The exponential decay of eigenfunctions for tight-binding Hamiltonians via landscape and dual landscape functions
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Annales Henri Poincar´ e
The exponential decay of eigenfunctions for tight-binding Hamiltonians via landscape and dual landscape functions Wei Wang and Shiwen Zhang Abstract. We consider the discrete Schr¨ odinger operator H = −Δ + V on a cube M ⊂ Zd , 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 Arnold et al. (Commun Partial Differ Equ 44:1186–1216, 2019) for Rd to a tight-binding Hamiltonian on Zd 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.
Contents 1. 2.
Introduction and main results Exponential decay of eigenfunctions of Agmon type 2.1. Preliminaries 2.2. Landscape theory on discrete lattices 2.3. Agmon estimates 2.4. Dual landscape for the high-energy modes 3. Dirichlet boundary condition 4. More numerical experiments Acknowledgements References
W. Wang, S. Zhang
Ann. Henri Poincar´e
1. Introduction and main results Localization of eigenfunctions is one the most important topics in mathematics and condensed matter physics. The term localization, roughly speaking, refers to the phenomenon that the eigenfunctions of an elliptic operator concentrate on a narrow region in space and are (exponentially) small outside the region. Take the one-electron model of condensed matter physics for example, for which the spectral and transport properties of the material are described by a (one-particle) Schr¨ odinger operator H = −Δ + V . The Laplacian Δ describes the kinetic energy of a free particle, and the potential V the presence of the external field. For example, the choice of a periodic function V can be used to describe a perfect crystal. The pioneering work of Anderson [7] back to 1958 says that randomness causes the states to (exponentially) localize due to the disorder of the background media. In the past several decades, the localization in a disordered system has attracted a lot of interest [2]. There are tremendously many beautiful results, e.g., [4,5,15,19,20,24,29] in the discrete setting, [14,17,21,22] in the continuum setting, which are far from a complete list. In 2012, a new theory was proposed by Filoche and Mayboroda [23] to study the location of localized state. They introduced the concept of the landscape, which is the solution u to Hu = 1 for a Schr¨ odinger operator H = −Δ + V on a finite doma
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