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Abstract <p>In this work, an inverse problem for the quadratic pencil of the Sturm–Liouville operator with an impulse in the finite interval is considered. It is shown that some information on eigenfunctions at some internal point <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7243_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(b\in\left(\frac{1}{2},1\right)\)</EquationSource> <!--ContMath2460209Khalili-m1--> </InlineEquation> and parts of two spectra uniquely determine the potential functions and all parameters in the boundary conditions. Moreover we prove that the potential functions on the whole interval and the parameters in the boundary conditions can be established from one spectrum and the potentials prescribed on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7243_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left(\frac{1}{2},1\right)\)</EquationSource> <!--ContMath2460209Khalili-m2--> </InlineEquation>.</p>

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On the Determination of the Quadratic Pencil of the Sturm–Liouville Operator with an Impulse

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • Y. Khalili,
  • D. Baleanu

摘要

Abstract

In this work, an inverse problem for the quadratic pencil of the Sturm–Liouville operator with an impulse in the finite interval is considered. It is shown that some information on eigenfunctions at some internal point \(b\in\left(\frac{1}{2},1\right)\) and parts of two spectra uniquely determine the potential functions and all parameters in the boundary conditions. Moreover we prove that the potential functions on the whole interval and the parameters in the boundary conditions can be established from one spectrum and the potentials prescribed on \(\left(\frac{1}{2},1\right)\) .