Higher-order localization landscape theory of Anderson localization

For a Hamiltonian $\widehat{H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying $\widehat{H}{u}_n(\vec{r})=u_{n-1}(\vec{r})$ with $u_0(\vec{r})=1$. For $n\to \mathrm{\infty }$, $1\mathrm{/}{v}_n(\vec{r})=u_{n-1}(\vec{r})\mathrm{/}{u}_n(\vec{r})$ converges to the lowest eigenenergy $E_1$ of $\widehat{H}$ whereas $u_{\mathrm{\infty }}(\vec{r})$ yields the corresponding wave function ${\psi }_1(\vec{r})$. For large but finite $n$, $v_n(\vec{r})$ can be approximated by a piecewise constant function $v_n(\vec{r})\simeq v_n^{(m)}$ for $\vec{r}\in {\mathrm{\Omega }}_m$ and yields progressively improving estimations of eigenenergies $E_m=1\mathrm{/}{v}_n^{(m)}$ of locally fundamental eigenstates ${\psi }_m(\vec{r})\propto u_n(\vec{r})$ in spatial domains ${\mathrm{\Omega }}_m$. 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, quasiperiodic potential, as well as for the uncorrelated random potential in two dimensions.