On the eigenvalues of the Robin Laplacian with a complex parameter · arXivDesk
1908.06041Aug 16, 2019Revised and expanded version. More details on the analytic eigenvalue curves are given, the list of references has been considerably expanded, a mistake in the proof of Theorem 1.5 has been corrected, and a new theorem (Theorem 1.6) and section (Section 8) have been added
On the eigenvalues of the Robin Laplacian with a complex parameter
We study the spectrum of the Robin Laplacian with a complex Robin parameter α on a bounded Lipschitz domain Ω. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on α∈C, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of α: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case α∈R
Nearby in the stack
. For the asymptotics of the eigenvalues as
α→∞
in
C
, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that every Robin eigenvalue either diverges to
∞
in
C
or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where
Ω
is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension
d≥2
all eigenvalues converge to the Dirichlet spectrum if
Reα
remains bounded from below as
α→∞
, while if
Reα→−∞
, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like