No abstract

We consider a discrete Schrödinger operator H=-Δ+V acting in ℓ2 (ℤd+1), with potential V supported by the subspace ℤd×{0}. We prove that σ (-Δ)=[-2 (d+1), 2(d+1)] is contained in the absolutely continuous spectrum of H. For this we develop a scattering theory for H. We emphasize the fact that this result applies to arbitrary potentials, so it depends on the structure of the problem rather than on a particular choice of the potential.

Using the conjugate operator method of Mourre we study the spectral theory of a class of unbounded Jacobi matrices. We especially focus on the case where the off-diagonal entries a n = n α (1 + o(1)) and diagonal ones b n = λn α (1 + o(1)) with α > 0, λ ∈ R.