resolvent Formula
  5. Motivation for Using the Resolvent
  6. Complementary Homogeneous Solution
  7. Particular Solution
  8. Constant Tuning
  9. From Resolvents to Maps
  10. Eigenvalue Counting Formulas
  11. Single Split
  12. $M_1$ Construction
  13. $M_2$ Construction
  14. An Evans Function Relation
  15. Double Split on One Wire
  16. ...and 6 more sections

Key Result

Theorem 1.5

Suppose $\Omega$ is split into $\Omega_1$ and $\Omega_2$ at some non-vertex point $s_1$, and let $\Gamma_1$ and $\Gamma_2$ be the new boundary conditions as constructed in Remark rem:extraops. Additionally, suppose $\lambda\in\rho(H^{\Omega_1}_{\Gamma_1})\cap \rho(H^{\Omega_2}_{\Gamma_2})$, and let where $E^\Omega_\Gamma$, $E^{\Omega_1}_{\Gamma_1}$, and $E^{\Omega_2}_{\Gamma_2}$ are the Evans fun

Figures (7)

  • Figure 1: A partition of $\Omega$ into $\Omega_1$ and $\Omega_2$ at $s_1$.
  • Figure 2: Two example partitions of $\Omega$ into three subgraphs.
  • Figure 3: Splitting $\Omega$ into three subgraphs with two cuts on one wire.
  • Figure 4: Splitting $\Omega$ into three subgraphs with two cuts on seperate wires.
  • Figure 5: A single split with $s_1=\frac{1}{3}$ and $\nu = -10$ for $\lambda\in [5,60]$. The red curve is $10E^{\Omega_2}_{\Gamma_2}$, the blue is $15E^{\Omega_1}_{\Gamma_1}$, the purple curve is $\frac{1}{25}(M_1 +M_2)$, and the black curve is $E$. These objects are vertically rescaled to accentuate the zeros and poles.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 29 more