Bögli–Kennedy–Lang conjecture on complex Robin eigenvalues

Let $\Omega \subset \mathbb{R}^d$, $d\geq 2$, be a bounded Lipschitz domain, and let $\alpha \in \mathbb{C}$ tend to infinity. The boundary parameter $\alpha$ defines the Robin Laplacian on $\Omega$, and the Dirichlet Laplacian is the Laplacian with zero boundary conditions.

Bögli–Kennedy–Lang conjecture. If $\operatorname{Re}\alpha \to -\infty$, then there exists an infinite family of analytic branches of absolutely divergent eigenvalues such that, if $\partial\Omega$ is of class $C^1$, their eigenvalues behave like

$$-\alpha^2+o(\alpha^2),$$

while otherwise they behave like

$$-C\alpha^2+o(\alpha^2),$$

where $C\geq 1$ may depend on $\Omega$ and on the corresponding eigenvalue curve. Every other eigenvalue converges to an eigenvalue of the Dirichlet Laplacian. If $\operatorname{Re}\alpha$ remains bounded from below, then every eigenvalue converges to an eigenvalue of the Dirichlet Laplacian.

This conjecture extends the known asymptotic picture for real Robin parameters to large complex parameters, where the Robin Laplacian is no longer self-adjoint and variational methods are unavailable. The paper recalls it as a conjecture from the cited earlier work; its resolution is not established in the supplied text.