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Über dieses Buch

This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory, written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research, with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress, provide additional motivation, and develop the necessary tools. Moreover, the authors share their views on future developments, hypotheses, and unsolved problems.

Inhaltsverzeichnis

Frontmatter

Quantum Semiconductor Models

We give an overview of analytic investigations of quantum semiconductor models, where we focus our attention on two classes of models: quantum drift diffusion models, and quantum hydrodynamic models. The key feature of those models is a quantum interaction term which introduces a perturbation term with higher-order derivatives into a system which otherwise might be seen as a fluid dynamic system. After a discussion of the modeling, we present the quantum drift diffusion model in detail, discuss various versions of this model, list typical questions and the tools how to answer them, and we give an account of the state-of-the-art of concerning this model. Then we discuss the quantum hydrodynamic model, which figures as an application of the theory of mixed-order parameter-elliptic systems in the sense of Douglis, Nirenberg, and Volevich. For various versions of this model, we give a unified proof of the local existence of classical solutions. Furthermore, we present new results on the existence as well as the exponential stability of steady states, with explicit description of the decay rate.
Li Chen, Michael Dreher

Large Coupling Convergence: Overview and New Results

In this paper we present a couple of old and new results related to the problem of large coupling convergence. Several aspects of convergence are discussed, namely norm resolvent convergence as well as convergence within Schatten-von Neumann classes. We also discuss the rate of convergence with a special emphasis on the optimal rate of convergence, for which we give necessary and sufficient conditions. The collected results are then used for the case of Dirichlet operators. Our method is purely analytical and is supported by a wide variety of examples.
Hichem BelHadjAli, Ali Ben Amor, Johannes F. Brasche

Smooth Spectral Calculus

A smooth spectral theory is presented in an abstract Hilbert space framework. The main assumption (of smoothness) is the Hölder continuity of the derivative of the spectral measure (density of states). A Limiting Absorption Principle (LAP) is derived on the basis of continuity properties of Cauchytype integrals. This abstract theory is then extended to include short-range perturbations and sums of tensor products. Applications to partial differential operators are presented. In the context of partial differential operators the spectral derivative is closely related to trace operators on compact (for elliptic operators) or non-compact manifolds.
Matania Ben-Artzi

Spectral Analysis and Geometry of Sub-Laplacian and Related Grushin-type Operators

In this article, we discuss three topics in the area of sub-Riemannian geometry and analysis.
Wolfram Bauer, Kenro Furutani, Chisato Iwasaki

Zeta Functions of Elliptic Cone Operators

This paper is an overview of aspects of the singularities of the zeta function, equivalently, of the small time asymptotics of the trace of the heat semigroup, of elliptic cone operators. It begins with a brief description of classical results for regular differential operators on smooth manifolds, and includes a concise introduction to the theory of cone differential operators. The later sections describe recent joint work of the author with J. Gil and T. Krainer on the existence of the resolvent of elliptic cone operators and the structure of its asymptotic behavior as the modulus of the spectral parameter tends to infinity within a sector in C on which natural ray conditions on the symbol of the operator are assumed. These ideas are illustrated with examples.
Gerardo A. Mendoza

Pseudodifferential Operators on Manifolds: A Coordinate-free Approach

The main aim of the paper is to demonstrate the advantage of a coordinate-free approach to the theory of pseudodifferential operators. We explain how one can define symbols and construct a symbolic calculus without using local coordinates, briefly review some known definitions and results, and discuss possible applications and further developments.
Peter McKeag, Yuri Safarov

Backmatter

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