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<p>A quantum graph with the second-order differential operator on edges and the Kirchhoff coupling condition at non-boundary vertices is considered. It is assumed that the graph consists of a compact subgraph and a finite number of semi-infinite leads. We study completeness of the system of resonance states in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1681_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> on the compact subgraph related to the complement of maximal incoming and outgoing subspaces. It is proved that the completeness takes place if and only if each vertex having infinite leads attached is unbalanced, i.e. it has different numbers of finite and infinite edges attached. A relation between the completeness of the resonance states and the Weyl asymptotics of the resonances is established. The completeness proof is based on the factorization of the characteristic function in the Sz-Nagy functional model for the problem in question.</p>

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Completeness of Resonance States and Weyl Asymptotics of Resonances for Quantum Graphs

  • 期刊论文

  • 发表日期：

    2025年3月18日

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  • I. Y. Popov,
  • I. V. Blinova,
  • A. I. Popov

摘要

A quantum graph with the second-order differential operator on edges and the Kirchhoff coupling condition at non-boundary vertices is considered. It is assumed that the graph consists of a compact subgraph and a finite number of semi-infinite leads. We study completeness of the system of resonance states in \(L_2\) L 2 on the compact subgraph related to the complement of maximal incoming and outgoing subspaces. It is proved that the completeness takes place if and only if each vertex having infinite leads attached is unbalanced, i.e. it has different numbers of finite and infinite edges attached. A relation between the completeness of the resonance states and the Weyl asymptotics of the resonances is established. The completeness proof is based on the factorization of the characteristic function in the Sz-Nagy functional model for the problem in question.