Higher-order localization landscape theory of Anderson localization

For a Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions ${u}_{n}(\mathbf{r})$ obeying $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}{u}_{n}(\mathbf{r})={u}_{n\ensuremath{-}1}(\mathbf{r})$ with ${u}_{0}(\mathbf{r})=1$. For $n\ensuremath{\rightarrow}\ensuremath{\infty},1/{v}_{n}(\mathbf{r})={u}_{n\ensuremath{-}1}(\mathbf{r})/{u}_{n}(\mathbf{r})$ converges to the lowest eigenenergy ${E}_{1}$ of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ whereas ${u}_{\ensuremath{\infty}}(\mathbf{r})$ yields the corresponding wave function ${\ensuremath{\psi}}_{1}(\mathbf{r})$. For large but finite $n,{v}_{n}(\mathbf{r})$ can be approximated by a piecewise constant function ${v}_{n}(\mathbf{r})\ensuremath{\simeq}{v}_{n}^{(m)}$ for $\mathbf{r}\ensuremath{\in}{\mathrm{\ensuremath{\Omega}}}_{m}$ and yields progressively improving estimations of eigenenergies ${E}_{m}=1/{v}_{n}^{(m)}$ of locally fundamental eigenstates ${\ensuremath{\psi}}_{m}(\mathbf{r})\ensuremath{\propto}{u}_{n}(\mathbf{r})$ in spatial domains ${\mathrm{\ensuremath{\Omega}}}_{m}$. 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, as well as for the uncorrelated random potential in two dimensions.