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• W2967302013 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, 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 $$alpha rightarrow infty $$ 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 along every analytic curve of eigenvalues, the Robin eigenvalues either diverge absolutely in $${mathbb {C}}$$ or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large $$alpha $$ . We also 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 $$dge 2$$ all eigenvalues converge to the Dirichlet spectrum if $${mathrm{Re}},alpha $$ remains bounded from below as $$alpha rightarrow infty $$ , while if $${mathrm{Re}},alpha rightarrow -infty $$ , then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like $$-alpha ^2$$ ." @default.
• W2967302013 created "2019-08-22" @default.
• W2967302013 creator A5046380709 @default.
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• W2967302013 date "2022-01-29" @default.
• W2967302013 modified "2023-10-04" @default.
• W2967302013 title "On the eigenvalues of the Robin Laplacian with a complex parameter" @default.
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• W2967302013 doi "https://doi.org/10.1007/s13324-022-00646-0" @default.
• W2967302013 hasPublicationYear "2022" @default.
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