We study the spectrum of the Robin Laplacian with a complex Robin parameter α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} on a bounded Lipschitz domain Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}. 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\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in {\mathbb {C}}$$\end{document}, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}: 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\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in {\mathbb {R}}$$\end{document}. For the asymptotics of the eigenvalues as α→∞\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \rightarrow \infty $$\end{document} in C\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}$$\end{document}, 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\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}$$\end{document} or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}. We also give a comprehensive treatment of the special cases where Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension d≥2\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 2$$\end{document} all eigenvalues converge to the Dirichlet spectrum if Reα\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm{Re}}\,\alpha $$\end{document} remains bounded from below as α→∞\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \rightarrow \infty $$\end{document}, while if Reα→-∞\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm{Re}}\,\alpha \rightarrow -\infty $$\end{document}, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like -α2\documentclass[12pt]{minimal}
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\begin{document}$$-\alpha ^2$$\end{document}.
