Introduction to spectral theory: selfadjoint ordinary differential operators

Selfadjoint Ordinary Differential Operators


Author: Boris Moiseevich Levitan,Ishkhan Saribekovich Sargsi͡an

Publisher: American Mathematical Soc.

ISBN: 082181589X

Category: Mathematics

Page: 525

View: 5025

This monograph is devoted to the spectral theory of the Sturm- Liouville operator and to the spectral theory of the Dirac system. In addition, some results are given for nth order ordinary differential operators. Those parts of this book which concern nth order operators can serve as simply an introduction to this domain, which at the present time has already had time to become very broad. For the convenience of the reader who is not familar with abstract spectral theory, the authors have inserted a chapter (Chapter 13) in which they discuss this theory, concisely and in the main without proofs, and indicate various connections with the spectral theory of differential operators.

Spectral Theory of Ordinary Differential Operators


Author: Joachim Weidmann

Publisher: Springer

ISBN: 3540479120

Category: Mathematics

Page: 304

View: 1066

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.

Spectral Theory of Canonical Differential Systems. Method of Operator Identities


Author: L.A. Sakhnovich

Publisher: Birkhäuser

ISBN: 3034887132

Category: Mathematics

Page: 202

View: 7761

Theorems of factorising matrix functions and the operator identity method play an essential role in this book in constructing the spectral theory (direct and inverse problems) of canonical differential systems. Includes many varied applications of the general theory.

Ordinary Differential Equations and Dynamical Systems


Author: Gerald Teschl

Publisher: American Mathematical Soc.

ISBN: 0821883283

Category: Mathematics

Page: 356

View: 7336

This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincare-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

Introduction to the Spectral Theory of Polynomial Operator Pencils


Author: A. S. Markus

Publisher: American Mathematical Soc.

ISBN: 0821890824

Category: Polynomial operator pencils

Page: 250

View: 5045

This monograph contains an exposition of the foundations of the spectral theory of polynomial operator pencils acting in a Hilbert space. Spectral problems for polynomial pencils have attracted a steady interest in the last 35 years, mainly because they arise naturally in such diverse areas of mathematical physics as differential equations and boundary value problems, controllable systems, the theory of oscillations and waves, elasticity theory, and hydromechanics. In this book, the author devotes most of his attention to the fundamental results of Keldysh on multiple completeness of the eigenvectors and associate vectors of a pencil, and on the asymptotic behavior of its eigenvalues and generalizations of these results. The author also presents various theorems on spectral factorization of pencils which grew out of known results of M. G. Krein and Heinz Langer. A large portion of the book involves the theory of selfadjoint pencils, an area having numerous applications. Intended for mathematicians, researchers in mechanics, and theoretical physicists interested in spectral theory and its applications, the book assumes a familiarity with the fundamentals of spectral theory of operators acting in a Hilbert space.

Theory of Linear Operators in Hilbert Space


Author: N. I. Akhiezer,I. M. Glazman

Publisher: Courier Corporation

ISBN: 0486318656

Category: Mathematics

Page: 378

View: 5480

This classic textbook by two mathematicians from the USSR's prestigious Kharkov Mathematics Institute introduces linear operators in Hilbert space, and presents in detail the geometry of Hilbert space and the spectral theory of unitary and self-adjoint operators. It is directed to students at graduate and advanced undergraduate levels, but because of the exceptional clarity of its theoretical presentation and the inclusion of results obtained by Soviet mathematicians, it should prove invaluable for every mathematician and physicist. 1961, 1963 edition.

An Introduction to Classical and P-adic Theory of Linear Operators and Applications


Author: Toka Diagana

Publisher: Nova Publishers

ISBN: 9781594544248

Category: Mathematics

Page: 116

View: 8448

This book provides the reader with a self-contained treatment of the classical operator theory with significant applications to abstract differential equations, and an elegant introduction to basic concepts and methods of the rapidly growing theory of the so-called p-adic operator theory.

Associations' Publications in Print


Author: N.A

Publisher: N.A

ISBN: 9780835214926

Category: Associations, institutions, etc

Page: N.A

View: 7524

1981- in 2 v.: v.1, Subject index; v.2, Title index, Publisher/title index, Association name index, Acronym index, Key to publishers' and distributors' abbreviations.

Introduction to Quantum Graphs


Author: Gregory Berkolaiko,Peter Kuchment

Publisher: American Mathematical Soc.

ISBN: 0821892118

Category: Mathematics

Page: 270

View: 6167

A ``quantum graph'' is a graph considered as a one-dimensional complex and equipped with a differential operator (``Hamiltonian''). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., ``meso-'' or ``nano-scale'') system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.

Spectral Analysis of Differential Operators

Interplay Between Spectral and Oscillatory Properties


Author: Fedor S. Rofe-Beketov,Aleksandr M. Khol'kin,Ognjen Milatovic

Publisher: World Scientific

ISBN: 9812703454

Category: Mathematics

Page: 438

View: 2420

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."

The Asymptotic Distribution of Eigenvalues of Partial Differential Operators


Author: Yu Safarov,D. Vassilev

Publisher: American Mathematical Soc.

ISBN: 9780821845776

Category: Mathematics

Page: 354

View: 3075

As the subject of extensive research for over a century, spectral asymptotics for partial differential operators attracted the attention of many outstanding mathematicians and physicists. This work studies the eigenvalues of elliptic linear boundary value problems and has as its main content a collection of asymptotic formulas describing the distribution of eigenvalues with high sequential numbers. Asymptotic formulas are used to illustrate standard eigenvalue problems of mechanics and mathematical physics. The volume provides a basic introduction to the necessary mathematical concepts and tools, such as microlocal analysis, billiards, symplectic geometry and Tauberian theorems. It is self-contained and is aimed at graduate students, research mathematicians, applied mathematicians, engineers, and physicists interested in partial differential equations.

Monographic Series


Author: Library of Congress

Publisher: N.A


Category: Children's literature in series

Page: N.A

View: 5350

Semi-bounded Differential Operators, Contractive Semigroups and Beyond


Author: Alberto Cialdea,Vladimir Maz'ya

Publisher: Springer

ISBN: 331904558X

Category: Mathematics

Page: 252

View: 3457

​In the present book the conditions are studied for the semi-boundedness of partial differential operators which is interpreted in different ways. Nowadays one knows rather much about L2-semibounded differential and pseudo-differential operators, although their complete characterization in analytic terms causes difficulties even for rather simple operators. Until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. The goal of the present book is to partially fill this gap. Various types of semi-boundedness are considered and some relevant conditions which are either necessary and sufficient or best possible in a certain sense are given. Most of the results reported in this book are due to the authors.

Banach Algebra Techniques in Operator Theory


Author: Ronald G. Douglas

Publisher: Springer Science & Business Media

ISBN: 1461216567

Category: Mathematics

Page: 198

View: 2074

A discussion of certain advanced topics in operator theory, providing the necessary background while assuming only standard senior-first year graduate courses in general topology, measure theory, and algebra. Each chapter ends with source notes which suggest additional reading along with comments on who proved what and when, followed by a large number of problems of varying difficulty. This new edition will appeal to a whole new generation of students seeking an introduction to this topic.