Spectral and Scattering Theory for Ordinary Differential Equations

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.

• 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.

“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)