×

On self-adjoint realizations of sign-indefinite Laplacians. (English) Zbl 1449.35191

Let \(\Omega \subset \mathbb{R}^d\) be a domain divided onto two parts \(\Omega^{\pm}\) having a joint smooth surface. The author considers the spectral problem for the operator \(-\nabla \cdot h \nabla\) in \(L_2(\Omega)\) with the Dirichlet problem on \(\partial \Omega\). The function \(h\) is equal to unity in \(\Omega^{+}\) and \(-\mu <0\) in \(\Omega^{-}\). This case of different signs of \(h\) corresponds to “metamaterials”. The model case when \(\Omega^{\pm}\) are two rectangles having the joint side is investigated. The obtained results are extended to general case.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47A10 Spectrum, resolvent
47B25 Linear symmetric and selfadjoint operators (unbounded)
47G30 Pseudodifferential operators
PDFBibTeX XMLCite
Full Text: Link