Spectral Properties of Non-self-adjoint Operators

Spectral Properties of Non-self-adjoint Operators
Author: John L. Weir
Publisher:
Total Pages: 220
Release: 2010
Genre: Nonselfadjoint operators
ISBN:


Download Spectral Properties of Non-self-adjoint Operators Book in PDF, Epub and Kindle

The aim of this thesis is to study the spectral properties of non-self-adjoint operators via related self-adjoint operators. We consider two different prob-lems: one in which the spectra of a family of non-self-adjoint operators are identical to those of a family of self-adjoint operators and one in which the growth rates of semigroups generated by non-self-adjoint operators are bounded by the growth rates of semigroups generated by related self-adjoint operators. -- In the first problem, we consider a family of non-self-adjoint operators arising in the study of a problem in fluid mechanics in a paper written by Benilov, O'Brien and Sazonov, who argued from numerical and asymptotic evidence that the spectra of the operators are real. We show that the spectra of the operators are identical to the spectra of a family of self-adjoint operators and consist of infinitely many real eigenvalues which accumulate only at infinity. We make use of this correspondence to study certain other properties of the eigenvalues of the non-self-adjoint operators via the self-adjoint operators. In particular, we consider the asymptotic distribution of the eigenvalues for each fixed operator, and the behaviour of each eigenvalue as a small parameter tends to zero. -- In the second, we study the spectral asymptotics of large skew symmetric perturbations of a wide class of Schrodinger operators, generalizing some of the results obtained by Gallagher, Gallay and Nier for the one-dimensional quantum harmonic oscillator. We obtain bounds on the growth rates of the one-parameter semigroups generated by the perturbed operators in terms of the minima of the spectra of related self-adjoint operators. These self-adjoint operators are perturbations of the original Schrodinger operators by non-negative potentials, and we obtain lower bounds on the spectral minima in terms of the behaviour of the potentials at their zeros.

Spectral Theory of Self-Adjoint Operators in Hilbert Space

Spectral Theory of Self-Adjoint Operators in Hilbert Space
Author: Michael Sh. Birman
Publisher: Springer Science & Business Media
Total Pages: 316
Release: 2012-12-06
Genre: Mathematics
ISBN: 9400945868


Download Spectral Theory of Self-Adjoint Operators in Hilbert Space Book in PDF, Epub and Kindle

It isn't that they can't see the solution. It is Approach your problems from the right end that they can't see the problem. and begin with the answers. Then one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be com pletely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order" , which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators

Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators
Author: John Locker
Publisher: American Mathematical Soc.
Total Pages: 266
Release: 2000
Genre: Mathematics
ISBN: 0821820494


Download Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators Book in PDF, Epub and Kindle

Develops the spectral theory of an nth order non-self-adjoint two- point differential operator L in the complex Hilbert space L2[0,1]. The differential operator L is determined by an nth order formal differential l and by n linearly independent boundary values B1,.,Bn. Locker first lays the foundations of the spectral theory for closed linear operators and Fredholm operators in Hilbert spaces before developing the spectral theory of the differential operator L. The book is a sequel to Functional analysis and two-point differential operators, 1986. Annotation copyrighted by Book News, Inc., Portland, OR.

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
Author: Johannes Sjöstrand
Publisher: Springer
Total Pages: 496
Release: 2019-05-17
Genre: Mathematics
ISBN: 3030108198


Download Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations Book in PDF, Epub and Kindle

The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago. In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book. Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.

Spectral Theory and Its Applications

Spectral Theory and Its Applications
Author: Bernard Helffer
Publisher: Cambridge University Press
Total Pages: 263
Release: 2013-01-17
Genre: Mathematics
ISBN: 110703230X


Download Spectral Theory and Its Applications Book in PDF, Epub and Kindle

Introduces the basic tools in spectral analysis using numerous examples from the Schrödinger operator theory and various branches of physics.

Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients
Author: Aleksandr Kholkin
Publisher:
Total Pages: 0
Release: 2019
Genre: Electronic books
ISBN:


Download Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients Book in PDF, Epub and Kindle

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green,Äôs function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

Non-Selfadjoint Operators in Quantum Physics

Non-Selfadjoint Operators in Quantum Physics
Author: Fabio Bagarello
Publisher: John Wiley & Sons
Total Pages: 434
Release: 2015-07-24
Genre: Science
ISBN: 1118855264


Download Non-Selfadjoint Operators in Quantum Physics Book in PDF, Epub and Kindle

A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics, the book features: Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and perturbation theory Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics and condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.

Spectral Theory and Differential Operators

Spectral Theory and Differential Operators
Author: E. Brian Davies
Publisher: Cambridge University Press
Total Pages: 198
Release: 1995
Genre: Mathematics
ISBN: 9780521587105


Download Spectral Theory and Differential Operators Book in PDF, Epub and Kindle

This book could be used either for self-study or as a course text, and aims to lead the reader to the more advanced literature on partial differential operators.

Spectral Theory of Block Operator Matrices and Applications

Spectral Theory of Block Operator Matrices and Applications
Author: Christiane Tretter
Publisher: Imperial College Press
Total Pages: 297
Release: 2008
Genre: Mathematics
ISBN: 1848161123


Download Spectral Theory of Block Operator Matrices and Applications Book in PDF, Epub and Kindle

This book presents a wide panorama of methods to investigate the spectral properties of block operator matrices. Particular emphasis is placed on classes of block operator matrices to which standard operator theoretical methods do not readily apply: non-self-adjoint block operator matrices, block operator matrices with unbounded entries, non-semibounded block operator matrices, and classes of block operator matrices arising in mathematical physics.The main topics include: localization of the spectrum by means of new concepts of numerical range; investigation of the essential spectrum; variational principles and eigenvalue estimates; block diagonalization and invariant subspaces; solutions of algebraic Riccati equations; applications to spectral problems from magnetohydrodynamics, fluid mechanics, and quantum mechanics.

A Guide to Spectral Theory

A Guide to Spectral Theory
Author: Christophe Cheverry
Publisher: Springer Nature
Total Pages: 258
Release: 2021-05-06
Genre: Mathematics
ISBN: 3030674622


Download A Guide to Spectral Theory Book in PDF, Epub and Kindle

This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book. Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.