Eigenvalues of Robin Laplacians in infinite sectors

Abstract

AbstractFor , let denote the infinite planar sector of opening 2α, urn:x-wiley:0025584X:media:mana201600314:mana201600314-math-0003and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as urn:x-wiley:0025584X:media:mana201600314:mana201600314-math-0018and 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.