Spectral and Scattering Theory for Ordinary Differential Equations: Vol. I: Sturm-Liouville Equations

ISBN-13: 9783030590871 / Angielski / Miękka / 2020 / 379 str.

Christer Bennewitz; Malcolm Brown; Rudi Weikard

Recenzje

“The book is written in a fresh and fluent style, always starting with informal explanations before proceeding to precisely formulated definitions and theorems. Proofs are well structured and detailed without being pedantic. Each chapter has notes sketching the historical development, pointing to other approaches or indicating wider aspects. Exercises are added to each paragraph and even to some of the appendices. It provides lecturers of courses on ODEs or functional analysis with new material to demonstrate to students … .” (Hubert Kalf, zbMATH 1468.34001, 2021)

Spis treści

1 Introduction.- 2 Hilbert space.- 3 Abstract spectral theory.- 4 Sturm–Liouville equations.- 5 Left-definite Sturm–Liouville equations.- 6 Oscillation, spectral asymptotics and special functions.- 7 Uniqueness of the inverse problem.- 8 Scattering.- A Functional analysis.- B Stieltjes integrals.- C Schwartz distributions.- D Ordinary differential equations.- E Analytic functions.- F The Camassa–Holm equation.- References.- Symbol Index.- Subject Index.

O autorze

Christer Bennewitz is Emeritus Professor of Mathematics at Lund University. He previously worked at the University of Addis Abeba in Ethiopia, at Uppsala University in Sweden and at the University of Alabama at Birmingham in the US. His work is in the field of spectral theory for ordinary differential equations, in the last 25 years or so concentrating on inverse theory for problems not accessible to the Gelfand–Levitan theory.

Malcolm Brown is Professor of Computational Mathematics at Cardiff University. His work is focused on using both analytic and computational tools to get a better understanding of problems in the spectral theory of differential operators. Recently he has become interested in inverse problems, especially inverse spectral problems, and in questions of uniqueness and recovery that involve the Dirichlet-to-Neumann map.

Rudi Weikard is Professor of Mathematics at the University of Alabama at Birmingham. His recent work has been concentrated on inverse spectral and scattering theory for ordinary differential equations, for example, inverse resonance problems and Dirichlet-to-Neumann maps for quantum graphs. Previously he worked on analytic properties of the KdV hierarchy and the asymptotic behavior of large quantum systems.

Blurb

This graduate textbook offers an introduction to the spectral theory of ordinary differential equations, focusing on Sturm–Liouville equations.

Sturm–Liouville theory has applications in partial differential equations and mathematical physics. Examples include classical PDEs such as the heat and wave equations. Written by leading experts, this book provides a modern, systematic treatment of the theory. The main topics are the spectral theory and eigenfunction expansions for Sturm–Liouville equations, as well as scattering theory and inverse spectral theory. It is the first book offering a complete account of the left-definite theory for Sturm–Liouville equations.

The modest prerequisites for this book are basic one-variable real analysis, linear algebra, as well as an introductory course in complex analysis. More advanced background required in some parts of the book is completely covered in the appendices. With exercises in each chapter, the book is suitable for advanced undergraduate and graduate courses, either as an introduction to spectral theory in Hilbert space, or to the spectral theory of ordinary differential equations. Advanced topics such as the left-definite theory and the Camassa–Holm equation, as well as bibliographical notes, make the book a valuable reference for experts.