Schrödinger operators with empty singularly continuous spectra

Abstract

Let H be a semibounded perturbation of the Laplacian H₀ in L²(R^d). For an admissible function φ sufficient conditions are given for the completeness of the scattering system {φ(H), φ(H₀). If φ is the exponential function and if e^{−λ H} is an integral operator we denote the kernel of the difference D_λ= e^{−λ H} − e^{−λ H₀} by D_λ(x, y), λ> 0. The singularly continuous spectrum of H is empty if ∫_R^d dx ∫_R^d dy |D_λ(x,y)| (1+ |y|²)^α< ∞ for some α > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.