Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908674348285952 |
|---|---|
| author | Hsu, Yueh-Sheng Labbé, Cyril |
| author_facet | Hsu, Yueh-Sheng Labbé, Cyril |
| contents | We propose a simple construction of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is $\mathbf{R}$. To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ Hsu, Yueh-Sheng Labbé, Cyril Probability Spectral Theory 35J10, 60H15 (Primary) 47A10 (Secondary) We propose a simple construction of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is $\mathbf{R}$. To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators. |
| title | Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ |
| topic | Probability Spectral Theory 35J10, 60H15 (Primary) 47A10 (Secondary) |
| url | https://arxiv.org/abs/2401.17900 |