Eigenvalues of Robin Laplacians in infinite sectors

Magda Khalile and Konstantin Pankrashkin

Mathematische Nachrichten, 2018, vol. 291, issue 5-6, 928-965

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.

Date: 2018
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