# Sharp estimates for the integrated density of states in Anderson tight-binding models

@article{Desforges2020SharpEF, title={Sharp estimates for the integrated density of states in Anderson tight-binding models}, author={Perceval Desforges and Svitlana Mayboroda and Shiwen Zhang and Guy David and Douglas N. Arnold and Wei Wang and Marcel Filoche}, journal={Physical Review A}, year={2020} }

Recent work [1] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schrodinger operator throughout the spectrum, called the \emph{Landscape Law}. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the Landscape Law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be…

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## References

SHOWING 1-10 OF 48 REFERENCES

### Weak Disorder Localization and Lifshitz Tails: Continuous Hamiltonians

- Mathematics, Computer Science
- 2002

It is proved that, in the weak disorder regime, the spectrum in a neighborhood of size C \cdot \lambda $ of a non-degenerate simple band edge is exponentially and dynamically localized.

### Dual landscapes in Anderson localization on discrete lattices

- Physics
- 2015

The localization subregions of stationary waves in continuous disordered media have been recently demonstrated to be governed by a hidden landscape that is the solution of a Dirichlet problem…

### Singularities in spectra of disordered systems: an instanton approach for arbitrary dimension and randomness

- Mathematics, Computer Science
- 1989

The precise form of the Lifshitz tail is derived here, by means of a field-theoretic description, and of instanton calculus, which provides new results for an arbitrary distribution of potentials, in arbitrary dimension.

### Weak Disorder Localization and Lifshitz Tails

- Mathematics
- 2002

Abstract: This paper is devoted to the study of localization of discrete random Schrödinger Hamiltonians in the weak disorder regime. Consider an i.i.d. Anderson model and assume that its left…

### The Parabolic Anderson Model

- Mathematics
- 2016

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model…

### Universal mechanism for Anderson and weak localization

- PhysicsProceedings of the National Academy of Sciences
- 2012

It is demonstrated that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape, which partitions the system into weakly coupled subregions.

### Localization of eigenfunctions via an effective potential

- MathematicsCommunications in Partial Differential Equations
- 2019

Abstract We consider the localization of eigenfunctions for the operator on a Lipschitz domain Ω and, more generally, on manifolds with and without boundary. In earlier work, two authors of the…

### The Parabolic Anderson Model: Random Walk in Random Potential

- Mathematics
- 2016

This is a comprehensive survey on the research on the parabolic Anderson model the heat equation with random potential or the random walk in random potential of the years 1990 2015. The investigation…

### The energy spectrum of disordered systems

- Physics
- 1964

Abstract (by Editor) A detailed report is given of the theoretical work carried out by the author during recent years on problems connected with the energy spectrum of a disordered solid. The…