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<p>By regarding the nodes of a Sturm–Liouville problem as functionals of the weights, we prove the sharp bounds for the locations of such nodes if the weights are negative and admit fixed <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2901_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norms. The proof is based on a strong continuity of the nodes in the weight and the known results on sharp lower and upper estimates for Dirichlet eigenvalues. We obtain the explicit expressions for sharp bounds and the results are given as elementary functions.</p>

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Sharp Bounds for All Nodes of a Sturm–Liouville Problem with Negative Weights

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Jifeng Chu,
  • Ke Jiang,
  • Gang Meng

摘要

By regarding the nodes of a Sturm–Liouville problem as functionals of the weights, we prove the sharp bounds for the locations of such nodes if the weights are negative and admit fixed \(L^1\) L 1 -norms. The proof is based on a strong continuity of the nodes in the weight and the known results on sharp lower and upper estimates for Dirichlet eigenvalues. We obtain the explicit expressions for sharp bounds and the results are given as elementary functions.