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On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators

Netanel Levi

TL;DR

This work addresses fractal continuity properties of spectral measures for one-dimensional Schrödinger operators, constructing explicit half-line and line examples that separate fractal dimensions between whole-line and half-line restrictions. By leveraging transfer-matrix analysis, subordinacy theory, and boundary behavior of Borel transforms, the authors realize a half-line operator with essential spectrum $[-2,2]$ whose restricted spectral measures have packing dimension $0$, a line operator with Hausdorff dimension $1$ on $[-2,2]$ yet zero-dimensional half-line restrictions, and a line operator with a positive-measure set $A$ that is invisible to all half-line boundary conditions. The results demonstrate sharp, decoupled fractal-continuity phenomena across line vs half-line restrictions and rank-one perturbations, shedding light on the interplay between spectral singularity structure and transfer-matrix dynamics in 1D quantum systems.

Abstract

We construct examples of one-dimensional Schrödinger operators that illustrate the subtle nature of fractal continuity properties of spectral measures. First, we present half-line operators whose spectral measures have packing dimension zero for all boundary conditions. Second, we exhibit a whole-line operator whose spectral measure has Hausdorff dimension one, while every half-line restriction (under any boundary condition) has spectral measure of Hausdorff dimension zero. Finally, for the same whole-line operator, we prove the existence of a Borel set that carries positive spectral measure, yet has measure zero with respect to the spectral measure of the positive half-line restriction for every boundary condition.

On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators

TL;DR

This work addresses fractal continuity properties of spectral measures for one-dimensional Schrödinger operators, constructing explicit half-line and line examples that separate fractal dimensions between whole-line and half-line restrictions. By leveraging transfer-matrix analysis, subordinacy theory, and boundary behavior of Borel transforms, the authors realize a half-line operator with essential spectrum whose restricted spectral measures have packing dimension , a line operator with Hausdorff dimension on yet zero-dimensional half-line restrictions, and a line operator with a positive-measure set that is invisible to all half-line boundary conditions. The results demonstrate sharp, decoupled fractal-continuity phenomena across line vs half-line restrictions and rank-one perturbations, shedding light on the interplay between spectral singularity structure and transfer-matrix dynamics in 1D quantum systems.

Abstract

We construct examples of one-dimensional Schrödinger operators that illustrate the subtle nature of fractal continuity properties of spectral measures. First, we present half-line operators whose spectral measures have packing dimension zero for all boundary conditions. Second, we exhibit a whole-line operator whose spectral measure has Hausdorff dimension one, while every half-line restriction (under any boundary condition) has spectral measure of Hausdorff dimension zero. Finally, for the same whole-line operator, we prove the existence of a Borel set that carries positive spectral measure, yet has measure zero with respect to the spectral measure of the positive half-line restriction for every boundary condition.
Paper Structure (12 sections, 24 theorems, 34 equations)

This paper contains 12 sections, 24 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectral measures of self-adjoint operators
  4. Hausdorff and packing measures
  5. Half-Line Operators
  6. Certain examples of half-line operators
  7. Operators with packing-dimension zero for every rank-one perturbation
  8. Operators with Hausdorff-dimension zero for every rank-one perturbation
  9. Line Operators
  10. Some general theory
  11. Proof of Theorem \ref{['main_thm_2']}
  12. Proof of Theorem \ref{['main_thm_3']}

Key Result

Theorem 1.1

There exists a half-line Schrödinger operator $H$ such that $\Sigma = \sigma_{\text{ess}}(H) = [-2,2]$, and for every $\theta \in [0,\pi)$, the restriction of $\mu_\theta$ to $[-2,2]$ has packing dimension zero.

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 39 more