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Abstract <p> We study the Sturm–Liouville operator generated in the Hilbert space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2790_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2[0,+\infty )\)</EquationSource> </InlineEquation> by a differential expression containing the Diracdelta function with zero boundary condition. We prove that the eigenvalues <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2790_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _n\)</EquationSource> </InlineEquation> of this operator satisfy certain inequalities. Theproblem on the location of the first eigenvalue <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2790_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1\)</EquationSource> </InlineEquation> dependingon the parameters of the differential expression is solved. In particular, we obtain conditionsunder which <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2790_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1\)</EquationSource> </InlineEquation> is negative.</p>

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Distribution of Spectrum of the Weber Operator Perturbed by the Dirac Delta Function

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • A. S. Pechentsov

摘要

Abstract

We study the Sturm–Liouville operator generated in the Hilbert space \(L^2[0,+\infty )\) by a differential expression containing the Diracdelta function with zero boundary condition. We prove that the eigenvalues \(\lambda _n\) of this operator satisfy certain inequalities. Theproblem on the location of the first eigenvalue \(\lambda _1\) dependingon the parameters of the differential expression is solved. In particular, we obtain conditionsunder which \(\lambda _1\) is negative.