数理解析研究所講究録
第 1902 巻 2014 年 70-81

70

Eﬃcient multi-particle localization bounds
in continuous random media
Victor Chulaevsky’ Yuri Suhov\dagger

Abstract
We study multi-particle interactive quantum disordered systems in the
Euclidean space of arbitrary dimension and in the so-called quantum (ormetric) graphs, subject to a Gaussian random potential field with continuous argument. We prove exponential and strong dynamical localizationin presence of a nontrivial interaction between the particles. The obtainedbounds on the decay of the eigenfunction correlators are uniform in finite
or infinite volumes, so the Anderson localization phenomenon is eﬀectively proved in finite volumes of arbitrarily large size. In earlier works,
the multi-particle localization bounds were established only in the entire(infinitely extended) physical configuration space.

Introduction. The model and results

1

Until recently, the rigorous Anderson localization theory focused on singleparticle models. (In the physical community, notable papers on multi-particle
systems with interaction appeared as early as in 2005-2006; see [3, 17 Initial
rigorous results on multi-particle lattice localization were presented in [8-10]
and [2]; continuous models have been considered in [4], [6]. Further progress
was shown in [18, 19], [21].
In the present paper we consider an interactive multi-particle Anderson models in configuration spaces of two types:(1) a Euclidean space$\mathbb{R}^{d}$of arbitrary dimension $d\geq 1$ ;(2) a quantum graph associated with a combinatorial graph with polynomial$d\geq 2$ , but
growth of balls; formally, we focus on the graphs over
our technique easily extends to more general underlying graphs.
$\mathbb{Z}^{d}\mapsto \mathbb{R}^{d},$The main method used is a new modification of the multi-particle multi-scale
analysis (MPMSA). More precisely, in contrast to earlier works [6, 10, 21], we
use the fixed-energy multi-particle variant of the multi-scale analysis, and theninfer from the fixed-energy estimates their energy-interval counterparts required
for the proof of spectral and dynamical localization. This results in a simpler
and more straightforward scale induction than in earlier works.
The results are summarized as follows.
*Universit\’e de Reims, D\’epartement de Math\’ematiques, Moulin de la Housse, B.P. 1039,

51687 Reims Cedex 2, France
Statistical Laboratory, DPMMS, University of Cambridge, Wilberforce Road, Cambridge,
CB3 OWB, UK$\dagger$V. Chulaevsky, Y. Suhov$\bullet$$\bullet$

For an $N$ -particle system we prove uniform localization bounds valid for
finite or infinite regions of the physical, single-particle configuration space.
All previously published results (except for [9] where two-particle systems
were considered) provided much less eﬃcient bounds in finite-size systems.
In our models, the external random potential is generated by a random
field with continuous argument (in the Euclidean space or, respectively, ona metric (a.k. . quantum) graph. Such model, in the single-particle setting, was studied in the works [14, 15] where spectral (but not dynamical)
localization was established. Generally speaking, this class of random potentials has been studied much less than the so-called alloy-type potentials
with random scatterer amplitudes.
$a$$\bullet$$\bullet$

1.1

Working with lower-unbounded random potentials, we are able to relax
the positivity assumption on the interaction energy, assumed in [6, 19, 21].

Compared to the works [6, 19, 21], we relax the assumption of Independence
At Distance (IAD) and allow for strongly mixing random potentials withcorrelations of infinite range.

The $N$ -particle Hamiltonian in$\mathbb{R}^{d}$

We consider a system of $N>1$ distinguishable quantum particles in the Eu$d\geq 1$ , with positions, so theclidean space
configuration space is. The configurations will be denoted by$x=(x_{1},\ldots, x_{N})\in(\mathbb{R}^{d})^{N}$ .
e.g.,lettres,
boldface lowercase
It is convenient touse both inand inthe $\max$-norms, viz.
$x_{j}=(x_{j}^{(1)}, \ldots, x_{j}^{(d)})\in \mathbb{R}^{d}$$\mathbb{R}^{d},$$(\mathbb{R}^{d})^{N}\cong \mathbb{R}^{Nd}$$\mathbb{R}^{d}$$(\mathbb{R}^{d})^{N}$$|x|= \max_{1\leq i\leq d}|x^{(i)}|, |x|=\max_{1\leq j\leq N}|x_{j}|.$Sometimes we also make use of the Euclidean distance$|x|_{2}.$We introduce the mappings: $x=(x_{1}, \ldots, x_{N})\mapsto(x_{j}, j\in.\mathcal{J})$ , for the, called the partial projections.
index subsets
Each particle is subject to the (common) external random potential, given
by a sample of a Gaussian random field (r.f.), relative to someprobability space, with a.s. continuous samples. (See Sect. 1.3 and 1.4
where the assumptions on the r.f.
are listed.)
The Hamiltonian of the $N$ -particle system inis assumed to have the form
$H^{(N)}(\omega)=-\Delta+V(x;\omega)+U(x)$ , namely$\Pi_{\mathcal{J}}$$\mathcal{J}\subset\{1, .

.

.

, N\}$$V:\mathbb{R}^{d}\cross\Omegaarrow \mathbb{R}$$(\Omega, \mathfrak{F}, \mathbb{P})$$V$$\mathbb{R}^{d}$$H^{(N)}(\omega)=-\Delta+\sum_{1\leq j\leq N}V(x_{j};\omega)+\sum_{1\leq j<k\leq N}U^{(2)}(|x_{j}-x_{k}$where$\Delta$

is the standard Laplacian in(1.1)$(\mathbb{R}^{d})^{N},$$\Delta=\sum_{1\leq j\leq N1}\triangle_{J}\prime,$

and the two-body interaction potential
generating the interaction energy
operator $U(x)$ is measurable, bounded and compactly supported.
is well-defined as a selfWith probability 1, the random Hamiltonian, e.g., with the domain
adjoint operator in the Hilbert space$U^{(2)}$$H(\omega)$$\mathcal{H}^{(N)}\cong L^{2}(\mathbb{R}^{Nd})$

71

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Localization bounds for multi-particle systems

formed by compactly supported smooth functions. This follows from the standard results on Schr\"odinger operators with the potential of tempered growth.
, the total potential energy
This is well-known to be true1, if for$W(x;\omega)$ satisfies $|W(x;\omega)|\leq C(|x|_{2}^{2}+1)^{1/2}$ , and this (indeed, much stronger)
growth bound follows from our assumptions on the Gaussian r.f. $V.$
Apart from the $\max$-norm distance, it will be convenient to use also the, defined as follows:
symmetrized $\max$ -distance$\omega\in\Omega$$\mathbb{P}-a.e.$$\rho_{S}$$\rho_{S}(x, y)=\min_{\pi\in \mathfrak{S}_{N}}|\pi(x)-y|$Here the symmetric group
positions:$\mathfrak{S}_{N}$acts on$\mathbb{R}^{N}$

(1.2)

.

by permutations of the particle$\pi(x_{1}, \ldots, x_{N})=(x_{\pi(1)}, , \ldots, x_{\pi(N)})$.

The reason for using the symmetrized distance is that the quantum particles are
considered as distinguishable, while the Hamiltonian is permutation-symmetric,
thus leaves invariant the subspace of symmetric functions. As a result, one
cannot achieve eﬃcient decay bounds on the (localized) eigenfunctions in the. In fact, even if the interaction potential
original, $\max$-norm distance inwere not permutation-symmetric, the symmetry of the external potential would
make it diﬃcult to rule out
between arbitrarily distant points ofacting on
the orbits of the symmetric groupby permutatio1ls
of the positions, of radius
We will work with the balls (in the $\max$-norm),$L>0$ , centered at $x=(x_{1}, \ldots, x_{N})$ :$(\mathbb{R}^{d})^{N}$$\prime\dagger_{resonances"}$$\mathfrak{S}_{N}$$(x_{1}, \ldots, x_{N})$$x_{j}.$$B_{L}^{(N)}(x)\subset(\mathbb{R}^{d})^{N}$$N$$B_{L}^{(N)}(x) :=\{y :|x-y|\leq L\}=j=1\cross B_{L}(x_{j})$

,

(1.3)

where $B_{L}(x)=\{y\in \mathbb{R}^{d}:|y-x|<L\}.$
Our techniques apply also to a large class of quantum graphs of tempered
growth, but for the sake of notational brevity, we consider in the present paper
the $N$ -particle systems in the quantum graph embedded in the Euclidean space.
Specifically, we consider the usual (combinatorial) oriented countable graphwith the vertex set
and the edge set
formed by the pairsof nearest neighbors in :$\Gamma^{(1)}=\mathbb{Z}^{d}\mapsto \mathbb{R}^{d}$$\mathcal{E}^{(1)}$$\mathbb{Z}^{d}$$\mathbb{Z}^{d}\cross \mathbb{Z}^{d}\ni e=\langle x, y\rangle\in \mathcal{E}^{(1)}\Leftrightarrow|x-y|_{2}=1$$(recall: the$ graph $is$ oriented, $so the$ edge

associate with each edge

$e=\langle x,$$y\rangle$$\bullet$the initial point$\bullet$the terminal point $\tau_{e}=y,$$\bullet$the edge interval $I_{e}\cong(0,1)$ .

$\langle x, y\rangle is an$ordered pair) . Further, we:$\iota_{e}=x,$open interval $(0,1)$ , but theis merely a replica
Formally speaking,embeddingallows one to identify it with the aﬃne interval inof$\{t\cdot
x+(1-t)\cdot
y,$$t\in(O,$the form$of\cdot the$$I_{e}$$\mathbb{Z}^{d}\mapsto \mathbb{R}^{d}$$\mathbb{R}^{d}$$1$

1 See e.g., [7].

V. Chulaevsky, Y. Suhov

A combinatorial graph equipped with the association
is called metricgraph; the term “quantum grap emerged in the wake of numerous researches,in theoretical and mathematical physics, of real quantum systems with the configuration space that can be eﬃciently reduced to a collection of interconnectedthin tubular fragments. Naturally, the edge intervals need not be of unit length;
for exemple, the metric graphs with random edge lengths have also been considered in the literature (cf., e.g., [20]). A complete specification of a quantum, but since wegraph should include, therefore, the mapping
consider only the model with the identical, unit lengths, the underlying combinatorial graph structure
is suﬃcient, and we keep this shorter (albeit
slightly abusive) notation referring to a quantum graph, and omit
The Hilbert space of single partiele quantum states in the quantum graphis the direct sum$erightarrow I_{e}$$h’$$\mathcal{I}:e\mapsto I_{e},$$e\in \mathcal{E}^{(1)}$$(\Gamma^{(1)}, \mathcal{E}^{(1)})$$\mathcal{I}.$$(\Gamma^{(1)}, \mathcal{E}^{(1)})$$\mathcal{H}^{(1)}=\bigoplus_{e\in \mathcal{E}(1)}L^{2}(I_{e})\cong\bigoplus_{e\in \mathcal{E}^{(1)}}L^{2}(0,1)$

It is convenient to define the Laplace operator
hand via the bilinear form$\triangle$.

on the quantum graph at$( \phi, \psi)\mapsto-\sum_{e\in \mathcal{E}(1)}(\phi’)\psi’)_{L^{2}(I_{e})}=-\sum_{e\in \mathcal{E}(1)}\int_{I_{e}}\overline{\phi’(x)}\psi’(x)dx,$

well-defined, e.g., for the smooth, compactly supported functions. Given, referred to as thea measurable, locally bounded function $V$ :external potential, the Schr\"odinger operator $H=-\Delta+V$ on the quantum
graph, with the potential , is also defined via its bilinear form:$\phi,$$\psi$$\sqcup_{e\in \mathcal{E}^{(1)}}I_{e}arrow \mathbb{R}$$V$$\mathfrak{h}[\phi, \psi]=(\phi, \psi)\mapsto-\sum_{e\in \mathcal{E}(1)}((\phi’, \psi’)_{L^{2}(I_{e})}+(\phi, V\psi)_{L^{2}(I_{e})})$$=- \sum_{e\in \mathcal{E}(1)}\int_{I_{e}}(\overline{\phi’(x)}\psi’(x)+V(x)\overline{\phi(x)}\psi(x))dx,$

for the smooth, compactly supported functions. To obtain a self-adjoint(or rather, conextension of $H$ , one usually considers the Kirchhoﬀtact) conditions. First, it is convenient to compactify the edge intervals
andconsider the closed intervals, 1 . Next, given a test function , assume
that it extends to a continuous function on the union of . The embedding
makes these abstract construction more straightforward: the collection of the unit edges between the vertices ofis endowed with the metric, so one can simply identify
induced by the Euclidean distance inwith this subset of , which has the structure of (singular) Riemannian manifold of dimension 1; the singularity comes of course from the vertex points which
do not have a neighborhood homeomorphic to an open interval. Making use of, 1 , we will write
meaning the limiting
the identification
andvalue at the left and, respectively, right endpoint of the interval , 1
The Kirchhoﬀ conditions require that for each vertex point of the quantum
graph, the test function be continuous at , and that$\phi,$$\psi$$\prime\prime boundary"$$I_{e}$$\overline{I}_{e}\cong[0$$]$$\phi$$\overline{I}_{e}$$\mathbb{Z}^{d}\mapsto \mathbb{R}^{d}$$\mathbb{Z}^{d}$$\mathbb{R}^{d}$$\bigcup_{e\in \mathcal{E}^{(1)}}\overline{I}_{e}$$\mathbb{R}^{d}$$a$$\overline{I}_{e}rightarrow[0$$]$$\phi(0_{e})$$\phi(1_{e})$

$[0$

$v$

$\phi$$v$$\sum_{e:\iota_{e}=v}\phi’(0_{e})-\sum_{e:\tau_{e}=v}\phi’(1_{e})=0.$

$].$

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Localization bounds for multi-particle systems

It can be shown that $H$ can be defined as a self-adjoint Schr\"odinger operator with
Kirchhoﬀ conditions, for slowly growing potentials . We work with Gaussian
random fields on the quantum graphs embedded in the Euclidean space, and it
follows from general results going back to the pioneering work by Fernique [13]. sample $V(x;\omega)$ of a continuous Gaussian field
grows
that
on
not faster than $c_{1}(\omega)\ln^{c_{2}}(|x|+1)$ .
The notion of quantum graph and the key constructions used in the localization theory on quantum graphs have been developed by Sabri in his recent $PhD$
project. Following [21], we will the $N$ -particle analog of the quantum graph the$N$ -graph.
Specifically, the configuration space of an $N$ -particle system on a conven, with identical, unit lengths
tional (single-particle) quantum graph$N$of the edges, is the collection of open -dimensional unit cubes$(0,1)^{N}$ , labeled by the $N$-tuples $e=(e_{1}, \ldots, e_{n})\in(\mathcal{E}^{(1)})^{N}$
The kinetic energy operator of the $N$-particle system is the Laplace operator on the union of
the $N$-edges, and the potential energy operator (including the external random
potential and the interaction potential) is the operator of multiplication by thepotential energy function. It is shown in [21] that the $N$ -particle Hamiltonian
is well-defined for interactions of finite range and bounded external potentials.
This result can be easily extended to the external random potentials of slow
growth, including the Gaussian random fields growing at logarithmic rate. For
further details, we refer the reader to the paper [21].
$V$$\mathbb{P}-a.e$$V(\cdot;\omega)$$\Gamma^{(1)}$$(\Gamma^{(1)}, \mathcal{E}^{(1)})$$\sigma_{e}=\sigma_{e}^{(N)}\cong$

Some basic facts on Gaussian fields

1.2

For brevity, we consider in this subsection only the random fields in a Euclidean
space. Working with Gaussian r.f. on a metric graph requires a more generalframework of random fields on metric spaces $(X, \rho)$ endowed with the structure, whereof a measure spaceis generated by the Borel sets in $X$(cf., e.g., [1]), but such a generalization is quite straightforward.
A convenient way to define a Gaussian r.f. with regular samples inis as, which, in general, has to be measurable
follows. Fix a function :and positive definite: for any measurable function :$(X, \mathfrak{B}, mes)$$\mathfrak{B}$$\mathbb{R}^{d}$$C$$\mathbb{R}^{d}\cross \mathbb{R}^{d}arrow \mathbb{R}$$\phi$$\mathbb{R}^{d}arrow \mathbb{C}$$\langle\phi, \phi\rangle_{C}:=\int_{R^{d}xN^{d}}C(x, y)\overline{\phi(x)}\phi(y)dxdy\geq 0,$

andunless $\phi=0$ a.e.
We assume that is continuous (for we need the samples of the r.f. to becontinuous).
Then there exists a Gaussian r.f.
inwith zero mean and the covariance function $\mathbb{E}[V(x;\omega)V(y;\omega)]=C(x, y)$ . Its local regularity properties, e.g.,a.s. local boundedness and a.s. continuity, can be studied with the help of its
restrictions to bounded Borel sets (e.g., cubes), the a.s. local boundedness and
For the Gaussian r.f. in a cube
a.s. continuity can be inferred from the continuity of the covariance function;in fact, this result can be extended2 to a large class of Gaussian r.f. in metricspace satisfying some entropy condition (cf. [1, Condition (iii) in Theorem$\langle\phi,$$\phi\rangle_{C}>0$

$C$

$V$

$\mathbb{R}^{d}$$Q\subset \mathbb{R}^{d}.$$Q\subset \mathbb{R}^{d}$$Q$

2See, e.g., the original papers by Fernique [13], Talagrand [23] and the discussion in themonograph by Adler [1], where an extensive bibliography can be found.

V. Chulaevsky, Y. Suhov

4.16]). For brevity, we refer to it as the entropy-regularity, but do not introduce
it formally, for it would take one on a rather long technical detour.
In particular, for stationary Gaussian r.f. on a compact group, and for a
restriction of a stationary G.r. . to a compact subset of a non-compact group $Q,$
both the a.s. boundedness and the a.s. continuity of the samples are equivalent
to the entropy-regularity condition on(cf. [1, Theorem 4.16]).
Next, given a bounded Borel set, introduce the Hilbert space
formed by measurable functions with support in and finite value ofwith the scalar product
$f$

$Q$

$Q\subset \mathbb{R}^{d}$$\mathcal{L}_{C}^{2}(Q)$$Q$$\phi$$\langle\phi,$$\phi\rangle c,$$\langle\phi, \psi\rangle=\int_{Q\cross Q}C(x, y)\overline{\phi(x)}\psi(y)dxdy$

and the norm denoted by
Given an orthonormal basis $\{\eta_{i}, i=1, 2, . . .\}$ in
in in a norm-convergent series$\Vert\cdot\Vert_{C}.$$\mathcal{L}_{C}^{2}(Q)$, one can expand

$V$

$Q$

$V(x; \omega)=\sum_{i\geq 1}c_{i}(\omega)\eta_{i}(x) , x\in Q$

(1.4)

,

. Our crucial
with uncorrelated (hence, independent) Gaussian r.v.
eigenvalue concentration (EVC) bound, resulting in the new, eﬃcient decaybounds on the EF correlators, is based on the decomposition$c_{i}=c_{i}^{(Q)}$$V(x;\omega)=\xi_{Q}(\omega)1_{Q}(x)+\tilde{V}(x;\omega)$

(1.5)

,

which is obtained as a particular case of the expansion (1.4) with $\eta_{1}(x)=$ Const(note that a proper normalization is required for ).
$\eta_{1}$1.3

Assumptions for the Hamiltonian in$\mathbb{R}^{d}$(V1) The centered Gaussian r.f.
generating the external:potential is translation invariant, has varianceand a.s. continuous samples. It satisfies the strong mixing condition with the Rosenblatt coeﬃcient$\delta>0$ (this includes the case where
has finite-range correlations). There exists $C’>0$ such that$V$$\mathbb{R}^{\’{a}}\cross\Omegaarrow \mathbb{R}$$\sigma^{2}>0$$\alpha(r)\leq Ce^{-r^{\delta}},$$V$$\mathbb{E}[(\int_{\{|x|\leq L\}}V(x;\omega)dx)^{2}]\geq C’L^{d}, L\geq 1$

.

(1.6)

Remark 1.1. The role of the condition (1.6) is to ensure the growth of thevariance of the integral of over large balls, resulting in a controllable, satisfiguring in the key
factory growth of the normalization constant in the $r.v.$
decomposition (1.5).
$V$$\xi_{Q}$(U1) The interaction energy,$U(x)=U(x_{1}, \ldots, x_{N})=\sum_{1\leq i<j\leq N}U^{(2)}(|x_{\iota’}-x_{j}$is generated by a two-body interaction potential
bounded and compactly supported:$U^{(2)}$:$\exists r_{0}\in(0, +\infty)\forall r\geq r_{0} U^{(2)}(r)=0.$$\mathbb{R}+arrow \mathbb{R}$which is75

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Localization bounds for multi-particle systems

1.4

Assumptions for the Hamiltonian on a quantum graphis -translation invariant,(V2) The centered Gaussian r.f.
samples. It satisfies theanda.s.
continuous
has uniformly bounded varianceO.
strong mixing condition with the Rosenblatt coeﬃcient$C’>0$
There exists
such that$V:\Gamma^{(N)}\cross\Omegaarrow \mathbb{R}$$\mathbb{Z}^{d}$$\alpha(r)\leq Ce^{-r^{\delta}},$$\mathbb{E}[(\int_{\{|x|\leq L\}}V(x;\omega)d\mu(x))^{2}]\geq C’L^{d}, L\geq 1$

Here $d\mu(x)$ is the measure on the underlying quantum graphthe Lebesgue measures on the edge intervals$I_{e},$.

$\Gamma^{(1)}$$\delta>$(1.7), induced by$e\in \mathcal{E}^{(1)}.$(U2) The interaction energy is generated by a two-body interaction potential:
which is bounded and compactly supported:$U^{(2)}$$\mathbb{R}+arrow \mathbb{R}$$\exists r_{0}\in(0, +\infty)\forall r\geq r_{0} U^{(2)}(r)=0.$2

Main results

2.1

Localization in a Euclidean space

Theorem 1. Let be given integers $d\geq 1,$ $N^{*}\geq 2$ , a stationary Gaussian random field with continuous argument, :, satisfying the assumptions$(Vl)$ , and an interaction potential
satisfying the assumptions $(Ul).$ Consider the $N$ -particle random Anderson Hamiltonian $H^{(N)}(\omega)=-\Delta+V(x;\omega)+$$U(x)$ of the form (1.1).
Then for all $1\leq N\leq N^{*}$ , for any $m>0$ and $\nu>0$ there exists $E^{*}>-\infty$
such that, for$V$$\mathbb{R}^{d}\cross\Omegaarrow \mathbb{R}$$U^{(2)}$$\mathbb{P}-a.e.$(A)$H^{(N)}(\omega)$$\omega,$has pure point spectrum in $(-\infty, E^{*}$ ];(B) all eigenfunctions
exponentially:of$\psi_{j}(\omega)$$H^{(N)}(\omega)$with eigenvalues in $(-\infty, E^{*}$ ] decay$|\psi_{j}(x, \omega)|\leq C_{j}(\omega)e^{-m|x|},$for some$C_{j}(\omega)\in(0, \infty)$;(C) $H^{(N)}(\omega)$ features strong dynamical localization in $(-\infty, E^{*}$ ]: there is a non-random number $\kappa>0$ such that$\forall x, y\in(\mathbb{R}^{d})^{N} \Upsilon_{x_{:}y}:=\mathbb{E}[:.]$$\leq e^{-\nu(\rho_{S}(x,y))^{\kappa}}$

Here $P_{(-\infty,E^{*}]}(H^{(N)})$ is the spectral projection for the operator $H^{(N)}$ on theinterval $(-\infty, E^{*}$ ].
In contrast to the works [6, 19], we do not assume the non-negativity of theinteraction potential. The latter condition has been used (cf. [6, 18, 19]) in the
proofs of multi-particle Anderson localization at $/extreme”$ energies, viz. in
the Lifshitz tails zone. In fact, in absence of the strong disorder condition, theV. Chulaevsky, Y. Suhov

positivity of the interaction is a convenient (and so far, the only) tool allowing
one to establish the initial-scale MSA bounds for multi-particle systems, reducing them to their single-particle counterparts. The specificity of the Gaussian(and, more generally, lower-unbounded) potentials is that the localization mechanism at suﬃciently low, negative energies is quite similar to that responsible
for the onset of localization under strong disorder, regardless of how small is the(nonzero) amplitude of the random potential.
But a more important improvement comes from the fact that the decayof the EF correlators is now proved with respect to the (symmetrized) normdistance, and not in the Hausdorﬀ distance, used explicitly in [2] (in the latticesystems) and in [18, 19] (in lattice and continuous systems). Implicitly, thedecay estimates related to the Hausdorﬀ distance appeared also in our earlier
papers [6, 10]. While the probabilistic decay estimates for the eigenfunctions
and their correlators ultimately give rise to spectral and dynamical localization in an infinitely extended configuration space, they do not imply directly
reasonable localization bounds in arbitrarily large, but bounded regions of the(single-particle) configuration space. This makes them much less suitable forthe applications to the physical models, e.g., in the solid state physics, where adisordered sample has always a finite size.

Remark 2.1. Theorem 1 establishes strong dynamical localization for $N\leq N^{*},$
including the case where $N=1$ . Earlier, Fischer et al. [15] proved spectral 10with Gaussian random
calization for the 1-particle Anderson Hamiltonian in$\mathbb{R}^{d}$potential (of continuous argument, as in our model), under a slightly more
restrictive hypothesis on the covariance function C. They also used the fixedenergy MSA induction. The derivation of spectral localization was obtained, as
in the paper by von Dreifus and Klein [12] (devoted to the 1-particle Anderson lattice model with Gaussian potential), with the help of the Simon Wolf
criterion [22]. Unfortunately, the latter deep, remarkable result$\bullet$$\bullet$does not lead directly to the energy-interval localization bounds, suﬃcientfor the proof of the dynamical localization, andhas not been extended so far to the multi-particle systems.

Note also that our technique allows to infer some energy-interval estimates fromthe fixed-energy analysis performed in [15], thus proving strong dynamical 10calization under the hypotheses of [15], but the resulting decay rate of theeigenfunction correlators would be significantly slower than in Theorem 1.

Localization in a quantum graph

2.2

Theorem 2. Let be given integers $d\geq 1,$ $N^{*}\geq 2$ . Consider the $N$ -graph-stationary Gaussian random
over the single-particle quantum graph$(V2)$ , an interaction potenassumptions, satisfying thefield :
$N$
satisfying the assumptions $(U2)$ , and the -particle random Anderson
tial
Hamiltonian $H^{(N)}(\omega)=-\Delta+V(x;\omega)+U(x)$ .
Then for all $1\leq N\leq N^{*}$ , for any $m>0$ and $\nu>0$ , there exists $E^{*}>-\infty$
such that, for$\Gamma^{(N)}$$\Gamma^{(1)},$$V$$a\mathbb{Z}^{d}$$\Gamma^{(1)}\cross\Omegaarrow \mathbb{R}$$U^{(2)}$$\mathbb{P}-a.e.$(A)$H^{(N)}(\omega)$$\omega,$has pure point spectrum in $(-\infty, E^{*}$ ];

77

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Localization bounds for multi-particle systems(B) all eigenfunctions
exponentially:of$\psi_{j}(\omega)$with eigenvalues in$H^{(N)}(\omega)$$(-\infty, E^{*}$] decay$|\psi_{j}(x, \omega)|\leq C_{j}(\omega)e^{-m|x|},$(C)for some$C_{j}(\omega)\in(0, \infty)$$H^{(N)}(\omega)$features strong dynamical localization in

random number$\kappa>0$;$(-\infty, E^{*}$]: there is a non-such that$\forall x, y\in\Gamma^{(N)} T_{x_{:}y} :=\mathbb{E}[\sup_{t\in \mathbb{R}}\langle 1_{x}|P_{(-\infty,E^{*}]}(H^{(N)})e^{-itH^{(N)}}|1_{y}\rangle]$$\leq e^{-\nu}(\rho_{S}(x,y))^{\kappa}$

Here, comparing our results with those obtained in the work by Sabri [21],we make similar remarks.
First of all, our decay bounds are established with respect to the (symmetrized) norm distance. Secondly, the external potential does not have thediscrete, alloy-type structure, but is generated by a random field with continuous argument. Finally, we do not assume the non-negativity of the interaction
potential.

3

The main strategy of the proofs

Although the geometric nature of the configuration space is quite diﬀerent forthe multi-particle systems in a Euclidean space and in a quantum graph, themain distinction of the techniques used in our work resides in a fairly generalprobabilistic argument, for which the Gaussian random potentials represent thesimplest case.

3.1

Eﬃcient multi-particle EVC bounds

Definition 3.1. An $N$ -particle ballin the $\max$ -norm distance is called
weakly separatedfrom
iﬀ there exists a single-particle ball of diameter$\leq 2NL$ and index subsets(possibly
such that
and$B_{L}^{(N)}(x)$$B_{L}^{(N)}(y)$$\mathcal{J}_{1},$$Q$$\mathcal{J}_{2}\subset\{1, .

.

.

, N\}$$|\mathcal{J}_{1}|>|\mathcal{J}_{2}|$$\mathcal{J}_{2}=\emptyset)$$\Pi_{\mathcal{J}_{1}}B_{L}^{(N)}(x)\cup\Pi_{\mathcal{J}_{2}}B_{L}^{(N)}(y)\subset Q,$(3.1)$\Pi_{\mathcal{J}_{1}^{C}}B_{L}^{(N)}(x)\cup\Pi_{\mathcal{J}_{2}^{C}}B_{L}^{(N)}(y)\cap Q=\emptyset.$,
A pair of balls
is called weakly separated iﬀ at least one ofthese balls is weakly separated from the other.
$B_{L}^{(N)}(x)$$B_{L}^{(N)}(y)$Lemma 3 (Cf. [5, Lemma 3.7]). Any pair of balls$\rho_{S}(x, y)>3NL$ is weakly separated.

$B_{L}^{(N)}(x)$,$B_{L}^{(N)}(y)$with,
We will $cal13NL$-distant the ballswith $\rho_{S}(x, y)>3NL.$
brevity,For
we formulate below one statement, which applies to the modelin the Euclidean space and in the quantum graph. In both cases, the main
conditions ((U1) & (V1) or, respectively, (V2) & (U2)) upon the potentials
andare implicitly assumed.
$B_{L}^{(N)}(x)$$B_{L}^{(N)}(y)$$V$$U^{(2)}$

V. Chulaevsky, Y. Suhov

Theorem 4. Letandbe two weakly separated balls. Denoteby,andthe spectra of the respective Hamiltonians. Fix $\beta\in(0,1)$ .
Then for all $L>0$ large enough, the following bound holds true:$B_{L}^{(N)}(y)$$B_{L}^{(N)}(x)$$\Sigma_{x_{:}L}(\omega)$$\Sigma_{y,L}(\omega)$$H_{B_{L}(x)}(\omega)$$H_{B_{L}(y)}(\omega)$$\mathbb{P}\{$

dist ((3.2)$\Sigma_{x_{:}L}(\omega), \Sigma_{y_{j}L}(\omega))\leq e^{-L^{\beta}}\}\leq e^{-L^{\beta/2}}$

The estimate (3.3) is a fairly straightforward adaptation to Gaussian r.f.
with continuous argument of an EVC bound given in [5, Lemma 3.8] in the caseof lattice Hamiltonians.
On account of Lemma 3, we infer from Lemma 4 the key EVC estimate (cf.
[5, Theorem 2.1]).
Theorem 5. Letandbe two $3NL$ -distant cubes, and,the spectra of the respective Hamiltonians
Then for all $L>0$ large enough, the following bounds holds true:$B_{L}^{(N)}(y)$$B_{L}^{(N)}(x)$$H_{B_{L}(x)}(\omega)$$\Sigma_{y_{j}L}(\omega)$$\mathbb{P}\{$

dist ($\Sigma_{x,L}(\omega)$$H_{B_{L}(y)}(\omega)$

,

.

(3.3)

$\Sigma_{x.L}(\omega), \Sigma_{y_{:}L}(\omega))\leq e^{-L^{\beta}}\}\leq e^{-L^{\beta/2}}$

The main tool for the proof of the above Wegner-type estimate is the decomposition of a Gaussian r.f. in a subsetof the underlying configuration
space of finite measure:$Q$$V(x;\omega)1_{Q}(x)=\xi_{Q}(\omega)1_{Q}(x)+\tilde{V}_{Q}(x;\omega)$

(3.4)

,

).
where the random variable
is independent of the ’ fluctuation“ field$:=$
Such a decomposition can be obtained by expanding the restriction$VrQ$ in an orthogonal series
over an orthonormal basis, within the Hilbert space
Therefore, conditional on, the sample mean
admits (Gaussian) bounded probability density. Given two $3NL$ -distant cubes, such a decomposition,with a proper choice of the subset depending upon $x,$ and , gives rise to,,a simple representation of eigenvalues of the operators
and ultimately to the upper bound (3.3).
$\tilde{V}$$\omega$$\xi_{Q}$$V_{Q}$$V_{Q}= \sum_{n\geq 0}c_{n}(\omega)\eta_{n}(x)$$\{\eta_{n}(\cdot)\}$$\eta 0=con\mathcal{S}t1_{Q}.$$\mathcal{L}_{C}^{2}(Q)-$$V_{Q}$$\xi_{Q}$

$Q$

$y$

$L$

$H_{B_{L}(x)}(\omega)$

3.2$H_{B_{L}(y)}(\omega)$

The scaling scheme

The exponential decay of the EFs with eigenvalues in the localization energy, can be proved with the help
band $(-\infty, E^{*}$ ], with suﬃciently large negativeof the MPMSA procedure simpler that the one used in [6, 10], with scale lengths$1<\alpha<2.$
defined by the recursive equation
More precisely, instead of the variable-energy scaling analysis with $L_{k+1}=$, used in [6, 10] and essentially going back to its single-particle variant
developed by von Dreifus and Klein [11], we carry our first the fixed-energy
MSA induction with a multiplicative growth of the scale lengths: $L_{k+1}=YL_{k},$. The advantages of the multiplicative length scale sequence havebeen demonstrated by Germinet and Klein [16]; in particular, it leads easily toa sub-exponential decay of the EF correlators.
On the other hand, a second MSA induction is still required, with exponen, to prove exponential decay of the localized
tial growth of the scales,$E^{*}$$L_{k+1}=[L_{k}^{\alpha}],$$[L_{k}^{\alpha}]$$\mathbb{N}\ni Y\geq 2$$L_{k+1}=[L_{k}^{\alpha}]$

79

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Localization bounds for multi-particle systems

eigenfunctions, for the multiplicative scheme proves only the sub-exponential
decay of EFs.
Note that Klein and Nguyen [18, $19|$ , adapting to the multi-particle systems
the bootstrap MSA developed by Germinet and Klein [16] for the single-particle
models, proved both exponential decay of eigenfunctions and sub-exponential
decay of EF correlators in the localization energy zone, in lattice and continuous
systems. As usual in the bootstrap MSA, one starts with fairly weak assump, and then infers
tions on the localization properties in a finite cube of size
from them much stronger properties at arbitrary large scales . This requires
the total of four interconnected multi-scale analyses. However, the EVC bounds,
and as a result, the decay bounds are obtained in [18, 19] with respect to the
Hausdorﬀ and not the norm distance.
It is worth noticing that Klein and Nguyen addressed in [19] a hard analyticalproblem related to the lack of the so-called complete covering condition for thealloy-type external random potential (considered in their work); such a condition
was assumed in [6], and this made substantially simpler the crucial EVC bound.
Klein and Nguyen adapted to the multi-particle setting the quantitative \’uniquecontinuation principle, thus extending the MPMSA technique to a much larger
class of alloy potentials than in [6]. In the present work, we simply do not
encounter such a functional-analytical problem. Pictorially, the Gaussian r.f.
with continuous argument is ’ omnipresent “ in the configuration space, while analloy-type potential may aﬀect only a subset thereof. The representation (3.4)
provides a very clear formalization of this informal argument.
In our work, we do not aim at the sharpest probabilistic decay estimates
but focus on more eﬃcient decay bounds in terms of the (symmetrized) normdistance. This allows us to make use of only two multi-scale analyses-one withthe multiplicative scales $L_{k+1}=YL_{k}$ and another one with the faster growing
scalesThe main ノ EVC bound provided by Theorem 5 and valid for $al13NL$-distantcubes (in the symmetrized norm-distance) has another advantage: it simplifies the multi-particle scale induction, which becomes much closer to its singleparticle counterpart than in [6, 10, 21] and in [18, 19].
$L_{0}$$L_{k}$$L_{k+1}=[L_{k}^{\alpha}].$Acknowledgements
VC thanks the Gakushuin University of Tokyo, the Kyoto University and the
RIMS (Tokyo) for the warm hospitality in December 2013, and Prof. F. Nakano,
S. Kotani and N. Minami for stimulating discussions of localization propertiesof disordered quantum systems. YS thanks IME USP, Brazil, for the warm
hospitality during the academic year of 2013-4.

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