08 Fakultät Mathematik und Physik

Now showing 1 - 10 of 21

Open Access
Long wave approximation over and beyond the natural time scale
(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)
Open Access
Existence and non-existence of breather solutions on necklace graphs
(2023) Kielwein, Tobias; Schneider, Guido (Prof. Dr.)
Open Access
Invasion phenomena in pattern-forming systems admitting a conservation law structure
(2021) Hilder, Bastian; Schneider, Guido (Prof. Dr.)
Open Access
Approximation of a two‐dimensional Gross-Pitaevskii equation with a periodic potential in the tight‐binding limit
(2024) Gilg, Steffen; Schneider, Guido
The 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.
Open Access
Virtual levels of multi-particle quantum systems and their implications for the Efimov effect
(2020) Bitter, Andreas; Weidl, Timo (Prof. TeknD)
Open Access
Mathematische Modellierung von wellenoptischer Absorption beim Laserschneiden
(2022) Wörner, Maximilian; Schneider, Guido (Prof. Dr.)
Open Access
On the uniqueness of the Calderón problem and its application in electrical impedance tomography
(2023) Pombo, Ivan; Wirth, Jens (Prof. Dr.)
This thesis addresses several questions about the uniqueness and reconstruction of the conductivity γ from knowledge of the boundary information encapsulated in the Dirichlet-to-Neumann map Λγ . This problem is well-known in the literature as Calder ́on problem. In two dimensions, we extend the uniqueness of Calder ́on problem in two dimensions for complex conductivities with curves of discontinuity based on the stationary phase method and the introduction of new exponentially growing solutions. In three dimensions, we extend the result established by Nachman for real conductivities with two derivatives, by noting that most of the proof holds with the need of extending some of the results to encapsulate the complex case. Moreover, we establish a methodology to recover the complex conductivity from small complex frequencies, but some open questions are left about this reconstruction process. Furthermore, we reduce the differentiability condition for uniqueness to hold. We have shown that the Dirichlet-to-Neumann map uniquely determines complex conductivities with one derivative. Our approach is completely novel and introduces a quaternionic analysis approach to deal with the problem in three dimensions. With the quaternionic framework we also introduce a possible path to show uniqueness for real conductivities in L∞. This is a step in the direction of a complete answer to Calder ́on’s question in three dimensions. This problem is also relevant for practical applications, in particular medical imaging where it is used in Electrical Impedance Tomography (EIT). For practical implementations, a reconstruction algorithm is required to transform the boundary measurements into a conductivity profile. We use iterative methods to obtain a reconstruction method and our goal is to provide a simple and effective way to compute the required Jacobian matrix This approach is based in automatic differentiation (AD) tools . We show that AD can be used to efficiently and effectively compute the Jacobian matrix of a numerical method that simulates the voltages measurements. Further, we show that this computation is as effective as analytical closed-forms applied in general iterative method in order to reconstruct the conductivity profile.
Open Access
Quantization of algebras defined by ultradifferentiable group actions
(2022) Brinker, Jonas; Wirth, Jens (apl. Prof.)
Open Access
From short-range to contact interactions in two-dimensional many-body quantum systems
(2022) Griesemer, Marcel; Hofacker, Michael
Quantum systems composed of N distinct particles in R2with two-body contact interactions of TMS type are shown to arise as limits-in the norm resolvent sense-of Schrödinger operators with suitably rescaled pair potentials.
Open Access
Absence of the Efimov effect in dimensions one and two
(2021) Barth, Simon; Weidl, Timo (Prof. TeknD)

08 Fakultät Mathematik und Physik

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