Spectral theory for domains in R of finite measure

Let Omega be a measurable subset of R(n) of finite positive Lebesgue measures. The following two problems are considered: (i) Find commuting self-adjoint extensions of the minimal operators -i( partial differential)/ partial differentialx(k), k = 1,..., n (Omega open). (ii) Find a set Lambda subset R(n) such that the functions e(lambda) = exp(ilambda(1)x(1) +... + ilambda(n)x(n)) for lambda in Lambda form an orthonormal basis for L(2)(Omega). The problems are known to be equivalent under mild regularity conditions on Omega, and existence holds in two cases: (i) there is a connected open set Omega' such that the symmetric difference OmegaDeltaOmega' is a null set and Omega' is a fundamental domain for a discrete total subgroup; and (ii)Omega = [unk](ainR) (a + [unk]), disjoint union neglecting null sets, in which [unk] is a fundamental domain and R is a "set of representors" for a finite group of translations. Case i is equivalent to a function theoretic condition of Forelli, and case ii is established when the existence of a discrete covariance group is assumed. Generalizations of the geometric results i and ii for spectral sets in arbitrary Lie groups are indicated.