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Abstract <p>In this paper we consider a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(3 \times 3\)</EquationSource> <!--RusMath2570050Rasulov-m1--> </InlineEquation> operator matrix <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{A}}_{\mu }}\)</EquationSource> <!--RusMath2570050Rasulov-m2--> </InlineEquation> with a spectral parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu &gt; 0\)</EquationSource> <!--RusMath2570050Rasulov-m3--> </InlineEquation> related with the Hamiltonian of a system with nonconserved and no more than three particles on a one-dimensional lattice. Essential and discrete spectra of the operator matrix <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{A}}_{\mu }}\)</EquationSource> <!--RusMath2570050Rasulov-m4--> </InlineEquation> are described. It is established that the operator matrix <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{A}}_{\mu }}\)</EquationSource> <!--RusMath2570050Rasulov-m5--> </InlineEquation> has at most four simple eigenvalues outside of the essential spectrum. Spectral estimates for the lower and upper bounds of the operator matrix <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10506_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{A}}_{\mu }}\)</EquationSource> <!--RusMath2570050Rasulov-m6--> </InlineEquation> are obtained using cubic numerical range, Gershgorin enclosures, and classical perturbation theory.</p>

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Spectral Estimates for the Bounds of an Operator Matrix of Order Three

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • T. H. Rasulov,
  • M. Sh. Sharipova

摘要

Abstract

In this paper we consider a \(3 \times 3\) operator matrix \({{\mathcal{A}}_{\mu }}\) with a spectral parameter \(\mu > 0\) related with the Hamiltonian of a system with nonconserved and no more than three particles on a one-dimensional lattice. Essential and discrete spectra of the operator matrix \({{\mathcal{A}}_{\mu }}\) are described. It is established that the operator matrix \({{\mathcal{A}}_{\mu }}\) has at most four simple eigenvalues outside of the essential spectrum. Spectral estimates for the lower and upper bounds of the operator matrix \({{\mathcal{A}}_{\mu }}\) are obtained using cubic numerical range, Gershgorin enclosures, and classical perturbation theory.