Michael Demuth - Böcker

The spectral theory of Schrodinger operators, in particular those with random potentials, continues to be a very active field of research. This work focuses on various known criteria in the spectral theory of selfadjoint operators in order to identify the spectrum and its components a la Lebesgue decomposition.Key features and topics:* Well-developed exposition of criteria that are especially useful in determining the spectra of deterministic and random Schrodinger operators occurring in quantum theory * Systematically uses measures and their transforms (Fourier, Borel, wavelet) to present a unifying theme * Establishes criteria for identifying the spectrum * Examines a series of applications to show point spectrum and continuous spectrum in some models of random operators * Presents a series of spectral-theoretic results for the perturbed operators introduced in earlier chapters with examples of localization and delocalization in the theory of disordered systems * Presents modern criteria (using wavelet transform, eigenfunction decay) that could be used to do spectral theory * Unique work in book form combining the presentation of the deterministic and random cases, which will serve as a platform for further research activities This concise unified presentation is aimed at graduate students and researchers working in the spectral theory of Schrodinger operators with either fixed or random potentials in particular.However, given the large gap that this book fills in the literature, it will serve a wider audience of mathematical physicists because of its contribution to works in spectral theory.

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.

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.

This volume presents self-contained survey articles on modern research areas written by experts in their fields. The topics are located at the interface of spectral theory, theory of partial differential operators, stochastic analysis, and mathematical physics. The articles are accessible to graduate students and researches from other fields of mathematics or physics while also being of value to experts, as they report on the state of the art in the respective fields.

This volume presents self-contained survey articles on modern research areas written by experts in their fields. The articles are accessible to graduate students and researches from other fields of mathematics or physics while also being of value to experts, as they report on the state of the art in the respective fields.

This volume focuses on recent developments in non-linear and hyperbolic equations. It will be a most valuable resource for researchers in applied mathematics, the theory of wavelets, and in mathematical and theoretical physics. Nine up-to-date contributions have been written on invitation by experts in the respective fields. The book is the third volume of the subseries "Advances in Partial Differential Equations".

Partial differential equations constitute an integral part of mathematics. They lie at the interface of areas as diverse as differential geometry, functional analysis, or the theory of Lie groups and have numerous applications in the applied sciences. A wealth of methods has been devised for their analysis. Over the past decades, operator algebras in connection with ideas and structures from geometry, topology, and theoretical physics have contributed a large variety of particularly useful tools. One typical example is the analysis on singular configurations, where elliptic equations have been studied successfully within the framework of operator algebras with symbolic structures adapted to the geometry of the underlying space. More recently, these techniques have proven to be useful also for studying parabolic and hyperbolic equations. Moreover, it turned out that many seemingly smooth, noncompact situations can be handled with the ideas from singular analysis. The three papers at the beginning of this volume highlight this aspect. They deal with parabolic equations, a topic relevant for many applications. The first article prepares the ground by presenting a calculus for pseudo differential operators with an anisotropic analytic parameter. In the subsequent paper, an algebra of Mellin operators on the infinite space-time cylinder is constructed. It is shown how timelike infinity can be treated as a conical singularity.

This volume contains the proceedings of "PDE 2000", the international conference on partial differential equations held July 24 -28, 2000, in Clausthal. The confer- ence took place during the EXPO 2000 and was sponsored by the Land Nieder- sachsen, the Deutsche Forschungsgemeinschaft, the Bergstadt Clausthal-Zellerfeld and the Kreissparkasse Clausthal-Zellerfeld. This conference continues a series: Ludwigfelde 1976, Reinhardsbrunn 1985, Holz- hau 1988, Breitenbrunn 1990, Lambrecht 1991 (proceedings in Operator Theory: Advances and Applications, Vol. 57, Birkhauser Verlag 1992), Potsdam 1992 and 1993, Holzhau 1994 (proceedings in Operator Theory: Advances and Applications, Vol. 78, Birkhauser Verlag 1995), Caputh 1995 and Potsdam 1996 (proceedings in Mathematical Research, Vol. 100, Akademie Verlag 1997). The intention of the organizers was to bring together specialists from different areas of modern analysis, mathematical physics and geometry, to discuss not only the recent progress in their own fields but also the interaction between these fields.The special topics of the conference were spectral and scattering theory, semiclas- sical and asymptotic analysis, pseudodifferential operators and their relation to geometry, as well as partial differential operators and their connection to stochas- tic analysis and to the theory of semigroups. The scientific advisory board of the conference in Clausthal consisted of M. Ben- Artzi (Jerusalem), Chen Hua (Peking), M. Demuth (Clausthal), T. Ichinose (Kanazawa), 1. Rodino (Thrin), B.-W. Schulze (Potsdam) and J. Sjostrand (Paris).

A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated.A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as compactness or trace class properties of differences of Feynman-Kac semigroups, preservation of absolutely continuous and/or essential spectra and completeness of scattering systems.The unified approach provides a new viewpoint of and a deeper insight into the subject. The book is aimed at advanced students and researchers in mathematical physics and mathematics with an interest in quantum physics, scattering theory, heat equation, operator theory, probability theory and spectral theory.

This volume contains the proceedings of the International Conference on "Par- tial Differential Equations" held in HolzhaujErzgebirge, Germany, July 3~9, 1994. The conference was sponsored by the Max-Planck-Gesellschaft, the Deutsche For- schungsgemeinschaft, the Land Brandenburg and the Freistaat Sachsen. It was initiated by the Max-Planck-Research Group "Partielle Differential- gleichungen und Komplexe Analysis" at the University of Potsdam as one of the annual meetings of the research group. This conference is part of a series begun by the former Karl-Weierstraf3-Institute of Mathematics in Berlin, with the confer- ences in Ludwigsfelde 1976, Reinhardsbrunn 1985, Holzhau 1988 (proceedings in the Teubner Texte zur Mathematik 112, Teubner-Verlag 1989), Breitenbrunn 1990 (proceedings in the Teubner Texte zur Mathematik 131, Teubner-Verlag 1992), and Lambrecht 1991 (proceedings in Operator Theory: Advances and Applications, Vol. 57, Birkhiiuser Verlag 1992); subsequent conferences took place in Potsdam in 1992 and 1993 under the auspices of the Max-Planck-Research Group "Partielle Differentialgleichungen und Komplexe Analysis" at the University of Potsdam.It was the intention of the organizers to bring together specialists from differ- ent areas of modern analysis, geometry and mathematical physics to discuss not only recent progress in the respective disciplines but also to encourage interaction between these fields. The scientific advisory board of the Holzhau conference consisted of S. Al- beverio (Bochum), L. Boutet de Monvel (Paris), M. Demuth (Clausthal), P. Gilkey (Eugene), B. Gramsch (Mainz), B. Helffer (Paris), S.T. Kuroda (Tokyo), B.-W. Schulze (Potsdam).

Vorwort Mit Investmentfonds Geld verdienen Fonds sind in aller Munde. Man redet über Investment. Sie haben auch schon davon gehört? Investmentfonds sind im Gespräch. Und jeder hat etwas zu sagen. Der Investmentfonds-Branche wird von Experten und solchen, die es gerne wären, ein Jahrzehnt des Booms vorhergesagt. Allein 1993 er­ zielten deutsche Publikumsfonds sowie Luxemburger Fonds deutscher Provenienz mit mehr als 37 Milliarden DM erneut ein rekordverdächti­ ges Mittelaufkommen. Nicht zu vergessen die ausländischen Invest­ mentgesellschaften, die verstärkt auf den deutschen Markt drängen und für große Anlegerkreise eine lukrative Alternative darstellen. In­ vestmentfonds sind mittlerweile ein fester Baustein der privaten Ver­ mögensplanung und erreichen immer größere Bevölkerungskreise. Und wenn es denn boomt, dann wollen Sie doch bestimmt auch ein Wörtchen mitreden, oder? Und deshalb werden Sie in diesem Buch ei­ niges über die Vielfalt der Fondswelt erfahren und hoffentlich für sich auch einige brauchbare Strategien finden, denn natürlich sollen Sie nicht nur mitreden, sondern hoffentlich auch mitverdienen können. Investmentfonds in Katalogen, Fondsboutiquen in den Einkaufszentren Ihrer Stadt oder gar Fondskauf am Automaten? Zugegeben! Noch kön­ nen Sie keine Fonds am Automaten kaufen, aber die klassischen Fondsverkäufer, die bislang als Berater in Banken oder Versicherungen saßen, haben Konkurrenz bekommen. Immer mehr unabhängige Vermögensverwalter und -berater bieten Ihren Kunden Fondsanteile zum Erwerb an. Und in zwischenzeitlich bundesweit mehr als dreißig Fondsboutiquen hat der Kunde mit oder ohne Beratung die Qual der Wahl zwischen mehr als 1.200 Fonds.

Vorwort Options scheine - auch Warrants genannt - haben seit Beginn des Börsen­ booms 1985 national wie international eine imposante Renaissance erlebt. Im Sog weltweit steigender Aktienkurse stiegen auch die Kurse von Op­ tionsscheinen in oft schwindelerregende Höhen. Kursgewinne von 100 Prozent und mehr in wenigen Tagen waren und sind keine Seltenheit. Doch bereits im Frühjahr 1986 erlebten viele Options schein-Inhaber eine deutliche Korrektur bei deutschen Aktien-Optionsscheinen. Sie zeigte vie­ len Anlegern, daß die Börsen keine Einbahnstraßen sind. Im Herbst 1986 folgten dann drastische Kursverluste in japanischen DM-Optionsscheinen, als der japanische Nikkei-Dow-Jones mehr als 10 000 Punkte verlor. Viele Optionsscheine wurden wertlos. Auch im Börsencrash vom Oktober 1987 und im "Ein-Tages-Crash" vom 16.10.1989 verloren Optionsscheine dra­ matisch und bescherten ihren Inhabern überproportional hohe Verluste. Diese wären zu vermeiden gewesen - wenn rationale Aspekte die emotio­ nale Hektik bestimmt hätten. Denn schon nach wenigen Tagen konnten Mutige, die - antizyklisch und analytisch - in die richtigen Warrants ein­ gestiegen sind, satte Profite einstreichen. Die Golfkrise legte dann ab August 1990 alle internationalen Options­ schein-Märkte für nahezu ein Jahr lahm. Doch immer wieder erholten sich die Kurse der Options scheine und brachten ihren Anlegern gute Profite.

This volume focuses on recent developments in non-linear and hyperbolic equations. In the first contribution, the singularities of the solutions of several classes of non-linear partial differential equations are investigated. Applications concern the Monge-Ampere equation, quasi-linear systems arising in fluid mechanics as well as integro-differential equations for media with memory. There follows an article on L_p-L_q decay estimates for Klein-Gordon equations with time-dependent coefficients, explaining, in particular, the influence of the relation between the mass term and the wave propagation speed. The next paper addresses questions of local existence of solutions, blow-up criteria, and C^8 regularity for quasilinear weakly hyperbolic equations. Spectral theory of semibounded selfadjoint operators is the topic of a further contribution, providing upper and lower bounds for the bottom eigenvalue as well as an upper bound for the second eigenvalue in terms of capacitary estimates.

A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated.A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as compactness or trace class properties of differences of Feynman-Kac semigroups, preservation of absolutely continuous and/or essential spectra and completeness of scattering systems.The unified approach provides a new viewpoint of and a deeper insight into the subject. The book is aimed at advanced students and researchers in mathematical physics and mathematics with an interest in quantum physics, scattering theory, heat equation, operator theory, probability theory and spectral theory.