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Spectral Multiplicity Bounds for Jacobi Operators on Star-Like Graphs

Netanel Levi, Tal Malinovitch

TL;DR

This work addresses how large the multiplicity of the singular continuous spectrum can be for Jacobi operators on star-like graphs. By combining generalized eigenfunction expansions, subordinacy theory, and Borel-transform techniques, the authors prove the singular continuous multiplicity satisfies $N(E)\le m-1$ for $\mu_{\text{sc}}$-a.e. $E$, and they construct explicit examples achieving equality. The argument hinges on showing all generalized eigenfunctions in the singular part must be subordinate, then exploiting a dimension-counting mechanism to rule out a full $m$-dimensional invariant subspace; a key contradiction arises from restricting to branches. The paper also discusses sharpness, extensions to spherically homogeneous trees, finite-range operators, and a general pasting framework for gluing self-adjoint operators, highlighting both the reach and limitations of the approach.

Abstract

We study the spectral multiplicity of Jacobi operators on star-like graphs with $m$ branches. Recently, it was established that the multiplicity of the singular continuous spectrum is at most $m$. Building on these developments and using tools from the theory of generalized eigenfunction expansions, we improve this bound by showing that the singular continuous spectrum has multiplicity at most $m-1$. We also show that this bound is sharp, namely, we construct operators with purely singular continuous spectrum of multiplicity $m-1$.

Spectral Multiplicity Bounds for Jacobi Operators on Star-Like Graphs

TL;DR

This work addresses how large the multiplicity of the singular continuous spectrum can be for Jacobi operators on star-like graphs. By combining generalized eigenfunction expansions, subordinacy theory, and Borel-transform techniques, the authors prove the singular continuous multiplicity satisfies for -a.e. , and they construct explicit examples achieving equality. The argument hinges on showing all generalized eigenfunctions in the singular part must be subordinate, then exploiting a dimension-counting mechanism to rule out a full -dimensional invariant subspace; a key contradiction arises from restricting to branches. The paper also discusses sharpness, extensions to spherically homogeneous trees, finite-range operators, and a general pasting framework for gluing self-adjoint operators, highlighting both the reach and limitations of the approach.

Abstract

We study the spectral multiplicity of Jacobi operators on star-like graphs with branches. Recently, it was established that the multiplicity of the singular continuous spectrum is at most . Building on these developments and using tools from the theory of generalized eigenfunction expansions, we improve this bound by showing that the singular continuous spectrum has multiplicity at most . We also show that this bound is sharp, namely, we construct operators with purely singular continuous spectrum of multiplicity .

Paper Structure

This paper contains 13 sections, 16 theorems, 41 equations, 2 figures.

Table of Contents

  1. Intro
  2. Preliminaries
  3. Notation and setting
  4. Generalized eigenfunction expansion and direct integrals
  5. Subordinacy theory
  6. The Borel transform
  7. Proof of Theorem \ref{['main_thm']}
  8. Getting the contradiction
  9. Examples and further discussion
  10. Sharpness
  11. Spherically homogeneous trees and multiplicity of the spectrum
  12. Finite range operator on star-like graph
  13. General setting - pasting self-adjoint operators

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a star-like graph with $m$ branches and $J$ be a Jacobi operator on it, then for $\mu_{\text{sc}}$-almost every $E\in\mathbb{R}$, $N\left(E\right)\leq m-1$. Furthermore, for every $m$ there is a star like graph $\mathcal{G}_m$ with $m$ branches such that for $\mu_{\text{sc}}$-al

Figures (2)

  • Figure 1: Three examples of star-like graphs. The dashed lines represent copies of $\mathbb{N}$. (a) is a star-like graph with $3$ branches, where the compact component can be taken to be any finite subgraph that contains the inner triangle. (b) is also called a star graph with $3$ branches. The compact component can be taken to be any finite subgraph that contains the vertex $o$ and two of its neighbors. (c) is a star-like graph with $12$ branches. Graph (c) is also called the trimming of a $4$-regular tree. In the language of Section \ref{['branching_section']}, the branching sequence for this graph is $\left(3,3,1,1,1,1,\ldots\right)$. This figure is taken from levi2023subordinacy.
  • Figure 2: A star graph with 6 branches. Here, the vertices of $\mathcal{G}_1$ are $\left\{v_1,v_2,v_3\ldots\right\}$ and the potential $b$ is given by $b\left(v_i\right)=V_0\left(i\right)$. In addition, $b\left(o\right)=0$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 33 more