On self-adjoint realizations of sign-indefinite Laplacians

Konstantin Pankrashkin (1)

1 LMO - Laboratoire de Mathématiques d'Orsay

Let Ω ⊂ R d be a domain and Σ a hypersurface cutting Ω into two parts Ω ±. For µ > 0, consider the function h whose value is (−µ) in Ω − and 1 in Ω +. In the present contribution we discuss the construction and some properties of the self-adjoint realizations of the operator L = −∇ · (h∇) in L 2 (Ω) with suitable (e.g. Dirichlet) on the exterior boundary. We give first a detailed study for the case when Ω ± are two rectangles touching along a side, which is based on operator-valued differential operators, in order to see in an elementary but an abstract level the principal effects such as a loss of regularity and unusual spectral properties. Then we give a review of available approaches and results for more general geometric configurations and formulate some open problems.

Domaines: Théorie spectrale [math.SP]

Konstantin Pankrashkin. On self-adjoint realizations of sign-indefinite Laplacians. 2018. ⟨hal-01960406⟩