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Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge

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Abstract

We consider a metric graph consisting of two edges, one of which has length \(\varepsilon \) which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter \(\varepsilon \). In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, \(\varepsilon ^0\), \(\varepsilon ^{-2/3}\) or \(\varepsilon ^{-1}\). These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge.

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Notes

  1. We also considered other descriptions of the vertex conditions, such as those listed in [5, Thm. 1.4.4] as well as the parametrization introduced in [13]. The parametrization we use in (2.2) results in the least cumbersome classification of the asymptotic behaviors, Table 1.

  2. Intuitively, the derivative \(u'_\varepsilon \) does not change very much over a short edge; since \(u'_\varepsilon (0)=0\), we also expect \(u'_\varepsilon \) to be close to 0 on the other end.

  3. The name “non-resonance” was chosen due to an analogy to Sommerfeld radiation condition for resonances, as it seeks to exclude the situation where non-zero values on the short edges occur in the absence of any input from the order 1 edges.

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Acknowledgements

The authors thank the anonymous referee for numerous improving suggestions. The research by D.I. Borisov was supported by Russian Science Foundation, grant no. 23-11-00009, https://rscf.ru/project/23-11-00009/.

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Authors and Affiliations

  1. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    Gregory Berkolaiko & Marshall King

  2. Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa, Russia, 450008

    Denis I. Borisov

  3. University of Hradec Králové, Rokitanského 62, 50003, Hradec Králové, Czech Republic

    Denis I. Borisov

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  1. Gregory Berkolaiko
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Berkolaiko, G., Borisov, D.I. & King, M. Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge. Anal.Math.Phys. 13, 90 (2023). https://doi.org/10.1007/s13324-023-00853-3

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  • DOI: https://doi.org/10.1007/s13324-023-00853-3

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