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Hardcover ISBN:  9781470453664 
Product Code:  SURV/245 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470454302 
Product Code:  SURV/245.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470453664 
eBook ISBN:  9781470454302 
Product Code:  SURV/245.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 245; 2019; 250 ppMSC: Primary 47; 34
In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular SturmLiouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded selfadjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on selfadjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained.
In this monograph the authors summarize some of these directions discussing selfadjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasidifferential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, selfadjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and selfadjoint characterizations are extended to twointerval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
ReadershipGraduate students and researchers interested in differential equations, in particular, the SturmLiouville problem.

Table of Contents

Differential equations and expressions

First order systems

Quasidifferential expressions and equations

The Lagrange identity and maximal and minimal operators

Deficiency indices

Symmetric, selfadjoint, and dissipative operators

Regular symmetric operators

Singular symmetric operators

Selfadjoint operators

Selfadjoint and symmetric boundary conditions

Solutions and spectrum

Coefficients, the deficiency index, spectrum

Dissipative operators

Twointerval problems

Twointerval symmetric domains

Twointerval symmetric domain characterization with maximal domain functions

Other topics

Green’s function and adjoint problems

Notation

Topics not covered and open problems


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In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular SturmLiouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded selfadjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on selfadjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained.
In this monograph the authors summarize some of these directions discussing selfadjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasidifferential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, selfadjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and selfadjoint characterizations are extended to twointerval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
Graduate students and researchers interested in differential equations, in particular, the SturmLiouville problem.

Differential equations and expressions

First order systems

Quasidifferential expressions and equations

The Lagrange identity and maximal and minimal operators

Deficiency indices

Symmetric, selfadjoint, and dissipative operators

Regular symmetric operators

Singular symmetric operators

Selfadjoint operators

Selfadjoint and symmetric boundary conditions

Solutions and spectrum

Coefficients, the deficiency index, spectrum

Dissipative operators

Twointerval problems

Twointerval symmetric domains

Twointerval symmetric domain characterization with maximal domain functions

Other topics

Green’s function and adjoint problems

Notation

Topics not covered and open problems