Sturm-Liouville Theory: Past and Present
The conference commemorating the 200th anniversary of the birth of C. F. Sturm took place at Geneva in September 2003. This book contains twelve survey articles, which are expanded versions of contributions to the conference. They show how Sturm's ideas have been developing up to the present day.
Three articles were written by N. Everitt. The first one describes improvements of the classical Sturm and Liouville (S-L) results made by H. Weyl, M. Stone and E. Titchmarsh. The second one presents a catalogue of more than fifty examples of differential equations of S-L type and their properties. The third paper (written jointly with C. Benewitz) refers to the eigenfunction expansion problem and methods based on function theory.
D. Hinton and B. Simon describe various aspects and generalizations of the Sturm oscillation theorem. Investigations of spectral properties of S-L operators have stimulated a great deal of research in operator theory and vice versa. This fact can be followed in the contributions written by D. Gilbert (the link between asymptotics of solutions and spectral properties and, in particular, the concept of subordinacy), Y. Last (discrete and continuous Schrödinger operators), R. del Rio (influence of boundary conditions upon spectral properties) and J. Weidmann (approximation of singular problems by regular ones). Results on the inverse spectral theory for systems of S-L equations are described by M. Malamud. There are also two contributions devoted to nonlinear equations; bifurcation phenomena are presented by C-N. Chen and the overview on evolution of zero sets for solutions of nonlinear parabolic equations is written by V. Galaktionov and P. Harwin.
The level of presentation of all the surveys is accessible to graduate students. They will find here comments on the current state of research and information about the most important ordinary differential equations. Since the articles are also accompanied by many references, reading of the book is a good starting point for research in differential equations and/or functional analysis. Any mathematician will find the historical contextual information useful.