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

✍ Scribed by Vladislav V. Kravchenko

• Publisher: Springer International Publishing;Birkhäuser
• Year: 2020
• Tongue: English
• Leaves: 155
• Series: Frontiers in Mathematics
• Edition: 1st ed.
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-xi
Introduction (Vladislav V. Kravchenko)....Pages 1-4
Front Matter ....Pages 5-5
Preliminaries on Sturm-Liouville Equations (Vladislav V. Kravchenko)....Pages 7-13
Direct and Inverse Sturm-Liouville Problems on Finite Intervals (Vladislav V. Kravchenko)....Pages 15-18
Direct and Inverse Sturm-Liouville Problems on a Half-Line (Vladislav V. Kravchenko)....Pages 19-22
Quantum Scattering Problem on the Half-Line (Vladislav V. Kravchenko)....Pages 23-24
Scattering Problem on the Line (Vladislav V. Kravchenko)....Pages 25-28
Inverse Scattering Transform Method (Vladislav V. Kravchenko)....Pages 29-31
Front Matter ....Pages 33-33
Main Transmutation Operators (Vladislav V. Kravchenko)....Pages 35-40
Construction of Transmutations and Series Representations for Solutions (Vladislav V. Kravchenko)....Pages 41-60
Series Representations for the Kernel A(x, t) and for the Jost Solution (Vladislav V. Kravchenko)....Pages 61-74
Front Matter ....Pages 75-75
Sturm–Liouville Problems on Finite Intervals (Vladislav V. Kravchenko)....Pages 77-87
Spectral Problems on Infinite Intervals (Vladislav V. Kravchenko)....Pages 89-98
Front Matter ....Pages 99-99
The Inverse Sturm–Liouville Problem on a Finite Interval (Vladislav V. Kravchenko)....Pages 101-116
Solution of the Inverse Problem on the Half-Line (Vladislav V. Kravchenko)....Pages 117-123
Solution of the Inverse Quantum Scattering Problem on the Half-Line (Vladislav V. Kravchenko)....Pages 125-130
Solution of the Inverse Scattering Problem on the Line (Vladislav V. Kravchenko)....Pages 131-143
Back Matter ....Pages 145-154

✦ Subjects

Mathematics; Analysis; Operator Theory