Higher-order localization landscape theory of Anderson localization

For a Hamiltonian H containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions uₙ(r) obeying H uₙ(r) = uₙ₋₁(r) with u₀(r) = 1. For n → ∞, 1/vₙ(r) = uₙ₋₁(r)/uₙ(r) converges to the lowest eigenenergy E₁ of H whereas u∞(r) yields the corresponding wave function ψ₁(r). For large but finite n, vₙ(r) can be approximated by a piecewise constant function vₙ(r) vₙ⁽ᵐ⁾ for r ∈ Ωₘ and yields progressively improving estimations of eigenenergies Eₘ = 1/vₙ⁽ᵐ⁾ of locally fundamental eigenstates ψₘ(r) ∝ uₙ(r) in spatial domains Ωₘ. These general results are illustrated by a number of examples in one dimension: box potential, sequence of randomly placed infinite potential barriers, smooth and spatially uncorrelated random potentials, quasiperodic potential.