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

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 analysis 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, we refine this strategy, and prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).
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Pré-publication, Document de travail
IF_PREPUB. Comments: Continues arXiv:1803.00449 on Gelfand's approach to Sturm's theorem, with a small overl.. 2018
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https://hal.archives-ouvertes.fr/hal-01832618
Contributeur : Pierre Bérard <>
Soumis le : dimanche 8 juillet 2018 - 11:48:32
Dernière modification le : vendredi 13 juillet 2018 - 01:06:19

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  • HAL Id : hal-01832618, version 1
  • ARXIV : 1807.03990

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Pierre Bérard, Bernard Helffer. Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented. IF_PREPUB. Comments: Continues arXiv:1803.00449 on Gelfand's approach to Sturm's theorem, with a small overl.. 2018. 〈hal-01832618〉

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