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Doctoral Thesis
DOI
10.11606/T.45.2016.tde-29082016-175729
Document
Author
Full name
Alex Javier Hernandez Ardila
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2016
Supervisor
Committee
Pava, Jaime Angulo (President)
Ferreira, Ademir Pastor
Lopes, Orlando Francisco
Ramirez, José Felipe Linares
Siciliano, Gaetano
Title in Portuguese
Estabilidade de ground state para a equação de Schrödinger logarítmica com potenciais do tipo delta
Keywords in Portuguese
Delta potencial
Delta-derivada potencial
Equação de Schrödinger logarítmica
Estabilidade orbital
Ground state
Abstract in Portuguese
Na primeira parte do trabalho estudamos a equação de Schrödinger logarítmica com um delta potencial; $V(x)=-\gamma \,\delta(x)$, onde $\delta$ é a distribuição de Dirac na origem e o parâmetro real $\gamma$ descreve a intensidade do potencial. Estabelecemos a existência e unicidade das soluções do problema de Cauchy associado em um espaço de funções adequado. No caso do potencial atrativo ($\gamma>0$), calculamos de forma explícita o seu único ground state e mostramos a sua estabilidade orbital.\\ A segunda parte trata detalhadamente da equação de Schrödinger logarítmica com um delta derivada potencial; $V(x)=-\gamma\, \delta^{\prime}(x)$. A boa colocação global para o problema de Cauchy é verificada em um espaço de funções adequado. No caso do potencial atrativo ($\gamma>0$), o conjunto dos ground states é completamente determinado. Mais precisamente: se $0<\gamma\leq2$, então há um único ground state e é uma função ímpar; se $\gamma>2$, então existem dois ground states não-simétricos. Em adição, provamos que cada ground state é orbitalmente estável através de uma abordagem variacional. Finalmente, usando a teoria de extensão de operadores simétricos, também mostramos um resultado de instabilidade para $\gamma>2$.
Title in English
Stability of the ground states for a logarithmic Schrödinger equation with delta-type potentials
Keywords in English
Delta potential
Delta-prime potential
Ground state
Logarithmic Schrödinger equation
Orbital stability
Abstract in English
The first part of this thesis deals with the logarithmic Schrödinger equation with a delta potential; $V(x)=-\gamma \,\delta(x)$, where $\delta$ is the Dirac distribution at the origin and the real parameter $\gamma$ is interpreted as the strength of the potential. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive potential case ($\gamma>0$), we explicitly compute the unique ground state and we show their orbital stability .\\ The second part deals with the case of the logarithmic Schrödinger equation with a delta prime potential; $V(x)=-\gamma\, \delta^{\prime}(x)$. Global well-posedness is verified for the Cauchy problem in a suitable functional space. In the attractive potential case ($\gamma>0$), the set of the ground state is completely determined. More precisely: if $0<\gamma\leq2$, then there is a single ground state and it is an odd function; if $\gamma>2$, then there exist two non-symmetric ground states. Moreover, we show that every ground state is orbitally stable via a variational approach. Finally, by applying the theory of extensions of symetric operators, we also prove a result of instability for $\gamma>2$.
 
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Publishing Date
2016-08-31
 
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