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, 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. 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 along every analytic curve of eigenvalues, the Robin eigenvalues either diverge absolutely in C or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large α. We also 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 Re α remains bounded from below as α → ∞, while if Re α → −∞, then there is an infinite family of divergent eigenvalue curves, each of which behaves asymptotically like −α 2 .

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as δ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to ∞ in C: for each vertex v with a Robin parameter α ∈ C for which Re α → −∞ sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like −α 2 / deg v 2 , while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at v; if Re α remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh-Weyl M-functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s).

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as \delta conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to \infty in \mathbb C : for each vertex v with a Robin parameter \alpha \in \mathbb C for which Re \alpha \to -\infty sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like -\alpha^2/\mathrm {deg} v^2 , while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at v ; if \mathrm {Re} \alpha remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh–Weyl M -functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s).

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