A version of Gordon's theorem for multi-dimensional Schrödinger operators

We consider discrete Schrödinger operators in H = Δ + V in ℓ^2(Z^d) with d ≥ 1, and study the eigenvalue problem for these operators. It is shown that the point spectrum is empty if the potential V is sufficiently well approximated by periodic potentials. This criterion is applied to quasiperiodic V and to so-called Fibonacci-type superlattices

Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.Comment: 22 page