Sturm-Liouville Theory: Past and Present W. O. Amrein, A. M. Hinz, and D. B. Pearson (Editors) Birkauser (2005) 335 pp., price USD 89.95 ISBN: 3764370661.
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: intermediate.

This text is a collection of survey articles written to commemorate the bicentennial of the birth of C. F. Sturm in 1803 in Geneva. The subtitle, "Past and Present," is apt. While a few of the twelve articles are purely historical in content (e.g., "Sturm-Liouville Theory 1900 to 1950" by W. N. Everitt), most give a gentle transition from the classical to the modern. Indeed, the reader comes away from the book with more than a suspicion that Sturm foresaw more than he ever wrote down.

Sturm's name is most often associated with the simple version of Sturm-Liouville theory taught to undergraduates in a differential equations course. This is merely one simple aspect of Sturm's oeuvre, which includes foundational results in ODEs, PDEs, spectral theory, special functions, and operator theory. The historical aspects of Sturm's life and work are interesting and comprehensive. Though dedicated to Sturm, the work of Liouville receives some coverage, as does that of Weyl, Titchmarsh, and others.

The articles in this compilation are, on the whole, very well-written and of high pedagogical quality, much more so than most proceedings from conferences. This reviewer [who does not work in Sturm-Liouville theory] was struck by how well the authors argued the case that this is a fascinating subject, rich in both intrinsic beauty and extrinsic merit. The curious reader will find among the many excellent articles in this volume the following:

• The article by B. Simon on "Sturm Oscillation and Comparison Theorems" gives what is advertised as a `celebratory and pedagogical' treatment of Sturm's oscillation theory. It is a tight and pleasant expository article on orthogonal polynomials and Jacobi matrices in the analysis of Sturm-Liouville operators and difference equations.

• The article by V. Galaktionov and P. Harwin on "Sturm's Theorems on Zero Sets in Nonlinear Parabolic Equations" is a compendium of the research, classical and modern, flowing from Sturm's theorems on the evolution of zeros in scalar parabolic PDE's. The topics implicated in this staccato listing are surprisingly broad.

• An article by C.-N. Chen on "A Survey of Nonlinear Sturm-Liouville Equations" focuses on bifurcation results for nonlinear Sturm-Liouville equations.

• W. N. Everitt has compiled a detailed listing of differential equations of Sturm-Liouville type along with pointers to the literature. This comprehensive list contains more than fifty types of equations, from the classical [Bessel, Airy, Legendre] to the modern [fuel cell, Plum, Zettl], the latter of which are important in modeling and also in numerical analysis.

It is nice to see such a book published: it is useless as a course textbook, will never generate much in sales, and was obviously a labor of love for the authors and editors. It is also an engaging, well-written, and informative text which should please specialists and non-specialists alike. Researchers in dynamical systems and differential equations have much to thank Sturm for, and much to find in this volume.