Simons Collaboration on Localization of Waves Annual Meeting 2020
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Organizers
Svitlana Mayboroda, University of Minnesota
Marcel Filoche, Ecole Polytechnique
The first annual meeting of the Localization of Waves Collaboration sponsored by the Simons Foundation hosted 138 world-class scientists (including two Fields medalists and two Nobel Prize winners in physics) in New York City to explore profound connections between disorder, geometric complexity, the behavior of waves and recent advances in mathematical analysis for important applications in physics involving wave localization. The meeting successfully introduced the greater Simons Community to the Filoche-Mayboroda ‘Landscape Law’ and sparked exciting and fruitful conversations across multiple disciplines.
Collaboration Consultant professor Shuji Nakamura (University of California, Santa Barbara) kicked off the first discussion with a brief presentation, “History of InGaN-based LED” and explained how the first high-efficient InGaN blue LEDs were realized, overcoming the large density of defects present in the materials. He ended his talk by introducing the audience to the role and the importance of wave localization in nitride-based semiconductors. Founding PI Marcel Filoche’s (École Polytechnique) talk “The Localization Landscape: Order Through Disorder” showed the existence of an underlying structure called ‘localization landscape’ for wave localization in all system types. Professor Filoche showed the implementation of this tool into a semi-classical drift-diffusion transport model of semiconductor devices as a catalyst for discussion. In his talk “Delocalization of Eigenvectors of Random Matrices,” Collaboration Consultant Terence Tao (University of California, Los Angeles) explained a number of techniques that have been successfully used in recent years to establish delocalization results for eigenvalues of various random matrix ensembles such as Wigner matrices. Founding PI Sir Richard Friend (University of Cambridge) presented how localization of electronic states affects efficient LED and solar cells operations in his talk “Organic Electronics.” He shared progress that sparked discussion on the transport phenomena in organic semiconductors and how excited states and spin play fundamental roles in this phenomena. Collaboration director and founding PI Svitlana Mayboroda showed that behind each possibly disordered system, the localization landscape structure exists, which predicts the location and shape of the localized eigenfunctions, governs their long-range exponential decay and provides accurate bounds for the counting function of their eigenvalues in her presentation “The Hidden Landscape of Wave Localization.” Professor Charles Fefferman (Princeton University) shared how the properties of graphene arise in the spectral theory of certain Schrödinger operators in the plane in his talk “The One Electron Model of Graphene.” Day one concluded with PI and speaker dinner at Park Avenue Winter.
Day two began with a talk by Sir Michael Berry (University of Bristol) entitled “Coherent Destructive Interference: Superoscillations and Wave Geometry” that explained how functions can vary faster than their fastest Fourier components, presenting examples of almost-destructive interference in waves of all kinds (optical, acoustic, quantum, ocean tides, etc.). Discussions led beautifully into the next talk on the complexity of the topology of a hyper-surface in higher dimensions. Professor Peter Sarnak (Princeton University) presented “The Topologies of Nodal Sets of Random Monochromatic Waves” to show there is a corresponding universal law for its distribution over connected components. The conference concluded with Professor Steven Kivelson’s (Stanford University) talk “How Interactions Change the Physics of Wave Localization.” Professor Kivelson argued both from a theoretical perspective and by invoking experimental results.
Fifteen posters, representing collaborative subgroup teams, were on display throughout the meeting to showcase recent advancements. Mathematics deliverables included Agmon metric and Agmon distance assessed with the effective potential, new bounds to the counting function (the Landscape Law) and challenging concepts in geometrical measure theory such as the definition of a new harmonic measure for low-dimensional sets. New numerical techniques speeding up 10 times the computation of localized eigenfunctions were presented. Physics subgroup projects include the computation of Lyapunov exponents in 1-D cold atom systems (to test the effective mobility edge), a theory of hopping transport using the Agmon metric based on the effective potential experimental probe of charge carrier localization in disordered nitride alloys and the observation of excitonic localization in organic semiconductors. This meeting also served as a springboard for the launch of a social media campaign to promote the Collaboration via @SimonsWave.
The annual meeting helped solidify year-two goals and objections for the Collaboration. Svitlana Mayboroda will continue to lead efforts to study laws governing the density of states in disordered potentials. Together with Guy David and David Jerison, they will continue to investigate the geometry of interfaces, minimizers, soap films and level sets of solutions and eigenfunctions. The goal is to establish sharp equivalence between geometric features of boundaries and interfaces and the appropriate regularity of the Green function and the landscape. Marcel Filoche, Svitlana Mayboroda, Guy David and David Jerison will also study higher energy effects using the localization landscape: mobility edge, correlations, localization at higher energies and transition to delocalization, including the role of the modified differential operator and the transition from classical to quantum regime. For the Poisson-Schrödinger system and the Hartree-Fock approximation, James Speck, Claude Weisbuch and Douglas Arnold will continue to address key challenges such as computing the absorption curves for semiconductors via localization landscapes of electrons and holes in 3-D. Douglas Arnold will continue to direct the computational efforts which play an important role in almost all the collaboration’s activities. In particular, he is working collaboratively to develop forms of a posteriori error estimation and adaptivity which maximally exploit localization behavior. Led by Alain Aspect, the collaboration will continue to investigate the cold atom model system, with a goal of theoretically justifying the landscape approach and measuring the mobility edge and critical exponents of the Anderson transition. Marcel Filoche, Claude Weisbuch and James Speck will lead the research efforts for nitride-based semiconductors, model quantum transport, undertake systematic numerical and experimental studies of localization in order to determine accurate bandgaps in InGaN alloys, to understand the origin of the low efficiency of green LED simulations, and to validate the computed 3-D Urbach tails. Richard Friend will continue to lead the efforts on organic semiconductors, developing experimental studies (including the study of temperature-dependent Urbach energies) and localization landscape models including quantum transport.
We are grateful to the Simons Foundation for the opportunity to delve into this exciting topic and look forward to sharing results of our efforts at the next Simons Foundation Annual meeting in 2021.
Shuji Nakamura
Materials and ECE Departments
Solid State Lighting and Energy Center
University of California, Santa Barbara
History of InGaN-based LED
In the 1970s and ’80s, efficient blue and green light-emitting diodes (LED) were the last missing elements for solid-state display and lighting technologies due to the lack of suitable materials. By that time, III-nitride alloys were regarded as the least decent candidates due to various ‘impossible’ difficulties. However, a series of unexpected breakthroughs in the 1990s changed people’s view. Finally, in 1993, the first high-efficient InGaN blue LEDs were invented and commercialized. Nowadays, InGaN-based LEDs have become the most widely used light source in many applications.
Marcel Filoche
Directeur de recherche CNRS
Physique de la Matière Condensée, Ecole Polytechnique
The Localization Landscape: Order Through Disorder
Standing waves in disordered or complex systems can be subject to an intriguing phenomenon which has puzzled physics and mathematical communities for more than 60 years, called wave localization. This phenomenon consists of a concentration of the wave energy in a very restricted subregion of the entire domain and has been observed in mechanics, acoustics and quantum physics. We will show the existence of an underlying structure for wave localization in all system types. This structure, called ‘localization landscape,’ is the solution to a Dirichlet problem associated to the wave equation. Going further, the landscape also defines an ‘effective localization potential,’ providing a new insight into the confinement of the waves in disordered media. This potential allows us to predict the localization region, the energies of the localized modes, the density of states and the long-range decay of the wave functions.
As an example, we present here the implementation of this tool into a semiclassical drift-diffusion transport model of semiconductor devices. We will show how this novel model enables us to account for quantum effects at the nanoscale and to compute light absorption and light emission between localized quantum states, granting an acceleration of the computation time of full LED simulations (carrier transport and light emission) by several orders of magnitude compared to the Schrödinger-Poisson drift-diffusion (SP-DD) type approach.
Terence Tao
Department of Mathematics
University of California, Los Angeles
Delocalization of Eigenvectors of Random Matrices
We survey a number of techniques that have been successfully used in recent years to establish delocalization results for eigenvalues of various random matrix ensembles, such as Wigner matrices.
Richard Friend
Cavendish Laboratory
University of Cambridge
Organic Electronics
Pi-conjugated organic molecules and polymers now provide a set of well-performing semiconductors that support devices, including light-emitting diodes (LEDs) as used in smartphone displays and lighting, field-effect transistors (FETs) and photovoltaic diodes (PVs). These are attractive materials to manufacture, particularly for these large-area applications, but as Friend will explore in this talk, their electronic properties are very different from standard semiconductors such as silicon. Firstly, electronic overlap between adjacent molecules is relatively poor, and this often drives localization of electronic states. Secondly, dielectric screening is weak so that Coulomb interactions between charges and spin exchange energies are large. Management of transport and of excited state spin is fundamental for efficient LED and solar cells operation. I will discuss in particular recent progress in the control of emissive spin singlet excited states and non-emissive spin triplet excited states.
Svitlana Mayboroda
Northrop Professor
Department of Mathematics
University of Minnesota
The Hidden Landscape of Wave Localization
Complexity of the geometry, randomness of the potential and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization, a phenomenon of confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk, we show that behind a possibly disordered system, there exists a structure, referred to as a ‘landscape function,’ which predicts the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and delivers accurate bounds for the corresponding eigenvalues. In particular, we establish non-asymptotic estimates from above and below on the integrated density of states of the Schroedinger operator using a counting function for the minima of the localization landscape. The results are deterministic and rely on a new uncertainty principle. Narrowing down to the context of disordered potentials, Mayboroda derives the best currently available bounds on the integrated density of states for the Anderson model.
Charles Fefferman
Herbert E. Jones University Professor of Mathematics
Princeton University
The One Electron Model of Graphene
Many remarkable properties of graphene arise already in the spectral theory of certain Schroedinger operators in the plane. The relevant potentials have the ‘honeycomb’ symmetries of the tiling of the plane by regular hexagons. Fefferman’s talk presents results and unsolved problems regarding such Schroedinger operators (joint work with Michael Weinstein and James Lee Thorp).
Michael Berry
Emeritus Professor of Physics
University of Bristol, UK
Coherent Destructive Interference: Superoscillations and Wave Geometry
In physics, the mathematical phenomenon of superoscillations, in which functions vary faster than their fastest Fourer components (‘faster than they should’), is associated with almost-destructive interference. It occurs near phase singularities of waves of all kinds (optical, acoustic, quantum, ocean tides, etc.). Superoscillations are associated with quantum weak measurements. They are a compact way to represent fractals (e.g., the Weierstrass nondifferentiable function) to specified resolution. In light represented by scalar waves and in many contexts in quantum physics, superoscillations are rather common, but in vector light, represented by electric fields — and more so when magnetic fields are included — they are unexpectedly rare. One scheme for sub-wavelength imaging is based on superoscillations. Superoscillations in red light can escape as gamma radiation.
Peter Sarnak
Institute for Advanced Study
Princeton University
The Topologies of Nodal Sets of Random Monochromatic Waves
The topology of a hyper-surface in P^n(R) of high degree can be very complicated. However, if we choose such an algebraic hyper-surface — or nodal set of a monchromatic wave — at random, there is a corresponding universal law for its distribution over connected components. Little is known about these laws, and aspects appear to be dramatically different for n=2 and n>2. The zero sets of monochromatic waves are a model for nodal sets of eigenfunctions of quantizations of chaotic Hamiltonians.
Steven Kivelson
Institute for Theoretical Physics
Stanford University
How Interactions Change the Physics of Wave Localization
‘Anderson localization’ typically refers to a property of noninteracting quantum particles in a random potential, or more generally to the solution of a wave equation in a random medium. In physical systems, particles interact with each other and this — even if the interactions are in some sense arbitrarily weak — can qualitatively change the nature of the resulting phases of matter. Kivelson will discuss — both from a theoretical perspective and by invoking the results of experiment — some examples in which interactions fundamentally change the physically interesting properties of a macroscopic collection of quantum particles (i.e., electrons) in a random medium.
Observing Exciton Localization Through Temperature- and Pressure-Dependence in Molecular Crystals
Antonios Alvertis (Cambridge University), Richard Friend (Cambridge University)
Subgroup: Organics
Excitons in molecular crystals are central to optoelectronic devices. While most theoretical studies of excitons have focused on small molecular clusters, excitons in the solid state can delocalize over a large number of molecules. In this work, we study the excitons of the acene series of molecular crystals and show that exciton delocalization strongly affects the dependence of exciton energies on structural changes. Two manifestations of this effect appear through exciton pressure- and temperature-dependence. For delocalized excitons, increasing pressure (reducing volume) leads to a strong red shift of exciton energies, and conversely thermal expansion (increasing volume) should lead to a strong blue shift of these exciton energies with increasing temperature. However, we experimentally find that exciton energies are largely temperature-independent, apparently contradicting the observed strong pressure dependence. We develop a microscopic theory and show that this is due to the competition of thermal expansion with exciton-phonon interactions. Delocalized excitons couple to temperature-activated low-frequency phonons more prominently than localized excitons, almost perfectly cancelling the effect of thermal expansion in each case. For increasing exciton localization, the effect of high-frequency phonons that are active due to quantum fluctuations becomes more important and needs to be accounted for, in order to achieve predictive power for exciton energies.
The Localization Landscape of the Aubry-André Model: A Deterministic Approach to Randomness
The Aubry-André potential is a pseudo-periodic potential which exhibits similar confining behavior as random potentials. We investigate the properties of the localization landscape applied to these pseudo-periodic potentials and calculate a closed formula for the landscape.
The Landscape Law for Predicting Lipshitz Tails in Random Potentials
Douglas N. Arnold (University of Minnesota),Guy David (Univ Paris-Saclay), Perceval Desforges (École Polytechnique), Marcel Filoche (École Polytechnique), Svitlana Mayboroda (University of Minnesota)
A very recent study has shown the existence of a function (called the landscape law) that provides lower and upper bounds for the counting function of the Schrödinger equation. In this study, we present simulations for this landscape law and determine the sharpest values for the constants involving in the inequalities bounding the actual counting function.
Direct Experimental Evidence of Landscape-Predicted Localized States in Disordered InGaN Quantum Wells By Scanning Tunneling Luminescence Spectroscopy
Marcel Filoche (École Polytechnique), Wiebke Hahn (École Polytechnique), Yves Lassailly (École Polytechnique), Jean-Marie Lentali (École Polytechnique), Shuji Nakamura (University of California – Santa Barbara), Jacques Peretti (École Polytechnique), James S. Speck (University of California – Santa Barbara), Claude Weisbuch (École Polytechnique/University of California – Santa Barbara), Yuh-Renn Wu (National Taiwan University)
Subgroup: Nitride semiconductors
Nitride ternary alloys (such as InGaN and AlGaN) are affected by intrinsic compositional fluctuations on short range (nm scale) due to random positioning of alloy atoms on the crystal lattice which induces disorder. We use scanning tunneling electroluminescence (STL) to detect the radiative recombination of electrons locally injected by a scanning tunneling microscope tip in a GaN/InGaN/GaN single quantum well. Fluctuations in the emission spectrum are observed when changing the injecting tip position by a few nm. Analyzing the linewidth and peak energy of the luminescence spectrum as a function of the tip position we evidence localization effects induced by alloy disorder with 5 nm resolution.
Local Computation for Localized Eigenstates
Douglas N. Arnold (University of Minnesota), Kaibo Hu (University of Minnesota)
We exploit local nature of eigenstates to develop reliable and efficient algorithms for computing all localized eigenfunctions under a certain energy level. The algorithm is based on division of the computational domain into smaller overlapping subdomains and solving local eigenvalue problems numerically on each subdomain. The proposed algorithm can be efficient when a large-scale computation for the global problem is very expensive.
Elliptic Operators with a BMO Anti-Symmetric Part
Steve Hofmann (University of Missouri), Linhan Li (University of Minnesota), Svitlana Mayboroda (University of Minnesota), Jill Pipher (Brown University)
Subgroup: GMT
We aim to develop a comprehensive elliptic theory for divergence-form second-order elliptic operators whose anti-symmetric part is in the John-Nirenberg space BMO. The lack of boundedness invalidates many of the arguments in the classical elliptic theory, but we are able to conclude that the BMO anti-symmetric part does not hurt.
Efficient Computation of Localized Eigenmodes
Douglas N. Arnold (University of Minnesota), Tyson Loudon (University of Minnesota)
Subgroup: Absorption Curves
An efficient algorithm for computing localized eigenpair is proposed which combines high degree local corrections and low dimensional global corrections. The method is demonstrated to have the same accuracy as using the finite element method, and yet the most expensive part of the proposed algorithm is the same cost as a linear finite element eigenvalue solve.
Nitride Alloy Disorder Characterization and Fundamental Impact on Light Emitting Diodes
Bastien Bonef (University of California – Santa Barbara), Guillaume Lheureux (University of California – Santa Barbara), Cheyenne Lynsky (University of California – Santa Barbara), James S. Speck (University of California – Santa Barbara), Claude Weisbuch (École Polytechnique/University of California – Santa Barbara), Yuh-Renn Wu (National Taiwan University)
Subgroup: Nitride semiconductors
Landscape theory introduces a novel method to account for quantum disorder effects into the classical drift-diffusion model of semiconductor transport through the localization landscape theory. Quantum confinement and quantum tunneling in the disordered system change dramatically the energy barriers acting on the perpendicular transport of heterostructures. This model solves the carrier dynamics with quantum effects self-consistently and provides a computationally much faster solver when compared with the Schrödinger equation resolution. The current-voltage characteristics modeled by three-dimensional simulation of a full nitride-based light-emitting diode (LED) structure with compositional material fluctuations closely match the experimental behavior of high-quality blue LEDs. The model allows also a fine analysis of the quantum effects involved in carrier transport through such complex heterostructures. Compared to blue LEDs, green LEDs suffer from poor power conversion efficiency as a result of large excess forward voltage. Identifying sources of excess voltage and barriers to carrier transport in green LEDs is a clear path forward to improving the device wall plug efficiency (WPE). Using a combined experimental and computational approach, we demonstrated that polarization barriers in green LEDs contribute to excess voltage. These data support prior results that point to piezoelectric polarization, resulting from lattice mismatch between the QWs and quantum barriers (QBs), as a barrier to efficient carrier transport into the QWs. Artificially turning off the polarization fields in the simulation does not entirely suppress this effect. We propose theoretical evidence for sequential filling of the quantum wells in multi-quantum well nitride LEDs with high indium content
Computing the Spectral Function of Cold Atoms Condensates with the Localization Landscape
Douglas N. Arnold (University of Minnesota), Alain Aspect (Institut D’Optique Graduate School), Dominique Delande (Sorbonne Univ), Marcel Filoche (École Polytechnique), Vincent Josse (Institut D’Optique Graduate School), Pierre Pelletier (École Polytechnique), Svitlana Mayboroda (University of Minnesota)
Subgroup: Cold Atoms
In ultracold atoms physics, the spectral function is a key quantity providing insights into the experimental results. From a theory standpoint, it describes the scattering properties of the atoms. However, theoretical predictions in disordered media are still limited in their scope. This poster presents new advancement on this problem through the lens of the localization landscape theory, as well as recent direct measurements of the spectral function.
Exponential decay of Green’s function for the Schrödinger operator and the Filoche-Mayboroda landscape function
Svitlana Mayboroda (University of Minnesota), Bruno Poggi Cevallos (University of Minnesota)
Subgroup: GMT
We establish estimates for the Dirichlet and Neumann resolvents and Green’s function associated to the Schrödinger operator L = -div A∇ + V on a smooth bounded domain, which decay exponentially (with no dependence on size of the domain) in terms of the Filoche-Mayboroda effective potential 1/u, where u is the landscape function, the Neumann solution to the problem – div A∇u + Vu = 1. We compare this bound to other exponential decay bounds obtained via the Fefferman-Phong inequality.
Unipolar transport (from David Browne, through Micha, to new work from Clayton and Morteza)Disordered transport in unipolar semiconducting materials: III-Nitride Alloy Structures, Localization Landscape Theory and Experimental Results
David Browne (University of California – Santa Barbara), Micha Fireman (University of California – Santa Barbara), Morteza Monavarian (University of California – Santa Barbara), Kai Shek Qwah (University of California – Santa Barbara), James S. Speck (University of California – Santa Barbara), Claude Weisbuch (École Polytechnique/University of California – Santa Barbara), Yuh-Renn Wu (National Taiwan University)
Subgroup: Unipolar Heterostructures
“We report on our experimental and theoretical findings with regards to carrier transport through unipolar III-Nitride heterostructures. These studies were conducted to determine the effect of random alloy fluctuations on the carrier transport behaviour within InGaN and AlGaN layers, commonly found in nitride devices. Unipolar heterostructures serve as great test structures to experimentally investigate the carrier transport. An isotype structure would eliminate the recombination process from analysis, which considerably simplifies the picture and helps in understanding the carrier transport mechanisms in heterojunctions. It was found that n-type InGaN heterostructures were shown to display diode-like current voltage behavior due to the polarization-induced conduction band barrier in the quantum well region. It was also found that in order to capture this voltage behaviour in simulations, alloy fluctuations had to be incorporated. On the other hand, undoped AlGaN failed to provide a barrier to electron transport in n-type unipolar structures. This was also attributed to the random alloy fluctuations within AlGaN, providing percolative pathways for the electrons.
Recently, a similar study was conducted on p-type AlGaN heterostructures. The results show that even a thin UID Al x Ga 1-x N (x = 14%, 13 nm) introduces an asymmetric barrier to the hole transport. However, p-type doping of the AlGaN layer results in a drastic drop in the potential barrier to hole transport in both directions. These studies show the effectiveness of unipolar heterostructures as test vehicles to further our understanding in nitride electronic and optoelectronic devices.”
Experimental Evidence of the Impact of Localization on Electron Transport in Disordered Semiconductor Ternary Alloys by Low-Energy Photo-Emission Spectroscopy
Marcel Filoche (École Polytechnique), Yves Lassailly (École Polytechnique), Jacques Peretti (École Polytechnique), Shuji Nakamura (University of California – Santa Barbara), Mylène Sauty (École Polytechnique), James S. Speck (University of California – Santa Barbara), Abel Thayil (École Polytechnique), Claude Weisbuch (École Polytechnique/University of California – Santa Barbara)
InGaN is the nitride semiconductor material which allows light emission in the visible range for all LED lamps. As a ternary alloy, it exhibits an intrinsic compositional disorder, which affects its optical and electronic properties. We use low-energy photoemission spectroscopy to experimentally probe the influence of disorder-induced localization on photon absorption and electron transport in this material. We observe a strong dependence of the quantum yield with excitation energy and with temperature. This is explained by a freezing of electron transport at low temperature, as confirmed by the evolution of the energy distribution of the extracted electrons with temperature.
Transport of Localized Charge Carriers in Disordered Media
Marcel Filoche (École Polytechnique), Jean-Marie Lentali (École Polytechnique), James S. Speck (University of California – Santa Barbara), Abel Thayil (École Polytechnique), Claude Weisbuch (École Polytechnique/University of California – Santa Barbara), Yuh-Renn Wu (National Taiwan University)
Accounting for the effects of disorder in semiconductor devices, especially carrier localization, is a real challenge as it requires to compute quantum effects at the nanoscale in devices where the overall dimensions are typically of the order of 100 nm or more. The localization landscape theory, introduced first in 2012 and later applied to nitride-based alloys enables us to account for such effects. In this theory, an effective potential (the reciprocal of the landscape) predicts the regions of localization of the eigenstates, their corresponding energies, and more globally the local density of states without having to explicitly solve the Schrodinger equation. We present here a model for electronic transport based on that theory. We detail the mathematical structure of this model which incorporates hopping between the localized states, and then analyze numerical simulations of transport in disordered semiconductor alloys.
Eigenmode Prediction with Landscape Function in 3D
Douglas N. Arnold (University of Minnesota), Wei Wang (University of Minnesota)
Landscape function performs well in computing the eigenmode and numerical results have been presented in 1D and 2D. In this work, we display the computation in 3D, which show the effectiveness of the landscape function in predicting eigenvalues and eigenfunctions.
Landscape Theory for Discrete Schrödinger Operators
Douglas N. Arnold (University of Minnesota), Marcel Filoche (Ecole Polytechnique), Svitlana Mayboroda (University of Minnesota), Wei Wang (University of Minnesota), Shiwen Zhang (University of Minnesota)
Subgroup: Absorption Curves (Urbach Tail)
We consider a discrete Schrödinger operator on a cube M, with periodic boundary condition. We study a hidden landscape function, defined as the solution of an inhomogeneous problem with uniform right-hand side. We show that the reciprocal of the landscape function acts as an effective potential, which can be used to predict the location of the localized eigenfunctions and eigenvalues of the operator. Explicit bounds on the exponential decay of the eigenfunctions of the system near both the bottom and top of the spectrum are obtained. Moreover, using a counting function for the minima of the effective potential, we obtain estimates for the integrated density of states non-asymptotically. Our results are deterministic and are independent of the size of the cube. We provide numerical experiments to confirm the results on the Agmon type of localization. We also derive the best possible Lifschitz tail bounds on the integrated density of states for the Anderson model.
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