Self-adjoint indefinite Laplacians

Abstract

Let $\Omega_-$ and $\Omega_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, 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 $L_{\mu}=-\nabla\cdot h_\mu \nabla$ in $L^2(\Omega_-\cup\Omega_+)$ 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 $\mu\ne 1$, 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 $n\ge 3$, 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.