Espectro absolutamente contínuo do operador Laplaciano
Autor(a) principal: | |
---|---|
Data de Publicação: | 2018 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFSCAR |
Texto Completo: | https://repositorio.ufscar.br/handle/ufscar/9903 |
Resumo: | Let $\Omega$ be a periodic waveguide in $\mathbb R^3$, we denote by $-\Delta_\Omega^D$ and $-\Delta_\Omega^N$ the Dirichlet and Neumann Laplacian operators in $\Omega$, respectively. In this work we study the absolutely continuous spectrum of $-\Delta_\Omega^j$, $j \in \{D,N\}$, on the condition that the diameter of the cross section of $\Omega$ is thin enough. Furthermore, we investigate the existence and location of band gaps in the spectrum $\sigma(-\Delta_\Omega^j)$, $j \in \{D,N\}$. On the other hand, we also consider the case where $\Omega$ is a twisting waveguide (bounded or unbounded) and not necessarily periodic. In this situation, by considering the Neumann Laplacian operator $-\Delta_\Omega^N$ in $\Omega$, our goal is to find the effective operator when $\Omega$ is ``squeezed''. However, since in this process there are divergent eigenvalues, we consider $-\Delta_\Omega^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting because we find different effective operators in each situation. In the case where $\Omega$ is periodically twisted and thin enough, we obtain information on the absolutely continuous spectrum of $-\Delta_\Omega^N$ (restricted to that subspaces) and existence and location of band gaps in its structure. |
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Mamani, Carlos Ronal MamaniVerri, Alessandra Aparecidahttp://lattes.cnpq.br/8794549732815622http://lattes.cnpq.br/7491471460040429ca0d2f76-c0d7-4f88-a2e8-9a999601379d2018-05-08T14:25:23Z2018-05-08T14:25:23Z2018-04-06MAMANI, Carlos Ronal Mamani. Espectro absolutamente contínuo do operador Laplaciano. 2018. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2018. Disponível em: https://repositorio.ufscar.br/handle/ufscar/9903.https://repositorio.ufscar.br/handle/ufscar/9903Let $\Omega$ be a periodic waveguide in $\mathbb R^3$, we denote by $-\Delta_\Omega^D$ and $-\Delta_\Omega^N$ the Dirichlet and Neumann Laplacian operators in $\Omega$, respectively. In this work we study the absolutely continuous spectrum of $-\Delta_\Omega^j$, $j \in \{D,N\}$, on the condition that the diameter of the cross section of $\Omega$ is thin enough. Furthermore, we investigate the existence and location of band gaps in the spectrum $\sigma(-\Delta_\Omega^j)$, $j \in \{D,N\}$. On the other hand, we also consider the case where $\Omega$ is a twisting waveguide (bounded or unbounded) and not necessarily periodic. In this situation, by considering the Neumann Laplacian operator $-\Delta_\Omega^N$ in $\Omega$, our goal is to find the effective operator when $\Omega$ is ``squeezed''. However, since in this process there are divergent eigenvalues, we consider $-\Delta_\Omega^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting because we find different effective operators in each situation. In the case where $\Omega$ is periodically twisted and thin enough, we obtain information on the absolutely continuous spectrum of $-\Delta_\Omega^N$ (restricted to that subspaces) and existence and location of band gaps in its structure.Seja $\Omega$ um tubo periódico em $\mathbb R^3$, denote por $-\Delta_D^\Omega$ e $-\Delta^N_\Omega$ os operadores Laplacianos de Dirichlet e Neumann em $\Omega$, respectivamente. Neste trabalho, estudamos o espectro absolutamente contínuo de $-\Delta^j_\Omega$, $j\in\{D,N\}$, sob a condição de que o diâmetro da seção transversal de $\Omega$ é suficientemente pequeno. Além disso, investigamos a existência e a localização de lacunas no espectro $\sigma(-\Delta^j_\Omega)$, $j\in \{D,N\}$. Por outro lado, também consideramos o caso em que $\Omega$ é apenas um tubo torcido (limitado ou ilimitado), não necessariamente periódico. Nesta situação, considerando o Laplaciano de Neumann $-\Delta^N_\Omega$ em $\Omega$, nosso objetivo é encontrar o operador efetivo quando $\Omega$ é ``espremido''. No entanto, já que neste processo existam autovalores divergentes, consideramos $-\Delta^N_\Omega$ atuando em subespaços específicos do espaço de Hilbert inicial. A estratégia é interessante porque encontramos operadores efetivos diferentes em cada situação. No caso em que $\Omega$ é periodicamente torcido e suficientemente fino, obtemos também informações sobre o espectro absolutamente contínuo de $-\Delta^N_\Omega$ (restrito a tais subespaços) e a existência e a localização de lacunas na sua estrutura do seu espectro.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)porUniversidade Federal de São CarlosCâmpus São CarlosPrograma de Pós-Graduação em Matemática - PPGMUFSCarTubos periódicosLaplaciano de DirichletLaplaciano de NeumannEspectro absolutamente contínuoLacunas espectraisPeriodic waveguideDirichlet LaplacianNeumann LaplacianAbsolutely continuos spectrumBand gapsCIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISEEspectro absolutamente contínuo do operador LaplacianoAbsolutely continuous spectrum of the Laplacian operatorinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisOnline600600005cffdc-c40f-4078-ad93-cb1878b1c89finfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFSCARinstname:Universidade Federal de São Carlos (UFSCAR)instacron:UFSCARLICENSElicense.txtlicense.txttext/plain; charset=utf-81957https://repositorio.ufscar.br/bitstream/ufscar/9903/6/license.txtae0398b6f8b235e40ad82cba6c50031dMD56ORIGINALMAMANI_Carlos_2018.pdfMAMANI_Carlos_2018.pdfapplication/pdf607084https://repositorio.ufscar.br/bitstream/ufscar/9903/7/MAMANI_Carlos_2018.pdf1244b34cd2c07fa65216ed5b0038b6cfMD57TEXTMAMANI_Carlos_2018.pdf.txtMAMANI_Carlos_2018.pdf.txtExtracted texttext/plain158904https://repositorio.ufscar.br/bitstream/ufscar/9903/8/MAMANI_Carlos_2018.pdf.txt5d17ab0e45cf410f3045536f0dd04c63MD58THUMBNAILMAMANI_Carlos_2018.pdf.jpgMAMANI_Carlos_2018.pdf.jpgIM Thumbnailimage/jpeg6128https://repositorio.ufscar.br/bitstream/ufscar/9903/9/MAMANI_Carlos_2018.pdf.jpg33f87c8e23d17e48f6c6a1ec22579da3MD59ufscar/99032023-09-18 18:31:14.319oai:repositorio.ufscar.br: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Repositório InstitucionalPUBhttps://repositorio.ufscar.br/oai/requestopendoar:43222023-09-18T18:31:14Repositório Institucional da UFSCAR - Universidade Federal de São Carlos (UFSCAR)false |
dc.title.por.fl_str_mv |
Espectro absolutamente contínuo do operador Laplaciano |
dc.title.alternative.eng.fl_str_mv |
Absolutely continuous spectrum of the Laplacian operator |
title |
Espectro absolutamente contínuo do operador Laplaciano |
spellingShingle |
Espectro absolutamente contínuo do operador Laplaciano Mamani, Carlos Ronal Mamani Tubos periódicos Laplaciano de Dirichlet Laplaciano de Neumann Espectro absolutamente contínuo Lacunas espectrais Periodic waveguide Dirichlet Laplacian Neumann Laplacian Absolutely continuos spectrum Band gaps CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE |
title_short |
Espectro absolutamente contínuo do operador Laplaciano |
title_full |
Espectro absolutamente contínuo do operador Laplaciano |
title_fullStr |
Espectro absolutamente contínuo do operador Laplaciano |
title_full_unstemmed |
Espectro absolutamente contínuo do operador Laplaciano |
title_sort |
Espectro absolutamente contínuo do operador Laplaciano |
author |
Mamani, Carlos Ronal Mamani |
author_facet |
Mamani, Carlos Ronal Mamani |
author_role |
author |
dc.contributor.authorlattes.por.fl_str_mv |
http://lattes.cnpq.br/7491471460040429 |
dc.contributor.author.fl_str_mv |
Mamani, Carlos Ronal Mamani |
dc.contributor.advisor1.fl_str_mv |
Verri, Alessandra Aparecida |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/8794549732815622 |
dc.contributor.authorID.fl_str_mv |
ca0d2f76-c0d7-4f88-a2e8-9a999601379d |
contributor_str_mv |
Verri, Alessandra Aparecida |
dc.subject.por.fl_str_mv |
Tubos periódicos Laplaciano de Dirichlet Laplaciano de Neumann Espectro absolutamente contínuo Lacunas espectrais |
topic |
Tubos periódicos Laplaciano de Dirichlet Laplaciano de Neumann Espectro absolutamente contínuo Lacunas espectrais Periodic waveguide Dirichlet Laplacian Neumann Laplacian Absolutely continuos spectrum Band gaps CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE |
dc.subject.eng.fl_str_mv |
Periodic waveguide Dirichlet Laplacian Neumann Laplacian Absolutely continuos spectrum Band gaps |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE |
description |
Let $\Omega$ be a periodic waveguide in $\mathbb R^3$, we denote by $-\Delta_\Omega^D$ and $-\Delta_\Omega^N$ the Dirichlet and Neumann Laplacian operators in $\Omega$, respectively. In this work we study the absolutely continuous spectrum of $-\Delta_\Omega^j$, $j \in \{D,N\}$, on the condition that the diameter of the cross section of $\Omega$ is thin enough. Furthermore, we investigate the existence and location of band gaps in the spectrum $\sigma(-\Delta_\Omega^j)$, $j \in \{D,N\}$. On the other hand, we also consider the case where $\Omega$ is a twisting waveguide (bounded or unbounded) and not necessarily periodic. In this situation, by considering the Neumann Laplacian operator $-\Delta_\Omega^N$ in $\Omega$, our goal is to find the effective operator when $\Omega$ is ``squeezed''. However, since in this process there are divergent eigenvalues, we consider $-\Delta_\Omega^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting because we find different effective operators in each situation. In the case where $\Omega$ is periodically twisted and thin enough, we obtain information on the absolutely continuous spectrum of $-\Delta_\Omega^N$ (restricted to that subspaces) and existence and location of band gaps in its structure. |
publishDate |
2018 |
dc.date.accessioned.fl_str_mv |
2018-05-08T14:25:23Z |
dc.date.available.fl_str_mv |
2018-05-08T14:25:23Z |
dc.date.issued.fl_str_mv |
2018-04-06 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
MAMANI, Carlos Ronal Mamani. Espectro absolutamente contínuo do operador Laplaciano. 2018. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2018. Disponível em: https://repositorio.ufscar.br/handle/ufscar/9903. |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufscar.br/handle/ufscar/9903 |
identifier_str_mv |
MAMANI, Carlos Ronal Mamani. Espectro absolutamente contínuo do operador Laplaciano. 2018. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2018. Disponível em: https://repositorio.ufscar.br/handle/ufscar/9903. |
url |
https://repositorio.ufscar.br/handle/ufscar/9903 |
dc.language.iso.fl_str_mv |
por |
language |
por |
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600 600 |
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005cffdc-c40f-4078-ad93-cb1878b1c89f |
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info:eu-repo/semantics/openAccess |
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openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de São Carlos Câmpus São Carlos |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática - PPGM |
dc.publisher.initials.fl_str_mv |
UFSCar |
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Universidade Federal de São Carlos Câmpus São Carlos |
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