  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).
