Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

Abstract

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(∂/∂xj):j=1,…,d} with domain Cc∞(Ω) where the self-adjointness is defined relative to L2(Ω), and Ω is a given open subset of Rd. The measure on Ω is Lebesgue measure on Rd restricted to Ω. The problem originates with Segal and Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when Ω=I×Ω2 and I is an open interval. We then apply the results to the case when Ω is a d cube, Id, and we describe possible subsets Λ⊂Rd such that {eλ|Id:λ∈Λ} is an orthonormal basis in L2(Id).