Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

Ekanga, Trésor

doi:https://doi.org/10.1155/2020/5270541

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 5270541 | https://doi.org/10.1155/2020/5270541

Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

Trésor Ekanga¹

Academic Editor: Laurent Raymond

Received28 Apr 2019

Revised21 Aug 2019

Accepted06 Sept 2019

Published05 May 2020

Abstract

We study the multiparticle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multiparticle lower spectral edges are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multiparticle Anderson model at low energy. The proof based on the multiparticle multiscale analysis bounds needs the values of the external random potential to be independent and identically distributed, whose common probability distribution is at least Log-Hölder continuous.

1. Introduction

This paper follows our previous works [1, 2] on localization for multiparticle random lattice Schrödinger operators at low energy. Some other papers [3–10] analyzed multiparticle models in the regime including the strong disorder or the low energy and for different type of models such as the alloy-type Anderson model or the multiparticle Anderson model in quantum graphs [11].

In their work [10], Klein and Nguyen developed the continuum multiparticle bootstrap multiscale analysis of the Anderson model with alloy type external potential. The method of Klein and Nguyen is very close in the spirit to that of our work [2]. The results of [2] were the first rigorous mathematical proof of localization for many body interacting Hamiltonians near the bottom of the spectrum on the lattice. In the present paper we prove similar results in the continuum.

The work by Sabri [11], uses a different strategy in the course of the multiparticle multiscale analysis at low energy. The analysis is made by considering the Green functions, i.e., the matrix elements of the local resolvent operator instead of the norm of the kernel as it will be developed in this paper and this obliged the author to modify the standard Combes Thomas estimate and adapted it to matrix elements of the local resolvent. Also, our proof on the almost surely spectrum is completely different. The scale induction step in the multiparticle multiscale analysis as well as the strategy of the localization proofs is also different. Chulaevsky [6] used the results of Klein and Nguyen [10] and analyzed multiparticle random operators with alloy-type external potential with infinite range interaction at low energy.

Let us emphasize that the almost sure nonrandomness of the bottom of the spectrum of the multiparticle random Hamiltonian is the heart the problem of localization at low energy for multiparticle systems. In this work, we propose a very clear and constructive proof of this fact. We also prove the exponential localization in the max-norm and the strong dynamical localization near the bottom of the spectrum.

Our multiparticle multiscale analysis is more close in the spirit to its single particle counterpart developed by Stollmann [12] in the continuum case and by von Dreifus and Klein [13] in the lattice case.

Let us now discuss on the structure of the paper. In the next Section, we set up the model, give the assumptions and formulate the main results. In Section 3, we give two important results for our multiparticle multiscale analysis scheme, namely, the Wegner and the Combes Thomas estimates, one, important to bound the probability of resonances, while the other is used to bound the initial scale lengths estimates for energies near the bottom of the spectrum. In Section 4, we prove the initial length scale of the multiscale analysis. Section 5 is devoted the multiparticle multiscale induction step. In Section 6 we prove the variable energy multiparticle multiscale analysis result. Finally, in Section 7, We prove the main results.

2. The Model Assumptions and the Main Results

2.1. The Model

We fix at the very beginning the number of the particles . We are concerned with multiparticle random Schrödinger operators of the forms:

acting in . Sometimes, we will use the identification . Above, is the Laplacian on , represents the inter-particle interaction potential which acts as a multiplication operator in . Additional information in is given in the assumptions. is the multiparticle random external potential, also acting as multiplication operator on . For , and is an i.i.d. random stochastic process relative to the probability space with , and where is the common probability measure of the i.i.d. random variables . Explicitly, we have that for any

Observe that the noninteracting Hamiltonian can be written as a tensor product

where, acting on . We will also consider random Hamiltonian ; defined similarly. Denote by the max-norm in .

2.2. Assumptions

(I) Short-Range Interaction. Fix any . The potential of inter-particle interaction is bounded, nonnegative and of the form

where is a compactly supported function such that

The external random potential is an i.i.d. random field relative to and is defined by for . The common probability distribution function, , of the i.i.d. random variables , associated to the measure is defined by:

(P) Log-Hölder Continuity Condition. The random potential field is i.i.d., of nonnegative values and the corresponding probability distribution function is log-Hölder continuous: more precisely,

Note that this last condition depends on the parameter which will be introduced in Section 3.

2.3. The Results

For any we denote by the spectrum of and the infimum of .

Theorem 1. Let . Under assumptions and we have with probability one:Consequently,

Theorem 2. Under the assumptions and , there exists bigger than such that with -probability one:(i)the spectrum of in is nonempty and pure point,(ii)any eigenfuction corresponding to eigenvalues in is exponentially decaying at infinity in the max-norm.

Theorem 3. Assume that the hypotheses and hold true, then there exists bigger than and a positive such that for any bounded and any we haveis finite, where , is the spectral projection of onto the interval and is a compact domain.

Some parts of the rest of the text overlap with the paper [14] but for the reader convenience we give all the details of the arguments.

3. Input for the Multiparticle Multiscale Analysis and Geometry

3.1. Geometric Facts

According to the general structure of the multiscale analysis, we work with rectangular domains. For , we denote by the -particle cube, i.e.,

and given , we define the rectangle

where are the cubes of side length , center at points . We also define

and introduce the characteristic functions:

The volume of the cube is . We denote the restriction of the Hamiltonian to by

We denote the spectrum of by and its resolvent by

Let be a positive constant and consider . A cube , will be called -nonsingular (-NS) if and

where

Otherwise, it is called -singular (-S).

Let us introduce the following:

Definition 4. Let , and . (A)A cube is called -resonant (-R) if Otherwise, it is called -nonresonant (-R).(B)A cube is called -completely nonresonant (-CNR), if it does not contain any -R cube of size . In particular is itself -NR.

We will also make use of the following notion,

Definition 5. A cube is -separable from if there exists a nonempty subset such thatA pair is separable if and if one of the cube is -separable from the other.

Lemma 6. Let .(A)For any , there exists a collection of -particle cubes with , , such that if satisfies andthen the cubes and are separable.(B)Let be an -particle cube. Any cube withis -separable from for some .

Proof. See Appendix A.

3.2. The Multiparticle Wegner Estimates

We state below the Wegner estimates directly in a form suitable for our multiparticle multiscale analysis using assumption .

Theorem 7. Assume that the random potential satisfies assumption , then(A) For any (B)where , depends only on the fixed number of particles and the configuration dimension .

Proof. See the articles [15, 16].

We also give the Combes–Thomas estimates in

Theorem 8. Let be a Schrödinger operator on , and . Set . If is less than , then for any , we have thatfor all .

Proof. See the proof of Theorem 1 in [17].

We define the mass depending on the parameters , , and the initial length scale in the following way:

We recall below the geometric resolvent and the eigenfunction decay inequalities.

Theorem 9 (geometric resolvent inequality (GRI)). For a given bounded . There is a positive constant such that for , , and , the following inequality holds true:

Proof. See [12], Lemma 2.5.4.

Theorem 10 (eigenfunctions decay inequality (EDI)). For every , and every polynomially bounded function :

Proof. See Section 2.5 and Proposition 3.3.1. in [12].

4. The Initial Bounds of the Multiparticle Multiscale Analysis

In this Section, we denote by the bottom of the spectrum of the Hamiltonian i.e., . We give the following bound from the single-particle localization theory.

Theorem 11. Under the hypotheses and , for any positive , there exists a positive such thatfor all .

Proof. See the book by Peter Stollmann [12].

Now, in the following statement, we show that the same result holds true for the multiparticle random Hamiltonian.

Theorem 12. Under the hypotheses and , for any positive there exists a positive such thatfor all .

Proof. We denote by the multiparticle random Hamiltonian without interaction. Observe that, since the interaction potential is nonnegative we havewhere and the are the eigenvalues of the single-particle random Hamiltonians . So, if , then for example and this implies the required probability bound of the assertion.

We are now ready to prove our initial length scale estimate of the multiparticle multiscale analysis given below.

Recall that the positive parameter is defined by .

Theorem 13. Assume that the hypotheses and hold true. Then there exists a positive such thatfor large enough.

Proof. Set . If the first eigenvalue satisfies , then for all energy , we have:Thus using the Combes–Thomas estimate Theorem 8. Thus for large enough depending on the dimension , we getNow, since , for large enough, we have thatThe above analysis then implies thatYielding the required result.

Below, we develop the induction step of the multiscale analysis and although the text overlaps with the paper [14], for the reader convenience we also give the detailed of the proofs of some important results.

5. Multiscale Induction

In the rest of the paper, we assume that and is the interval from the previous Section.

Recall the following facts from [2]: Consider a cube with . We define

and

Definition 14. Let be a constant and . We define the sequence recursively as followsLet be a positive constant. We also introduce the following property, namely the multiscale analysis bounds at any scale length and for any pair of separable cubes and .
(DS. k, n, N).where .

In both the single-particle and the multiparticle systems, given the results on the multiscale analysis property (DS.k, n, N) above, one can deduce the localization results see for example the papers [13, 18] for those concerning the single-particle case and [2, 7] for multiparticle systems. We have the following

Definition 15 (fully/partially interactive). An -particle cube is called fully interactive (FI) ifand partially interactive (PI) otherwise.

The following simple statement clarifies the notion of PI cubes.

Lemma 16. If a cube is PI, then there exists a subset with such that

Proof. See Appendix B.

If is a PI cube by the above Lemma, we can write it as

with,

where , and . Throughout, when we write a PI cube in the form (5.7) we implicitly assume that the projections satisfy (5.8). Let be the decomposition of the PI cube and and be the eigenvalues and the corresponding eigenfunctions of and respectively. Next, we can choose the eigenfunctions of as tensor products:

The eigenfunctions appearing in subsequent arguments and calculation will be assumed normalized. Now, we turn to geometric properties of FI cubes.

Lemma 17. Let , and consider two FI cubes and with . Then

Proof. See Appendix C.

Given an -particle cube and , we denote by (i) the maximal number of pairwise separable -singular PI cubes ; (ii) by , the maximal number of (not necessary separable) -singular PI-cubes contain in with and for all ; (iii) the maximal number of -singular FI cubes with for all (Note that by Lemma 17; two FI cubes and with are automatically separable); (iv) ; (v) ; (vi) the maximal number of -singular cubes with and for all ; (vii) the maximal number of pairwise separable -singular cube ;

Clearly,

5.1. Pairs of Partially Interactive Cubes

Let be a PI-cube. We also write for any point , in the same way as . So the corresponding Hamiltonian is written in the form:

or in compact form:

We denote by and the corresponding Green functions respectively. Introduce the following notions:

Definition 18 (see [10]). Let and . Consider a PI cube . Then is called -highly nonresonant (-HNR) if (i) for all , the cube is -CNR. (ii) for all , the cube is -CNR.

Definition 19 ((E,m)-tunnelling). Let , and a positive . Consider a PI cube . Then is called (i) left-tunnelling (-LT) if such that contains two separable -S cubes and with . Otherwise it is called nonleft-tunnelling (-NLT). (ii) right-tunnelling (-RT) if such that contains two separable -S cubes and with . Otherwise it is called nonright-tunnelling (-NRT). (iii) -tunnelling (-T) if either it is -LT or -RT. Otherwise it is called -nontunnelling (-NT).

We reformulate and prove Lemma 3.18 from [10] in our context.

Lemma 20. Let . If a PI cube is not -HNR, then(i) either there exist , such that the -particle rectangle is -R,(ii) or there exist , such that the -particle rectangle is -R.

Proof. By Definition 18, if is not -HNR then either (a) there exists such that is not -CNR or (b) there exists such that is not -CNR. Let us first focus on case (a). Since is not -CNR there exist , such that and is -R. So . Therefore, there exists such that . Now consider , since the cube is PI, we have , henceThus is -R. The same arguments shows that case (ii) arises when (b) occurs.

Lemma 21. Let and be a PI cube. Assume that is -NT and -HNR. Then is -NS.

Proof. Let be the decomposition of the PI cube . Let and be the eigenvalues and corresponding eigenfunctions of and respectively. Then, we can choose the eigenvectors and corresponding eigenvalues of as followsBy the assumed -HNR property of the cube , for all eigenvalues one has is -CNR. Next, by assumption of -NT, does not contain any pair of separable -S cubes of radius therefore by Lemma 26, and the cube is also -NS, yieldingThe same analysis for also givesFor any , thus either or consider first the latter case. Then we haveBut by definitionFor Indeed, setting ,and for sufficiently large, hence Thus, is -NS. Finally, the case is similar.

Lemma 22. Let and assume property for any . Then for any PI cube one has

Proof. Consider a PI cube . By Definition 19, we have that the eventis contained in the unionNow, since and we have . So for any , . Further, using property we haveA similar argument also shows thatso thatThe assertion follows by observing that for provided is large enough and .

Theorem 23. Let . There exists such that if and if for holds true for any then holds true for any pair of separable PI cubes and .

Proof. Let and be two separable PI cubes. Consider the events:If then or is -HNR, then it must be -T: otherwise it would have been -NS by Lemma 21. Similarly, if is -HNR, then it must be -T. This implies thatTherefore,Next by combining Theorem 7 and Lemma 20 we obtain that . Finally

For subsequent calculations and proofs, we give the following two Lemmas.

Lemma 24. If with then . Similarly if then .

Proof. See Appendix D.

Lemma 25. With the above notations, assume that holds true for all then

Proof. See Appendix E.

5.2. Pairs of Fully Interactive Cubes

Our aim now is to prove for a pair of fully interactive cubes n and . We adapt to the continuum a very crucial and hard result obtained in the paper [2] and which generalized to multiparticle systems some previous work by von Dreifus and Klein [13] on the lattice and Stollmann [12] in the continuum for single particle models.

Lemma 26. Let with and . Suppose that(i) is -CNR.(ii) .Then there exists such that if we have that is -NS.

Proof. Since , there exist at most cubes of side length contained in that are -S with centers at distance . Therefore, we can find with .such that, if , then the cube is -NS.
We do an induction procedure in and start with . We estimate . Suppose that have been choosen for , we have two cases:
Case (a) is -NS. In this case, we apply the (GRI) Theorem 9 and obtainWe replace in the above analysis with and we getwhere is choosen in such a way that the norm in the right hand side in the above equation is maximal. Observe that . We therefore obtainwith .
Case (b) is -S. Thus, there exists such that . We apply again the (GRI) this time with and and obtain
We have almost everywhereHence, by choosing is such a way that the right hand side is maximal, we getSince , and the cubes are disjoint, we obtain thatso that the cube must be -NS. We therefore perform a new step as in case (a) and obtainwith and .
Summarizing, we get withwith . After iterations with steps of case (a) and steps of case (b), we obtainNow since we have thatSo can be made arbitrarily small if and hence is large enough. We also have for For large hence . Using the (GRI), we can iterate if . Thus, we can have at least steps of case (a) withuntil the induction eventually stop. Since , we can bound from below:which yieldsTherefore,Finally, by -nonresonance of and since we can cover by small cubes , equation (5.50) with instead of , yieldswherewithwe obtainif for some large enough. Since Therefore, we can computeprovided for some large enough . Finally, we obtain that . This proves the result.

The main result of this subsection is Theorem 28 below. We will need the following preliminary result.

Lemma 27. Given , asssume that property holds true for all pairs of separable FI cubes. Then for any

Proof. See the proof in Appendix F.

Theorem 28. Let . There exists such that if and if for (i) for all holds true,(ii) holds true for all pairs of FI cubes,then holds true for any pairs of separable FI cubes and .

Above we use the convention means no assumption.

Proof. Consider a pair of separable FI cubes and and set . DefineLet . If , then either or is -CNR and . The cube cannot be -CNR: indeed, by Lemma 26 it would be -NS. So the cube is -CNR and -S. This implies again by Lemma 26 thatTherefore, , so that , henceand By Theorem 7. Now let us estimate and similarly . Sincethe inequality implies that either or, . Therefore, by Lemmas 25 and 27 with (),where we used that , . Finally

5.3. Mixed Pairs of Cubes

Finally, it remains only to derive in case (III), i.e., for pairs of -particle cubes where one is PI while the other is FI.

Theorem 29. Let . There exists such that if and if for (i) holds true all ,(ii) holds true for all ,(iii) holds true for all pairs of FI cubes,then holds true for any pair of separable cubes and where one is PI while the other is FI.

Proof. Consider a pair of separable -particle cubes , and suppose that is PI while is FI. Set and introduce the eventsLet then, for all either is -CNR or is -CNR and is )-NT. The cube cannot be -CNR. Indeed by Lemma 21 it would have been -NS. Thus the cube is -CNR, so by Lemma 26: otherwise would be -NS. Therefore, . Consequently,Recall that the probabilities and have already been estimated in Sections 5.1 and 5.2. We therefore obtain

6. Conclusion: The Multiparticle Multiscale Analysis

Theorem 30. Let and , where . satisfy and respectively. There exists a positive such that for any property holds true for all provided is large enough.

Proof. We prove that for each , property is valid. To do so, we use an induction on the number of particles . For the property holds true for all by the single-particle localization theory [12]. Now suppose that for all holds true for all , we aim to prove that holds true for all . For , the property is valid using Theorem 13. Next, suppose that holds true for all , then by combining this last assumption with above, one can conclude that:(i) holds true for all and for all pairs of PI cubes using Theorem 23.(ii) holds true for all and for all pairs of FI cubes using Theorem 28. (iii) holds true for all and for all pairs of MI cubes using Theorem 29. Hence, Theorem 30 is proven.

7. Proofs of the Results

7.1. Proof of Theorem 1.

Let . We aim to prove almost surely. Assumption implies that is nonnegative and assumption also implies that is nonnegative. Since, , we get that almost surely . It remains te see that almost surely.

Let . Define,

where positive is the range of the interaction . We also define the following sequence in ,

where . Using the identification , we can also write with each , . Obviously, each term of the sequence belongs to . For , set,

We have that almost surely see for example [12]. So, if we set for

for all . Now put

We also have that . Let , for this , By the Weyl criterion, there exist Weyl sequences related to and each operator . By the density property of compactly supported functions , in , we can directly assume that each is of compact support, i.e., for some integer large enough. Set

and put, . We translate each function to have support contained in the cube . Next consider the sequence defined by the tensor product,

We have that and we aim to show that, is a Weyl sequence for and . For any :

Indeed, for the values of inside the cube the interaction potential vanishes and for those values outside that cube, equals zero too. Therefore,

which tends to zero as tends to infinity because, for all each as , since is a Weyl sequence for and . This completes the proof.

7.2. Proof of Theorem 2

Using the multiparticle multiscale analysis bounds in the continuum property , we extend to multiparticle systems the strategy of Stollmann [12].

For and an integer , using the notations of Lemma 6

and define

where the positive parameter is to be chosen later. We can easily check that

Moreover, if , then the cubes and are separable by Lemma 6. Now, also define

with . Now property combined with the cardinality of imply

Since, (in fact ), we get is finite. Thus, setting

by the Borel Cantelli Lemma and the countability of we have that . Therefore, it suffices to pick and prove the exponential decay of any nonzero eigenfunction of .

Let be a polynomially bounded eigenfunction satisfying (EDI) (see Theorem 10). Let with positive (if there is no such , we are done.) The cube cannot be -NS for infinitely many . Indeed, given an integer , if is -NS then by (EDI) and the polynomial bound on we get

and the last term tends to as tends to infinity in contradiction with the choice of . So there is an integer finite such that the cube is -S. At the same time, since , there exists such that if does not occur. We conclude that for all , for all , is -NS. Let and choose positive such that

so that

for .(1)Since, ,

(2)Since ,

Thus,

Now, setting , the assumption linking and implies that

Because . Let , recall that this implies that all the cubes with centers in and side length are -NS. Thus, for any , we choose such that . Therefore,

Up to a set of Lebesgue measure zero, we can cover by at most cubes

By choosing which gives a maximal norm, we get

so that

Thus, by an induction procedure, we find a sequence in with the bound

Since and , we can iterate at least times until, we reach the boundary of . Next, using the polynomial bound on , we obtain:

We can conclude that given with , we can find such that if , then

if . This completes the proof of the exponential localization in the max-norm.

7.3. Proof of Theorem 3

For the proof of the multiparticle dynamical localization given the multiparticle multiscale analysis in the continuum, we refer to the paper by Boutet de Monvel et al. [19].

Appendix

A. Proof of Lemma 6

(A) Consider positive and . is called an -cluster if the union

cannot be decomposed into two nonempty disjoint subsets. Next, given two configurations , we proceed as follows: (1) We decompose the vector into maximal -clusters (each of diameter ) with . (2) Each position corresponds to exactly one cluster (3) If there exists such that , then the cubes and are separable. (4) If (3) is wrong, then for all , . Thus for all , such that . Now for any there exists such . Therefore for such , by hypothesis there exists such that . Next let so that . We have that

since .

Notice that above we have the bound because is a center of the -cluster Hence for all must belong to one of the cubes for the -positions . Set . For any choice of at most possibilities; must belong to the cartesian product of cubes of side length i.e., an -dimensional cube of size , the assertion then follows.

(B) Set and consider a cube with . Then there exist such that . Consider the maximal connected component of the union containing . Its diameter is bounded by . We have

now, since

then

Recall that and

for some such that . Finally, we get

and the latter quantity is strictly positive. This implies that is separable from .

B. Proof of Lemma 16

Set and assume that . If the union of cubes , were not decomposable into two (or more) disjoint groups, then, it would be connected hence its diameter would be bounded by hence which contradicts the hypothesis. Therefore, there exists an index subset such that for all and , this implies that

C. Proof of Lemma 17

If for some positive

then there exists such that . Since both cubes are fully interactive,

By the triangle inequality, for any , and , we have

Therefore, for any ,

which proves the claim.

D. Proof of Lemma 24

Assume that is less than (i.e.,there is no pair of separable cubes of radius in but . Then must contain at least cubes , which are not separable but satisfy for all . On the other hand, by Lemma 6 there are at most cubes , such that any cube with , is separable from . Hence for all . But since for all there must be at most one center per cube , . Hence we come to a contradiction

The same analysis holds true if we consider only PI cubes.

E. Proof of Lemma 25

Suppose that , then by Lemma 24 i.e., there are at least two separable -S PI cubes , inside . The number of possible pairs of centers such that

is bounded by . Then, setting

with .

F. Proof of Lemma 27

Suppose there exist pairwise separable fully interactive cubes , . Then by Lemma 17 for any pair , the corresponding random Hamiltonians and are independent and so are their spectra and their Green functions. For we consider the events:

then by assumption , we have for

and by independence of the events

To complete the proof, note that the total number of different families of cubes , is bounded by

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

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Copyright © 2020 Trésor Ekanga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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