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Sturm-Liouville theory. (English) Zbl 1103.34001
Mathematical Surveys and Monographs 121. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3905-5/hbk). xi, 328 p. \$ 84.00 (2005).

Since its inception over 150 years ago, numerous works on Sturm-Liouville problems have been published, and many monographs have covered and continue to cover aspect of Sturm-Liouville. The monograph under review is a welcome addition to that literature; it is solely dedicated to Sturm-Liouville problems

$-{\left(p{y}^{\text{'}}\right)}^{\text{'}}+qy=\lambda wy\phantom{\rule{2.em}{0ex}}\left(1\right)$

over an interval $J\subset ℝ$, subject to boundary conditions, where $1/p$, $q$, $w$ are locally integrable functions, and it covers more topics of this theory than any other book: from the very classical results to the most recent ones. Not every aspect of Sturm-Liouville theory can be covered in one volume; for example, inverse problems are not discussed. However, the detailed comments at the end of each chapter provide ample references to the literature on virtually every aspect of Sturm-Liouville theory. Indeed, the comprehensive bibliography of nearly 650 journal articles and monographs is an invaluable resource for everyone working in Sturm-Liouville theory.

The book is divided into five chapters: existence and uniqueness problems; regular boundary value problems; oscillation and singular existence problems; singular boundary value problems; examples and other topics.

Each part can be read independently, and the prerequisites are modest. This makes the book also ideal reading for anyone interested in applying Sturm-Liouville theory. All stated results are either proved or a precise bibiliographic reference to the proof is given.

In Part 1, the initial value problem for $-{\left(p{y}^{\text{'}}\right)}^{\text{'}}+qy=f$ is studied by considering the first-order system

${Y}^{\text{'}}=PY+F,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where

$P=\left(\begin{array}{cc}0& 1/p\\ q& 0\end{array}\right),\phantom{\rule{1.em}{0ex}}F=\left(\begin{array}{c}0\\ f\end{array}\right),\phantom{\rule{1.em}{0ex}}Y=\left(\begin{array}{c}y\\ p{y}^{\text{'}}\end{array}\right)·$

In this system, $1/p=0$, i. e., $p=\infty$ is allowed on a set of positive measure. This results in some exotic examples in the spectral theory dealt with in later parts.

In Part 2, the spectrum of regular eigenvalue problems is discussed. This is done in a straightforward classical manner without the use of functional analysis, but includes the Prüfer transformation approach. It is shown that the general form of (2) allows for problems with finite spectrum of exactly $n$ eigenvalues for any positive integer $n$.

Part 3 is concerned with oscillation theory and the limit-point, limit-circle classification, which lays the basis for Part 4. After a short introduction to general singular boundary value problems, Part 4 focusses on the selfadjoint case, both for definite and indefinite problems. It is here where functional analysis and operator theory is introduced. In particular, the author includes a brief introduction to Krein spaces for the study of indefinte problems.

Finally, in Part 5, numerous examples are considered. Among others, they are classified according to singular/regular, oscillatory/nonoscillatory, limit-point/limit-circle.

In summary, this monograph offers a wealth of information on Sturm-Liouville theory and is an ideal textbook for a course in this field, serves as an indispensible source for every researcher working in this area, is ideally suited for self-study due to its detailed proofs and comprehensive bibliography, and is recommended to any applied scientist who wants to use Sturm-Liouville theory.

##### MSC:
 34-02 Research monographs (ordinary differential equations) 34B20 Weyl theory and its generalizations 34B24 Sturm-Liouville theory 34L05 General spectral theory for OD operators 47B25 Symmetric and selfadjoint operators (unbounded) 47B50 Operators on spaces with an indefinite metric