Periodic Differential Operators

Periodic Differential Operators
Author: B. Malcolm Brown
Publisher: Springer Science & Business Media
Total Pages: 220
Release: 2012-10-30
Genre: Mathematics
ISBN: 3034805284

Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.

Periodic Integral and Pseudodifferential Equations with Numerical Approximation

Periodic Integral and Pseudodifferential Equations with Numerical Approximation
Author: Jukka Saranen
Publisher: Springer Science & Business Media
Total Pages: 461
Release: 2013-03-09
Genre: Mathematics
ISBN: 3662047969

An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.

Impulsive Differential Equations

Impulsive Differential Equations
Author: Drumi Bainov
Publisher: Routledge
Total Pages: 238
Release: 2017-11-01
Genre: Mathematics
ISBN: 1351439103

Impulsive differential equations have been the subject of intense investigation in the last 10-20 years, due to the wide possibilities for their application in numerous fields of science and technology. This new work presents a systematic exposition of the results solving all of the more important problems in this field.

Spectral Theory of Ordinary Differential Operators

Spectral Theory of Ordinary Differential Operators
Author: Joachim Weidmann
Publisher: Springer
Total Pages: 310
Release: 2006-11-15
Genre: Mathematics
ISBN: 3540479120

These notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.

Stability & Periodic Solutions of Ordinary & Functional Differential Equations

Stability & Periodic Solutions of Ordinary & Functional Differential Equations
Author: T. A. Burton
Publisher: Courier Corporation
Total Pages: 370
Release: 2014-06-24
Genre: Mathematics
ISBN: 0486150453

This book's discussion of a broad class of differential equations includes linear differential and integrodifferential equations, fixed-point theory, and the basic stability and periodicity theory for nonlinear ordinary and functional differential equations.

Spectral Analysis of Differential Operators

Spectral Analysis of Differential Operators
Author: Fedor S. Rofe-Beketov
Publisher: World Scientific
Total Pages: 466
Release: 2005
Genre: Mathematics
ISBN: 9812703454

This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic SchrAdinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other monographs. The only prerequisites needed to read it are basics of ordinary differential equations and operator theory. It should be accessible to graduate students, though its main topics are of interest to research mathematicians working in functional analysis, differential equations and mathematical physics, as well as to physicists interested in spectral theory of differential operators."

Floquet Theory for Partial Differential Equations

Floquet Theory for Partial Differential Equations
Author: P.A. Kuchment
Publisher: Birkhäuser
Total Pages: 363
Release: 2012-12-06
Genre: Science
ISBN: 3034885733

Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389]. They arise in many physical and technical applications [177, 178, 272]. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for integration of nonlinear differential equations. This has led to significant progress in this traditional area [27, 71, 72, 111 119, 250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393, 403, 404]. At the same time, many theoretical and applied problems lead to periodic partial differential equations. We can mention, for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107, 123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417], hydrodynamics [179, 180], elasticity theory [395], the theory of guided waves [87-89, 208, 300], homogenization theory [29, 41, 348], direct and inverse scattering [175, 206, 216, 314, 388, 406-408], parametric resonance theory [122, 178], and spectral theory and spectral geometry [103 105, 381, 382, 389]. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations. The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267].

Spectral Theory of Differential Operators

Spectral Theory of Differential Operators
Author: T. Suslina
Publisher: American Mathematical Soc.
Total Pages: 318
Release: 2008-01-01
Genre: Mathematics
ISBN: 9780821890776

"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.

Spectral Analysis of Differential Operators

Spectral Analysis of Differential Operators
Author: Fedor S. Rofe-Beketov
Publisher: World Scientific
Total Pages: 463
Release: 2005
Genre: Science
ISBN: 9812562761

- Detailed bibliographical comments and some open questions are given after each chapter - Indicates connections between the content of the book and many other topics in mathematics and physics - Open questions are formulated and commented with the intention to attract attention of young mathematicians

Spectra of Random and Almost-Periodic Operators

Spectra of Random and Almost-Periodic Operators
Author: Leonid Pastur
Publisher: Springer
Total Pages: 0
Release: 2011-12-10
Genre: Science
ISBN: 9783642743481

In the last fifteen years the spectral properties of the Schrodinger equation and of other differential and finite-difference operators with random and almost-periodic coefficients have attracted considerable and ever increasing interest. This is so not only because of the subject's position at the in tersection of operator spectral theory, probability theory and mathematical physics, but also because of its importance to theoretical physics, and par ticularly to the theory of disordered condensed systems. It was the requirements of this theory that motivated the initial study of differential operators with random coefficients in the fifties and sixties, by the physicists Anderson, 1. Lifshitz and Mott; and today the same theory still exerts a strong influence on the discipline into which this study has evolved, and which will occupy us here. The theory of disordered condensed systems tries to describe, in the so-called one-particle approximation, the properties of condensed media whose atomic structure exhibits no long-range order. Examples of such media are crystals with chaotically distributed impurities, amorphous substances, biopolymers, and so on. It is natural to describe the location of atoms and other characteristics of such media probabilistically, in such a way that the characteristics of a region do not depend on the region's position, and the characteristics of regions far apart are correlated only very weakly. An appropriate model for such a medium is a homogeneous and ergodic, that is, metrically transitive, random field.