Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913438467358720 |
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| author | Egidi, Michela Mugnolo, Delio Seelmann, Albrecht |
| author_facet | Egidi, Michela Mugnolo, Delio Seelmann, Albrecht |
| contents | We develop a Logvinenko--Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on $L^2$-norms of eigenfunctions -- and linear combinations thereof -- in terms of their $L^2$- and $W^{1,2}$-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_10441 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs Egidi, Michela Mugnolo, Delio Seelmann, Albrecht Spectral Theory Analysis of PDEs Functional Analysis Optimization and Control 34B45 (05C50 35P15 81Q35) We develop a Logvinenko--Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on $L^2$-norms of eigenfunctions -- and linear combinations thereof -- in terms of their $L^2$- and $W^{1,2}$-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs. |
| title | Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs |
| topic | Spectral Theory Analysis of PDEs Functional Analysis Optimization and Control 34B45 (05C50 35P15 81Q35) |
| url | https://arxiv.org/abs/2304.10441 |