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Hardcover ISBN:  9781470476663 
Product Code:  COLL/67 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470478247 
Product Code:  COLL/67.E 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
Hardcover ISBN:  9781470476663 
eBook ISBN:  9781470478247 
Product Code:  COLL/67.B 
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Book DetailsColloquium PublicationsVolume: 67; 2024; 927 ppMSC: Primary 34; 47
This book provides a detailed treatment of the various facets of modern Sturm–Liouville theory, including such topics as Weyl–Titchmarsh theory, classical, renormalized, and perturbative oscillation theory, boundary data maps, traces and determinants for Sturm–Liouville operators, strongly singular Sturm–Liouville differential operators, generalized boundary values, and Sturm–Liouville operators with distributional coefficients. To illustrate the theory, the book develops an array of examples from Floquet theory to shortrange scattering theory, higherorder KdV trace relations, elliptic and algebrogeometric finite gap potentials, reflectionless potentials and the Sodin–Yuditskii class, as well as a detailed collection of singular examples, such as the Bessel, generalized Bessel, and Jacobi operators. A set of appendices contains background on the basics of linear operators and spectral theory in Hilbert spaces, Schatten–von Neumann classes of compact operators, selfadjoint extensions of symmetric operators, including the Friedrichs and Krein–von Neumann extensions, boundary triplets for ODEs, Kreintype resolvent formulas, sesquilinear forms, Nevanlinna–Herglotz functions, and Bessel functions.
ReadershipGraduate students and researchers interested in ordinary differential operators.

Table of Contents

Introduction

A bit of physical motivation

Preliminaries on ODEs

The regular problem on a compact interval $[a,b]\subset\mathbb{R}$

The singular problem on $(a,b)\subseteq \mathbb{R}$

The spectral function for a problem with a regular endpoint

The 2 x 2 spectral matrix function in the presence of two singular interval endpoints for the problem on $(a,b)\subseteq\mathbb{R}$

Classical oscillation theory, principal solutions, and nonprinicpal solutions

Renormalized oscillation theory

Perturbative oscillation criteria and perturbative Hardytype inequalities

Boundary data maps

Spectral zeta functions and computing traces and determinants for SturmLiouville operators

The singular problem on $(a,b)\subseteq\mathbb{R}$ revisited

Fourcoefficient SturmLiouville operators and distributional potential coefficients

Epilogue: Applications to some partial differnetial equations of mathematical physics

Basic facts on linear operators

Basics of spectral theory

Classes of bounded linear operators

Extensions of symmetric operators

Elements of sesquilinear forms

Basics of NevanlinnaHerglotz functions

Bessel functions in a nutshell

Bibliography

Author index

List of symbols

Subject index


Additional Material

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This book provides a detailed treatment of the various facets of modern Sturm–Liouville theory, including such topics as Weyl–Titchmarsh theory, classical, renormalized, and perturbative oscillation theory, boundary data maps, traces and determinants for Sturm–Liouville operators, strongly singular Sturm–Liouville differential operators, generalized boundary values, and Sturm–Liouville operators with distributional coefficients. To illustrate the theory, the book develops an array of examples from Floquet theory to shortrange scattering theory, higherorder KdV trace relations, elliptic and algebrogeometric finite gap potentials, reflectionless potentials and the Sodin–Yuditskii class, as well as a detailed collection of singular examples, such as the Bessel, generalized Bessel, and Jacobi operators. A set of appendices contains background on the basics of linear operators and spectral theory in Hilbert spaces, Schatten–von Neumann classes of compact operators, selfadjoint extensions of symmetric operators, including the Friedrichs and Krein–von Neumann extensions, boundary triplets for ODEs, Kreintype resolvent formulas, sesquilinear forms, Nevanlinna–Herglotz functions, and Bessel functions.
Graduate students and researchers interested in ordinary differential operators.

Introduction

A bit of physical motivation

Preliminaries on ODEs

The regular problem on a compact interval $[a,b]\subset\mathbb{R}$

The singular problem on $(a,b)\subseteq \mathbb{R}$

The spectral function for a problem with a regular endpoint

The 2 x 2 spectral matrix function in the presence of two singular interval endpoints for the problem on $(a,b)\subseteq\mathbb{R}$

Classical oscillation theory, principal solutions, and nonprinicpal solutions

Renormalized oscillation theory

Perturbative oscillation criteria and perturbative Hardytype inequalities

Boundary data maps

Spectral zeta functions and computing traces and determinants for SturmLiouville operators

The singular problem on $(a,b)\subseteq\mathbb{R}$ revisited

Fourcoefficient SturmLiouville operators and distributional potential coefficients

Epilogue: Applications to some partial differnetial equations of mathematical physics

Basic facts on linear operators

Basics of spectral theory

Classes of bounded linear operators

Extensions of symmetric operators

Elements of sesquilinear forms

Basics of NevanlinnaHerglotz functions

Bessel functions in a nutshell

Bibliography

Author index

List of symbols

Subject index