Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu}=- ablacdot h_mu abla$ in $L^2(Omega_-cupOmega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Sigma$) and study some properties of its unique self-adjoint extension $mathcal{L}_mu:=overline{L_mu}$. If $mu e 1$, then $mathcal{L}_mu$ simply coincides with $L_mu$ and has compact resolvent. If $n=2$, then $mathcal{L}_1$ has a non-empty essential spectrum, $sigma_mathrm{ess}(mathcal{L}_{1})={0}$. If $nge 3$, the spectral properties of $mathcal{L}_1$ depend on the geometry of $Sigma$. In particular, it has compact resolvent if $Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Sigma$. We discuss some links between the resulting self-adjoint operator $mathcal{L}_mu$ and some effects observed in negative-index materials.
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