Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

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. e 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.


Introduction
is paper follows our previous works [1,2] on localization for multiparticle random lattice Schrödinger operators at low energy. Some other papers [3][4][5][6][7][8][9][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. e method of Klein and Nguyen is very close in the spirit to that of our work [2]. e 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.
e work by Sabri [11], uses a different strategy in the course of the multiparticle multiscale analysis at low energy. e 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 omas estimate and adapted it to matrix elements of the local resolvent. Also, our proof on the almost surely spectrum is completely different. e 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 omas 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

e Model.
We fix at the very beginning the number of the particles 푁 ≥ 2. We are concerned with multiparticle random Schrödinger operators of the forms: acting in 2 R 푑 푁 . Sometimes, we will use the identification R 푁 ≅ R . Above, Δ is the Laplacian on R , U represents the inter-particle interaction potential which acts as a multiplication operator in 2 R 푁푑 . Additional information in U is given in the assumptions. V is the multiparticle random external potential, also acting as multiplication operator on 2 R 푁푑 . For x = 푥 1 , . . . , 푥 푁 ∈ R 푑 푁 , V(x) = 푉 푥 1 + ⋅ ⋅ ⋅ + 푉 푥 푁 and 푉(푥, 휔), 푥 ∈ Z is an i.i.d.

Assumptions.
(I) Short-Range Interaction. Fix any 푛 = 1, . . . , 푁. e potential of inter-particle interaction U is bounded, nonnegative and of the form where Φ : R + → R is a compactly supported function such that e external random potential 푉 : Z × 훺 → R is an i.i.d. random field relative to (훺, B, P) and is defined by , of the i.i.d. random variables 푉(푥, ⋅), ∈ Z associated to the measure is defined by: (P) Log-Hölder Continuity Condition. e random potential field 푉(푥, 휔); 푥 ∈ Z 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. Consequently, Theorem 2. Under the assumptions (I) and (P), there exists * bigger than (푁) 0 such that with P-probability one: (i) the spectrum of H (푁) (휔) in 퐸 (푁) 0 , 퐸 * is nonempty and pure point, (ii) any eigenfuction corresponding to eigenvalues in 퐸 (푁) 0 , 퐸 * is exponentially decaying at infinity in the max-norm.
Theorem 3. Assume that the hypotheses (I) and (P) hold true, then there exists * bigger than (푁) 0 and a positive 푠 * (푁, 푑) such that for any bounded K ⊂ Z and any 푠 ∈ 0, 푠 * we have 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 Advances in Mathematical Physics and given 퐿 : 푖 = 1, . . . , 푛 , we define the rectangle where (1) 퐿 푖 are the cubes of side length 2퐿 , center at points ∈ Z . We also define and introduce the characteristic functions: We denote the restriction of the Hamiltonian H (푛) to C (푛) (u) by We denote the spectrum of H (푛) C (푛) (u) by H (푛) C (푛) (u) and its resolvent by Let be a positive constant and consider ∈ R. A cube where Otherwise, it is called (퐸, 푚)-singular ((퐸, 푚)-S).
Let us introduce the following: We will also make use of the following notion, and if one of the cube is J -separable from the other.
퐿 y ⊂ R 푛푑 be an -particle cube. Any cube Proof. See Appendix A. ☐

e Multiparticle Wegner Estimates.
We state below the Wegner estimates directly in a form suitable for our multiparticle multiscale analysis using assumption (P).
Theorem 7. Assume that the random potential satisfies assumption (P), then where 푝 ≥ 6푁푑, depends only on the fixed number of particles and the configuration dimension .
Proof. See the proof of eorem 1 in [17]. ☐ We define the mass depending on the parameters , , and the initial length scale in the following way: (3.14) Advances in Mathematical Physics 4 푖 = 1, . . . , 푛. So, if (푛) 0 ≤ −1/2 , then for example (1) 1 ≤ −1/2 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.
Theorem 13. Assume that the hypotheses (I) and (P) hold true. en there exists a positive * such that for 퐿 ≥ 0 large enough. Now, since 훾(푚, 퐿, 푛) = 푚 1 + 퐿 −1/8 푁−푛 ≤ 2 푁 푚, for 퐿 ≥ 0 large enough, we have that e above analysis then implies that Yielding 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.

Multiscale Induction
In the rest of the paper, we assume that 푛 ≥ 2 and 0 is the interval from the previous Section.
We recall below the geometric resolvent and the eigenfunction decay inequalities.

The Initial Bounds of the Multiparticle Multiscale Analysis
In this Section, we denote by 퐸 푛 0 (휔) the bottom of the spectrum of the Hamiltonian H (푛) . We give the following bound from the single-particle localization theory. 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. Proof. We denote by H (푛) 0 (휔) the multiparticle random Hamiltonian without interaction. Observe that, since the interaction potential U is nonnegative we have (1) 푖 are the eigenvalues of the single-particle random Hamiltonians 퐻 (1) Advances in Mathematical Physics of the PI cube C (푛) 퐿 (u) and 휆 , 휑 and 휇 , 휙 be the eigenvalues and the corresponding eigenfunctions of H ( ὔ ) and respectively. Next, we can choose the eigenfunctions e eigenfunctions appearing in subsequent arguments and calculation will be assumed normalized. Now, we turn to geometric properties of FI cubes.
Proof. See Appendix C. ☐ Given an -particle cube C (푛) 퐿 (u) and ∈ R, we denote by Clearly,

Pairs of Partially Interactive
Cubes. Let , in the same way as (5.11) Recall the following facts from [2]: Consider a cube C (푛) . We define and Definition 14. Let 퐿 0 ≥ 3 be a constant and 훼 = 3/2. We define the sequence 퐿 : 푘 ≥ 1 recursively as follows Let 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 C (푛) 퐿 (u) and where 푝 ≥ 6푁푑.
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 C (푛) ⊂ Z 푛푑 is called fully interactive (FI) if and partially interactive (PI) otherwise. e following simple statement clarifies the notion of PI cubes. where u ὔ = u J = 푢 , 푗 ∈ J , u ὔὔ = 푢 ; 푗 ∈ J ὔ = card J and ὔὔ = card J . roughout, when we write a PI cube C (푛) 퐿 (u) in the form (5.7) we implicitly assume that the projections

Advances in Mathematical Physics 8
We replace in the above analysis x with x ℓ and we get where x ℓ+1 is choosen in such a way that the norm in the right hand side in the above equation is maximal. Observe that |x ℓ − x ℓ+1 | = 퐿 푘 /3. We therefore obtain We apply again the (GRI) this time with C (푛) 퐿 푘+1 (x) and C (푛) 2퐿 푘 x 푖 0 and obtain We have almost everywhere Hence, by choosing x is such a way that the right hand side is maximal, we get 2퐿 푘 x 푖 0 and the cubes C (푛) 2퐿 x 푖 are disjoint, we obtain that so that the cube C (푛) 퐿 ( x) must be (퐸, 푚)-NS. We therefore perform a new step as in case (a) and obtain with x ℓ+1 ∈ x and | x − x ℓ+1 | = 퐿 푘 /3.
Summarizing, we get x ℓ+1 with For subsequent calculations and proofs, we give the following two Lemmas.
Proof. See Appendix D.

Pairs of Fully Interactive Cubes.
Our aim now is to prove ( .푘 + 1, 푛, 푁) for a pair of fully interactive cubes C (푛) 퐿 푘+1 (x)n and C (푛) 퐿 푘+1 y . 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.
e main result of this subsection is eorem 28 below. We will need the following preliminary result.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that they have no conflicts of interest.