Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.14902 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909579252596736 |
|---|---|
| author | Pfeiffer, Paul Täufer, Matthias |
| author_facet | Pfeiffer, Paul Täufer, Matthias |
| contents | We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14902 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator Pfeiffer, Paul Täufer, Matthias Analysis of PDEs Mathematical Physics Optimization and Control 35Pxx, 35A23, 93B05, 82B44 We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model. |
| title | Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator |
| topic | Analysis of PDEs Mathematical Physics Optimization and Control 35Pxx, 35A23, 93B05, 82B44 |
| url | https://arxiv.org/abs/2309.14902 |