Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
We resolve both a conjecture and a problem of Z. Shen from the 90's regarding non-asymptotic bounds on the eigenvalue counting function of the magnetic Schrödinger operator L=-(∇-ia)+V with a singular or irregular magnetic field B on R, n≥3. We do this by constructing a new landscape function f...
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| Tipo de documento: | artigo |
| Data de publicação: | 2024 |
| País: | España |
| Recursos: | Universitat Autònoma de Barcelona |
| Repositório: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglês |
| OAI Identifier: | oai:ddd.uab.cat:292043 |
| Acesso em linha: | https://ddd.uab.cat/record/292043 https://dx.doi.org/urn:doi:10.1016/j.aim.2024.109665 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Landscape function Magnetic Schrödinger operator Spectral theory Weyl's law Schrödinger operator Eigenvalue counting |
| Resumo: | We resolve both a conjecture and a problem of Z. Shen from the 90's regarding non-asymptotic bounds on the eigenvalue counting function of the magnetic Schrödinger operator L=-(∇-ia)+V with a singular or irregular magnetic field B on R, n≥3. We do this by constructing a new landscape function for L, and proving its corresponding uncertainty principle, under certain directionality assumptions on B, but with no assumption on ∇B. These results arise as applications of our study of the Filoche-Mayboroda landscape function u, a solution to the equation Lu=-divA∇u+Vu=1, on unbounded Lipschitz domains in R, n≥1, and 0≤V∈L , under a mild decay condition on the Green's function. For L, we prove a priori exponential decay of Green's function, eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight 1/u, which may degenerate. Similar a priori results hold for L. Furthermore, when n≥3 and V satisfies a scale-invariant Kato condition and a weak doubling property, we show that 1/u is pointwise equivalent to the Fefferman-Phong-Shen maximal function m(⋅,V) (also known as Shen's critical radius function); in particular this gives a setting where the Agmon distance with weight 1/u is not too degenerate. Finally, we extend results from the literature for L regarding exponential decay of the fundamental solution and eigenfunctions, to the situation of irregular magnetic fields with directionality assumptions. |
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