# Surface Localization in Impurity Band with Random Displacements and Long-Range Interactions

Authors: Victor Chulaevsky
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We study the random Schrödinger operators in a Euclidean space with the disorder generated by two complementary mechanisms: random substitution in a lower-dimensional layer and random displacements in the bulk, without additional assumptions regarding the reflection symmetry of the site potentials. The latter are assumed to be bounded and have a power-law decay. Complementing earlier results obtained in the strong disorder regime, we establish spectral and strong dynamical localization in the impurity zone near the bottom of spectrum for arbitrarily weak amplitudes of the random displacements, provided the concentration of impurities is sufficiently small.


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## Body

## 1. Introduction

### 1.1. The Model

By a random displacements Hamiltonian inRd one usually calls a random Schrödinger operator of the form(1)Hω=-hΔ+V0x+∑a∈Zduax-Xaω,where h>0, Zd⊂Rd is a discrete lattice (usually periodic), and Xa(ω) are random positions of the “scatterers” labeled by the lattice sites a and characterized by the local potentials (“scatterer potentials”) ua. In particular, all ua may be identical. Here V0(x) is a nonrandom component of the potential generated by the media, which one may choose to be trivial (V0(x)≡0) or not. Generally speaking, such models are more difficult to study than the random alloy models with Hamiltonians (2)-hΔ+∑a∈Zcaωuax-a,where the randomness resides in the IID (independent and identically distributed) amplitudes ca(ω), and the latter are usually assumed to have a Lipschitz- or Hölder-continuous probability distributions.There is a considerable wealth of mathematical research papers and monographs on alloy models and their finite-difference analogs, while the number of rigorous results on the random displacements models (RDM, in short) remains quite limited.Both in the class of alloy models and the RDM, one usually assumesu to be compactly supported; this is of course a conscious simplification of the physical reality where the fundamental interactions have infinite range. Indeed, a rigorous study of long-range models is substantially more complicated due to the long-range correlations induced by u.In the present paper, we treat a model which combines these two features: the randomness is provided only by the displacementsXa(ω) and the potentials u decay polynomially at infinity. To be more precise, we also introduce an additional disorder mechanism, random substitutions of scatterer potentials, as described below.Klopp [1] studied the random displacements model with the help of powerful techniques of the semiclassical analysis, hence under the assumption that the kinetic energy amplitude h in -hΔ is sufficiently small. Since multiplication of an operator by a nonzero constant does not change the nature of its spectral components (absolutely continuous, singular continuous, and pure point spectra), the assumption h≪1 in H=-hΔ+V(ω) is equivalent to the assumption h-1≫1 in the rescaled operator h-1H(ω)=-Δ+h-1V(ω); the latter is usually called thestrong disorder regime. The semiclassical approach has a number of important advantages; in particular, it resulted in [1] in the proof of a Wegner-type estimate and of the Initial Length Scale (ILS) estimate based on the Lifshitz tails phenomenon (cf. [2–4]). However, an adaptation of these technically involved methods to the random displacements models with an infinite-range, slowly decaying interaction potential u is far from being straightforward, and it is beyond the scope of the present paper.Lott and Stolz [5] and Baker et al. [6, 7] studied the spectral minimizers for the displacements model, in the context, and obtained important analytic (deterministic) results in this direction. Further progress was obtained by Klopp et al. [8] who described the energy minimizing configurations in the displacements model, assuming that the site potential is reflection-symmetric in each coordinate variable xi=0, 1≤i≤d (see the precise assumptions (A1) and (A2) in [8]), and proved Anderson localization in their model for energies near the bottom of the spectrum.We consider the model where the local potentials are of two different types. Physically, it corresponds to a disordered media sample with different types of atoms (or ions, when some electrons are mobile; for simplicity we will always speak of “atoms” bearing the respective local potentials). Generally speaking, one can consider the models with an arbitrary numberK>1 of atoms, but the most important feature of our model is that in some sublattice Zd0↪Zd=Zd0×Zd1 some randomly selected atoms are of one particular type with potentials taking values below those of all other types (negative, in our model).A realistic example is given by a thin film (some recent CPU chips are built on 14 nm films, with just a few atoms across it) where one thin filament is “doped” by randomly placed atoms producing a low energy band: below the main energy band of the bulk material. Similarly, in 3 dimensions, one can think of a surface of a crystal with randomly places impurities; such a layer, producing a low energy band, can be sandwiched between two much thicker samples of the bulk material. A thorough examination of our arguments in the subsequent sections evidences that the main results can be easily extended to a model where the impurities are randomly placed inside a finite-width layer and not just a monolayer formed by a sublatticeZd0 of nonzero codimension. Our technique allows one to prove exponential spectral and strong dynamical localization in the impurity zone produced solely by randomness in local displacements within a subsample of the bulk material (located at a finite distance from the impurity layer).From the point of view of experimental physics and new technologies, an approach relying only on random displacements provides a solid base for the emergence of Anderson localization, where more traditional random alloy models rely crucially on the fluctuations of the strength of the local potentialsωau(x-a), with ωa having a continuous (even absolutely continuous) probability distribution. It is quite clear from the basic facts of the electrostatic interactions that the amplitude of the potential is proportional to the electric charge of a given ion, and the latter is aninteger multiple of the electron charge e, so it cannot have a continuous distribution. Of course, the popular alloy models implicitly refer to a complex structure of the ionic potentials, yet the question of whether or not the latter could result in a nice probability distribution of local potential amplitudes is far from obvious, and this is where the random displacements mechanism proves much more pertinent and universal.To formalize the above considerations, we introduce the extended samplesω=(ω,ϑ) on a probability space (Ω,B,P), where ω=(ωa)a∈Z and ϑ=(ϑa)a∈Zd0 are mutually independent random fields on the entire lattice Z and on the sublattice Zd0, respectively, and each of them is IID (has independent and identically distributed random values). To make explicit some geometrical constraints we introduce for the sake of clarity, we assume that the intersite distances in Zd are greater than or equal to 1 (e.g., Zd=Zd), but in the case of more general lattices it suffices to adapt intermediate geometrical parameters. Apart from the difference in the index sets of ωa and ϑa (Z versus Zd0), the respective common marginal probability distributions are quite different:(U1)
ϑa, a∈Zd0, are Bernoulli random variables with Pϑa=1=p=1-Pϑa=0∈(0,1).(U2)
ωa∈Rd, a∈Z, have a bounded, compactly supported probability density ρ0(x), x∈Rd. Specifically, we assume for the sake of clarity of presentation that there exist 0<r1<r2<1/2 such that
(U2.1)
supp⁡ρ0⊂Br2(0),(U2.2)
ρ0(x)≥c1Br1(0)(x), c>0.Given the random displacementsωa, we denote by Xa≔a+ωa the random positions of the (centers of) the atoms forming the media.Next, we consider two types of the local potentials,u0 and u1. Simply put, the potential V induced by any sample without “impurities” (potentials u1) is nonnegative, while an impurity at a∈Zd0 (implemented by a site potential u0(·-a)) produces anegative potential well in some ball Br0(a), and we assume that this negative well cannot be destroyed by any configuration of atom types and displacements occurring outside some neighborhood of the site a.Specifically, we assume thatu0 and u1 fulfill the following conditions:(U3)
Forr≥1/2 and some A>2d,
(U3.1)
|u1(r)|≤C1r-A, C1∈(0,+∞);(U3.2)
u0(r)=r-A.Fromu1(r) with r≥1/2 we only need an upper bound (U3.1) on its decay rate, while more detailed analytical information is required for u0(r) and its gradient, which explains an explicit choice made in (U3.2).The main distinction betweenu0 and u1 is explained by the following hypothesis (recall Xa=a+ωa):(U4)
There exist someE(-)<0 and r0>0 such that for and all x∈Br0(0) one has(3)u1x+∑a≠0supωa≤r2⁡maxk=0,1⁡ukx-Xa⁡≤E-,whileu0(x)≥0 for all x∈Rd.In addition, we assume that the kinetic energy amplitudeh (cf. (1)) is sufficiently small, depending upon u0, so that the negative wells (starting from a single one) produce deterministically a nontrivial negative spectrum of H.Since (U3.1) gives a deterministic bound on the decay|u| (in fact, it could be assumed in addition to the condition of nonpositivity everywhere), and u0(r)=r-A outside a ball of radius 1/2, it suffices to take u1(r) negative and sufficiently large in Br0(0) to create a deep negative well. Further, the variational estimates show that such a well around an impurity produces negative spectrum, at least for the amplitude h of the Laplacian small enough.The main point of the present work is the analysis of the spectral band produced by theu0-wells.Introduce the alloy potential inRd of the form(4)Vx;ω=VBx;ω+VSx;ω(here (B) stand for “Bulk” and (S) for “Surface”), with(5)VBx;ω=∑a∈Zdu0x-Xa,VSx;ω,ϑ=∑a∈Zd0:ϑa=0u0x-Xa+∑a∈Zd0:ϑa=1u1x-Xa(recall Xa=a+ωa).Once the selection of the potential typesϑa∈0,1 is made, the only randomness in the potential sample V(x;ω) is the one coming from the displacements Xa(ω).The assumptions on smallness of the support of the probability measureρ0 (of the displacements ωa) had been made before in several works on the random displacements models (RDM) where it was actually crucial that the supports of the local potentials, even after admissible displacements of their centers within neighborhoods of the lattice sites a, do not overlap, for otherwise the subtle functional-analytic results used in earlier proofs would break down. It will be clear from our presentation that we can afford supp⁡ρ0⊂Br¯(0) with any fixed r¯<+∞, at the price of a more elaborate analysis and some additional assumptions on the site potentials (either a “hard core” condition or smallness of the positive potential u0, in order not to destroy the impurity wells). This follows from our strategy of proof of the crucial eigenvalue concentration (or Wegner-type) estimate which relies only on the randomness of ωa with a filling an affine sublattice parallel to Zd0 at some finite distance r2 from the latter. The reader will see that the displacements of all remaining random positions Xa=a+ωa can be (and often will be) rendered nonrandom by conditioning. However, this also makes more cumbersome some important geometrical arguments, and for this reason we postpone possible generalizations to a forthcoming work.Kirsch et al. [9] adapted the variable-energy (or energy-interval) MSA approach developed by Fröhlich et al. [10] (cf. also [11]), reformulated by von Dreifus and Klein [12], to the alloy models with a polynomial decay of the interaction potential u(r)=r-A, under the assumption A>4d. However, their techniques apply directly to the models featuring the so-called monotonicity condition: a variation of any random parameters (scatterer amplitudes, in their case) has to produce a variation of the potential with fixed sign (either positive or negative), while the displacements of the scatterers result in sign-indefinite variations of the induced potential.An important issue one has to face in the class of the random displacements models is the proof of the ILS estimate for an arbitrarily small disorder amplitude, without the strong disorder condition. The main point of the present paper is that this issue can be addressed in a relatively simple way for the surface models where a layer of the sample producing a spectral band, separated by a gap from the bulk spectrum, is subject both to the random displacements disorder and to the substitution disorder (due to a random selection of the lattice positions for the impurities).The latter kind of disorder is “discrete” and not generated by variation of continuous parameters. Such a situation usually renders the localization analysis very difficult; the first result in this area was obtained in a remarkable work by Bourgain and Kenig [13] in 2001, for the random alloys with Bernoulli distribution of random scatterers amplitudes. Germinet and Klein [14] have later extended the Bourgain–Kenig approach to arbitrary nontrivial distributions, but the technique here remains very complex and the localization bounds (for the eigenfunction correlators) remain quite modest. In our paper, we use a “discrete” substitution disorder solely for the proof of the Initial Length Scale estimate, which is already quite difficult to obtain otherwise. To the author’s best knowledge, there is no prior result for the weak-disorder localization at low energy in the random displacements model with slowly decaying interaction.Moreover, we show that the moderate rate of decay of the Green functions obtained at the first stage of the multiscale analysis can be significantly enhanced by a “soft” argument, so that despite a relatively slow, power-law decay of interaction (hence, a relatively strong long-distance correlation in the random environment) the decay of eigenfunction correlators is at least fractional-exponential.The materials with an impurity band supported by specific atoms injected into a sample play important role in experimental physics and technology. In our paper, we thus show that even a relatively weak disorder induced by structural factors (substitution and local displacements of the atoms/ions) can produce strong dynamical localization in presence of a realistic, slow decay of the media-electron interaction at large distances.


### 1.2. Local Hamiltonians and Useful Notation

Given a cubeB3l(u), with l≥1, we call Bl(u) thecore and the annulus BLk(u)∖Blk(u) theshell of B3l(u). Further, given a cube BYL(u) with L=3l, Y≥2, we can cover the shell of BYL(u) by (3Y)d cores of radius lk. The respective cores Blk(x) will be called admissible.We define the “boundary” of a cubeBLk(u) as follows: ∂BLk(u)≔BLk(u)∖BLk-2(u).The spectrum of an operatorH will be denoted by Σ(H) and its resolvent set by ϱ(H).


### 1.3. Main Results

Theorem 1.
Assume that the conditions (U1)–(U4) are fulfilled. Then there existp^>0 and an interval I^⊂R such that the following holds for the random displacements model where p≡Pϑa=1∈(0,p^).(A)
With probability one,H(ω) has pure point spectrum in the interval I^, and all the eigenfunctions ψ of H(ω) with eigenvalues E∈I^ decay exponentially at infinity: for some m>0 one has (6)ψxy;ω;ϑ≤Cψω,ϑe-my-x∀y∈Z.(B)
Letχx, x∈Rd, be the indicator function of a unit cube centered at x. There exist C,ϱ∈(0,+∞) such that for all x≠y and for any bounded Borel function ϕI^ with supp⁡ϕI^⊂I^ and ϕI^∞≤1 one has (7)EχxϕI^Hωχy≤Ce-x-yϱ.The smallness of the concentration of impuritiesp in the “surface” layer is used only in the proof of the Initial Length Scale estimate (cf. Lemma 19).


## 1.1. The Model

By a random displacements Hamiltonian inRd one usually calls a random Schrödinger operator of the form(1)Hω=-hΔ+V0x+∑a∈Zduax-Xaω,where h>0, Zd⊂Rd is a discrete lattice (usually periodic), and Xa(ω) are random positions of the “scatterers” labeled by the lattice sites a and characterized by the local potentials (“scatterer potentials”) ua. In particular, all ua may be identical. Here V0(x) is a nonrandom component of the potential generated by the media, which one may choose to be trivial (V0(x)≡0) or not. Generally speaking, such models are more difficult to study than the random alloy models with Hamiltonians (2)-hΔ+∑a∈Zcaωuax-a,where the randomness resides in the IID (independent and identically distributed) amplitudes ca(ω), and the latter are usually assumed to have a Lipschitz- or Hölder-continuous probability distributions.There is a considerable wealth of mathematical research papers and monographs on alloy models and their finite-difference analogs, while the number of rigorous results on the random displacements models (RDM, in short) remains quite limited.Both in the class of alloy models and the RDM, one usually assumesu to be compactly supported; this is of course a conscious simplification of the physical reality where the fundamental interactions have infinite range. Indeed, a rigorous study of long-range models is substantially more complicated due to the long-range correlations induced by u.In the present paper, we treat a model which combines these two features: the randomness is provided only by the displacementsXa(ω) and the potentials u decay polynomially at infinity. To be more precise, we also introduce an additional disorder mechanism, random substitutions of scatterer potentials, as described below.Klopp [1] studied the random displacements model with the help of powerful techniques of the semiclassical analysis, hence under the assumption that the kinetic energy amplitude h in -hΔ is sufficiently small. Since multiplication of an operator by a nonzero constant does not change the nature of its spectral components (absolutely continuous, singular continuous, and pure point spectra), the assumption h≪1 in H=-hΔ+V(ω) is equivalent to the assumption h-1≫1 in the rescaled operator h-1H(ω)=-Δ+h-1V(ω); the latter is usually called thestrong disorder regime. The semiclassical approach has a number of important advantages; in particular, it resulted in [1] in the proof of a Wegner-type estimate and of the Initial Length Scale (ILS) estimate based on the Lifshitz tails phenomenon (cf. [2–4]). However, an adaptation of these technically involved methods to the random displacements models with an infinite-range, slowly decaying interaction potential u is far from being straightforward, and it is beyond the scope of the present paper.Lott and Stolz [5] and Baker et al. [6, 7] studied the spectral minimizers for the displacements model, in the context, and obtained important analytic (deterministic) results in this direction. Further progress was obtained by Klopp et al. [8] who described the energy minimizing configurations in the displacements model, assuming that the site potential is reflection-symmetric in each coordinate variable xi=0, 1≤i≤d (see the precise assumptions (A1) and (A2) in [8]), and proved Anderson localization in their model for energies near the bottom of the spectrum.We consider the model where the local potentials are of two different types. Physically, it corresponds to a disordered media sample with different types of atoms (or ions, when some electrons are mobile; for simplicity we will always speak of “atoms” bearing the respective local potentials). Generally speaking, one can consider the models with an arbitrary numberK>1 of atoms, but the most important feature of our model is that in some sublattice Zd0↪Zd=Zd0×Zd1 some randomly selected atoms are of one particular type with potentials taking values below those of all other types (negative, in our model).A realistic example is given by a thin film (some recent CPU chips are built on 14 nm films, with just a few atoms across it) where one thin filament is “doped” by randomly placed atoms producing a low energy band: below the main energy band of the bulk material. Similarly, in 3 dimensions, one can think of a surface of a crystal with randomly places impurities; such a layer, producing a low energy band, can be sandwiched between two much thicker samples of the bulk material. A thorough examination of our arguments in the subsequent sections evidences that the main results can be easily extended to a model where the impurities are randomly placed inside a finite-width layer and not just a monolayer formed by a sublatticeZd0 of nonzero codimension. Our technique allows one to prove exponential spectral and strong dynamical localization in the impurity zone produced solely by randomness in local displacements within a subsample of the bulk material (located at a finite distance from the impurity layer).From the point of view of experimental physics and new technologies, an approach relying only on random displacements provides a solid base for the emergence of Anderson localization, where more traditional random alloy models rely crucially on the fluctuations of the strength of the local potentialsωau(x-a), with ωa having a continuous (even absolutely continuous) probability distribution. It is quite clear from the basic facts of the electrostatic interactions that the amplitude of the potential is proportional to the electric charge of a given ion, and the latter is aninteger multiple of the electron charge e, so it cannot have a continuous distribution. Of course, the popular alloy models implicitly refer to a complex structure of the ionic potentials, yet the question of whether or not the latter could result in a nice probability distribution of local potential amplitudes is far from obvious, and this is where the random displacements mechanism proves much more pertinent and universal.To formalize the above considerations, we introduce the extended samplesω=(ω,ϑ) on a probability space (Ω,B,P), where ω=(ωa)a∈Z and ϑ=(ϑa)a∈Zd0 are mutually independent random fields on the entire lattice Z and on the sublattice Zd0, respectively, and each of them is IID (has independent and identically distributed random values). To make explicit some geometrical constraints we introduce for the sake of clarity, we assume that the intersite distances in Zd are greater than or equal to 1 (e.g., Zd=Zd), but in the case of more general lattices it suffices to adapt intermediate geometrical parameters. Apart from the difference in the index sets of ωa and ϑa (Z versus Zd0), the respective common marginal probability distributions are quite different:(U1)
ϑa, a∈Zd0, are Bernoulli random variables with Pϑa=1=p=1-Pϑa=0∈(0,1).(U2)
ωa∈Rd, a∈Z, have a bounded, compactly supported probability density ρ0(x), x∈Rd. Specifically, we assume for the sake of clarity of presentation that there exist 0<r1<r2<1/2 such that
(U2.1)
supp⁡ρ0⊂Br2(0),(U2.2)
ρ0(x)≥c1Br1(0)(x), c>0.Given the random displacementsωa, we denote by Xa≔a+ωa the random positions of the (centers of) the atoms forming the media.Next, we consider two types of the local potentials,u0 and u1. Simply put, the potential V induced by any sample without “impurities” (potentials u1) is nonnegative, while an impurity at a∈Zd0 (implemented by a site potential u0(·-a)) produces anegative potential well in some ball Br0(a), and we assume that this negative well cannot be destroyed by any configuration of atom types and displacements occurring outside some neighborhood of the site a.Specifically, we assume thatu0 and u1 fulfill the following conditions:(U3)
Forr≥1/2 and some A>2d,
(U3.1)
|u1(r)|≤C1r-A, C1∈(0,+∞);(U3.2)
u0(r)=r-A.Fromu1(r) with r≥1/2 we only need an upper bound (U3.1) on its decay rate, while more detailed analytical information is required for u0(r) and its gradient, which explains an explicit choice made in (U3.2).The main distinction betweenu0 and u1 is explained by the following hypothesis (recall Xa=a+ωa):(U4)
There exist someE(-)<0 and r0>0 such that for and all x∈Br0(0) one has(3)u1x+∑a≠0supωa≤r2⁡maxk=0,1⁡ukx-Xa⁡≤E-,whileu0(x)≥0 for all x∈Rd.In addition, we assume that the kinetic energy amplitudeh (cf. (1)) is sufficiently small, depending upon u0, so that the negative wells (starting from a single one) produce deterministically a nontrivial negative spectrum of H.Since (U3.1) gives a deterministic bound on the decay|u| (in fact, it could be assumed in addition to the condition of nonpositivity everywhere), and u0(r)=r-A outside a ball of radius 1/2, it suffices to take u1(r) negative and sufficiently large in Br0(0) to create a deep negative well. Further, the variational estimates show that such a well around an impurity produces negative spectrum, at least for the amplitude h of the Laplacian small enough.The main point of the present work is the analysis of the spectral band produced by theu0-wells.Introduce the alloy potential inRd of the form(4)Vx;ω=VBx;ω+VSx;ω(here (B) stand for “Bulk” and (S) for “Surface”), with(5)VBx;ω=∑a∈Zdu0x-Xa,VSx;ω,ϑ=∑a∈Zd0:ϑa=0u0x-Xa+∑a∈Zd0:ϑa=1u1x-Xa(recall Xa=a+ωa).Once the selection of the potential typesϑa∈0,1 is made, the only randomness in the potential sample V(x;ω) is the one coming from the displacements Xa(ω).The assumptions on smallness of the support of the probability measureρ0 (of the displacements ωa) had been made before in several works on the random displacements models (RDM) where it was actually crucial that the supports of the local potentials, even after admissible displacements of their centers within neighborhoods of the lattice sites a, do not overlap, for otherwise the subtle functional-analytic results used in earlier proofs would break down. It will be clear from our presentation that we can afford supp⁡ρ0⊂Br¯(0) with any fixed r¯<+∞, at the price of a more elaborate analysis and some additional assumptions on the site potentials (either a “hard core” condition or smallness of the positive potential u0, in order not to destroy the impurity wells). This follows from our strategy of proof of the crucial eigenvalue concentration (or Wegner-type) estimate which relies only on the randomness of ωa with a filling an affine sublattice parallel to Zd0 at some finite distance r2 from the latter. The reader will see that the displacements of all remaining random positions Xa=a+ωa can be (and often will be) rendered nonrandom by conditioning. However, this also makes more cumbersome some important geometrical arguments, and for this reason we postpone possible generalizations to a forthcoming work.Kirsch et al. [9] adapted the variable-energy (or energy-interval) MSA approach developed by Fröhlich et al. [10] (cf. also [11]), reformulated by von Dreifus and Klein [12], to the alloy models with a polynomial decay of the interaction potential u(r)=r-A, under the assumption A>4d. However, their techniques apply directly to the models featuring the so-called monotonicity condition: a variation of any random parameters (scatterer amplitudes, in their case) has to produce a variation of the potential with fixed sign (either positive or negative), while the displacements of the scatterers result in sign-indefinite variations of the induced potential.An important issue one has to face in the class of the random displacements models is the proof of the ILS estimate for an arbitrarily small disorder amplitude, without the strong disorder condition. The main point of the present paper is that this issue can be addressed in a relatively simple way for the surface models where a layer of the sample producing a spectral band, separated by a gap from the bulk spectrum, is subject both to the random displacements disorder and to the substitution disorder (due to a random selection of the lattice positions for the impurities).The latter kind of disorder is “discrete” and not generated by variation of continuous parameters. Such a situation usually renders the localization analysis very difficult; the first result in this area was obtained in a remarkable work by Bourgain and Kenig [13] in 2001, for the random alloys with Bernoulli distribution of random scatterers amplitudes. Germinet and Klein [14] have later extended the Bourgain–Kenig approach to arbitrary nontrivial distributions, but the technique here remains very complex and the localization bounds (for the eigenfunction correlators) remain quite modest. In our paper, we use a “discrete” substitution disorder solely for the proof of the Initial Length Scale estimate, which is already quite difficult to obtain otherwise. To the author’s best knowledge, there is no prior result for the weak-disorder localization at low energy in the random displacements model with slowly decaying interaction.Moreover, we show that the moderate rate of decay of the Green functions obtained at the first stage of the multiscale analysis can be significantly enhanced by a “soft” argument, so that despite a relatively slow, power-law decay of interaction (hence, a relatively strong long-distance correlation in the random environment) the decay of eigenfunction correlators is at least fractional-exponential.The materials with an impurity band supported by specific atoms injected into a sample play important role in experimental physics and technology. In our paper, we thus show that even a relatively weak disorder induced by structural factors (substitution and local displacements of the atoms/ions) can produce strong dynamical localization in presence of a realistic, slow decay of the media-electron interaction at large distances.


## 1.2. Local Hamiltonians and Useful Notation

Given a cubeB3l(u), with l≥1, we call Bl(u) thecore and the annulus BLk(u)∖Blk(u) theshell of B3l(u). Further, given a cube BYL(u) with L=3l, Y≥2, we can cover the shell of BYL(u) by (3Y)d cores of radius lk. The respective cores Blk(x) will be called admissible.We define the “boundary” of a cubeBLk(u) as follows: ∂BLk(u)≔BLk(u)∖BLk-2(u).The spectrum of an operatorH will be denoted by Σ(H) and its resolvent set by ϱ(H).


## 1.3. Main Results

Theorem 1.
Assume that the conditions (U1)–(U4) are fulfilled. Then there existp^>0 and an interval I^⊂R such that the following holds for the random displacements model where p≡Pϑa=1∈(0,p^).(A)
With probability one,H(ω) has pure point spectrum in the interval I^, and all the eigenfunctions ψ of H(ω) with eigenvalues E∈I^ decay exponentially at infinity: for some m>0 one has (6)ψxy;ω;ϑ≤Cψω,ϑe-my-x∀y∈Z.(B)
Letχx, x∈Rd, be the indicator function of a unit cube centered at x. There exist C,ϱ∈(0,+∞) such that for all x≠y and for any bounded Borel function ϕI^ with supp⁡ϕI^⊂I^ and ϕI^∞≤1 one has (7)EχxϕI^Hωχy≤Ce-x-yϱ.The smallness of the concentration of impuritiesp in the “surface” layer is used only in the proof of the Initial Length Scale estimate (cf. Lemma 19).


## 2. Eigenvalue Concentration and Comparison Estimates

### 2.1. Stollmann’s Estimate

Recall some well-known notions and results from the eigenvalue concentration analysis of random operators introduced and used in earlier works [4, 15–18].Definition 2.
Let a finite nonempty index setJ, a Euclidean space RJ≅R|J| with coordinates indexed by the elements of J, and a probability measure μ on R be given. Introduce the product measure μ⊗J on RJ. A function Φ:RJ→R is called diagonally monotone iff (i)
q=(q1,…,qJ)↦Φ(q) is monotone in each variable xj, 1≤j≤J,(ii)
for allq∈RJ and t≥0 one has (8)Φx+t,…,t-Φx≥Ct.From this point on, speaking of a diagonally monotone functionΦ, we always assume that the respective index J, Euclidean space RJ, and a constant C>0 are given. To indicate an explicit value of C, we sometimes say that Φ is diagonally C-monotone.Lemma 3 (Stollmann’s lemma).
LetΦ:RJ→R be a diagonally C-monotone function and μ⊗J a product measure on RJ. Let Sμ:0,+∞→0,1 be the continuity modulus of μ (otherwise called the Levy concentration function of μ): (9)Sμ:s⟼supa∈R⁡μa,a+s. Then the following estimate holds true: (10)supa∈R⁡Pq∈RJ:Φq∈a,a+s≤C-1JSμs.Definition 4.
Let a Hilbert spaceH, a Euclidean space RJ, J≥1, and a probability measure μ on R be given. A family of self-adjoint operators H(q) in H, q∈RJ, is called diagonally monotone iff (11)Hq+r≥Bq∀q∈RJ∀r∈R+J, in the sense of quadratic forms, and for any vector f∈H with f=1 the function Φf:RJ→R defined by (12)Φfq=Hqf,f is diagonally monotone.By a simple application of the min-max principle for the self-adjoint operators, we have the following useful result.Lemma 5 (Cf. [16]).
In the general setting of Definition4, the following holds true: (A)
IfH(q) is a diagonally monotone operator family in H with compact resolvent, then all eigenvalues EiH(q) of H(q) are also diagonally monotone.(B)
IfH(q) is diagonally monotone and K is another self-adjoint operator in H independent of q, then H(q)+K is also diagonally monotone.Corollary 6 (Cf. [17, 18]).
Consider a parametric family of Schrödinger operatorsHΛ(q)=-ΔΛ+V(x,q) in a cube Λ⊂Rd, d≥1, with a lower-bounded potential V:Λ×RJ→E∗,+∞ depending upon a vector-valued parameter q∈RJ, 1≤|J|<+∞. Let a probability measure μ on R with continuity modulus sμ be given, and introduce the product measure μ⊗J on RJ and the corresponding probability measure induced on the operator families V(·,q) and H(q). If the operator family V(·,q) is diagonally monotone, then for any interval [a,b]⊂R and any E∈[a,b] one has (13)Pdist⁡ΣHq,E≤ϵ≤CE∗,bJΛsμϵ.


### 2.2. Stable Eigenvalue Concentration Estimate

Definition 7.
Let an integerk≥0 and a cube B=BLk(u) be given. A sample ω is called(i)
(E,ϵ)-NR (nonresonant) in B iff (14)dist⁡ΣHBω,E≥ϵ;(ii)
(E,ϵ)-CNR (completely nonresonant) in B iff for all j<Lk+1/Lk(15)dist⁡ΣHBjLkuω,E≥ϵ.Until Section3 (where we formulate an Initial Length Scale estimate, Lemma 19), the positions of the sublattice sites a with ϑa=1 (impurities) will be considered fixed, so we do not rely on the substitution disorder due to the sample ϑa,a∈Zd0 and work with the displacements sample ωa,a∈Z, without repeating it every time again, and often drop the ϑ-dependence from notation.Lemma 8.
Fix anyk≥0 and a cube B=BLk(u)⊂Rd. Fix τ>1. Then for any ϵ≥Lk-(A-d)τ one has (16)Pωis  notE,ϵ-CNR  inB≤Cg-1Lkd+1ϵ.Definition 9.
Let a cubeB=BLk(u) and a real number τ>1 be given. Denote B¯=BLτ(u). The sample ωB¯ is called(i)
(E,ϵ)-SNR (strongly NR) in B iff for any complementary subsample ωB¯⊥ the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-NR in B,(ii)
(E,ϵ)-SCNR (strongly CNR) in B iff for any complementary subsample ωB¯⊥ the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-CNR in B.A suitable value ofτ>1 will be fixed later, in Section 4, and the notations B≡BLk(u), B¯≡BLτ(u), ωB¯, ωB¯⊥ will appear in a number of formulae and statements below. Usually the values of k, Lk∈N, u∈Z, and τ>1 will be clear from the context.In earlier works by Kirsch et al. [19] and by Boutet de Monvel and Stollmann [20], some functional-analytical facts pertaining to random alloy models in general and to surface models in particular have been established.Mutatis mutandis, some relevant definitions, arguments, and results from [19, 20] will be used below in the context of the surface model with random displacements disorder.In Proposition10, one refers to a certain subset W⊂Rd, which is formed in our case by the “wells” around the impurities, so V(x;ω,ϑ)<E(-) for x∈W (cf. (3)), and P is the exterior of the balls B1/2(a) around the impurities a (i.e., with ϑa=1). Given a cube B⊂Rd, denote by B~0 the section B∩Rd0 by the impurity layer, and consider the finite set of scatterer indices (17)AB=a′=a+ed0+1,witha∈B~0. With this choice of AB, for each a∈B~0 there exists a site a′ outside the subset W but at a uniformly bounded distance from a∈B~0. Then we can write HB=H~B+V~, where H~B=-Δ+U~ with (18)U~x=∑a∉ABuϑax-Xa,V~x=∑a∈ABuϑax-Xa.Proposition 10 (cf. [20, Lemma 2.1]).
LetB⊂Rd be an open cube, H be a Schrödinger operator in B with Dirichlet boundary conditions on the border of B, and P⊂W be such that dist⁡(P,Wc)≔κ>0. Then there exists C=C(κ) such that the following holds.
LetV0 and V be uniformly locally summable in power p (in Lp) with p=2 if d≤3 and p>d/2 if d>3. Denote H0=-Δ+V0, G0(λ)=(H0-λ)-1, and H=-Δ+V, G(λ)=(H-λ)-1. Assume that Φ is an eigenfunction of H with eigenvalue μ∈ϱ(H0). Then(19)χWΦ≥Φ1+CG0λ+G0λ∇.Further, combining Proposition11 and the Combes–Thomas estimate [21] for the under-the-barrier decay of the Green functions, one comes to the following.Proposition 11 (cf. [19, Prop. 2.1], [20, Prop. 2.2]).
Under the assumptions and with notations of Proposition10, there exist E1>E0 and C>0 such that for every cube B=BL(x), every ω∈Ω, and every eigenfunction Φ of HB(ω) with eigenvalue E∈[E0,E1] we get(20)χWΦ≥CΦ.Proposition 12 (cf. [20, Prop. 2.3]).
(21) P Σ H B ∩ J ϵ ≠ ⌀ ≤ C B 2 J ϵ .Proof.
LetEn=En(HB) be the eigenvalues of HB numerated in the increasing order, counting multiplicity. We are mainly concerned with En∈I^ (in the impurity band), which has two important consequences.
(i) Firstly, due to boundedness of the random potential, the operatorHB(ω) is a finite norm perturbation of the Laplacian -ΔB, and for the eigenvalues of the latter one has the well-known Weyl asymptotic formula [22]; see, for example, the discussion in [4, Section 4.1.8] in the context of the Wegner estimates. Specifically, it follows from [4, Theorem 4.1.25 and Corollary 4.1.26] that the number of eigenvalues of HB in I^ admits uniform, nonrandom upper bounds by OB.
(ii) Secondly, inequality (20) shows that for such En most of the “mass” of the respective eigenfunction Φn is in the subset W. Therefore, En is monotone increasing in each parameter qa, since increasing qa decreases the distance between the ath scatterer and any point x∈W (in the “surface” layer), hence increasing the potential induced in W. This is the main reason why we consider in this paper a surface random displacements model as opposed to a “bulk” RDM. From this point on, we recover a complete analogy with the conventional surface alloy disorder studied in [20] (and not displacements disorder). As a result, only a few technical adaptations of the well-understood techniques from [19, 20] are required; we describe these adaptations below.
Specifically, notice thatEn(q+te) is continuously differentiable, with the derivative which can be calculated with the help of the first-order perturbation formula: (22)ddtt=sEnH~B+tV~=V~Φs,Φs≥C>0 (we used again (20)). We conclude that V~ (and therefore H~B+tV~, too) is diagonally monotone. Now the claim follows from the Stollmann Lemma 3.Theorem 13.
(23) P ω B ¯ is  not E , ϵ -CNR in B ≤ C g - 1 L 2 d + 1 ϵ .Proof.
The(E,ϵ)-CNR property is violated in a cube B=BLk+1(u) if one of the smaller concentric cubes BL(u) with L∈Lk,Lk+1 is not (E,ϵ)-NR, so the claim follows from Proposition 12 by counting the number of elements L∈Lk,Lk+1.Theorem 14.
(24) P ω B ¯ is  not E , ϵ -SCNR in B ≤ C g - 1 L 2 d + 1 ϵ .Proof.
Letω=(ωB¯′,ωB¯′⊥). According to Definition 9, if ωB¯ is not (E,ϵ)-SCNR, then for some integer L∈Lk,Lk+1ωB¯ is not (E,ϵ)-SNR in B′≔BL(u). In other words, there exists a complementary subsample ωB¯⊥ such that the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-resonant in B′. Fix any L∈Lk,Lk+1 and the corresponding cube B′.
Next, consider any configurationωB¯⊥ and take its zero-extension (0B¯,ωB¯⊥) to the entire space; in other words, we consider ωB¯⊥ as a sample on its own but do not put any scatterers inside B¯. We obtain by a simple calculation the following:(25)1B′gV0,ωB¯′⊥≤ConstgL-A-dτ.
Further, take any subsampleωB¯, and identify it with its own zero-extension (ωB¯,0B¯c) to the entire space, and let λi(ωB¯) be the eigenvalues of HB(ωB¯). (One could take any complementary subsampleωB¯⊥;we take the zero-extension.) Then the eigenvalues Ei(ω) of HB(ω), with any ω=ωB¯⊔ωΛ⊥, obey (26)Eiω-λiωB¯≤gCL-A-dτ≤ϵ. Thus (27)PωB¯:∃ωB¯⊥Eiω-E≤ϵ≤PωB¯:λiωB¯-E≤2ϵ. Now the claim stems from Theorem 13.


### 2.3. Eigenvalue Comparison Estimate

Theorem 15.
Let two sublattice sitesa,b∈Zd0 with |a-b|>2Lτ be given, and denote Ba=BL(a), Bb=BL(b). Then for any ϵ>0(28)Pdist⁡ΣHBaω,ΣHBbω≤ϵ≤CLA+1+2dϵ.The estimate given by Theorem15 is not as accurate as its counterpart for a single cube (Theorem 13), and the proof is much less involved, but it is worth noticing that its role in the localization analysis is quite different, too. Optimality of the eigenvalueconcentration (EVC) bound from Theorem 13 is the key to the proof of the exponential decay of the Green functions at fixed energy under thesharp condition A>d. Physically speaking, this gives rise to the absence of diffusion (see the discussion in the seminal paper by Fröhlich and Spencer [11]). On the mathematical level, Martinelli and Scoppola [23] showed, with the help of a nice application of Chebyshev’s inequality in the extended disorder-energy space, that a fast decay of Green’s functions implies absence of the absolutely continuous spectrum in the localization zone. However, the analysis of decay of the eigenfunctions and of their correlators requires a spectral reduction, or derivation of energy-interval (a.k.a. variable energy) MSA estimates, and the latter requires eigenvaluecomparison analysis for distant cubes. For that purpose alone, an eigenvalue comparison estimate need not actually be very accurate, which is not surprising, but what comes as a rather unexpected bonus is that the comparison estimate allows one to significantly boost the results of the first scaling analysis in a “soft” way, as we shall see in Section 5.Theorem15 will not be used in Sections 3 and 4.Proof of Theorem15.
DenoteBa=BL(a) and Bb=BL(b). Consider an eigenvalue Eia(ω) of HBa(ω) and Ejb(ω) of HBb(ω). Condition on all random amplitudes ωc,c∈Zd∖ωa and identify ωa with a real parameter s. Then HBa as well as HBb and their eigenvalues Eia, Ejb depend upon a single parameter s≡ωa. Our goal is to compare their sensitivity to the variations of s. To this end, consider the smooth operator family HBa(s)=H~Ba+sUa(s), where (29)H~Ba=-h2Δ+∑c≠aωcu·-c+sUas,Uax,s=ux-a1Bax. Similarly define (30)HBbs=H~Bb+sUbs,Ubx,s=ux-a1Bbx. Then (31)maxz∈Bb⁡Ubz≤M~L-A<CdL-A≤minz∈Ba⁡Uaz. Therefore, (32)Ua1Ba≥CdL-A1Ba,Ub1Ba≤M~L-A1Bb.Furthermore, computing the gradient of the potential x↦x-A, for M~ large enough we get that (33)ddsEi′s-Ej′′s≥M~-A-1-Cd-A-1L-A-1≥M~-A-12L-A-1>0. The probability measure of the random variable Eia(ωa)-Ejb(ωa) is therefore the image of μV (the measure of ωa) by the strictly monotone mapping s↦(Eia(s)-Ejb(s)), with the derivative lower-bounded by CL-A-1>0, so the Lipschitz continuity of μV is preserved up to a factor C-1LA+1. Now the claim follows by counting the number of eigenvalues in I, as in Proposition 12.


## 2.1. Stollmann’s Estimate

Recall some well-known notions and results from the eigenvalue concentration analysis of random operators introduced and used in earlier works [4, 15–18].Definition 2.
Let a finite nonempty index setJ, a Euclidean space RJ≅R|J| with coordinates indexed by the elements of J, and a probability measure μ on R be given. Introduce the product measure μ⊗J on RJ. A function Φ:RJ→R is called diagonally monotone iff (i)
q=(q1,…,qJ)↦Φ(q) is monotone in each variable xj, 1≤j≤J,(ii)
for allq∈RJ and t≥0 one has (8)Φx+t,…,t-Φx≥Ct.From this point on, speaking of a diagonally monotone functionΦ, we always assume that the respective index J, Euclidean space RJ, and a constant C>0 are given. To indicate an explicit value of C, we sometimes say that Φ is diagonally C-monotone.Lemma 3 (Stollmann’s lemma).
LetΦ:RJ→R be a diagonally C-monotone function and μ⊗J a product measure on RJ. Let Sμ:0,+∞→0,1 be the continuity modulus of μ (otherwise called the Levy concentration function of μ): (9)Sμ:s⟼supa∈R⁡μa,a+s. Then the following estimate holds true: (10)supa∈R⁡Pq∈RJ:Φq∈a,a+s≤C-1JSμs.Definition 4.
Let a Hilbert spaceH, a Euclidean space RJ, J≥1, and a probability measure μ on R be given. A family of self-adjoint operators H(q) in H, q∈RJ, is called diagonally monotone iff (11)Hq+r≥Bq∀q∈RJ∀r∈R+J, in the sense of quadratic forms, and for any vector f∈H with f=1 the function Φf:RJ→R defined by (12)Φfq=Hqf,f is diagonally monotone.By a simple application of the min-max principle for the self-adjoint operators, we have the following useful result.Lemma 5 (Cf. [16]).
In the general setting of Definition4, the following holds true: (A)
IfH(q) is a diagonally monotone operator family in H with compact resolvent, then all eigenvalues EiH(q) of H(q) are also diagonally monotone.(B)
IfH(q) is diagonally monotone and K is another self-adjoint operator in H independent of q, then H(q)+K is also diagonally monotone.Corollary 6 (Cf. [17, 18]).
Consider a parametric family of Schrödinger operatorsHΛ(q)=-ΔΛ+V(x,q) in a cube Λ⊂Rd, d≥1, with a lower-bounded potential V:Λ×RJ→E∗,+∞ depending upon a vector-valued parameter q∈RJ, 1≤|J|<+∞. Let a probability measure μ on R with continuity modulus sμ be given, and introduce the product measure μ⊗J on RJ and the corresponding probability measure induced on the operator families V(·,q) and H(q). If the operator family V(·,q) is diagonally monotone, then for any interval [a,b]⊂R and any E∈[a,b] one has (13)Pdist⁡ΣHq,E≤ϵ≤CE∗,bJΛsμϵ.


## 2.2. Stable Eigenvalue Concentration Estimate

Definition 7.
Let an integerk≥0 and a cube B=BLk(u) be given. A sample ω is called(i)
(E,ϵ)-NR (nonresonant) in B iff (14)dist⁡ΣHBω,E≥ϵ;(ii)
(E,ϵ)-CNR (completely nonresonant) in B iff for all j<Lk+1/Lk(15)dist⁡ΣHBjLkuω,E≥ϵ.Until Section3 (where we formulate an Initial Length Scale estimate, Lemma 19), the positions of the sublattice sites a with ϑa=1 (impurities) will be considered fixed, so we do not rely on the substitution disorder due to the sample ϑa,a∈Zd0 and work with the displacements sample ωa,a∈Z, without repeating it every time again, and often drop the ϑ-dependence from notation.Lemma 8.
Fix anyk≥0 and a cube B=BLk(u)⊂Rd. Fix τ>1. Then for any ϵ≥Lk-(A-d)τ one has (16)Pωis  notE,ϵ-CNR  inB≤Cg-1Lkd+1ϵ.Definition 9.
Let a cubeB=BLk(u) and a real number τ>1 be given. Denote B¯=BLτ(u). The sample ωB¯ is called(i)
(E,ϵ)-SNR (strongly NR) in B iff for any complementary subsample ωB¯⊥ the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-NR in B,(ii)
(E,ϵ)-SCNR (strongly CNR) in B iff for any complementary subsample ωB¯⊥ the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-CNR in B.A suitable value ofτ>1 will be fixed later, in Section 4, and the notations B≡BLk(u), B¯≡BLτ(u), ωB¯, ωB¯⊥ will appear in a number of formulae and statements below. Usually the values of k, Lk∈N, u∈Z, and τ>1 will be clear from the context.In earlier works by Kirsch et al. [19] and by Boutet de Monvel and Stollmann [20], some functional-analytical facts pertaining to random alloy models in general and to surface models in particular have been established.Mutatis mutandis, some relevant definitions, arguments, and results from [19, 20] will be used below in the context of the surface model with random displacements disorder.In Proposition10, one refers to a certain subset W⊂Rd, which is formed in our case by the “wells” around the impurities, so V(x;ω,ϑ)<E(-) for x∈W (cf. (3)), and P is the exterior of the balls B1/2(a) around the impurities a (i.e., with ϑa=1). Given a cube B⊂Rd, denote by B~0 the section B∩Rd0 by the impurity layer, and consider the finite set of scatterer indices (17)AB=a′=a+ed0+1,witha∈B~0. With this choice of AB, for each a∈B~0 there exists a site a′ outside the subset W but at a uniformly bounded distance from a∈B~0. Then we can write HB=H~B+V~, where H~B=-Δ+U~ with (18)U~x=∑a∉ABuϑax-Xa,V~x=∑a∈ABuϑax-Xa.Proposition 10 (cf. [20, Lemma 2.1]).
LetB⊂Rd be an open cube, H be a Schrödinger operator in B with Dirichlet boundary conditions on the border of B, and P⊂W be such that dist⁡(P,Wc)≔κ>0. Then there exists C=C(κ) such that the following holds.
LetV0 and V be uniformly locally summable in power p (in Lp) with p=2 if d≤3 and p>d/2 if d>3. Denote H0=-Δ+V0, G0(λ)=(H0-λ)-1, and H=-Δ+V, G(λ)=(H-λ)-1. Assume that Φ is an eigenfunction of H with eigenvalue μ∈ϱ(H0). Then(19)χWΦ≥Φ1+CG0λ+G0λ∇.Further, combining Proposition11 and the Combes–Thomas estimate [21] for the under-the-barrier decay of the Green functions, one comes to the following.Proposition 11 (cf. [19, Prop. 2.1], [20, Prop. 2.2]).
Under the assumptions and with notations of Proposition10, there exist E1>E0 and C>0 such that for every cube B=BL(x), every ω∈Ω, and every eigenfunction Φ of HB(ω) with eigenvalue E∈[E0,E1] we get(20)χWΦ≥CΦ.Proposition 12 (cf. [20, Prop. 2.3]).
(21) P Σ H B ∩ J ϵ ≠ ⌀ ≤ C B 2 J ϵ .Proof.
LetEn=En(HB) be the eigenvalues of HB numerated in the increasing order, counting multiplicity. We are mainly concerned with En∈I^ (in the impurity band), which has two important consequences.
(i) Firstly, due to boundedness of the random potential, the operatorHB(ω) is a finite norm perturbation of the Laplacian -ΔB, and for the eigenvalues of the latter one has the well-known Weyl asymptotic formula [22]; see, for example, the discussion in [4, Section 4.1.8] in the context of the Wegner estimates. Specifically, it follows from [4, Theorem 4.1.25 and Corollary 4.1.26] that the number of eigenvalues of HB in I^ admits uniform, nonrandom upper bounds by OB.
(ii) Secondly, inequality (20) shows that for such En most of the “mass” of the respective eigenfunction Φn is in the subset W. Therefore, En is monotone increasing in each parameter qa, since increasing qa decreases the distance between the ath scatterer and any point x∈W (in the “surface” layer), hence increasing the potential induced in W. This is the main reason why we consider in this paper a surface random displacements model as opposed to a “bulk” RDM. From this point on, we recover a complete analogy with the conventional surface alloy disorder studied in [20] (and not displacements disorder). As a result, only a few technical adaptations of the well-understood techniques from [19, 20] are required; we describe these adaptations below.
Specifically, notice thatEn(q+te) is continuously differentiable, with the derivative which can be calculated with the help of the first-order perturbation formula: (22)ddtt=sEnH~B+tV~=V~Φs,Φs≥C>0 (we used again (20)). We conclude that V~ (and therefore H~B+tV~, too) is diagonally monotone. Now the claim follows from the Stollmann Lemma 3.Theorem 13.
(23) P ω B ¯ is  not E , ϵ -CNR in B ≤ C g - 1 L 2 d + 1 ϵ .Proof.
The(E,ϵ)-CNR property is violated in a cube B=BLk+1(u) if one of the smaller concentric cubes BL(u) with L∈Lk,Lk+1 is not (E,ϵ)-NR, so the claim follows from Proposition 12 by counting the number of elements L∈Lk,Lk+1.Theorem 14.
(24) P ω B ¯ is  not E , ϵ -SCNR in B ≤ C g - 1 L 2 d + 1 ϵ .Proof.
Letω=(ωB¯′,ωB¯′⊥). According to Definition 9, if ωB¯ is not (E,ϵ)-SCNR, then for some integer L∈Lk,Lk+1ωB¯ is not (E,ϵ)-SNR in B′≔BL(u). In other words, there exists a complementary subsample ωB¯⊥ such that the full sample ω=(ωB¯,ωB¯⊥) is (E,ϵ)-resonant in B′. Fix any L∈Lk,Lk+1 and the corresponding cube B′.
Next, consider any configurationωB¯⊥ and take its zero-extension (0B¯,ωB¯⊥) to the entire space; in other words, we consider ωB¯⊥ as a sample on its own but do not put any scatterers inside B¯. We obtain by a simple calculation the following:(25)1B′gV0,ωB¯′⊥≤ConstgL-A-dτ.
Further, take any subsampleωB¯, and identify it with its own zero-extension (ωB¯,0B¯c) to the entire space, and let λi(ωB¯) be the eigenvalues of HB(ωB¯). (One could take any complementary subsampleωB¯⊥;we take the zero-extension.) Then the eigenvalues Ei(ω) of HB(ω), with any ω=ωB¯⊔ωΛ⊥, obey (26)Eiω-λiωB¯≤gCL-A-dτ≤ϵ. Thus (27)PωB¯:∃ωB¯⊥Eiω-E≤ϵ≤PωB¯:λiωB¯-E≤2ϵ. Now the claim stems from Theorem 13.


## 2.3. Eigenvalue Comparison Estimate

Theorem 15.
Let two sublattice sitesa,b∈Zd0 with |a-b|>2Lτ be given, and denote Ba=BL(a), Bb=BL(b). Then for any ϵ>0(28)Pdist⁡ΣHBaω,ΣHBbω≤ϵ≤CLA+1+2dϵ.The estimate given by Theorem15 is not as accurate as its counterpart for a single cube (Theorem 13), and the proof is much less involved, but it is worth noticing that its role in the localization analysis is quite different, too. Optimality of the eigenvalueconcentration (EVC) bound from Theorem 13 is the key to the proof of the exponential decay of the Green functions at fixed energy under thesharp condition A>d. Physically speaking, this gives rise to the absence of diffusion (see the discussion in the seminal paper by Fröhlich and Spencer [11]). On the mathematical level, Martinelli and Scoppola [23] showed, with the help of a nice application of Chebyshev’s inequality in the extended disorder-energy space, that a fast decay of Green’s functions implies absence of the absolutely continuous spectrum in the localization zone. However, the analysis of decay of the eigenfunctions and of their correlators requires a spectral reduction, or derivation of energy-interval (a.k.a. variable energy) MSA estimates, and the latter requires eigenvaluecomparison analysis for distant cubes. For that purpose alone, an eigenvalue comparison estimate need not actually be very accurate, which is not surprising, but what comes as a rather unexpected bonus is that the comparison estimate allows one to significantly boost the results of the first scaling analysis in a “soft” way, as we shall see in Section 5.Theorem15 will not be used in Sections 3 and 4.Proof of Theorem15.
DenoteBa=BL(a) and Bb=BL(b). Consider an eigenvalue Eia(ω) of HBa(ω) and Ejb(ω) of HBb(ω). Condition on all random amplitudes ωc,c∈Zd∖ωa and identify ωa with a real parameter s. Then HBa as well as HBb and their eigenvalues Eia, Ejb depend upon a single parameter s≡ωa. Our goal is to compare their sensitivity to the variations of s. To this end, consider the smooth operator family HBa(s)=H~Ba+sUa(s), where (29)H~Ba=-h2Δ+∑c≠aωcu·-c+sUas,Uax,s=ux-a1Bax. Similarly define (30)HBbs=H~Bb+sUbs,Ubx,s=ux-a1Bbx. Then (31)maxz∈Bb⁡Ubz≤M~L-A<CdL-A≤minz∈Ba⁡Uaz. Therefore, (32)Ua1Ba≥CdL-A1Ba,Ub1Ba≤M~L-A1Bb.Furthermore, computing the gradient of the potential x↦x-A, for M~ large enough we get that (33)ddsEi′s-Ej′′s≥M~-A-1-Cd-A-1L-A-1≥M~-A-12L-A-1>0. The probability measure of the random variable Eia(ωa)-Ejb(ωa) is therefore the image of μV (the measure of ωa) by the strictly monotone mapping s↦(Eia(s)-Ejb(s)), with the derivative lower-bounded by CL-A-1>0, so the Lipschitz continuity of μV is preserved up to a factor C-1LA+1. Now the claim follows by counting the number of eigenvalues in I, as in Proposition 12.


## 3. Impurity Band: Location and Spectral Gap

An upper bound on the ground state energyE∗ can be obtained by the variational principle.


## 4. Multiscale Analysis

Givenα>1 and 1<L0∈N, define recursively an integer sequence Lk(34)Lk≔Lk-1α,k≥1.Further, let m0>0, and set for k≥0(35)mk≔m0∏j=0k1-ηj,ηk≔2Lk-cα,c>0,with c>0 to be specified in the proof of Corollary 21. Clearly, with L0 large enough we have a convergent product ∏j≥0(1-ηj)≥1/2, so mk≥m0/2.GivenA>2d and τ>1, we define two positive sequences(36)δk≔e-mkLk,ϵk≔Lk-A-dτ.Definition 16.
Let an integerk≥0 and a cube B=BLk(u) be given. Denote ∂BLk(u)≔BLk(u)∖BLk-2(u). A sample ω is called(1)
(E,δ)-NS (nonsingular) in B iff E∉Σ(HB(ω)) and(37)1∂BLkGBLkE,ω1BLk/3≤3Lk-dδ.Definition 17.
Letτ>1 and a cube B=BLk(u), k≥0 be given. Denote B¯=B1/2Lkτ(u) and introduce the decomposition ω=(ωB¯,ωB¯⊥). Let P be one of the properties (E,δ)-NS, (E,ϵ)-NR, or (E,ϵ)-CNR relative to B. We will say that a subsample ωB¯ has a strong (or stable) property P iff for any complementary subsample ωB¯⊥ the full sample ω=(ωB¯,ωB¯⊥) has property P in B. Respectively, the three aforementioned notations are replaced by (E,δ)-SNS, (E,ϵ)-SNR, and (E,ϵ)-SCNR.Note that an eventωB¯has strong propertyP is measurable with respect to the σ-algebra FB¯ generated by {Xa,a∈B¯}; hence any collection of such events relative to disjoint cubes BL(ui),i∈1,M is independent.Definition 18.
(i) A cubeBLk+1(u) is called (E,δk,τ,K)-good iff it contains no collection of K or more cubes {BLk(xi),1≤i≤Sk+1} with pairwise Lkτ-distant centers, neither of which is (E,δk)-NS.
(ii) The cubeBLk+1(u) is called (E,δk,τ,K)-strongly good ((E,δk,τ,K)-S-good) iff it contains no collection of K cubes {BLk(xi),1≤i≤K}, with pairwise Lkτ-distant centers, neither of which is (E,δk)-SNS.Lemma 19 (ILS estimate).
Letτ>1. There exist L∗∈N and an interval I=[-3E,-2E] with E>0, such that for any L0≥L∗, m0>0, s>0 and the impurity concentration ρ>0 small enough, one has, with δ0=em0L0,(38)supE∈I⁡PBxis  notE,δ0,τ-SNS≤L0-s.Proof.
FixL0∈N; consider a cube B=BL0(u) and the local Hamiltonian HB(ω). Observe that the condition(39)θa=0∀a∈BL0+r0 (absence of impurity atoms in the augmented cube BL0+r0(u)) implies that (40)infx∈B⁡Vx,ω,ϑ≥-E; thus (41)dist⁡E,ΣHBω,ϑ,E≥E∀E∈-3E,-2E. By the Combes–Thomas estimate [21], we then have for any E∈I≡[-3E,-2E](42)1∂BGBE1B1/3Lku≤CE-1e-cELk, for some C,c∈(0,+∞). With an appropriate choice of L∗, this implies the (E,δ0)-nonsingularity of BL0(u), where δ0=e-m0L0 and, for example, m0=1/2cE>0.
Further,(43)P∃a∈BL0+r0u:θa≠0≤BL0+r0uρ, and the RHS can be made arbitrarily small by taking the concentration of impurities ρ small enough. In particular, it can be made smaller than L0-s, once L0 is fixed.The next statement is a standard result of the MSA, essentially going back to [12, Lemma 4.2] and streamlined in [24, Section 5] and later in [25, Lemma 2].Lemma 20 (conditions for nonsingularity).
Consider a cubeB=BLk+1(u) and suppose that(i)
B is (E,ϵk+1)-NR with ϵk+1≥δk1-c, for some ϵk+1,δk,c∈(0,1);(ii)
B is (E,δk,τ,K)-good, with K≥0 such that(44)N≔Lk+1Lk-10KLkτ≥1.Then B is (δkN+c,E)-NS.In our caseδk=e-mkLk and ϵk+1=e-Oln⁡Lk+1; hence the condition ϵk+1≥δk1-c in hypothesis (ii) is fulfilled for any c∈(0,1) and large L0.Corollary 21 (conditions for strong nonsingularity).
Let a cubeB=BLk+1(u), k≥0, be given and suppose that(i)
B is (E,ϵk+1)-SNR;(ii)
B is (E,δk,τ,K)-S-good, with K≥0 such that (44) holds.
ThenB is (E,δk+1)-SNS.Proof.
DenoteB¯=BLk+1τ(u). One has to show that, with a fixed sample ωB¯⊥ satisfying the hypotheses (i)-(ii), the cube B is (E,δk+1)-NS for the sample (ωB¯,ωB¯⊥) regardless of the complementary sample ωB¯⊥.
First, notice that the condition (i) is already stable with respect toωB¯⊥.
Next, by (ii) there exist at mostK-1 cubes B1/2Lkτ(xi) such that any ball BLk(x) with x∉⋃i=1KBLkτ(xi) is (E,δk)-SNS. The support of ωB¯⊥ lies outside all the cubes B1/2Lkτ(xi); hence the distant sample ωB¯⊥ cannot affect the strong nonsingularity property of the cubes B1/2Lkτ(xi). Applying Lemma 20, we see that the cube B is (E,δ~k+1)-NS with(45)-ln⁡δ~k+1=Nk+1mkLk-C′ln⁡Lk+1+ln⁡3Lk+1d-ln⁡3Lk+1d≥LkYk+1mk1-10KLkτLkα-C′′ln⁡Lk+1mkLk+1+ln⁡3Lk+1d≥Lk+1mk1-ηk+ln⁡3Lk+1d,where ηk is as in (35), with c=α-τ>0. Thus B is (E,δk+1)-SNS.Until the end of this section, we will need to examine only the nonsingularity properties of the cubes having nonempty intersection with the sublatticeZd0 (the impurity layer), since we are concerned only with energies E∈I^, and any cube outside this layer has energies above I^. Moreover, since we are free to choose on any scale the partition of Rd into a union of cubes of size Lk, we cover first Zd0 by cubes BLk(a) with a∈Zd0 and then decompose the rest of the space into Lk-cubes having empty intersection with Zd0. We will not repeat this fact every time again.Lemma 22.
Let two real numbersA>d and τ>1 be given. Consider a cube BLk+1(x) and let ϵk+1=Lk+1-(A-d)τ. Then (46)PωB¯:∃ωB¯⊥BLk+1xis  notE,ϵk+1-SCNR≤12Lk+1-A-dτ+2d+1.Proof.
By Definition17, if BLk+1(z), with z∈{x,y}, is not (E,ϵk+1)-SCNR for some fixed E, then for some R∈Lk,Lk+1 the cube BR(x) is not (E,ϵk+1)-SNR. Even the largest among them, BLk+1(z), is surrounded by a belt of width Lk+1τ. By Theorem 14, (47)PωB¯:∃ωB¯⊥dist⁡ΣHBRz,E≤2ϵk+1≲L2d+1ϵk+1,so the claim follows by counting the number of values R∈Lk,Lk+1.Lemma 23.
LetA=3d+4γ with γ>0 and τ>2d+1/2γ be given. Set(48)ϱ=γd+4γ,σ=γd-1,α=1+στ,s=A-dτ-2d+1; then(49)s-αd>γτ>0.Furthermore, assume that(50)PBLkxis  notE,δk-SNS≲Lk-sand let K=2M(1+σ)s/γ≤Lkα-τ. Then for L0 large enough(51)PBLk+1uis  notE,δk+1,K-S-good≤14Lk+1-Ms.Proof.
Equation (49) follows by a simple calculation:(52)s-αd=A-dτ-2d+1-1+στd=τd+3γ-1+σd-2d+1=τ3γ-σd-2d+1≥τ3γ-γ-2d+1>γτ.Further, by Definition 17, an event (53)Bx=BLkxisE,δk,K-S-bad or notE,ϵk+1-SNR is F(B1/2Lkτ(x))-measurable, and so for K disjoint cubes B1/2Lkτ(xi), 1≤i≤K, we obtain(54)P⋂i=1KBxi=∏i=1KPBxi≤pkK.By (52), we have s-αd/α=γτ/(1+σ)τ=γ/1+σ. Thus with K=2Ms(1+σ)γ-1, one has for the maximal number S(ω) of pairwise Lkτ-distant singular cubes B1/2Lkτ(xi) inside BLk+1(u): (55)PSω≥K≤CLk+1KdpkK≤CLk+1-Ks/α-d<14Lk+1-Ms.This proves inequality (51).For future use, note that one can take, for example,(56)M=γ41+σsLkα-τ>Lkστ/2.The last inequality holds for all k with L0 large enough.Lemma 24 (scale induction).
Assume that the bound(57)P∃E∈I^∃z∈x,yBLjzis  notE,δj-SNS≤Lj-sholds for j=k≥0. Then it also holds for j=k+1.Proof.
By Corollary21, if the cube BLk+1(z) is not (E,δk+1)-SNS, then either (i)
BLk+1(z) is not (E,ϵk+1)-SNR, or(ii)
BLk+1(z) is not (E,δk,K)-S-good.
By Lemmas22 and 23, the probabilities of both events are upper-bounded by 1/2Lk+1-s, so the claim follows.Since the starting point for the scale induction (the Initial Length Scale estimate) is provided by Lemma19, by induction on k we come to the conclusion of the multiscale analysis of our model.Theorem 25.
Consider the random HamiltonianH(ω)=-Δ+V(x,ω). Fix an interval I⊂R. For any A>2d there exist s∗=s∗(A)<+∞ and L∗∈N such that if L0≥L∗ and the inequality(58)supE∈I⁡PBLkxis  notE,δk-SNS≤Lk-sholds for k=0 with s≥s∗, then it also holds for all k≥1.


## 5. Derivation of Strong Dynamical Localization

### 5.1. Enhancement of the MSA Estimates

Theorem 26.
Under the assumptions and with notations of Theorem25, for any two cubes BLk(x), BLk(y) with |x-y|≥Lkτ one has for some ϱ>0(59)P∃E≤Lk-Lkϱ.Proof.
It suffices to modify the proof of Lemma23 (and of the Lemma 24 stemming from it) in the following way. Instead of the condition that BLk(z) be (E,ϵk)-strongly nonresonant we employ the condition of (E,ϵk)-nonresonance. As a result, the choice of ϵk+1 is no longer restricted to ϵk+1≥Lk-(A-d)τ: we can take an arbitrarily small value of ϵk+1. Indeed, the strong nonresonance is required for the scale induction, but now we enhance the MSA estimates scale by scale, individually for each Lk, without resorting to another scale induction.
Consequently, we can also replace a fixed, scale-independent parameterK (figuring in the condition “BLk+1(z) is (E,δk+1,K)-good” by a growing value). Setting K=Kk+1=1/2Lk1-τ/α=1/2Lk1-σ/α, the claim follows.


### 5.2. Strong Dynamical Localization

It is well-known by now that the energy-interval MSA estimates imply strong dynamical localization; the shortest derivation is due to Germinet and Klein [24] who operated with the eigenfunction correlators in the entire space. Their argument becomes particularly simple in the situation where one proves first the decay bound for the eigenfunction correlators in finite cubes, where it can be encapsulated in a fairly elementary functional-analytical lemma, as was shown in our prior papers, for example, [25, 26].Proposition 27 (cf. [26, Thm. 7], [25, Thm. 3]).
Suppose that a bound of the form(60)P∃E∈I^∃z∈x,y:BLkzis  notE,δk-NS≤fLk is established for some function f≥0 and all k≥0. Then for |x-y|=:R∈[3Lk,3Lk+1], one has (61)EχxϕIHωχy≤C1RdfRα-1+C2e-mR.Taking into account the MSA estimates from Theorem26, assertion (B) of Theorem 1 follows directly from Proposition 27.


## 5.1. Enhancement of the MSA Estimates

Theorem 26.
Under the assumptions and with notations of Theorem25, for any two cubes BLk(x), BLk(y) with |x-y|≥Lkτ one has for some ϱ>0(59)P∃E≤Lk-Lkϱ.Proof.
It suffices to modify the proof of Lemma23 (and of the Lemma 24 stemming from it) in the following way. Instead of the condition that BLk(z) be (E,ϵk)-strongly nonresonant we employ the condition of (E,ϵk)-nonresonance. As a result, the choice of ϵk+1 is no longer restricted to ϵk+1≥Lk-(A-d)τ: we can take an arbitrarily small value of ϵk+1. Indeed, the strong nonresonance is required for the scale induction, but now we enhance the MSA estimates scale by scale, individually for each Lk, without resorting to another scale induction.
Consequently, we can also replace a fixed, scale-independent parameterK (figuring in the condition “BLk+1(z) is (E,δk+1,K)-good” by a growing value). Setting K=Kk+1=1/2Lk1-τ/α=1/2Lk1-σ/α, the claim follows.


## 5.2. Strong Dynamical Localization

It is well-known by now that the energy-interval MSA estimates imply strong dynamical localization; the shortest derivation is due to Germinet and Klein [24] who operated with the eigenfunction correlators in the entire space. Their argument becomes particularly simple in the situation where one proves first the decay bound for the eigenfunction correlators in finite cubes, where it can be encapsulated in a fairly elementary functional-analytical lemma, as was shown in our prior papers, for example, [25, 26].Proposition 27 (cf. [26, Thm. 7], [25, Thm. 3]).
Suppose that a bound of the form(60)P∃E∈I^∃z∈x,y:BLkzis  notE,δk-NS≤fLk is established for some function f≥0 and all k≥0. Then for |x-y|=:R∈[3Lk,3Lk+1], one has (61)EχxϕIHωχy≤C1RdfRα-1+C2e-mR.Taking into account the MSA estimates from Theorem26, assertion (B) of Theorem 1 follows directly from Proposition 27.



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