On the eigenvalues of the Robin Laplacian with a complex parameter

We study the spectrum of Robin Laplacian with a complex parameter $$\alpha $$ on bounded Lipschitz domain $$\Omega . start by establishing number properties corresponding operator, such as generation properties, analytic dependence eigenvalues and eigenspaces \in {\mathbb {C}}$$ , basis eigenfunctions. Our focus, however, is bounds asymptotics for functions : we providing estimates numerical range associated which lead to new eigenvalue even in case {R}}$$ For \rightarrow \infty $${\mathbb place min–max characterisation Dirichlet–Neumann bracketing techniques commonly used real case, exploit duality between Dirichlet-to-Neumann map. use this show that along every curve eigenvalues, either diverge absolutely or converge Dirichlet spectrum, well classify all possible points accumulation large also give comprehensive treatment special cases where an interval, hyperrectangle ball. This leads conjecture general smooth dimension $$d\ge 2$$ if $${\mathrm{Re}}\,\alpha remains from below while -\infty then there family divergent curves, each behaves asymptotically like $$-\alpha ^2$$