Perturbation Theory for Linear Operators de Tosio Kato

Perturbation Theory for Linear Operators: Classics in Mathematics, cartea 132

Autor Tosio Kato

en Limba Engleză Paperback – 15 feb 1995

Analiza stabilității valorilor proprii și a spectrelor sub influența perturbărilor reprezintă nucleul acestei ediții revizuite, publicată în seria Classics in Mathematics. Perturbation Theory for Linear Operators rămâne tratatul definitiv care formalizează legătura dintre analiza funcțională și fizica cuantică. Ne-a atras atenția rigoarea cu care Tosio Kato construiește progresia materialului: primele două capitole sunt dedicate spațiilor finit-dimensionale, oferind o bază solidă înainte de a trece la complexitatea operatorilor în spații Banach și Hilbert.

Această a doua ediție nu este doar o republicare, ci o actualizare necesară. Putem afirma că rescrierea secțiunilor cheie despre stabilitatea operatorilor închiși și adăugarea notelor suplimentare reflectă evoluția rapidă a domeniului după prima ediție. Comparabil cu Linear Operators in Hilbert Spaces de Joachim Weidmann în ceea ce privește precizia tehnică, volumul lui Kato se distinge prin profunzimea tratamentului analitic al seriilor de perturbare și al estimărilor de eroare. Spre deosebire de A Short Introduction to Perturbation Theory for Linear Operators, care extrage doar fundamentele pentru un public mai larg, acest volum complet oferă instrumentarul exhaustiv necesar cercetării avansate.

Structura este logică și incrementală, pornind de la teoria elementară a operatorilor, trecând prin teoremele de stabilitate și culminând cu teoria spectrală. Recomandăm acest volum pentru modul în care reușește să unifice rigoarea matematică pură cu necesitățile practice ale fizicii teoretice, corectând erorile tipografice anterioare și oferind o bibliografie extinsă care ancorează textul în contextul științific modern.

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Specificații

ISBN-13: 9783540586616
ISBN-10: 354058661X
Pagini: 648
Ilustrații: XXI, 623 p.
Dimensiuni: 155 x 235 x 34 mm
Greutate: 0.96 kg
Ediția:2nd ed. 1995
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Classics in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

De ce să citești această carte

Este resursa fundamentală pentru oricine studiază stabilitatea sistemelor liniare. Cititorul câștigă acces la o metodologie sistematică de tratare a perturbărilor, esențială în mecanica cuantică și hidrodinamică. Se adresează matematicienilor și fizicienilor care au nevoie de o bază teoretică solidă, dincolo de simple metode computaționale, oferind demonstrații complete și context istoric prin notele autorului.

Despre autor

Tosio Kato (1917–1999) a fost un cercetător de elită, considerat pionier al fizicii matematice moderne. După studii de fizică teoretică la Universitatea Imperială din Tokyo și o carieră academică începută în Japonia, s-a alăturat Universității din California, Berkeley, în 1962, unde a devenit Profesor Emeritus. Contribuțiile sale fundamentale acoperă teoria operatorilor, mecanica cuantică și ecuațiile diferențiale cu derivate parțiale. Opera sa este marcată de o claritate excepțională și de capacitatea de a rezolva probleme matematice complexe ridicate de fenomenele fizice, lăsând o moștenire durabilă în analiza funcțională contemporană.

Descriere scurtă

In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the para graphs V-§ 4.5, VI-§ 4.3, and VIII-§ 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention. Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced. Despite considerable expansion, the bibliography i" not intended to be complete. Berkeley, April 1976 TosIO RATO Preface to the First Edition This book is intended to give a systematic presentation of perturba tion theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences.

Cuprins

One Operator theory in finite-dimensional vector spaces.- § 1. Vector spaces and normed vector spaces.- § 2. Linear forms and the adjoint space.- § 3. Linear operators.- § 4. Analysis with operators.- § 5. The eigenvalue problem.- § 6. Operators in unitary spaces.- Two Perturbation theory in a finite-dimensional space.- § 1. Analytic perturbation of eigenvalues.- § 2. Perturbation series.- § 3. Convergence radii and error estimates.- § . Similarity transformations of the eigenspaces and eigenvectors.- § 5. Non-analytic perturbations.- § 6. Perturbation of symmetric operators.- Three Introduction to the theory of operators in Banach spaces.- § 1. Banach spaces.- § 2. Linear operators in Banach spaces.- § 3. Bounded operators.- § 4. Compact operators.- § 5. Closed operators.- § 6. Resolvents and spectra.- Four Stability theorems.- §1. Stability of closedness and bounded invertibility.- § 2. Generalized convergence of closed operators.- § 3. Perturbation of the spectrum.- § 4. Pairs of closed linear manifolds.- § 5. Stability theorems for semi-Fredholm operators.- § 6. Degenerate perturbations.- Five Operators in Hilbert spaces.- § 1. Hilbert space.- § 2. Bounded operators in Hilbert spaces.- § 3. Unbounded operators in Hilbert spaces.- § 4. Perturbation of self adjoint operators.- § 5. The Schrödinger and Dirac operators.- Six Sesquilinear forms in Hilbert spaces and associated operators.- § 1. Sesquilinear and quadratic forms.- § 2. The representation theorems.- § 3. Perturbation of sesquilinear forms and the associated operators.- § 4. Quadratic forms and the Schrödinger operators.- § 5. The spectral theorem and perturbation of spectral families.- Seven Analytic perturbation theory.- § 1. Analytic families of operators.- § 2.Holomorphic families of type (A).- § 3. Selfadjoint holomorphic families.- § 4. Holomorphic families of type (B).- § 5. Further problems of analytic perturbation theory.- § 6. Eigenvalue problems in the generalized form.- Eight Asymptotic perturbation theory.- § 1. Strong convergence in the generalized sense.- § 2. Asymptotic expansions.- § 3. Generalized strong convergence of sectorial operators.- § 4. Asymptotic expansions for sectorial operators.- § 5. Spectral concentration.- Nine Perturbation theory for semigroups of operators.- § 1. One-parameter semigroups and groups of operators.- § 2. Perturbation of semigroups.- § 3. Approximation by discrete semigroups.- Ten Perturbation of continuous spectra and unitary equivalence.- §1. The continuous spectrum of a selfadjoint operator.- § 2. Perturbation of continuous spectra.- § 3. Wave operators and the stability of absolutely continuous spectra.- § 4. Existence and completeness of wave operators.- § 5. A stationary method.- Supplementary Notes.- Supplementary Bibliography.- Notation index.- Author index.

Recenzii

"The monograph by T. Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces. It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced. Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4). Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8). The fundamentals of semigroup theory are given in chapter 9. The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10.
The first edition is now 30 years old. The revised edition is 20 years old. Nevertheless it is a standard textbook for the theory of linear operators. It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located. In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory. However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field.
Zentralblatt MATH, 836

Notă biografică

Biography of Tosio Kato
Tosio Kato was born in 1917 in a village to the north of Tokyo. He studied theoretical physics at the Imperial University of Tokyo. After several years of inactivity during World War II due to poor health, he joined the Faculty of Science at the University of Tokyo in 1951. From 1962 he was Professor of Mathematics at the University of California, Berkeley, where he is now Professor Emeritus.
Kato was a pioneer in modern mathematical physics. He worked in te areas of operator theory, quantum mechanics, hydrodynamics, and partial differential equations, both linear and nonlinear.