# On the eigenvalues of the Robin Laplacian with a complex parameter

## TL;DR

This paper investigates the spectrum of the Robin Laplacian with a complex parameter, establishing properties, bounds, and asymptotics of eigenvalues, and relating their behavior to the Dirichlet spectrum and the complex parameter's growth.

## Contribution

It introduces new eigenvalue bounds for complex Robin parameters and analyzes asymptotic behavior using duality with the Dirichlet-to-Neumann map, extending understanding beyond real parameters.

## Key findings

- Eigenvalues either diverge or converge to the Dirichlet spectrum as the complex parameter grows.
- New bounds on the numerical range of the Robin Laplacian are established.
- Asymptotic behavior of eigenvalues is characterized for special domains and conjectured for general smooth domains.

## Abstract

We study the spectrum of the Robin Laplacian with a complex Robin parameter $\alpha$ on a bounded Lipschitz domain $\Omega$. 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 $\alpha \in \mathbb C$, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of $\alpha$: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case $\alpha \in \mathbb R$. For the asymptotics of the eigenvalues as $\alpha \to \infty$ in $\mathbb C$, 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 $\infty$ in $\mathbb C$ or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where $\Omega$ is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension $d\geq 2$ all eigenvalues converge to the Dirichlet spectrum if ${\rm Re}\, \alpha$ remains bounded from below as $\alpha \to \infty$, while if ${\rm Re}\, \alpha \to -\infty$, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like $-\alpha^2$.