08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Long wave approximation over and beyond the natural time scale(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)Item Open Access Spectral theory of quantum graphs(2012) Demirel, Semra; Weidl, Timo (Prof. TeknD)We study some spectral problems for quantum graphs with a potential. On the one hand we analyze the quantitative dependence of bound states of the Schrödingeroperator on the potential. On the other hand we generalize certain basic identities from the one-dimensional scattering theory to quantum graphs. The first paper is concerned with the study of the discrete negative spectra of quantum graphs. We use the method of trace identities (sum rules) to derive inequalities of Lieb-Thirring, Payne-Polya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false. The second paper deals with the scattering problem for the Schrödinger equation on the half-line with the Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type. In the third paper we generalize results from the half-line case to the full graph case. More precisely, we consider the Schrödinger problem on a star shaped graph with n edges joined at a single vertex. A trace formula is derived for the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula. Besides these three articles this thesis also contains some further results. The method of sum rules is applied to the modified Schrödinger operator with variable coefficients to obtain a Lieb-Thirring type inequality with optimal constant. Furthermore, Lieb-Thirring inequalities are studied for star shaped graphs by using variational arguments and the method of symmetry decomposition of the corresponding Hilbert space. In several cases this leads to optimal constants in the inequalities.Item Open Access Existence and non-existence of breather solutions on necklace graphs(2023) Kielwein, Tobias; Schneider, Guido (Prof. Dr.)Item Open Access The KdV and Whitham limit for a spatially periodic Boussinesq model(2016) Bauer, Roman; Schneider, Guido (Prof. Dr.)Wir betrachten die KdV-Approximation und die Whitham-Approximation für ein räumlich periodisches Boussinesq-Modell. Wir zeigen Abschätzungen der Differenz zwischen der KdV- beziehungsweise der Whitham-Approximation und echten Lösungen des ursprünglichen Modells, welche garantieren, dass diese Amplituden-Gleichungen korrekte Vorhersagen über die Dynamik des räumlich periodischen Boussinesq-Modells über die natürlichen Zeitskalen machen. Der Beweis basiert auf Blochwellenanalysis und Energieabschätzungen.Item Open Access Invasion phenomena in pattern-forming systems admitting a conservation law structure(2021) Hilder, Bastian; Schneider, Guido (Prof. Dr.)Item Open Access Ein Beitrag zur Modellierung der Eigendynamik elastischer Schalen mit Hamiltonschen Strukturen(2000) Durdevic, Ivica; Kirchgässner, Klaus (Prof. Dr.)In der vorliegenden Arbeit werden die Schwingungen dünner elastischer Schalen im Rahmen der linearisierten Elastodynamik im energieerhaltenden Fall untersucht. Insbesondere wird eine dimensionsreduzierte Beschreibung (mit Hilfe des Koiterschen Modells) der Schalenbewegungen gegeben, wenn der Dickenparameter sehr klein wird. Erstmals wird nachgewiesen, daß die Schwingungen der dreidimensional ausgedehnten Schale aus den Schwingungen der Mittelfläche, bis auf angebbare Fehler, rekonstruiert werden können. Dazu wird unterschieden, ob eine Schale im biege- oder dehnungsdominanten Zustand schwingt. Für den ersten Fall gelingt die Rechtfertigung des Koiterschen Modells durch die Einführung einer langsamen Zeit unabhängig von der Form der Mittelfläche. Für dehnungsdominante Schwingungen kann das Koitersche Modell nur für Schalen mit positiv gekrümmter Mittelfläche gerechtfertigt werden. Das mathematische Werkzeug für eine Rechtfertigung der Dimensionsreduktion in der Schalendynamik wird durch die Theorie der Hamiltonschen Systeme bereitgestellt. Wesentlich für den Erfolg ist der Ausbau einer Approximationstheorie in Hamiltonschen Systemen. In dieser Arbeit geschieht dies mit Hilfe der sogenannten Fast-Poisson-Abbildungen, für welche ein mathematisches Fundament gebaut wurde.Item Open Access The validity of the Nonlinear Schrödinger approximation in higher space dimensions(2014) Hermann, Alina; Schneider, Guido (Prof. Dr.)The goal of the present work is the proof of approximation results for the Nonlinear Schördinger approximation in higher space dimensions for dispersive systems. The focus is on systems with resonant quadratic terms, which can lead to some explosion before the end of the approximation interval. In higher space dimensions the resonance structure is much more complicated than in case of one space dimension. The proof of approximation results is based on normal form transforms and the use of time-dependent norms.Item Open Access Self-adjointness and domain of a class of generalized Nelson models(2017) Wünsch, Andreas; Griesemer, Marcel (Prof. Dr.)Item Open Access On polarons and multipolarons in electromagnetic fields(2013) Wellig, David; Griesemer, Marcel (Prof. Dr.)This dissertation is concerned with a system of so-called large polarons in electromagnetic fields. We are especially interested in the ground state energy in the case of strong interactions between electrons and phonons, the strength of which is described by a coupling constant, and in the existence of bound states of several polarons. For the description of the polarons we use the model of H. Fröhlich, as well as the approximative models of Pekar, and Pekar and Tomasevich. The conjecture, that the ground state energies of these models asymptotically coincide in the leading order of the coupling constant, was the starting point of this work. We prove this conjecture for a large class of external electromagnetic fields. A suitable scaling of the fields makes sure, that they already play a non-trivial role in the leading order. The asymptotic coincidence of the ground state energies allows us to trace back the question of binding of Fröhlich polarons in the case of large couplings to the corresponding question in the model of Pekar and Tomasevich. The transcription of this dissertation is divided into four chapters, of which the introduction is the first one. The Chapters 2, 3 and 4 constitute three independent publications. The Chapter 2 is dedicated to the Pekar functional with electromagnetic fields. We proof the existence of a minimizer in the case of a constant magnetic field and a vanishing electric field. The minimizer exists as well, if this field configuration is locally perturbed such that the minimal energy is lowered. From the existence of the minimizer of the Pekar functional we derive binding of two polarons in the model of Pekar and Tomasevich. In Chapter 3, we compare the ground state energy of the 1-particle Fröhlich model in the limit of strong couplings with the minimum of the corresponding Pekar functional. We prove the above mentioned conjecture in the case of a single polaron. This result, in connection with the results of Chapter 2, allows us to prove binding of two Fröhlich polarons in strong electromagnetic fields. In Chapter 4 the analysis of the previous chapter is extended to N-polaron systems. To do so, an estimate of the interaction energy of spatially divided clusters of polarons in electromagnetic fields is derived. This allows us to proof the asymptotic exactness of the minimal energy of the Pekar-Tomasevich functional for strong couplings, whereas, as in Chapter 3, the external fields are suitably rescaled. As an application binding for N polarons in constant strong magnetic fields is proved.Item Open Access Approximation of a two‐dimensional Gross-Pitaevskii equation with a periodic potential in the tight‐binding limit(2024) Gilg, Steffen; Schneider, GuidoThe Gross-Pitaevskii (GP) equation is a model for the description of the dynamics of Bose-Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x‐direction and a harmonic oscillator potential in the y‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.