Spectral theory for domains in Rn of finite measure

AUTOR(ES)
RESUMO

Let Ω be a measurable subset of Rn of finite positive Lebesgue measures. The following two problems are considered: (i) Find commuting self-adjoint extensions of the minimal operators -i∂/∂xk, k = 1,..., n (Ω open). (ii) Find a set Λ ⊂ Rn such that the functions eλ = exp(iλ1x1 +... + iλnxn) for λ ∈ Λ form an orthonormal basis for L2(Ω). The problems are known to be equivalent under mild regularity conditions on Ω, and existence holds in two cases: (i) there is a connected open set Ω′ such that the symmetric difference ΩΔΩ′ is a null set and Ω′ is a fundamental domain for a discrete total subgroup; and (ii)Ω = [unk]a∈R (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.

ACESSO AO ARTIGO

Documentos Relacionados