The Frisch–Parisi formalism for fluctuations of the Schrödinger equation

We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth backgro...

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Detalles Bibliográficos
Autores: Kumar, S., Ponce Vanegas, F., Roncal, L., Vega, L.
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2022
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1429
Acceso en línea:http://hdl.handle.net/20.500.11824/1429
Access Level:acceso abierto
Palabra clave:Schrödinger equation
Vortex filament equation
Talbot effect
Frisch--Parisi formalism
Multifractals
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spelling The Frisch–Parisi formalism for fluctuations of the Schrödinger equationKumar, S.Ponce Vanegas, F.Roncal, L.Vega, L.Schrödinger equationVortex filament equationTalbot effectFrisch--Parisi formalismMultifractalsWe consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth background. We prove that the Frisch--Parisi formalism holds for $H_\delta(t) = \int_{[0,t]}h_{\text{p}, \delta}(2s)\,ds$, which is morally a simplification of the Riemann's non-differentiable curve $R$. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to $R$.BERC 2022-2025, FJC2019-039804-I, RYC2018-025477-I, Ikerbasque, PGC2018-094522-B-I00202220222022info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1429reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Inglésinfo:eu-repo/grantAgreement/EC/H2020/669689info:eu-repo/grantAgreement/MINECO//SEV-2017-0718info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PGC2018-094528-B-I00info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/14292026-06-19T12:47:47Z
dc.title.none.fl_str_mv The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
title The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
spellingShingle The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
Kumar, S.
Schrödinger equation
Vortex filament equation
Talbot effect
Frisch--Parisi formalism
Multifractals
title_short The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
title_full The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
title_fullStr The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
title_full_unstemmed The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
title_sort The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
dc.creator.none.fl_str_mv Kumar, S.
Ponce Vanegas, F.
Roncal, L.
Vega, L.
author Kumar, S.
author_facet Kumar, S.
Ponce Vanegas, F.
Roncal, L.
Vega, L.
author_role author
author2 Ponce Vanegas, F.
Roncal, L.
Vega, L.
author2_role author
author
author
dc.subject.none.fl_str_mv Schrödinger equation
Vortex filament equation
Talbot effect
Frisch--Parisi formalism
Multifractals
topic Schrödinger equation
Vortex filament equation
Talbot effect
Frisch--Parisi formalism
Multifractals
description We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth background. We prove that the Frisch--Parisi formalism holds for $H_\delta(t) = \int_{[0,t]}h_{\text{p}, \delta}(2s)\,ds$, which is morally a simplification of the Riemann's non-differentiable curve $R$. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to $R$.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022
2022
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1429
url http://hdl.handle.net/20.500.11824/1429
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/EC/H2020/669689
info:eu-repo/grantAgreement/MINECO//SEV-2017-0718
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PGC2018-094528-B-I00
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
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