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We consider the problem of finding commuting self-adjoint extensions of thepartial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domainC_c^\infty(\Omega) where the self-adjointness is defined relative toL^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega isLebesgue measure on R^d restricted to \Omega. The problem originates with I.E.Segal and B. Fuglede, and is difficult in general. In this paper, we provide arepresentation-theoretic answer in the special case when \Omega=I\times\Omega_2and I is an open interval. We then apply the results to the case when \Omega isa d-cube, I^d, and we describe possible subsets \Lambda of R^d such that{e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormalbasis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km,  02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt,  61.44.B

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