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On the Spectra of Separable and Amenable Schrödinger Operators
Get PDFWe study the spectral theory of ergodic Schrödinger operators.The focus is on multi-dimensional Schrödinger operators that appear in quantum mechanics to model physical phenomena, e.g., electrons in a lattice or quasicrystal.
We present two papers: \emph{On the spectra of separable 2D almost Mathieu operators} and \emph{Spectral estimates of dynamically-defined and amenable operator families}.
The second paper is joint work with S.~Beckus.
This dissertation explains what was done in those papers and how it closely relates to the research focus---to address specific problems related to the multi-dimensional case.
The one-dimensional case is well developed since there are many results and many methods available, e.g., transfer matrices and Schrödinger cocycles.
The multi-dimensional case on the other hand is notoriously difficult to study.In part of this dissertation, we consider a separable 2D discrete Schrödinger operator generated by 1D Almost Mathieu Operators.For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval.
This complements a result by J.~Bourgain establishing that for fixed couplings the spectrum has gaps for some positive-measure Diophantine frequencies.
The result generalizes to separable multi-dimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine.
The proof is based on the study of the thickness of the spectrum of the Almost Mathieu Operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.In the other part of this dissertation, we consider a class of multi-dimensional Schrödinger operators.The Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the unimodular second-countable locally-compact Hausdorff group is amenable.
Together with S.~Beckus it is proved that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is -Hölder continuous.
Also, the behavior can be improved resulting in the spectrum map being Lipschitz continuous if the coefficients are instead locally-constant.
In 1990, the square root behavior was established by J.~Avron, P.~H.~M.~v.~Mouche, and B.~Simon for the Almost Mathieu Operator or, more generally, the quasiperiodic operators with Lipschitz continuous potentials.
Our result extends the square root behavior to a bigger class of operators such as magnetic discrete Schrödinger operators with finite range and with Lipschitz continuous coefficients
Spectral approximation of aperiodic Schrödinger operators
Get PDFDie vorliegende Arbeit befasst sich mit Stetigkeitseigenschaften der Spektren einer Operatorfamilie indiziert durch einen topologischen Raum. Speziell werden lineare beschränkte Operatoren betrachtet, die selbst-adjungiert, unitär bzw. normal sind. Der Abstand der zugehörigen Spektren wird durch die Hausdorffmetrik gemessen. Die (Hölder-)Stetigkeit wird für diese Klassen von Operatoren charakterisiert. Hierbei treten interessante Effekte in den quantitativen Abschätzungen auf, falls sich spektrale Lücken schließen. Genauer gesagt, die Rate der Konvergenz wird halbiert, falls sich spektrale Lücken schließen. Es stellt sich heraus, dass die entwickelte Theorie optimale Ergebnisse liefert. Weiterhin wird gezeigt, dass sich die Spektren genau dann stetig verhalten, wenn die zugehörigen C*-Algebren ein stetiges Feld von C*-Algebren definieren. Darauf basierend wird ein Werkzeug bereitgestellt, um die Stetigkeit für große Klassen von Operatoren nachzuweisen. Im Speziellen wird die Stetigkeit der Spektren für verallgemeinerte Schrödingeroperatoren assoziiert zu einem dynamischen System charakterisiert. Basierend auf diesem Resultat werden Klassen von Subshifts analysiert, die durch periodische Systeme approximiert werden können. Dies liefert aufgrund der vorher entwickelten Theorie periodische Approximationen der Schrödingeroperatoren. Hier stellt sich im mehrdimensionalen Fall heraus, dass lokale Symmetrien in den Mustern für substitutionelle Systeme sind ein hinreichendes Kriterium für periodische Approximationen sind. Dazu wird der Begriff eines Wörterbuches, bekannt im eindimensionalen Fall, konzeptionell eingeführt und weiterentwickelt. Es wird gezeigt, dass der Raum der Wörterbücher ausgestattet mit der lokalen Mustertopologie homöomorph zu dem topologischen Raum der Subshifts ist. Dies liefert ein hilfreiches Werkzeug zur Untersuchung dieser Systeme. Im Abschluss werden einige Beispiele diskutiert und die entwickelte Theorie auf diese Beispiele angewendet
Spectral Approximation of Schrödinger operators: Continuity of the Spectrum
No full textNon UBCUnreviewedAuthor affiliation: University of JenaGraduat
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
Generalized eigenfunctions and eigenvalues: A unifying framework for Shnol-type theorems
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