<p>A closed formula for the spectral determinant for the wave equation on a bounded interval, subject to Dirichlet boundary conditions and an <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2802_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-multiple of the Dirac <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2802_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-type damping, is derived. Depending on the choice of the branch cut of the logarithm used in its definition, the spectral determinant diverges either for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2802_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2802_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Spectral Determinant for the Wave Equation on an Interval with Dirac Damping

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发表日期：

2025年6月12日

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David Krejčiřík,
Jiří Lipovský

摘要

A closed formula for the spectral determinant for the wave equation on a bounded interval, subject to Dirichlet boundary conditions and an $\alpha $ $α$ -multiple of the Dirac $\delta $ $δ$ -type damping, is derived. Depending on the choice of the branch cut of the logarithm used in its definition, the spectral determinant diverges either for $\alpha =2$ $α = 2$ or $\alpha =-2$ $α = - 2$ .