2025/05/11 by Netanel Levi, Levi, Netanel · 1 citation
Mathematics · #47B36 #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2505.07129
openalex publication_date 2025/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.