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.

The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform.

Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.

The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform.

This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant. \begin{align*} \partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} The model was introduced for the case of a single particle by Lagoutière, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves. The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered. The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling. Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs. Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated. / Diese Arbeit befasst sich mit dem Modell einer skalaren partiellen Differentialgleichung mit singulärem Quellterm, das die Interaktion zwischen einem reibungsfreiem Fluid, dargestellt durch die Burgers Gleichung, und einer gegebenen, endlichen Menge von sich in dem Fluid bewegenden Partikeln beschreibt, die eine punktweise Zugkraft auf das Fluid auswirken und durch eine entsprechende Reibungskonstante charakterisiert sind. \begin{align*} \partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} Das Modell wurde für den Fall der Interaktion mit einem einzelnen Partikel durch Lagoutière, Seguin and Takahashi eingeführt, stellt einen ersten Schritt zu einem besseren Verständnis der Interaktion zwischen einem Fluid und Festkörpern auf dem Level der partiellen Differentialgleichungen dar und hat die einzigartige Eigenschaft, dass Entropielösungen und die Interaktion mit Schockwellen berücksichtigt werden. Das Modell wird zu einer beliebigen, endlichen Anzahl von Partikeln erweitert und Interaktionen wie das Verschmelzen und Spaltung von Partikeln werden behandelt. Existierende Theorie der Entropie-Zulässigkeit im Hinblick auf Interfaces und Erhaltungsgleichungen mit unstetiger Flussfunktion wird zusammengefasst, die Resultate werden verglichen und für die Regionen mit Partikelinteraktionen angepasst. Zu diesem Zweck wird die Theorie der Germs, eingeführt von Andreianov, Karlsen und Risebro, auf den vorliegenden Fall eines nicht-erhaltenden Interfaces erweitert. Für das Riemann Problem von auseinanderdriftenden Partikeln werden die exakten Lösungen berechnet und eine Analyse des Verhaltens von Entropielösungen über die von den Partikeln erzeugten Interface wird genutzt, um ein physikalisch sinnvolles und mit der Theorie eines einzelnen Partikels konsistentes Verhalten beim Verschmelzen und Spalten von Partikeln herzuleiten. Mit Hilfe einer expliziten Konstruktionsmethode, hergeleiteten L-infinity Beschränkungen, einer Approximation der Partikelpfade und Kompaktheitsargumenten wird gezeigt, dass das entsprechende Cauchy Problem wohlgestellt ist. Eindeutigkeit im Raum der schwachen Entropielösungen wird mit beinahe klassischen Argumenten der Theorie von Kruzkov sowie der Theorie von L1-dissipativen Germs gezeigt. Notwendige Grundlagen zu hyperbolischen Erhaltungsgleichungen, unter anderem die Theorie schwacher Lösungen, Schock- und Verdünnungswellen sowie die Rankine-Hugoniot Bedingung, werden in Grundzügen am Anfang der Arbeit wiederholt.

The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the wellposedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the additional feature of being star-shaped at zero, the authors develop a series of results with focus on continuity properties of the modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus of continuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space. This study concludes with a new perspective on a concavity problem for the modulus of continuity, recently addressed by two of the authors in "Some note on the modulus of continuity for ill-posed problems in Hilbert space", 2011.

Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.

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.

We analyze the mathematical models of two classes of physical phenomena. The first class of phenomena we consider is the interaction between one or more insulating rigid bodies and an electrically conducting fluid, inside of which the bodies are contained, as well as the electromagnetic fields trespassing both of the materials. We take into account both the cases of incompressible and compressible fluids. In both cases our main result yields the existence of weak solutions to the associated system of partial differential equations, respectively. The proofs of these results are built upon hybrid discrete-continuous approximation schemes: Parts of the systems are discretized with respect to time in order to deal with the solution-dependent test functions in the induction equation. The remaining parts are treated as continuous equations on the small intervals between consecutive discrete time points, allowing us to employ techniques which do not transfer to the discretized setting. Moreover, the solution-dependent test functions in the momentum equation are handled via the use of classical penalization methods. The second class of phenomena we consider is the evolution of a magnetoelastic material. Here too, our main result proves the existence of weak solutions to the corresponding system of partial differential equations. Its proof is based on De Giorgi's minimizing movements method, in which the system is discretized in time and, at each discrete time point, a minimization problem is solved, the associated Euler-Lagrange equations of which constitute a suitable approximation of the original equation of motion and magnetic force balance. The construction of such a minimization problem is made possible by the realization that, already on the continuous level, both of these equations can be written in terms of the same energy and dissipation potentials. The functional for the discrete minimization problem can then be constructed on the basis of these potentials. / Wir analysieren die mathematischen Modelle von zwei Arten physikalischer Phänomene. Die erste Art von Phänomenen, die wir betrachten, ist die Wechselwirkung zwischen einem oder mehreren isolierenden starren Körpern und einem elektrisch leitenden Fluid, das die Körper umgibt, sowie den elektromagnetischen Feldern in beiden Materialien. Wir untersuchen sowohl den Fall inkompressibler als auch kompressibler Fluide. In beiden Fällen liefert unser Hauptresultat die Existenz von schwachen Lösungen für das zugehörige System partieller Differentialgleichungen. Die Beweise dieser Resultate beruhen auf hybriden diskret-kontinuierlichen Approximationsmethoden: Teile der Systeme werden in der Zeit diskretisiert, um das Problem der lösungsabhängigen Testfunktionen in der Induktionsgleichung zu bewältigen. Die verbleibenden Gleichungen werden als kontinuierliche Gleichungen auf den kleinen Intervallen zwischen aufeinanderfolgenden diskreten Zeitpunkten behandelt, sodass wir Techniken anwenden können, die sich nicht auf das diskretisierte System übertragen lassen. Darüber hinaus wird das Problem der lösungsabhängigen Testfunktionen in der Impulsgleichung durch die Verwendung klassischer Penalisierungsmethoden gelöst. Die zweite Art von Phänomenen, die wir betrachten, ist die Entwicklung eines magnetoelastischen Materials. Auch hier beweist unser Hauptresultat die Existenz schwacher Lösungen für das zugehörige System partieller Differentialgleichungen. Der Beweis basiert auf der Methode von De Giorgi, bei der das System in der Zeit diskretisiert und in jedem diskreten Zeitpunkt ein Minimierungsproblem gelöst wird, dessen zugehörige Euler-Lagrange-Gleichungen eine geeignete Approximation an die ursprüngliche Bewegungsgleichung und mikromagnetische Gleichung darstellen. Die Konstruktion eines solchen Minimierungsproblems wird durch die Erkenntnis ermöglicht, dass diese beiden Gleichungen bereits im kontinuierlichen System mithilfe derselben Energie- und Dissipationspotenziale ausgedrückt werden können. Das Funktional für das diskrete Minimierungsproblem kann dann auf Grundlage dieser Potenziale konstruiert werden.

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sightings 1. Unitary *-module spaces Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13 a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13 b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15 c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16 2. Operators on unitary *-module spaces Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. Graph regularity Pragmatism between weak and (strong) regularity . . . . . . . . . 27 a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 b) The case C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 c) Graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Transition. Orthogonal complementability and topology Back to Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Graph regular operators on Hilbert C*-modules 4. Commutative case: Operators on C_0(X) Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Interjection. Unboundedness and graph regularity . . . . . . . . . . 55 5. Relation to adjointable operators Sources of graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6. Concrete C*-algebras Association relation and affiliation relation . . . . . . . . . . . . . . . . 61 7. Examples Graph regular operators that are not regular . . . . . . . . . . . . . 67 a) Position and momentum operators as graph regular operators on a fraction algebra related to the Weyl algebra . . 67 b) A graph regular but not regular operator on the group C*-algebra of the Heisenberg group . . . . . . . . . . . . . . . 69 c) Unbounded Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 70 8. Bounded transform The canonical regular operator associated to a graph regular operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9. Absolute value and polar decomposition . . . . . . . . . . . . . . . 79 10. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11. Special matrices of C*-algebras Counter examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Abstract and open questions . . . . . . . . . . . . . . . . . . . . . . . . . 89 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Dank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99