Spectral Theory of Ordinary Differential Equations - Introduction

Introduction

Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles.

In the 1920s John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the spectral measure. The same formula had also been obtained independently by E. C. Titchmarsh in 1946 (scientific communication between Japan and the United Kingdom had been interrupted by World War II). Titchmarsh had followed the method of the German mathematician Emil Hilb, who derived the eigenfunction expansions using complex function theory instead of operator theory. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the resolvent of the singular differential operator could be approximated by compact resolvents corresponding to Sturm–Liouville problems for proper subintervals. Another method was found by Mark Grigoryevich Krein; his use of direction functionals was subsequently generalised by I. M. Glazman to arbitrary ordinary differential equations of even order.

Weyl applied his theory to Carl Friedrich Gauss's hypergeometric differential equation, thus obtaining a far-reaching generalisation of the transform formula of Gustav Ferdinand Mehler (1881) for the Legendre differential equation, rediscovered by the Russian physicist Vladimir Fock in 1943, and usually called the Mehler–Fock transform. The corresponding ordinary differential operator is the radial part of the Laplacian operator on 2-dimensional hyperbolic space. More generally, the Plancherel theorem for SL(2,R) of Harish Chandra and Gelfand–Naimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of spherical functions for the isometry groups of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real semisimple Lie groups was strongly influenced by the methods Weyl developed for eigenfunction expansions associated with singular ordinary differential equations. Equally importantly the theory also laid the mathematical foundations for the analysis of the Schrödinger equation and scattering matrix in quantum mechanics.