Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2020
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915267858137088 |
|---|---|
| author | Karpeshina, Yulia Parnovski, Leonid Shterenberg, Roman |
| author_facet | Karpeshina, Yulia Parnovski, Leonid Shterenberg, Roman |
| contents | We consider Schrödinger operators $H=-Δ+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[λ_*,+\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_05881 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators Karpeshina, Yulia Parnovski, Leonid Shterenberg, Roman Mathematical Physics Spectral Theory 35J10, 35P05 (primary), 47A13, 47A55, 47B93, 81Q10 (secondary) We consider Schrödinger operators $H=-Δ+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[λ_*,+\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way. |
| title | Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators |
| topic | Mathematical Physics Spectral Theory 35J10, 35P05 (primary), 47A13, 47A55, 47B93, 81Q10 (secondary) |
| url | https://arxiv.org/abs/2010.05881 |