On a spectral theorem of Weyl

TL;DR

This paper provides a geometric proof of Weyl's theorem on the continuous spectrum of Sturm-Liouville operators, potentially enabling generalizations to higher dimensions beyond traditional Green's function methods.

Contribution

It introduces a geometric proof of Weyl's spectral theorem, offering a new approach that may extend to more complex, higher-dimensional operators.

Findings

Geometric proof of Weyl's theorem on Sturm-Liouville operators

Potential for generalization to higher-dimensional cases

Alternative approach not relying on Green's functions

Abstract

We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features of Green's functions for linear ordinary differential operators; ours might offer better prospects for generalization to higher dimensions, as required for example in noncommutative harmonic analysis.