Self-adjoint indefinite Laplacians

Abstract

Let Ω − and Ω + be two bounded smooth domains in R n , n ≥ 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 = −∇ · h µ ∇ in L 2 (Ω − ∪ Ω + ) 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 L µ := L µ . If µ = 1, then L µ simply coincides with L µ and has compact resolvent. If n = 2, then L 1 has a non-empty essential spectrum, σ ess (L 1 ) = {0}. If n ≥ 3, the spectral properties of 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 L µ and some effects observed in negativeindex materials.2010 Mathematics Subject Classification. 35J05, 47A10, 47B25, 47G30.

A strategy for self-adjointness of Dirac operators: Applications to the MIT bag model and $\delta$-shell interactions

“…It is worth mentionning that simultaneously, in [7], the authors recover similar results with a boundary triplet technique. This phenomenom is reminescent of similar questions in the context of negative-index materials investigated in [9,12].…”

“…It is worth mentionning that simultaneously, in [7], the authors recover similar results with a boundary triplet technique. This phenomenom is reminescent of similar questions in the context of negative-index materials investigated in [9,12]. This paper is inspired by the strategy developped in [8] about two-dimensional Dirac operators with graphene boundary conditions.…”

Spectral transition for Dirac operators with electrostatic $δ$-shell potentials supported on the straight line

“…However, in some cases it turns out that there is an abrupt change in the spectral properties -in other words, a spectral transition. A well known example in this regard is the Smilansky model [29,30] (see also [4,5,6]) or the indefinite Laplacian studied in [13,20]. A spectral transition was also observed for Dirac operators with singular potentials supported on bounded curves in R 2 and surfaces in R 3 .…”

Spectral analysis of polygonal cavities containing a negative-index material

“…A necessary and sufficient condition for S λ to be selfadjoint was shown in [BDR99]: it corresponds exactly to λ / ∈ σ ess (A). Besides for some geometric situations which exclude corner resonances, it is proved in [CPP19,Pan19] that S λ is selfadjoint with compact resolvent for λ / ∈ {0, Λ m /2}, whereas it is not closed but is essentially selfadjoint if λ = Λ m /2 (the case λ = 0 is not dealt with). This shows that {Λ m /2} ⊂ σ ess (S λ ) ⊂ {0, Λ m /2} in these situations, which supports the natural conjecture that σ ess (S λ ) = σ ess (A) \ {Λ m }.…”

“…They considered a scalar transmission problem (which involves only one contrast) and showed by an integral equation technique that in the case of a smooth interface, the transmission problem is well-posed if and only if the contrast is different from the critical value −1. The detailed study of this critical value of the contrast is achieved in [Ola95] and more recently in [CPP19], both for smooth interfaces. The case of a two-dimensional non-smooth interface was tackled about fifteen years after the pioneering work of Costabel and Stephan: it was understood in [BDR99] that in the presence of a corner, this critical value becomes a critical interval (which contains −1) depending on the angle of the corner.…”