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Mathematical analysis demonstrates eigenvalue ordering bounds between Dirichlet and Neumann spectra in Sturm-Liouville operators, highlighting the geometric role of potential barriers and wells.
In this paper, we study theorder of the Dirichlet and Neumann eigenvalues for theSturm-Liouville equation -(p(x)y')'+q(x)y=λρ(x) y. We prove that λ₁≥μ₂ (resp. λ ₁≤μ₂) for single-barrier q and single-well pρ in$[0,1]$ (resp. for single-well q and single-barrier pρ in $[0,1]$).
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Hedhly et al. (2026) studied this question.
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