The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials
The paper deals with the Dirichlet-to-Neumann operator \(D\) acting on functions defined on the Lipschitz boundary \(\Gamma=\partial \Omega\) of a bounded open set \(\Omega\subset \mathbb{R}^d\), defined by \(D\varphi=\psi\), \(\text{Dom}\,D\subset L_2(\Gamma)\), \(\psi\in L_2(\Gamma)\) for which there exists a \(u\in H^1(\Omega)\) such that \[ \begin{cases} \text{Tr}\,u =\varphi,\\ -\Delta u+qu=0,\,\,\,\text{weakly on} \,\,\Omega,\\ \partial_{\nu}u=\psi, \end{cases} \] where \(\partial_{\nu}\) is the weak normal derivative, and \(q:\Omega\to \mathbb{C}\) is a bounded measurable complex valued function. The authors investigate some properties of the operator \(D\), by using form methods and the theory of self-adjoint graphs.
scientific article; zbMATH DE number 6710661
scientific article