On the eigenvalues of the Robin Laplacian with a complex parameter
The aim of the paper is to study the properties of the Robin Laplacian eigenvalue problem \[ \begin{cases} -\Delta u=\lambda u, & \text{in }\Omega,\\ \frac{\partial u}{\partial\nu}=\alpha u, & \text{on }\partial\Omega, \end{cases} \] when the parameter \(\alpha\) is not necessarily in \(\mathbb R\) but is instead allowed to vary in \(\mathbb C\). Notice in particular that, for \(\alpha\in\mathbb C\setminus\mathbb R\) the problem is no longer selfadjoint, meaning that most of the results based on variational techniques must be readapted. For instance, while the problem still has a discrete spectrum solely consisting of eigenvalues, the classical Courant min-max characterization does not make any sense for a generic complex parameter. The authors prove various results on the problem, especially about the relation between the eigenvalues \(\lambda\) and the parameter \(\alpha\), based on operator-theoretic techniques. After proving that, for any \(\alpha\in\mathbb C\), the spectrum is always discrete, consisting of eigenvalues of finite multiplicity and accumulating only at the complex infinity, they show that the dependence \(\lambda(\alpha)\) is holomorphic apart from crossing points in the spectrum, computing the derivative for simple eigenvalues \[ \lambda'(\alpha)=\frac{\int_{\partial\Omega}\psi(\alpha)^2 d\sigma(x)}{\int_{\Omega}\psi(\alpha)^2 dx}, \] where \(\psi(\alpha)\) is an eigenfunction associated with \(\lambda(\alpha)\). Interestingly enough, they show that the eigenfunctions form an orthonormal basis of \(L^2(\Omega)\) if and only if \(\alpha\in\mathbb R\), whereas if \(\alpha\notin\mathbb R\) they still form a basis but of differen type (Riesz basis in one dimension, Abel basis in higher dimension). The authors then proceed on studying the numerical range, providing the interesting bound \[ \Re \lambda(\alpha)\ge -\frac{C_1^2}{4}|\Re \alpha|^2-C_2|\Re \alpha|, \] where, if \(\Omega\in C^2\), then one can choose \(C_1=2\). This complements several known bounds in the literature, providing a simplified proof of the asymptotics \(\lambda(\alpha)=-C|\alpha|^2+o(\alpha^2)\) for \(\alpha\in\mathbb R_{<0}\). Moreover, it is shown that when \(\alpha\to\infty_{\mathbb C}\) in a way that keeps the eigenvalues bounded, then the only accumulation points are the eigenvalues of the Dirichlet Laplacian. This answers a conjecture that was posed for the case \(\mathbb R \ni \alpha\to -\infty\), providing a more complete picture since here \(\alpha\in\mathbb C\). The authors also consider the one-dimensional case, where they provide a thorough investigation of the relation between \(\lambda\) and \(\alpha\) (in particular studying the dependence \(\alpha(\lambda)\) as well), and the Dirichlet-to-Neumann maps, that are a somewhat dual formulation of the problem and allow for a different point of view in the matter. The paper is also complemented by several particular cases where it is possible to explicitly compute eigenvalues and eigenfunctions, specifically the interval, hyperrectangles, and balls.
spectral theory of non-self-adjoint operators
scientific article