<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> be two distinct vertices of an equilateral tree <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$T_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi _N^{(i)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ϕ</mi> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$i=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) be the characteristic functions of the Sturm-Liouville problem on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$T_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> rooted at <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> with Neumann conditions at the root and let <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq8.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi _D^{(i)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ϕ</mi> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$i=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) be the characteristic functions of the Sturm-Liouville problem on <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$T_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> with Dirichlet conditions at the root. We prove that if attaching any tree to <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$T_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> at the vertices <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> leads to cospectral trees and <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$d(v_1)=d(v_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq15.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi _N(\lambda )^{(1)}\equiv \phi _N(\lambda )^{(2)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϕ</mi> <mi>N</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≡</mo> <msub> <mi>ϕ</mi> <mi>N</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq16.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi _D(\lambda )^{(1)}\equiv \phi _D(\lambda )^{(1)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϕ</mi> <mi>D</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≡</mo> <msub> <mi>ϕ</mi> <mi>D</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> (which means that the scattering is the same at <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1730_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$v_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>).</p>

Cospectral Trees Indistinguishable by Scattering

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2025年6月12日

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Mats-Erik Pistol,
Vyacheslav Pivovarchik

摘要

Let $v_1$ $v_{1}$ and $v_2$ $v_{2}$ be two distinct vertices of an equilateral tree $T_0$ $T_{0}$ . Let $\phi _N^{(i)}$ $ϕ_{N}^{(i)}$ ( $i=1,2$ $i = 1, 2$ ) be the characteristic functions of the Sturm-Liouville problem on $T_0$ $T_{0}$ rooted at $v_i$ $v_{i}$ with Neumann conditions at the root and let $\phi _D^{(i)}$ $ϕ_{D}^{(i)}$ ( $i=1,2$ $i = 1, 2$ ) be the characteristic functions of the Sturm-Liouville problem on $T_0$ $T_{0}$ with Dirichlet conditions at the root. We prove that if attaching any tree to $T_0$ $T_{0}$ at the vertices $v_1$ $v_{1}$ and $v_2$ $v_{2}$ leads to cospectral trees and $d(v_1)=d(v_2)$ $d (v_{1}) = d (v_{2})$ then $\phi _N(\lambda )^{(1)}\equiv \phi _N(\lambda )^{(2)}$ $ϕ_{N} {(λ)}^{(1)} \equiv ϕ_{N} {(λ)}^{(2)}$ and $\phi _D(\lambda )^{(1)}\equiv \phi _D(\lambda )^{(1)}$ $ϕ_{D} {(λ)}^{(1)} \equiv ϕ_{D} {(λ)}^{(1)}$ (which means that the scattering is the same at $v_1$ $v_{1}$ and $v_2$ $v_{2}$ ).