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Minimal Spacing of Eigenvalues on Fractals

Bernard Akwei

Abstract

On the unit interval (I), and the Sierpinski Gasket ($\mathcal{SG}$), the spectral decimation function of the Laplacian has similar properties that result in positive minimum spacing of eigenvalues. Other fractals, for example the level-3 Sierpinski Gasket, $\mathcal{SG}_3$, may not necessarily enjoy these properties. Our goal is to obtain an easy and sufficient criterion for positive infimum spacing of eigenvalues in the spectrum based on the properties of the spectral decimation function for the appropriate fractal. We also give a sufficient condition for zero infimum spacing of eigenvalues.

Minimal Spacing of Eigenvalues on Fractals

Abstract

On the unit interval (I), and the Sierpinski Gasket (), the spectral decimation function of the Laplacian has similar properties that result in positive minimum spacing of eigenvalues. Other fractals, for example the level-3 Sierpinski Gasket, , may not necessarily enjoy these properties. Our goal is to obtain an easy and sufficient criterion for positive infimum spacing of eigenvalues in the spectrum based on the properties of the spectral decimation function for the appropriate fractal. We also give a sufficient condition for zero infimum spacing of eigenvalues.

Paper Structure

This paper contains 8 sections, 4 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Approximations
  4. Spectral decimation
  5. Minimal Spacing of Eigenvalues
  6. Zero Infimum Spacing
  7. Positive Infimum Spacing
  8. Wielandt's Inequalities

Key Result

Theorem 2

Assume $\Delta$ admits spectral decimation with spectral decimation function $R$ and suppose $0,\zeta >0$ are fixed point of $R$ with $|R'(\zeta)|>\abs{R'(0)}>1$. Then $\inf\{|\lambda-\lambda'|:\lambda\neq\lambda',\lambda,\lambda'\in \sigma(\Delta)\}=0$.

Figures (1)

  • Figure 1: An image of $\mathcal{SG}_3$

Theorems & Definitions (12)

  • Definition 1
  • Theorem 2
  • proof
  • Example 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Example 6
  • Example 7
  • ...and 2 more