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.

18 pages. Minor changes in the notation

References in corpus (2)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Representation Theorems for indefinite quadratic forms without spectral gap

Cited by in corpus (4)

• Two-dimensional Dirac operators with singular interactions supported on closed curves
• Spectral analysis of metamaterials in curved manifolds
• Maxwell's equations with hypersingularities at a negative index material conical tip
• On two-dimensional Dirac operators with critical delta-shell interactions