Asymptotics of Robin eigenvalues on sharp infinite cones

Abstract

For ε > 0 and n ∈ N consider the infinite coneand the operator Q α ε acting as the Laplacian u → −∆u on Ω ε with the Robin boundary condition ∂ ν u = αu at ∂Ω ε , where ∂ ν is the outward normal derivative and α > 0. It is known from numerous earlier works that the essential spectrum of Q α ε is [−α 2 , +∞) and that the discrete spectrum is finite for n = 1 and infinite for n ≥ 2, but the behavior of individual eigenvalues with respect to the geometric parameter ε was only addressed for n = 1 so far. In the present work we consider arbitrary n ≥ 2 and look at the spectral asymptotics as ε becomes small, i.e. as the cone becomes "sharp" and collapses to its central axis. Our main result is as follows: if n ≥ 2, α > 0 and j ∈ N are fixed, then the jth eigenvalue E j (Q α ε ) of Q α ε behaves as