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arXiv · 2404.19575

On the non-existence of oscillation numbers in Sturm-Liouville theory

Angelo B. Mingarelli
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Abstract

We prove an old conjecture that relates the existence of non-real eigenvalues of Sturm-Liouville Dirichlet problems on a finite interval to the non-existence of oscillation numbers of its real eigenfunctions, [[6], p.104, Problems 3 and 5]. This extends to the general case, a previous result in [1], [2] where it was shown that the presence of even one pair of non-real eigenvalues implies the non-existence of a positive eigenfunction (or ground state). We also provide estimates on the Haupt and Richardson indices and Haupt and Richardson numbers thereby complementing the original Sturm oscillation theorem with the Haupt-Richardson oscillation theorem discovered over 100 years ago with estimates on the missing oscillation numbers of the real eigenfunctions observed.

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Angelo B. Mingarelli. 2024-04-30. On the non-existence of oscillation numbers in Sturm-Liouville theory. https://arxiv.org/abs/2404.19575

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