The spectrum of operators elliptic along the orbits of ℝⁿ actions

It is shown that a periodic elliptic operator on ℝⁿ has no eigenvalues off of the set of discontinuities of its spectral density function. The methods involve operator algebras and are based on a "spectral duality" principal first introduced by J. Bellisard and D. Testard. A version of the spectral duality theorem is proved which relates the point spectrum of a certain family of operators to the continuous spectrum of an associated family.