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.

When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.

This paper is concerned with the investigation of the nonlinear eigenvalue problem describing the natural oscillations of a plate with a load that elastically attached to it. We study properties of eigenvalues and eigenfunctions of this eigenvalue problem and prove the existence theorem for eigensolutions. Theoretical results are illustrated by numerical experiments.

When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.

The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\in\N$ and $X$ is a complex Banach space. The latter means that our matrix entries $a_{ij}$ are indexed by multiindices $i,j\in\Z^N$ and that every $a_{ij}$ is itself a bounded linear operator on $X$. Our main focus lies on the case $p=\infty$, where new results are derived, and it is demonstrated in both general theory and concrete operator equations from mathematical physics how advantage can be taken of these new $p=\infty$ results in the general case $p\in[1,\infty]$.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

John Lott berechnete eine Signatur mit ganzzahligen Werten für den Orbitraum einer kompakten, orientierbaren (4k + 1)-Mannigfaltigkeit mit einer halbfreien S1-Wirkung. Diese Signatur ist eine Homotopieinvariante für den Orbitraum. Allerdings konstruierte er keinen Operator vom Dirac-Typ, der die Signatur als Index besitzt. In dieser Arbeit konstruieren wir einen solchen Operator auf dem Orbitraum der S1-Wirkung, einem Thom-Mather stratifizierten Raum mit einem singulären Stratum von positiver Dimension, und wir zeigen, dass der Operator im wesentlichen eindeutig bestimmt ist. Ferner zeigen wir, dass sein Index mit Lotts Signatur übereinstimmt, zumindest wenn der stratifizierte Raum die sogenannte Witt-Bedingung erfüllt. Wirnennendiesen Operator den induzierten Dirac-Schrödinger Operator. Unsere Konstruktionsstrategie ist es, einen geeigneten S1-invarianten transversal elliptischen Operator erster Ordnung auf den S1-invarianten Differentialformen zu definieren, der den gesuchten Operator auf den Differentialformen des Orbitraums induziert. Die Witt-Bedingung, eine topologische Bedingung, welche in diesem Fall von der Kodimension der betrachteten Punktmenge abhängt, lässt verschiedene analytische Schlussfolgerungen zu. Insbesondere ist, wenn die Bedingung nicht erfüllt ist, der Hodge-de Rham Operator auf dem Quotientenraum nicht notwendigerweise essentiell selbstadjungiert und die Wahl einer Randbedingung ist daher notwendig. Diese Wahlfreiheit erscheint unnatürlich in Anbetracht der Tatsache, dass Lotts Signatur unabhängig von der Witt-Bedingung wohldefiniert ist. Der Dirac-Schrödinger Operator, der in dieser Arbeit konstruiert wird, unterschei- det sich vom Hodge-de Rham Operator durch einen Term nullter Ordnung, welcher sicherstellt, dass der Operator wesentlich selbstadjungiert ist. Außerdem antikommutiert dieser Term nullter Ordnung mit der Signatur-Involution, wodurch der gesamte Operator zerfällt und so der Index berechnet werden kann, auch wenn die Witt-Bedingung nicht erfüllt ist. / John Lott has computed an integer-valued signature for the orbit space of a compact orientable (4k + 1) manifold with a semi-free S1-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrödinger operator. The strategy of the construction is to “push down” an appropriate S1-invariant first order transversally elliptic operator to the quotient space. The Witt condition, a topological condition which in this case depends on the codi- mension of the fixed point set, has various analytic consequences. In particular, when not satisfied, the Hodge-de Rham operator on the quotient space does not need to be essentially self-adjoint and therefore a choice of boundary conditions is required. This choice freedom is not natural in view of the fact that Lott’s signature is well defined independently of the Witt condition. The Dirac-Schrödinger operator constructed in this Thesis differs from the Hodge-de Rham operator by a zero order term which ensures it to be essentially self-adjoint. Moreover, this zero order term anti-commutes with the chirality involution allowing the whole operator to split so that the index can be computed even if the Witt condition is not satisfied.

Ein metrischer Graph mit einem selbstadjungierten, negativen Laplace-Operator wird Quantengraph genannt. In dieser Arbeit werden Transporteigenschaften zufälliger Laplace-Operatoren betrachtet. Dazu wird die Multiskalenanalyse (MSA) von euklidischen Räumen auf metrische Graphen angepasst. Eine Überdeckung der metrischen Graphen wird aus gleichmäßig polynomiellem Wachstum und der gleichmäßigen Beschränkung der Kantenlängen gewonnen. Als Hilfsmittel für die MSA werden eine Combes-Thomas-Abschätzung und eine Geometrische Resolventenungleichung bewiesen. Zusammen mit einer Wegner-Abschätzung und der Existenz von verallgemeinerten Eigenfunktionen wird mittels der modifizierten MSA spektrale Lokalisierung (d.h. reines Punktspektrum) mit polynomiell fallenden Eigenfunktionen am unteren Rand des Spektrums für negative Laplace-Operatoren mit zufälligem Potential geschlossen. Dabei sind alle Randbedingungen, die eine nach unten beschränkten Operator liefern, wählbar. / We prove spectral localization for infinite metric graphs with a self-adjoint Laplace operator and a random potential. Therefor we adapt the multiscale analysis (MSA) from the euclidean case to metric graphs. In the MSA a covering of the graph is needed which is obtained from a uniform polynomial growth of the graph. The geometric restrictions of the graph contain a uniform bound on the edge lengths. As boundary conditions we allow all settings which give a lower bounded self-adjoint operator with an associated quadratic form. The result is spectral localization (i.e. pure point spectrum) with polynomially decaying eigenfunctions in a small interval at the ground state energy.

In this habilitation thesis a factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip is established. These operators are considered with matrix-valued symbols and are thought of acting on the vector-valued analogues of the Hardy and Lebesgue spaces. A factorization theory for pure Toeplitz operators and singular integral operators without flip is known since decades and provides necessary and sufficient conditions for Fredholmness and formulas for the defect numbers. In particular, the invertibility of such operators is equivalent to the existence of a certain type of Wiener-Hopf factorization. In this thesis an analogous theory for the afore-mentioned more general classes of operators is developed. It turns out that a completely different kind of factorization is needed. This kind of factorization is studied extensively, and a corresponding Fredholm theory is established. A connection with the Hunt-Muckenhoupt-Wheeden condition is made, and several examples and applications are given as well. / In dieser Habilitationsschrift wird eine Faktorisierungstheorie für Toeplitz plus Hankel-Operatoren und singuläre Integraloperatoren mit Flip aufgestellt. Diese Operatoren werden mit matrixwertigem Symbol betrachtet und sind auf den vektorwertigen Analoga der Hardy- und Lebesgue-Räumen definiert. Eine Faktorisierungstheorie für reine Toeplitz bzw. singuläre Integraloperatoren ohne Flip ist seit Jahrzehnten bekannt. Sie liefert notwendige und hinreichende Bedingungen für die Fredholmeigenschaft und Formeln für die Defektzahlen. Insbesondere ist die Invertierbarkeit derartiger Operatoren äquivalent zur Existenz einer bestimmten Art der Wiener-Hopf-Faktorisierung. In dieser Habilitationsschrift wird eine entsprechende Theorie für die erwähnten, allgemeineren Klassen von Operatoren aufgestellt. Es stellt sich heraus, dass eine völlig andere Art der Faktorisierung benötigt wird. Diese Art der Faktorisierung wird eingehend studiert und eine entsprechende Fredholmtheorie wird entwickelt. Ein Zusammenhang mit der Hunt-Muckenhoupt-Wheeden Bedingung wird hergestellt. Mehrere Beispiele und Anwendungen werden ebenfalls angegeben.