Anisotropic function spaces, anisotropic fractals, and spectral theory for related fractal semi-elliptic operators

Die Theorie der anisotropen Funktionenräume entwickelte sich parallel zur Theorie von isotropen Funktionenräumen. Wir verweisen insbesondere auf Arbeiten von S.M. Nikol'skii¸ O.V. Besov. Die anisotropen Funktionenräume erscheinen dann, wenn man Di®erentialoperatoren untersucht, deren maximale Ableitungsordnungen verschieden von Richtung zu Richtung sind, z.B. der Operator der Wärmeleitungsgleichung. In der vorliegenden Arbeit werden Zusammenhänge zwischen fraktaler Geometrie und der Fourieranalysis, der Theorie der Funktionenräume sowie der Spektraltheorie einiger Differentialoperatoren untersucht. Die Arbeit hat fünf Teile. Im ersten Kapitel stellen wir Grundlagen für anisotrope Besov-Räume zusammen. Das zweite Kapitel widmet sich einigen wichtigen Eigenschaften der anisotropen Besov-Räume. Das dritte Kapitel beschäftigt sich mit Zerlegungen (Atome, Wavelets) in anisotropen Funktionenräumen. Unser Hauptziel in diesem Kapitel ist, das anisotrope Gegenstück zu einem Resultat von H.Triebel(2003) zu beweisen. In Kapitel 4 geben wir die Definition der anisotropen d-Mengen; das sind z.B. anisotrope Cantor-Mengen. Wir studieren die Existenz und die Eigenschaften des Spur Operators tr¡, zwischen den Funktionenräumen, basierend auf Wavelet-Darstellungen aus Kapitel 3. Im letzten Kapitel betrachten wir den semi-elliptischen Differentialoperator. Diese Resultate werden abschließend mit ähnlichen, bereits bekannten (Farkas,2001) verglichen

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an essential use of the functional analytic framework developed by the authors in \cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial$ of the real line: $\mathfrak{a}=\zeta(\partial)$. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta(s)$ proposed here, the role played by the complex variable $s$ in the classical universality theorem is now played by the family of `truncated infinitesimal shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral operator in connection with a spectral reformulation of the Riemann hypothesis as an inverse spectral problem for fractal strings. This latter work provided an operator-theoretic version of the spectral reformulation obtained by the second author and H. Maier in \cite{LaMa2}. In the long term, our work (along with \cite{La5, La6}), is aimed in part at providing a natural quantization of various aspects of analytic number theory and arithmetic geometry

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Man\'e projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by the author as a part of the crash course in the Analysis of Nonlinear PDEs at Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9, 2012

Some open questions in "wave chaos"

The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Eigenvalues of singular measures and Connes’ noncommutative integration

In a domain \Omega subset R^N we consider compact, Birman–Schwinger type operators ofthe form T_{P;A}= A^*P A with P being a Borel measure in \Omega containing a singular part,and A being an order N/2 pseudodifferential operator. Operators are defined by means ofquadratic forms. For a class of such operators, we obtain a proper version of H. Weyl’s lawfor eigenvalues, with order not depending on dimensional characteristics of the measure. Theseresults lead to establishing measurability, in the sense of Dixmier–Connes, of such operatorsand the noncommutative version of integration over Lipschitz surfaces and rectifiable set

Sums of two dimensional spectral triples

We study countable sums of two dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. We make an explicit computation of the last module for the unit interval. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the number N(K) of eigenvalues bounded by K behaves, such that N(K)/K is bounded, but without limit for K growing.Comment: 27 page