On the eigenvalues of the Robin Laplacian with a complex parameter
type: Item
instance of: scholarly article
title: On the eigenvalues of the Robin Laplacian with a complex parameter
zbMATH Open document ID: 1490.35108
author: Sabine Bögli
author: James Bernard Kennedy
author: Robin Lang
published in: Analysis and Mathematical Physics
publication date: 2022-02-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1908.06041
review text: 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…
reviewed by: Davide Buoso
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 81Q12
zbMATH DE Number: 7474296
zbMATH Keywords: Laplacian
zbMATH Keywords: Robin boundary conditions
zbMATH Keywords: spectral theory of non-self-adjoint operators
zbMATH Keywords: estimates on eigenvalues
MaRDI profile type: Publication
OpenAlex ID: W2967302013
arXiv ID: 1908.06041
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On the eigenvalues of the Robin Laplacian with a complex parameter (DOI)
DOI: 10.1007/S13324-022-00646-0