Kravchenko V.V. Direct and Inverse Sturm-Liouville Problems: A Method of Solution

Springer, 2020. — 155 p. — (Frontiers in Mathematics). — ISBN: 978-3-030-47848-3.This book provides an introduction to the most recent developments in the theory and practice of direct and inverse Sturm-Liouville problems on finite and infinite intervals. A universal approach for practical solving of direct and inverse spectral and scattering problems is presented, based on the notion of transmutation (transformation) operators and their efficient construction. Analytical representations for solutions of Sturm-Liouville equations as well as for the integral kernels of the transmutation operators are derived in the form of functional series revealing interesting special features and lending themselves to direct and simple numerical solution of a wide variety of problems.
The book is written for undergraduate and graduate students, as well as for mathematicians, physicists and engineers interested in direct and inverse spectral problems.Preliminaries on Sturm-Liouville Equations
Direct and Inverse Sturm-Liouville Problems on Finite Intervals
Direct and Inverse Sturm-Liouville Problems on a Half-Line
Quantum Scattering Problem on the Half-Line
Scattering Problem on the Line
Inverse Scattering Transform Method
Main Transmutation Operators
Construction of Transmutations and Series Representations for Solutions
Series Representations for the Kernel A(x, t) and for the Jost Solution
Sturm–Liouville Problems on Finite Intervals
Spectral Problems on Infinite Intervals
The Inverse Sturm–Liouville Problem on a Finite Interval
Solution of the Inverse Problem on the Half-Line
Solution of the Inverse Quantum Scattering Problem on the Half-Line
Solution of the Inverse Scattering Problem on the Line