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Eigenvalues of Robin Laplacians in infinite sectors

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  • Magda Khalile
  • Konstantin Pankrashkin

Abstract

For α∈(0,π), let Uα denote the infinite planar sector of opening 2α, Uα={(x1,x2)∈R2:|arg(x1+ix2)| 0. The essential spectrum of Tαγ does not depend on the angle α and equals [−γ2,+∞), and the discrete spectrum is non†empty if and only if α 0, and the nth eigenvalue En(Tαγ) of Tαγ behaves as En(Tαγ)=−γ2(2n−1)2α2+O(1)and admits a full asymptotic expansion in powers of α2. The eigenfunctions are exponentially localized near the origin. The results are also applied to δ†interactions on star graphs.

Suggested Citation

  • Magda Khalile & Konstantin Pankrashkin, 2018. "Eigenvalues of Robin Laplacians in infinite sectors," Mathematische Nachrichten, Wiley Blackwell, vol. 291(5-6), pages 928-965, April.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:5-6:p:928-965
    DOI: 10.1002/mana.201600314
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