<p>We obtain high energy asymptotics of Titchmarsh–Weyl (Weyl) matrix valued functions (matrix functions) of the generalised canonical systems related to matrix string equations generalising in this way a seminal Gesztesy-Simon result for the scalar Schrödinger equation. The matrix valued analog <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1687_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\Phi _1(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the amplitude function satisfies in this case an interesting new identity. The high energy asymptotics of Weyl matrix functions are derived in the form of the Fourier-type transform of this <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1687_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\Phi _1(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Contractive Weyl matrix functions are treated in the Sects. 2–5. In the last Sects. 6 and 7, we consider a special case of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1687_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$2p \times 2p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>p</mi> <mo>×</mo> <mn>2</mn> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> canonical systems and Weyl matrix functions belonging to Herglotz class. The corresponding structured operators are studied as well. Several examples are given. Application to a procedure for solving an inverse problem is presented in the last section.</p>

Generalised Canonical Systems Related to Matrix String Equations: Corresponding Structured Operators and High-Energy Asymptotics of Weyl Functions

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2025年4月1日

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Alexander Sakhnovich

摘要

We obtain high energy asymptotics of Titchmarsh–Weyl (Weyl) matrix valued functions (matrix functions) of the generalised canonical systems related to matrix string equations generalising in this way a seminal Gesztesy-Simon result for the scalar Schrödinger equation. The matrix valued analog $\Phi _1(x)$ $Φ_{1} (x)$ of the amplitude function satisfies in this case an interesting new identity. The high energy asymptotics of Weyl matrix functions are derived in the form of the Fourier-type transform of this $\Phi _1(x)$ $Φ_{1} (x)$ . Contractive Weyl matrix functions are treated in the Sects. 2–5. In the last Sects. 6 and 7, we consider a special case of $2p \times 2p$ $2 p \times 2 p$ canonical systems and Weyl matrix functions belonging to Herglotz class. The corresponding structured operators are studied as well. Several examples are given. Application to a procedure for solving an inverse problem is presented in the last section.