<p>This study examines the Sturm-Liouville equation, focusing on cases where the spectral parameter is included in the boundary conditions. By employing the Hochstadt-Lieberman theorem and the Weyl function, we demonstrate that when the potential is known for the interval <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3347_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> </math></EquationSource> <EquationSource Format="TEX">$\left (0,\frac{\pi}{2}\right )$</EquationSource> </InlineEquation>, a single spectrum can specify the potential for the whole interval <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3347_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(0,\pi )$</EquationSource> </InlineEquation>. Furthermore, utilizing the Gesztesy-Simon theorem alongside the Weyl function technique, we establish that if the potential is known a priori on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3347_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(0,\pi /2(1 - \alpha ))$</EquationSource> </InlineEquation> as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3347_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\alpha \in (0, 1)$</EquationSource> </InlineEquation>, some of spectra can uniquely identify the potential across the entire interval <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3347_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(0, \pi )$</EquationSource> </InlineEquation>.</p>

Reconstruction of the potential in the Sturm-Liouville equation with spectral boundary conditions

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2025年8月18日

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Yasser Khalili,
Nematollah Kadkhoda

摘要

This study examines the Sturm-Liouville equation, focusing on cases where the spectral parameter is included in the boundary conditions. By employing the Hochstadt-Lieberman theorem and the Weyl function, we demonstrate that when the potential is known for the interval $(0, \frac{π}{2})$ $\left (0,\frac{\pi}{2}\right )$ , a single spectrum can specify the potential for the whole interval $(0, π)$ $(0,\pi )$ . Furthermore, utilizing the Gesztesy-Simon theorem alongside the Weyl function technique, we establish that if the potential is known a priori on $(0, π / 2 (1 - α))$ $(0,\pi /2(1 - \alpha ))$ as $α \in (0, 1)$ $\alpha \in (0, 1)$ , some of spectra can uniquely identify the potential across the entire interval $(0, π)$ $(0, \pi )$ .