The exponential decay of eigenfunctions for tight-binding Hamiltonians via landscape and dual landscape functions

Abstract

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ on a cube $M\subset \mathbb{Z}^d$, with periodic or Dirichlet (simple) boundary conditions. We use a hidden landscape function $u$, defined as the solution of an inhomogeneous boundary problem with uniform right-hand side for $H$, to predict the location of the localized eigenfunctions of $H$. Explicit bounds on the exponential decay of Agmon type for low energy modes are obtained. This extends the recent work of Agmon type of localization in [5] for $\mathbb{R}^d$ to a tight-binding Hamiltonian on $\mathbb{Z}^d$ lattice. Contrary to the continuous case, high energy modes are as localized as the low energy ones in discrete lattices. We show that exponential decay estimates of Agmon type also appear near the top of the spectrum, where the location of the localized eigenfunctions is predicted by a different landscape function. Our results are deterministic and are independent of the size of the cube. We also provide numerical experiments to confirm the conditional results effectively, for some random potentials.