Self-adjoint indefinite Laplacians

Abstract

Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mathcal{L}_μ$ simply coincides with $L_μ$ and has compact resolvent. If $n=2$, then $\mathcal{L}_1$ has a non-empty essential spectrum, $σ_\mathrm{ess}(\mathcal{L}_{1})=\{0\}$. If $n\ge 3$, the spectral properties of $\mathcal{L}_1$ depend on the geometry of $Σ$. In particular, it has compact resolvent if $Σ$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Σ$ 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 $Σ$. We discuss some links between the resulting self-adjoint operator $\mathcal{L}_μ$ and some effects observed in negative-index materials.