We study the nonlinear Schrödinger type equation - Δu + (λg(x) + l)u = f(u) on the whole space R^N. The nonlinearity f is assumed to be asymptotically linear and g(x) ≥ 0 has a potential well. We do not assume a limit for g(x) as lxl →∞ . Using variational techniques, we prove the existence of a positive solution for λ large. In the case where f is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as λ →∞ is also considered.

The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y. / Garyounis University and Libyan Cultural Affairs

We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra. / Master of Science / In quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.

Cette thèse est consacrée à l'étude de quelques régimes asymptotiques de l'équation de Schrödinger semi-classique, en présence d'une non-linéarité non-locale de type Hartree. Elle comporte 3 parties, sous forme de 4 chapitres et une annexe. L'objet de la première partie, constituée du premier et deuxième chapitre, est l'étude du comportement asymptotique du modèle précédent pour un noyau singulier autour de l'origine, pour une condition initiale asymptotiquement de type WKB, en régime faiblement non-linéaire. Dans le premier chapitre nous montrons que sous certaines conditions de régularité sur la condition initiale, la solution est encore de type WKB à l'ordre principal, un résultat que nous obtenons dans le cadre fonctionnel de l'algèbre de Wiener. Nous donnons une preuve alternative au résultat précédent dans le cas particulier de l'équation de Schrödinger-Poisson dans le cadre fonctionnel d'espace de Sobolev rescalé, où la considération de correcteurs est nécessaire pour construire une solution approchée et pouvoir décrire la solution à l'ordre principal. La deuxième partie de cette thèse, objet du troisième chapitre, est consacrée à l'étude de la propagation de paquets d'onde pour un système couplé d'équations de Hartree en régime semi-classique, en présence de potentiels extérieurs sous-quadratiques. Nous décrivons analytiquement et numériquement le comportement asymptotique à l'ordre principal des fonctions d'onde solution du système, lorsqu'elles sont soumises à une condition initiale en forme de paquets d'onde, pour différentes tailles de non-linéarité. La dernière partie est constituée du quatrième chapitre et de l'annexe. Dans le quatrième chapitre nous considérons le problème de Cauchy de l'équation de Hartree avec noyau homogène ou dont la transformée de Fourier est dans un espace de Lebesgue, dans le cadre fonctionnel de l'algèbre de Wiener. Nous montrons quelques résultats sur le caractère bien posé du problème pour les noyaux considérés, dans des espaces faisant intervenir l'algèbre de Wiener. Nous concluons par une annexe dans laquelle nous considérons le problème de Cauchy de l'équation de Schrödinger-Poisson, en présence d'un potentiel extérieur indépendant du temps, dans les espaces de Sobolev pondérés. Nous étendons des résultats déjà obtenus sur l'existence de solutions globales dans les espaces de Sobolev sans poids lorsque le potentiel extérieur est nul, en montrant l'existence de solutions globales en temps dans les espaces de Sobolev pondérés pour toute régularité. / This thesis is devoted to the study of some asymptotic regimes of the semi-classical Schrödinger equation, in the presence of a nonlocal nonlinearity of Hartree-type . The purpose of the first part, consisting of the first and second chapter is the study of the asymptotic behavior of the previous model with a singular kernel around the origin for an initial data asymptotically of WKB-type, in a weakly nonlinear regime. In the first chapter we show that under some regularity conditions on the initial data, the solution still is of WKB-type at leading order, a result that we get in the functional framework of the Wiener algebra . We give an alternative proof to the previous result in the particular case of the Schrödinger-Poisson equation in the functional framework of rescaled Sobolev space, where the consideration of correctors is necessary to construct an approximate solution to describe the solution at leading order.The second part of this thesis, the subject of the third chapter is devoted to the study the propagation of wave packets for a coupled system of Hartree equations in a semi-classical regime , in the presence of sub-quadratic external potentials. We describe analytically and numerically the asymptotic behavior of the leading order of the wave functions solution of the system, for an initial data in the form of wave packets for different sizes of nonlinearity.The final part consists of the fourth chapter and appendix.In the fourth chapter we consider the Cauchy problem of the Hartree equation with a homogeneous kernel or of Fourier transform in a Lebesgue space, in the functional framework of the Wiener algebra. We show some results on the well-posedness of the problem for the considered kernels, in spaces involving the Wiener algebra.We conclude with an appendix in which we consider the Cauchy problem for the Schrödinger-Poisson equation in the presence of a time independent external potential in the weighted Sobolev spaces. We extend the results already obtained on the existence of global solutions in Sobolev spaces without weight when the external potential is reduced to zero, by showing the existence of global solutions in time in the weighted Sobolev spaces for all regularity.

Orientador: Marcos Tadeu de Oliveira Pimenta / Resumo: Para uma classe de equações quasilineares de Schrödinger, estabelecemos a existência de soluções positivas e nodais pelo método de Nehari. / Abstract: For a class of Schrödinger quasilinear equations, we established the existence of positive and nodal solutions by the Nehari method. / Mestre

Neste trabalho, estudamos propriedades de continuação única para as soluções da equação tipo Schrödinger com um ponto interação centrado em x=0, \\partial_tu=i(\\Delta_Z+V)u, onde V=V(x,t) é uma função de valor real e -\\Delta_Z é o operador escrito formalmente como \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] sendo \\delta_0 a delta de Dirac centrada em zero e Z qualquer número real. Logo, usamos estes resultados para ver o possível fenômeno de concentração das soluções, que explodem, da equação de tipo Schrödinger não linear com um ponto de interação em x=0, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] com ho>5. Também, mostramos que para certas condições sobre o potencial dependente do tempo V, a equação linear em cima tem soluções não triviais. / In this work, we study unique continuation properties for solutions of the Schrödinger equations with an point interaction centered at $x=0$, \\begin\\label \\partial_tu=i(\\Delta_Z+V)u, \\end where $V=V(x,t)$ is real value function and $-\\Delta_Z$ is the operator formally written \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] and $\\delta_0$ is Dirac\'s delta centered at zero and $Z$ is a real number. Next, we use these results in order to study the possible profile of the concentration of blow up solutions for the non linear Schrödinger equation with a point interaction at $x=0$, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] with $ho>5$. Besides, we show that the equation above has non trivial solutions for some conditions on the time dependent potencial $V$.

QC 20100414

In 1973, Combes and Thomas discovered a general technique for showing exponential decay of eigenfunctions. The technique involved proving the exponential decay of the resolvent of the Schrödinger operator localized between two distant regions. Since then, the technique has been been applied to several types of Schrödinger operators. This dissertation will show that the Combes--Thomas method works well with trace, Hilbert--Schmidt and other trace-type norms. The first result we prove shows exponential decay on trace-type norms of a resolvent of a Schrödinger operator localized between two distant regions. We build on this result by applying the Combes--Thomas method again to prove polynomial and sub-exponential decay estimates on functions of Schrödinger operators localized between two distant regions.

In this Dissertation thesis the spectral theory of Schrödinger operators modeling quasicrystals in dimension one ist investigated. We allow for a large class of measures as potentials covering also point interactions. The main results can be stated as follows: If the potential can be very well approximated by periodic potentials, then the correspondig Schrödinger operator does not have any eigenvalues. If the potential is aperiodic and satisfies a certain finite local complexity condition, the absolutely continuous spectrum is absent. We also prove Cantor spectra of zero Lebesgue measure for a large class of (a randomized version of) the operator.

Submitted by Anderson dos Santos Gonzaga null (andersongonzaga25@yahoo.com.br) on 2018-01-16T17:28:55Z No. of bitstreams: 1 Gonzaga.dissertação.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) / Approved for entry into archive by Claudia Adriana Spindola null (claudia@fct.unesp.br) on 2018-01-16T17:58:17Z (GMT) No. of bitstreams: 1 gonzaga_as_me_prud.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) / Made available in DSpace on 2018-01-16T17:58:17Z (GMT). No. of bitstreams: 1 gonzaga_as_me_prud.pdf: 1264952 bytes, checksum: e682e5fd46c5a7d68506f3f9499cded5 (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesse trabalho teórico na área das equações diferenciais parciais elípticas, estudamos uma versão estacionária da equação de Schrödinger não-linear, com não-linearidade do tipo assintoticamente linear. O objetivo principal versa sobre obter resultados de existência de uma solução nodal radialmente simétrica. Ainda, sob algumas condições, buscamos também obter informações sobre o seu índice de Morse. / In this theoretical work in elliptic partial di erential equations, we study a stationary version for the nonlinear Schödinger equation with nonlinearity of the assymptotically linear type. The main objective is getting, some results of existence for a radially symmetric nodal solution. Moreover, under some conditions, we look also obtaining information about its Morse index.