On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915283111772160 |
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| author | Levi, Netanel |
| author_facet | Levi, Netanel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators Levi, Netanel Spectral Theory 47B36 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. |
| title | On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators |
| topic | Spectral Theory 47B36 |
| url | https://arxiv.org/abs/2505.07129 |