**AMS Chelsea Publishing**

Volume: 373;
2011;
393 pp;
Hardcover

MSC: Primary 34; 35; 47;

**Print ISBN: 978-0-8218-5316-0
Product Code: CHEL/373.H**

List Price: $61.00

AMS Member Price: $54.90

MAA Member Price: $54.90

**Electronic ISBN: 978-1-4704-1580-8
Product Code: CHEL/373.H.E**

List Price: $61.00

AMS Member Price: $48.80

MAA Member Price: $54.90

#### Supplemental Materials

# Sturm-Liouville Operators and Applications: Revised Edition

Share this page
*Vladimir A. Marchenko*

Translated by A. Iacob

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

The spectral theory of Sturm-Liouville operators is a classical domain of
analysis, comprising a wide variety of problems. Besides the basic results
on the structure of the spectrum and the eigenfunction expansion of
regular and singular Sturm-Liouville problems, it is in this domain that
one-dimensional quantum scattering theory, inverse spectral problems, and
the surprising connections of the theory with nonlinear evolution
equations first become related. The main goal of this book is to show what
can be achieved with the aid of transformation operators in spectral
theory as well as in their applications. The main methods and results in
this area (many of which are credited to the author) are for the first
time examined from a unified point of view.

The direct and inverse problems of spectral analysis and the inverse
scattering problem are solved with the help of the transformation
operators in both self-adjoint and nonself-adjoint cases. The asymptotic
formulae for spectral functions, trace formulae, and the exact relation
(in both directions) between the smoothness of potential and the
asymptotics of eigenvalues (or the lengths of gaps in the spectrum) are
obtained. Also, the applications of transformation operators and their
generalizations to soliton theory (i.e., solving nonlinear equations of
Korteweg-de Vries type) are considered.

The new Chapter 5 is devoted to the stability of the inverse
problem solutions. The estimation of the accuracy with which the
potential of the Sturm-Liouville operator can be restored from the
scattering data or the spectral function, if they are only known on a
finite interval of a spectral parameter (i.e., on a finite interval of
energy), is obtained.

#### Readership

Graduate students and research mathematicians interested in operator theory.

#### Table of Contents

# Table of Contents

## Sturm-Liouville Operators and Applications: Revised Edition

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the revised edition vii8
- Preface ix10
- The Sturm-Liouville equation and transformation operators 116
- The Sturm-Liouville boundary value problem on the half line 101116
- The boundary value problem of scattering theory 173188
- Nonlinear equations 307322
- Stability of inverse problems 363378
- References 389404
- Back Cover Back Cover1410