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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Autor: Poggi, Bruno|||0000-0002-7992-8578
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
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spelling Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fieldsPoggi, Bruno|||0000-0002-7992-8578Landscape functionMagnetic Schrödinger operatorSpectral theoryWeyl's lawSchrödinger operatorEigenvalue countingWe 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. 22024-01-0120242024-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/292043https://dx.doi.org/urn:doi:10.1016/j.aim.2024.109665reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengEuropean Commission https://doi.org/10.13039/501100000780 101018680Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00071open accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2920432026-06-06T12:50:31Z
dc.title.none.fl_str_mv Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
title Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
spellingShingle Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
Poggi, Bruno|||0000-0002-7992-8578
Landscape function
Magnetic Schrödinger operator
Spectral theory
Weyl's law
Schrödinger operator
Eigenvalue counting
title_short Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
title_full Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
title_fullStr Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
title_full_unstemmed Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
title_sort Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
dc.creator.none.fl_str_mv Poggi, Bruno|||0000-0002-7992-8578
author Poggi, Bruno|||0000-0002-7992-8578
author_facet Poggi, Bruno|||0000-0002-7992-8578
author_role author
dc.subject.none.fl_str_mv Landscape function
Magnetic Schrödinger operator
Spectral theory
Weyl's law
Schrödinger operator
Eigenvalue counting
topic Landscape function
Magnetic Schrödinger operator
Spectral theory
Weyl's law
Schrödinger operator
Eigenvalue counting
description 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.
publishDate 2024
dc.date.none.fl_str_mv 2
2024-01-01
2024
2024-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/292043
https://dx.doi.org/urn:doi:10.1016/j.aim.2024.109665
url https://ddd.uab.cat/record/292043
https://dx.doi.org/urn:doi:10.1016/j.aim.2024.109665
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv European Commission https://doi.org/10.13039/501100000780 101018680
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00071
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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