The theory of differential equations is one of the outstanding creations of the human mind. Its influence upon the development of physical science would be hard to exaggerate. The long history and many applications of the theory, however, make it almost impossible to write a balanced account of the subject. Thus authors of student texts are confronted with the choice between writing rather superficially on a range of topics or in more depth on some narrow field, in which they have a particular interest.

In this book I have given a simple introduction to the spectral theory of linear differential operators. This spectral theory is an outgrowth of fundamental work of David Hilbert between 1900 and 1910 on the analysis of integral operators on infinite-dimensional spaces – now called Hilbert spaces. However, like almost every important new development in mathematics, it was preceded by much related work, for example Poincare's analysis of the Dirichlet problem and associated eigenvalues (1890–6). One could maintain that the subject started with the seminal work of Fourier on the solution of the heat equation using series expansions in sines and cosines, which was published by the Académie Française in 1822. Fourier submitted this work in 1807, during the Napoleonic era, and an account of his misfortunes during the fifteen year period before publication is given by Korner (1988). I have included the names and dates associated with a few of the key ideas in the text; a much more comprehensive account may be found in Dieudonné (1981).