STABLE CONTINUOUS SPECTRA FOR DIFFERENTIAL OPERATORS OF ARBITRARY ORDER

Abstract

Perturbations of the continuous spectra of two self-adjoint operators H0 and H can be controlled by sandwiched differences of the form [Formula: see text] where ϕj(·) are bounded functions of the operators involved. If ϕj(H0) and ϕj(H) are integral operators, we give general integral conditions for the kernels, such that the continuous, the absolutely continuous and the singular continuous spectra of H0 and H coincide. The abstract operator theoretical conditions are applied for H0 in L 2(ℝd) of the form (-Δ)α/2, α∈ (0,2) and (-Δ)m, m∈ℕ. H is given as a perturbation of H0, where we allow perturbations by variable coefficients, potentials or obstacles.