Yafaev, D.R., University of Rennes, France

Scattering Theory: Some Old and New Problems

2000. XVI, 169 pp.
3-540-67587-6


Scattering theory is, roughly speaking, perturbation theory of self-adjoint operators on the (absolutely)
continuous spectrum. It has its origin in mathematical problems of quantum mechanics and is intimately
related to the theory of partial differential equations. Some recently solved problems, such as asymptotic
completeness for the Schr?dinger operator with long-range and multiparticle potentials, as well as open
problems, are discussed. We construct also potentials for which asymptotic completeness is violated. This
corresponds to a new class of asymptotic solutions of the time-dependent Schr?dinger equation. Special
attention is paid to the properties of the scattering matrix, which is the main observable of the theory. The
book is addressed to readers interested in a deeper study of the subject.

Keywords: Wave operators, scattering matrix, asymptotic completeness, n-body problem, long-range
potentials MSC-Classification : 25P25, 47A40, 81U05, 81U10

Contents: Part 1. The Schroedinger operator of two-particle systems 1. Basic notions 2. Short-range
interactions. Asymptotic completeness 3. Short-range interactions. Miscellaneous 4. Long-range
interactions. The scheme of smooth perturbations 5. The generalized Fourier transform 6. Long-range matrix
potentials Part 2. The scattering matrix 7. A stationary representarion 8. The short-range case 9. The
long-range cas 10. The relative scattering matrix Part 3. The multiparticle Schroedinger operator and related
problems 11. Setting the scattering problem 12. Resolvent equations 13. Asymptotic completeness. A
sketch of proof 14. The scattering matrix for multiparticle systems 15. New channels of scattering 16. The
Heisenberg model 17. Infinite obstacle scattering

Series: Lecture Notes in Mathematics.VOL. 1735

Turesson, B.O., University of Link?ping, Sweden

Nonlinear Potential Theory and Weighted Sobolev Spaces

2000. XIV, 173 pp.
3-540-67588-4

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev
space Wm,pw (), where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces
occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory
covers results concerning approximation, extension, and interpolation, Sobolev and Poincar? inequalities,
Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory,
several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various
concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi
continuity of Sobolev functions, Poincar? inequalities, and spectral synthesis theorems.

Keywords: Potential theory, Sobolev spaces

Contents: Introduction.- Preliminaries: Notation and conventions. Basic results concerning weights.-
Sobolev spaces: The Sobolev space Wm,pw (). The Sobolev space Vm,pw (). Hausdorff measures.
Isoperimetric inequalities. Some Sobolev type inequalities. Embeddings into Lq().- Potential theory: Norm
inequalities for fractional integrals and maximal functions. Meyers' Theory for Lp-capacities. Bessel and
Riesz capacities. Hausdorff capacities. Variational capacities. Thinness: The case 1< p
<?(Unendlichkeitszeichen). Thinness: The case p = 1.- Applications of potential theory to Sobolev spaces:
Quasicontinuity. Measures in the dual of Wm,pw (). Poincar? type inequalities. Spectral synthesis.-
References.- Index.

Series: Lecture Notes in Mathematics.VOL. 1736

Wakabayashi, S., University of Tsukuba, Ibaraki, Japan

Classical Microlocal Analysis in the Space of Hyperfunctions

2000. VIII, 367 pp.
3-540-67603-1

The book develops "Classical Microlocal Analysis" in the spaces of hyperfunctions and microfunctions,
which makes it possible to apply the methods in the distribution category to the studies on partial
differential equations in the hyperfunction category. Here "Classical Microlocal Analysis" means that it
does not use "Algebraic Analysis." The main tool in the text is, in some sense, integration by parts. The
studies on microlocal uniqueness, analytic hypoellipticity and local solvability are reduced to the problems
to derive energy estimates (or a priori estimates). The author assumes basic understanding of theory of
pseudodifferential operators in the distribution category.

Keywords: Microlocal analysis, hyperfunction, microlocal uniqueness, analytic hypoellipticity, local
solvability MSC-Classification : 35-02

Contents: Chapter 1 Hyperfunctions > 1.1 Function spaces > 1.2 Supports > 1.3 Localization > 1.4
Hyperfunctions > 1.5 Further applications of the Runge approximation theorem >Chapter 2 Basic calculus of
Fourier integral operators and pseudodifferential operators > 2.1 Preliminary lemmas > 2.2 Symbol classes >
2.3 Definition of Fourier integral operators > 2.4 Product formula of Fourier integral operators I > 2.5 Product
formula of Fourier integral operators II > 2.6 Pseudolocal properties > 2.7 Pseudodifferential operators in B >
2.8 Parametrices of elliptic operators >Chapter 3 Analytic wave front sets and microfunctions > 3.1 Analytic
wave front sets > 3.2 Action of Fourier integral operators on wave front sets > 3.3 The boundary values of
analytic functions > 3.4 Operations on hyperfunctions > 3.5 Hyperfunctions supported by a half-space > 3.6
Microfunctions > 3.7 Formal analytic symbols >Chapter 4 Microlocal uniqueness > 4.1 Preliminary lemmas >
4.2 General results > 4.3 Microhyperbolic operators > 4.4 Canonical transformation > 4.5 Hypoellipticity
>Chapter 5 Local solvability > 5.1 Preliminaries > 5.2 Necessary conditions on local solvability and
hypoellipticity > 5.3 Sufficient conditions on local solvability > 5.4 Some examples >Chapter A Proofs of
product formulae > A.1 Proof of Theorem 2.4.4 > A.2 Proof of Corollary 2.4.5 > A.3 Proof of Theorem 2.4.6 >
A.4 Proof of Corollary 2.4.7 > A.5 Proof of Theorem 2.5.3 >Chapter B A priori estimates > B.1 Grusin
operators > B.2 A class of operators with double characteristics

Series: Lecture Notes in Mathematics.VOL. 1737

Emery, M., Universite Louis Pasteur, Strasbourg, France
Nemirovski, A., Israel Institute of Technology, Haifa, Israel
Voiculescu, D., University of California, Berkeley, CA, USA
Bernard, P., Universite Blaise Pascal, Aubiere, France
(Ed.)

Lectures on Probability Theory and Statistics
Ecole d'Ete de Probabilites de Saint-Flour XXVIII - 1998

2000. XI, 349 pp.
3-540-67736-4

This volume contains lectures given at the Saint-Flour Summer School of Probability Theory during 17th
Aug. - 3rd Sept. 1998.
The contents of the three courses are the following:
- Continuous martingales on differential manifolds.
- Topics in non-parametric statistics.
- Free probability theory.
The reader is expected to have a graduate level in probability theory and statistics. This book is of interest
to PhD students in probability and statistics or operators theory as well as for researchers in all these fields.
The series of lecture notes from the Saint-Flour Probability Summer School can be considered as an
encyclopedia of probability theory and related fields.

Keywords: Martingales, Differential Manifold, Non-Parametric Statistics, Free Probability Theory, MSC 2000
: 46L10, 46L53

Contents: Martingales continues dans les variete differentiables. Introduction.- Variete vecteurs,
covecteurs, diffuseurs, codiffuseurs.- Semimartingales dans une variete et geometrie d'ordre 2.-
Connections et martingales.- Fonctions convexes et comportement des martingales.- Mouvements
browniens et applications harmoniques. Topics in Non-Parametric Statistics.- Estimating regression
functions from Hölder balls and from Sobolev balls.- Spatial adaptive estimation on Sobolev balls.-
Estimating signals satisfying differential inequalities.- Aggregation of estimates I, II.- Estimating functionals
I, II. Lectures on Free Probability Theory.- Introduction.- Noncommutative probability and operator
algebra background.- Addition and multiplication of freely independent noncommutative random variables.-
Multiplication variables.- Generalized canonical form, noncrossing partitions. Free independence with
amalgamation.- Some basic free processes.- Random matrices in the large N limit.- Free entropy.

Series: Lecture Notes in Mathematics.VOL. 1738

Franke, J., Humboldt-Universitat zu Berlin, Germany
Hardle, W., Humboldt-Universitat zu Berlin, Germany
Stahl, G., Bundesaufsichtsamt fur das Kreditwesen, Berlin, Germany
(Eds.)

Measuring Risk in Complex Stochastic Systems

2000. Approx. 270 pp.
0-387-98996-X

This collection of articles by leading researchers will be of interest to people working in the area of
mathematical finance.

Contents: Integrated Risk Management and Extreme Value Theory.- Coherent Allocation Capital for Credit
Portfolios.- A Simple Approach to Country Risk.- The Structure of Credit Risk.- Extreme Value Theory and
Risk Management: Basic Results.- Sensitivity of Values at Risk.- Extremes of ARCH Models.- Risk Exposure
and its Sensitivity to Model Misspecification.- Neural Networks and Applications in Finance.- Nonlinear
Approximation and Statistical Applications I.- Semiparametric Lower Bounds for Tail Index Estimation.-
Bandwith Choice for M-estimators in Projection Pursuit and Single Index Regression.- Semiparametric
Indirect Inference.- Change-point Problem in ARCH Models.- Change in Polynomial Regression and Related
Processes.

Series: Lecture Notes in Statistics.VOL. 147