Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented

TL;DR

This paper rigorously implements Gelfand's strategy, inspired by quantum mechanics, to analyze the zeros of linear combinations of eigenfunctions of Sturm-Liouville problems, extending classical results with a complete proof.

Contribution

The paper provides a rigorous, complete proof of Gelfand's strategy for analyzing zeros of eigenfunction combinations, including multiplicity considerations, building on Sturm's classical work.

Findings

Proves that zeros of eigenfunction combinations divide the interval into at most n parts.

Refines Gelfand's strategy to account for zero multiplicities.

Extends classical Sturm results with a rigorous, quantum-inspired approach.

Abstract

In the second section "Courant-Gelfand theorem" of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y"(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$divide the interval into at most $n$ connected components, and concludes that "the lack of a published formal text with a rigorous proof \dots is still distressing." Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions. In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).