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Author : Gerald Teschl
Publisher : American Mathematical Soc.
Page : 373 pages
File Size : 42,7 Mb
Release : 2000
Category : Mathematics
ISBN : 9780821819401
This volume serves as an introduction and reference source on spectral and inverse theory of Jacobi operators and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.
Author : Gerald Teschl
Publisher : American Mathematical Society(RI)
Page : 373 pages
File Size : 55,7 Mb
Release : 2014-05-22
Category : MATHEMATICS
ISBN : 1470412993
This volume serves as an introduction and reference source on spectral and inverse theory of Jacobi operators and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.
Author : Decio Levi,Raphaël Rebelo,Pavel Winternitz
Publisher : Springer
Page : 435 pages
File Size : 43,9 Mb
Release : 2017-06-30
Category : Science
ISBN : 9783319566665
This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.
Author : Fritz Gesztesy,Helge Holden,Johanna Michor,Gerald Teschl
Publisher : Cambridge University Press
Page : 438 pages
File Size : 41,5 Mb
Release : 2008-09-04
Category : Mathematics
ISBN : 9781139473774
As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.
Author : Alwyn Scott
Publisher : Routledge
Page : 2881 pages
File Size : 45,6 Mb
Release : 2006-05-17
Category : Reference
ISBN : 9781135455576
In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.
Author : Yurij M. Berezansky,Mykola E. Dudkin
Publisher : Springer Nature
Page : 489 pages
File Size : 43,5 Mb
Release : 2024-01-06
Category : Mathematics
ISBN : 9783031463877
This monograph presents the solution of the classical moment problem, the construction of Jacobi matrices and corresponding polynomials. The cases of strongly,trigonometric, complex and real two-dimensional moment problems are discussed, and the Jacobi-type matrices corresponding to the trigonometric moment problem are shown. The Berezansky theory of the expansion in generalized eigenvectors for corresponding set of commuting operators plays the key role in the proof of results. The book is recommended for researchers in fields of functional analysis, operator theory, mathematical physics, and engineers who deal with problems of coupled pendulums.
Author : Jan Janas,Pavel Kurasov,A. Laptev,Sergei Naboko,Günter Stolz
Publisher : Springer Science & Business Media
Page : 261 pages
File Size : 40,7 Mb
Release : 2007-04-29
Category : Mathematics
ISBN : 9783764381356
This volume contains lectures delivered at the International Conference Operator Theory and its Applications in Mathematical Physics (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland. The idea behind these lectures was to present interesting ramifications of operator methods in current research of mathematical physics.
Author : Vadim Adamyan,Yu.M. Berezansky,Israel Gohberg,Myroslav L. Gorbachuk,Valentyna Gorbachuk,Anatoly N. Kochubei,Heinz Langer,Gennadi Popov
Publisher : Springer Science & Business Media
Page : 518 pages
File Size : 47,7 Mb
Release : 2009-08-29
Category : Mathematics
ISBN : 9783764399214
This is the second of two volumes containing peer-reviewed research and survey papers based on talks at the International Conference on Modern Analysis and Applications. The papers describe the contemporary development of subjects influenced by Mark Krein.
Author : T. Suslina
Publisher : American Mathematical Soc.
Page : 318 pages
File Size : 50,8 Mb
Release : 2008-01-01
Category : Mathematics
ISBN : 0821890778
"This volume is dedicated to the eightieth birthday of Professor M. Sh. Birman. It contains original articles in spectral and scattering theory of differential operators, in particular, Schrodinger operators, and in homogenization theory. All articles are written by members of M. Sh. Birman's research group who are affiliated with different universities all over the world. A specific feature of the majority of the papers is a combination of traditional methods with new modern ideas."--BOOK JACKET.
Author : Pavel Exner,Benoit Grébert
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 42,8 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829004
This work contains contributions presented at the conference, QMath-8: Mathematical Results in Quantum Mechanics'', held at Universidad Nacional Autonoma de Mexico in December 2001. The articles cover a wide range of mathematical problems and focus on various aspects of quantum mechanics, quantum field theory and nuclear physics. Topics vary from spectral properties of the Schrodinger equation of various quantum systems to the analysis of quantum computation algorithms. The book should be suitable for graduate students and research mathematicians interested in the mathematical aspects of quantum mechanics.
Author : Andrew Knightly,Charles Li
Publisher : American Mathematical Soc.
Page : 392 pages
File Size : 40,6 Mb
Release : 2006
Category : Mathematics
ISBN : 9780821837399
The Fourier coefficients of modular forms are of widespread interest as an important source of arithmetic information. In many cases, these coefficients can be recovered from explicit knowledge of the traces of Hecke operators. The original trace formula for Hecke operators was given by Selberg in 1956. Many improvements were made in subsequent years, notably by Eichler and Hijikata. This book provides a comprehensive modern treatment of the Eichler-Selberg/Hijikata trace formulafor the traces of Hecke operators on spaces of holomorphic cusp forms of weight $\mathtt{k >2$ for congruence subgroups of $\operatorname{SL 2(\mathbf{Z )$. The first half of the text brings together the background from number theory and representation theory required for the computation. Thisincludes detailed discussions of modular forms, Hecke operators, adeles and ideles, structure theory for $\operatorname{GL 2(\mathbf{A )$, strong approximation, integration on locally compact groups, the Poisson summation formula, adelic zeta functions, basic representation theory for locally compact groups, the unitary representations of $\operatorname{GL 2(\mathbf{R )$, and the connection between classical cusp forms and their adelic counterparts on $\operatorname{GL 2(\mathbf{A )$. Thesecond half begins with a full development of the geometric side of the Arthur-Selberg trace formula for the group $\operatorname{GL 2(\mathbf{A )$. This leads to an expression for the trace of a Hecke operator, which is then computed explicitly. The exposition is virtually self-contained, withcomplete references for the occasional use of auxiliary results. The book concludes with several applications of the final formula.
Author : Alexander L. Sakhnovich,Lev A. Sakhnovich,Inna Ya. Roitberg
Publisher : Walter de Gruyter
Page : 356 pages
File Size : 44,9 Mb
Release : 2013-07-31
Category : Mathematics
ISBN : 9783110258615
This book is based on the method of operator identities and related theory of S-nodes, both developed by Lev Sakhnovich. The notion of the transfer matrix function generated by the S-node plays an essential role. The authors present fundamental solutions of various important systems of differential equations using the transfer matrix function, that is, either directly in the form of the transfer matrix function or via the representation in this form of the corresponding Darboux matrix, when Bäcklund–Darboux transformations and explicit solutions are considered. The transfer matrix function representation of the fundamental solution yields solution of an inverse problem, namely, the problem to recover system from its Weyl function. Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, system auxiliary to the N-wave equation and a system rationally depending on the spectral parameter are obtained in this way. The results on direct and inverse problems are applied in turn to the study of the initial-boundary value problems for integrable (nonlinear) wave equations via inverse spectral transformation method. Evolution of the Weyl function and solution of the initial-boundary value problem in a semi-strip are derived for many important nonlinear equations. Some uniqueness and global existence results are also proved in detail using evolution formulas. The reading of the book requires only some basic knowledge of linear algebra, calculus and operator theory from the standard university courses.
Author : Malcolm Brown,Fritz Gesztesy,Pavel Kurasov,Ari Laptev,Barry Simon,Gunter Stolz,Ian Wood
Publisher : Springer Nature
Page : 731 pages
File Size : 54,8 Mb
Release : 2023-09-21
Category : Mathematics
ISBN : 9783031311390
This volume is dedicated to the memory of Sergey Naboko (1950-2020). In addition to original research contributions covering the vast areas of interest of Sergey Naboko, it includes personal reminiscences and comments on the works and legacy of Sergey Naboko’s scientific achievements. Areas from complex analysis to operator theory, especially, spectral theory, are covered, and the papers will inspire current and future researchers in these areas.
Author : Vladimir Rabinovich,Steffen Roch,Bernd Silbermann
Publisher : Birkhäuser
Page : 404 pages
File Size : 40,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879118
This is the first monograph devoted to a fairly wide class of operators, namely band and band-dominated operators and their Fredholm theory. The main tool in studying this topic is limit operators. Applications are presented to several important classes of such operators: convolution type operators and pseudo-differential operators on bad domains and with bad coefficients.
Author : Nikolai K. Nikolski
Publisher : American Mathematical Soc.
Page : 458 pages
File Size : 43,9 Mb
Release : 2010-10-06
Category : Mathematics
ISBN : 9780821852651
Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, Vol. 92, AMS, 2002, this unique work combines four major topics of modern analysis and its applications: A. Hardy classes of holomorphic functions, B. Spectral theory of Hankel and Toeplitz operators, C. Function models for linear operators and free interpolations, and D. Infinite-dimensional system theory and signal processing. This volume contains Parts C and D. Function models for linear operators and free interpolations: This is a universal topic and, indeed, is the most influential operator theory technique in the post-spectral-theorem era. In this book, its capacity is tested by solving generalized Carleson-type interpolation problems. Infinite-dimensional system theory and signal processing: This topic is the touchstone of the three previously developed techniques. The presence of this applied topic in a pure mathematics environment reflects important changes in the mathematical landscape of the last 20 years, in that the role of the main consumer and customer of harmonic, complex, and operator analysis has more and more passed from differential equations, scattering theory, and probability to control theory and signal processing. This and the companion volume are geared toward a wide audience of readers, from graduate students to professional mathematicians. They develop an elementary approach to the subject while retaining an expert level that can be applied in advanced analysis and selected applications.