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)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(Ω) where the self-adjointness is defined relative to L^2(Ω), and Ωis a given open subset of R^d. The measure on Ωis Lebesgue measure on R^d restricted to Ω. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when Ω=I\timesΩ_2 and I is an open interval. We then apply the results to the case when Ωis a d-cube, I^d, and we describe possible subsets Λof R^d such that {e^(i2πλ\dot x) restricted to I^d:λ\inΛ} is an orthonormal basis in L^2(I^d).