Abstract
We prove that spectral and strong dynamical localization occurs in the one-dimensional multiparticle Anderson model with weak interaction in the continuous configuration space. To obtain these results, the interaction amplitude must be sufficiently small. The general strategy relies on an estimate in the framework of multiscale analysis. In fact, we prove that the multiscale analysis estimates for the single-particle model are unchanged in passing to multiparticle systems if the interparticle interaction is sufficiently small. The only condition imposed on the probability distribution of the external potential, which is a random field of independent identically distributed random quantities, is that it must be logarithmically continuous in the Hölder sense.
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Notes
We note that by Lemma 4.2, two FI cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j)})\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j')})\) for which \(| \mathbf{u} ^{(j)}- \mathbf{u} ^{(j')}|\ge7NL_k\) are automatically separable.
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Appendix: Proofs of the results
Appendix A.1. Proof of Theorem 1.1
To extend Stollmann’s strategy [16] to a multiparticle system, we use multiparticle multiscale analysis estimates in condition DS\((k,N)\).
For \( \mathbf{x} _0\in \mathbb{Z} ^{Nd}\) and an integer \(k\ge0\), using the notation in Lemma 2.1, we set
Let \( \boldsymbol{\Psi} \) be a polynomially bounded eigenfunction satisfying the inequality of eigenfunction decay (see Theorem 2.4). Let the norm \(\| \mathbf{1} _{ \mathbf{C} ^{(N)}_1( \mathbf{x} _0)} \boldsymbol{\Psi} \|\ne0\) for \( \mathbf{x} _0\in \mathbb{Z} ^{Nd}\) (if such \( \mathbf{x} _0\) does not exist, then the proof is complete). Then \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} _0)\) cannot be an \((E,m,h)\)-NS cube for infinitely many \(k\). Indeed, if \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} _0)\) is an \((E,m,h)\)-NS cube for a given \(k\ge0\), then by the inequality of eigenfunction decay and the polynomial estimate for \( \boldsymbol{\Psi} \), we obtain
Let \(\rho\in(0,1)\), We choose
Let \( \mathbf{x} \in \tilde A_{k+1}( \mathbf{x} _0)\). Because \(| \mathbf{x} - \mathbf{x} _0|\ge b_kL_k/(1-\rho)\), we have the estimate
By induction, we can thus find \( \mathbf{x} _1, \mathbf{x} _2,\dots, \mathbf{x} _n\) in \(\Gamma_k\cap A_{k+1}( \mathbf{x} _0)\) with the estimate
Using the polynomial estimate for \( \boldsymbol{\Psi} \), we obtain
Appendix A.2. Proof of Lemma 2.1
We prove the first statement in the lemma. Let \(L>0\), \( \varnothing \ne \mathcal{J} \subset \{1,\dots,n\}\), and \( \mathbf{y} \in \mathbb{Z} ^{nd}\). We call \(\{y_j\}_{j\in \mathcal{J} }\) an \(L\)-cluster if \(\bigcup_{j\in \mathcal{J} } C^{(1)}_L(y_j)\) cannot be decomposed into two nonempty disjoint subsets.
For two given configurations \( \mathbf{x} , \mathbf{y} \in \mathbb{Z} ^{nd}\), we proceed as follows:
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1.
We decompose the vector \( \mathbf{y} \) into maximum \(L\)-clusters \(\Gamma_1,\dots,\Gamma_M\) (the diameter of each cluster does not exceed \(2nL\)), where \(M\le n\).
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2.
Each point \(y_i\) corresponds to exactly one cluster \(\Gamma_j\), \(j=j(i)\in\{1,\dots,M\}\).
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3.
If there exists \(j\in\{1,\dots,M\}\) such that \(\Gamma_j\cap \Pi \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )= \varnothing \), then the cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{y} )\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )\) are separable. Otherwise, \(\Gamma_k\cap\Pi \mathbf{C} ^{(n)}_L( \mathbf{x} )\ne \varnothing \) for all \(k=1,\dots,M\). For each \(k=1,\dots,M\), there then exists \(i\in\{1,\dots,n\}\) such that \(\Gamma_k\cap C^{(1)}_L(x_i)\ne \varnothing \). Further, for each \(j=1,\dots,n\), there exists \(k\in\{1,\dots,M\}\) such that \(y_j\in\Gamma_k\). For this \(k\), there then exists \(i\in\{1,\dots,n\}\) such that \(\gamma_k\cap C^{(1)}_L(x_i)\ne \varnothing \). Now let \(z\in\Gamma_k\cap C^{(1)}_L(x_i)\), and hence \(|z-x_i|\le L\). By virtue of \(y_j\in\Gamma_k\), we have
$$|y_j-x_i|\le |y_j-z|+|z-x_i|\le 2nL-L+L=2nL.$$
We note that we have the estimate \(|y_j-z|\le nL-L\) because \(y_j\) is the center of the \(L\)-cluster \(\Gamma_k\). For each \(j=1,\dots,n\), the coordinate \(y_j\) in the \(n\)-dimensional configuration \((y_1,\dots,y_n)\) must then belong to one of the cubes \(C^{(1)}_{2nL}(x_i)\). We set \(\kappa(n)=n^n\). Whatever the choice from at most \(\kappa(n)\) possibilities for the element \( \mathbf{y} =(y_1,\dots,y_n)\), this element must belong to the Cartesian product of \(n\) cubes of side length \(2L\), i.e., an \(nd\)-dimensional cube of side length \(2nL\), whence follows the first statement in the lemma.
We prove the second statement. We set
Appendix A.3. Proof of Lemma 4.1
We introduce the notation \(R:=2L+r_0\) and assume that
Appendix A.4. Proof of Lemma 4.2
If
Appendix A.5. Proof of Lemma 4.5
We assume that \(M^{ \mathrm{sep} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),E)\) does not exceed 2 (i.e., there do not exist two separable cubes of radius \(L_k\) in \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\)) but \(M( \mathbf{C} ^{(n)}( \mathbf{u} ),E)\ge\kappa(n)+2\). Then \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\) must contain at least \(\kappa(n){+}2\) cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{v} _i)\), \(0\le i\le\kappa(n)+1\) that are separable, but in this case, \(| \mathbf{v} _i- \mathbf{v} _{i'}|\ge7NL_k\) for all \(i\ne i'\).
On the other hand, by Lemma 2.1, there exist at most \(\kappa(n)\) cubes \( \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _i)\) such that each cube \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )\) with a center \( \mathbf{x} \notin\bigcup_j \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _j)\) is separable from \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{v} _0)\). Hence, \( \mathbf{v} _i\in\bigcup_j \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _j)\) for all \(i=1,\dots,\kappa(n)+1\). But \(| \mathbf{v} _i- \mathbf{v} _{i'}|\ge7NL_k\) for \(i\ne i'\), and each cube \( \mathbf{C} ^{(n)}_{2nL_k}(y_j)\), \(j=1,\dots,\kappa(n)\), must contain at least one center \( \mathbf{v} _i\). This leads to the contradiction \(\kappa(n)+1\le\kappa(n)\).
The reasoning holds if we consider only PI cubes.
Appendix A.6. Proof of Lemma 4.6
Let \(M_{ \mathrm{PI} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),I)\ge\kappa(n)+2\). Then \(M^{ \mathrm{sep} }_{ \mathrm{PI} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),I)\ge2\) by Lemma 4.5, i.e., there are at least two separable \((E,m,h)\)-S PI cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j_1)}\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j_2)})\) in \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\). The number of possible pairs of centers \(\{ \mathbf{u} ^{(j_1)}, \mathbf{u} ^{(j_2)}\}\) such that
Appendix A.7. Proof of Lemma 4.8
We assume that there exist \(2\ell\) pairwise separable FI cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j)})\subset \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\), \(j=1,\dots,2\ell\). Then by Lemma 4.2, for any pair \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i-1)})\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i)})\), the corresponding random Hamiltonians \( \mathbf{H} ^{(n)}_{ \mathbf{C} ^{(n)}_{L_k} ( \mathbf{u} ^{(2i-1)})}\) and \( \mathbf{H} ^{(n)}_{ \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i)})}\) are independent, as also are their spectra and their Green’s functions. We consider the events
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Ekanga, T. Localization in multiparticle Anderson models with weak interaction. Theor Math Phys 206, 357–382 (2021). https://doi.org/10.1134/S0040577921030089
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DOI: https://doi.org/10.1134/S0040577921030089