3 - Weyl's work on singular Sturm–Liouville operators

Summary

Wie doch ein einziger Reicher so viele Bettler in Nahrung Setzt! Wenn die Könige baun, haben die Kärrner zu tun.

(Schiller, Kant und seine Ausleger)

Abstract

Up to the year 1910 there had been many significant mathematical contributions to the theory of linear ordinary differential, and of linear integral equations. Many of these advances were based on the original studies initiated by Sturm and Liouville commencing in 1829. In the closing years of the 19^th century the work lead by the Göttingen school of mathematics gave a much needed overview of these significant and varied contributions to mathematical analysis.

The contributions of Hermann Weyl, in and around the year 1910, to the theory of Sturm-Liouville theory heralded the modern analytical and spectral study of boundary value problems. In particular the paper written for Mathematischen Annalen in 1910 stands today as a landmark not only in Sturm-Liouville theory, but in the development of mathematical analysis in the 20^th century.

This paper discusses the work of Weyl, and indeed of the Göttingen school of mathematics, in introducing the now familiar terms of Sturm-Liouville theory; limit-point and limit-circle endpoint classifications; the point, continuous and essential spectra; singular eigenfunction expansions; and the interplay of these results with the development of quantum theory in physics.

Introduction

The investigation of second-order linear ordinary differential equations has a long and fascinating history, extending back to the middle of the 18^th century. It shaped the concept of a function, led to Cantor's set theory and influenced the theories of measure and integration. It was essential to solving the initial-boundary-value problems for partial differential equations, for example the heat and wave equations, by separation of the variables.