A Wegner Estimate for Multi-Particle Random Hamiltonians
We prove aWegner estimate for a large class of multi-particle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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| Цитувати: | A Wegner Estimate for Multi-Particle Random Hamiltonians / W. Kirsch // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 121-127. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1064972016-09-30T03:02:47Z A Wegner Estimate for Multi-Particle Random Hamiltonians Kirsch, W. We prove aWegner estimate for a large class of multi-particle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper. 2008 Article A Wegner Estimate for Multi-Particle Random Hamiltonians / W. Kirsch // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 121-127. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106497 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We prove aWegner estimate for a large class of multi-particle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper. |
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Article |
| author |
Kirsch, W. |
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Kirsch, W. A Wegner Estimate for Multi-Particle Random Hamiltonians Журнал математической физики, анализа, геометрии |
| author_facet |
Kirsch, W. |
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Kirsch, W. |
| title |
A Wegner Estimate for Multi-Particle Random Hamiltonians |
| title_short |
A Wegner Estimate for Multi-Particle Random Hamiltonians |
| title_full |
A Wegner Estimate for Multi-Particle Random Hamiltonians |
| title_fullStr |
A Wegner Estimate for Multi-Particle Random Hamiltonians |
| title_full_unstemmed |
A Wegner Estimate for Multi-Particle Random Hamiltonians |
| title_sort |
wegner estimate for multi-particle random hamiltonians |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/106497 |
| citation_txt |
A Wegner Estimate for Multi-Particle Random Hamiltonians / W. Kirsch // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 121-127. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT kirschw awegnerestimateformultiparticlerandomhamiltonians AT kirschw wegnerestimateformultiparticlerandomhamiltonians |
| first_indexed |
2025-07-07T18:34:04Z |
| last_indexed |
2025-07-07T18:34:04Z |
| _version_ |
1837014183386808320 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 1, pp. 121�127
A Wegner Estimate
for Multi-Particle Random Hamiltonians
Werner Kirsch
Institut f�ur Mathematik and SFB TR 12, Ruhr-Universit�at Bochum
D-44780 Bochum, Germany
E-mail:werner.kirsch@ruhr-uni-bochum.de
Received July 12, 2007
We prove a Wegner estimate for a large class of multi-particle Anderson
Hamiltonians on the lattice. These estimates will allow us to prove Anderson
localization for such systems. A detailed proof of localization will be given
in a subsequent paper.
Key words: Anderson model, localization, Wegner estimate.
Mathematics Subject Classi�cation 2000: 82B44, 35R60, 35P20.
Dedicated to V. Marchenko and L. Pastur
1. Introduction
Wegner estimates originate in the famous paper [10]. There, Wegner proved
among other things that the integrated density of states for the Anderson Hamil-
tonian has a bounded density provided the probability distribution of the random
potential itself has a bounded density. This implies in particular an upper bound
on the probability that an Anderson Hamiltonian on a �nite box has eigenvalues
close to a given energy E.
Wegner's estimates play a key role in the multiscale method to prove Anderson
localization (see, e.g. [5] or [4]). Only recently Bourgain and Kenig [1] proved
Anderson localization for a Bernoulli model without an a priori Wegner estimate;
they prove a Wegner-type estimate inductively within the multiscale scheme.
Wegner's original work was restricted to lattice models. However, the estimate
was also proven for the continuum (see [3] for a recent rather optimal result and
[9] for a review on this subject).
In this note we prove a Wegner estimate for a multi-particle Anderson model.
In a subsequent paper we will also do multiscale analysis for this model. The �rst
c
Werner Kirsch, 2008
Werner Kirsch
Wegner estimate for a multi-particle random Hamiltonian was proved by Zenk [11].
Chulaevsky and Suhov [2] develope a multiscale analysis for certain (1-d) two-body
Hamiltonians. In this paper the Wegner estimate requires strong conditions on
the probability density of the random potential (e.g. analyticity). It was one of
the motivations of the present note to avoid these strong assumptions.
The method of proof applied here is close to Wegner's original idea and was
developed from the paper [6]. Note that there is a re�nement of this method by
Stollmann [8] which is likely to work in the multi-particle case as well.
We note that the method presented in this paper will also work for alloy-type
models in the continuous case. The necessary changes can be read o� from the
paper [6]. However, in the continuous case we get the volume factor of the bound
with an exponent 2. This su�ces to do a multiscale analysis, but it gives no result
for the regularity of the integrated density of states.
2. Models and Results
We will deal with a system of N interacting particles on a lattice Zd. We con-
sider these particles on the full Hilbert space, disregarding Fermionic or Bosonic
symmetry. Physically speaking we deal with distinguishable particles. Since the
full Hilbert space is a direct sum of the irreducible subspaces with respect to SN -
symmetry (including the totally symmetric and the totally antisymmetric sub-
spaces) the Wegner estimates for Fermions and Bosons follow immediately from
the result on the full space.
The one-particle Hilbert space we consider is `2(Zd) and the Hilbert space for
N (distinguishable) particles is consequently `2(ZNd). Any (bounded) operator A
on these Hilbert spaces is uniquely de�ned through its matrix elements A(x; y) =
(Æx; A Æy); where Æz is the vector in `2 with component 1 at lattice site z and 0
otherwise.
We write the lattice site x 2 Z
Nd as x = (x1; : : : ; xN ), where xi 2 Z
d denotes
the coordinates of the ith particle.
Each particle (with coordinates �) is the subject to a random potential v!(�)
which is the same for all particles. The random potential v!(�) consists of in-
dependent identically distributed random variables. Throughout we assume that
the distribution of the v(�) has a bounded density �(v). We denote the underlying
probability measure by P and the expectation with respect to P by E .
The kinetic energy operator for one particle is given by
h0 u(�) =
X
jnj=1; n2Zd
u(� + n); � 2 Z
d; (2.1)
the single particle Hamiltonian is consequently
h! = h0 + v!: (2.2)
122 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
A Wegner Estimate for Multi-Particle Random Hamiltonians
If h is a one-particle operator acting in `2(Zd); we denote by h(i) the corre-
sponding operator on `2(ZNd) acting on the ith particle only, more precisely: if h
has matrix elements h(�; �); then
h(i) u(x1; : : : ; xn) =
X
�2Zd
h(xi; �)u(x1; : : : ; xi�1; �; xi+1; : : : ; xN ): (2.3)
In other words,
h(i) = 1`2(Zd)
: : :
1`2(Zd)| {z }
i�1 times
h
1`2(Zd)
: : :
1`2(Zd)| {z }
n�i�1 times
: (2.4)
The N-particle Hamiltonian without interaction is de�ned by
H!; 0 =
NX
i=1
h(i): (2.5)
The interaction term U can be a rather general function on ZNd. We assume it
to be bounded for simplicity. We also suppose that U is a deterministic function,
it would be su�cient for our purpose to have U independent of v!. In most cases
U is a pair potential of the form U(x) =
P
i6=j u(xi � xj).
The N-particle Hamiltonian with interaction U is then given by
H!;U = H!; 0 + U: (2.6)
We will deal with this operator restricted to a bounded (hence �nite) do-
main �. The number of elements of � will be denoted by j� j.
We call a subset R of Zd a rectangle if
R = f� 2 Z
d
j L� � �� �M� for � = 1 : : : N g: (2.7)
A rectangular domain in ZNd is a set � of the form
� = �1 � �2 � : : :� �N ; (2.8)
where the �i are rectangles in Z
d. We use the notation �i(�) = �i. We call
a rectangular domain � regular if for all i; j = 1; : : : ; N either �i \ �j = ; or
�i = �j.
For any subset � of ZNd we de�ne the operator H� = H�
!;U by its matrix
elements
H� (x; y) = H!;U (x; y) for x; y 2 �: (2.9)
The main result of this note is the following Wegner estimate for multi-particle
operators:
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 123
Werner Kirsch
Theorem 2.1. If � is a regular rectangular domain, then
P
�
dist
�
�
�
H�
�
; E
�
< �
�
� C jj � jj1 j� j �: (2.10)
The assumption of regularity of the set � can be avoided. However, the proof
is more transparent with this assumption. The proof of Anderson localization by
multiscale analysis, which we will present in a forthcoming paper, will deal with
regular domains only.
The proof of Theorem 2.1 implies also that the integrated density of states
has a bounded density. This result can also be read o� from the explicitly known
form of the integrated density of states (see [7]).
3. Proof
We prove Theorem 2.1. Let � = �1 � �2 � � � � �N . We may assume that
�1 = �2 = � � � = �K (3.1)
and �1 \ �i = ; for all i > K: (3.2)
We denote the eigenvalues of H� by En = En(H
�). We order them so that
E1 � E2 � : : : and repeat any eigenvalue according to its multiplicity. The
eigenvalue counting function is denoted by
N(H�; E) = #fEn(H
�) � Eg: (3.3)
We will need the following Lemma:
Lemma 3.1. Suppose (3.1) and (3.2) hold. Denote by v(�) the value of the
random potential v! evaluated at the lattice site � 2 Z
d
. Then
X
�2�1
@En(H
�)
@v(�)
= K: (3.4)
P r o o f. Set V (x) =
PN
i=1 v(xi) Then for � 2 �1:
@V
@v(�)
(x1; : : : ; xN ) =
KX
i=1
� xi : (3.5)
Hence for each (x1; : : : ; xN ) 2 � we have
X
�2�1
@V
@v(�)
(x1; : : : ; xN ) =
X
�2�1
KX
i=1
� xi = K: (3.6)
124 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
A Wegner Estimate for Multi-Particle Random Hamiltonians
Let us denote by n the normalized eigenfunction of H� for the eigenvalue
En = En(H
�). The Feynman�Hellman theorem tells us that
@En
@v(�)
= h n;
@V
@v(�)
n i =
X
x2�
j n(x) j
2 @V
@v(�)
: (3.7)
Thus from (3.6) we obtain
X
�2�1
@En
@v(�)
=
X
x2�
j n(x) j
2
� X
�2�1
@V (x)
@v(�)
�
(3.8)
= K
X
x2�
j n(x) j
2 = K (3.9)
since n is normalized.
Let ' be an increasing C1�function on R, 0 � ' � 1 with ' = 1 on (�;1) and
' = 0 on (�1;��).
Then:
P
�
dist
�
�
�
H�
�
; E
�
< �
�
(3.10)
� E
�
N(H�; E + �) � N(H�; E � �)
�
(3.11)
= E
�
tr
�
�(E��;E+�](H
�)
� �
(3.12)
� E
�
tr
�
'(H�
�E + 2�) � ' (H�
�E � 2�
��
(3.13)
� E
� Z 2�
�2�
tr
�
'0(H�
�E + t)
�
dt
�
(3.14)
by Lemma 3.1:
�
1
K
X
n
2�Z
�2�
E
�
'0
�
En(H
�) �E + t
� X
�2�1
@En(H
�)
@v(�)
�
dt (3.15)
�
X
n
2�Z
�2�
X
�2�1
E
� @ '
�
En(H
�) �E + t
�
@v(�)
�
dt: (3.16)
Since E is a product measure, we can split it into an integration over v(�) which
we write as
R
� �(v) dv and the expectation with respect to the other random
variables, which expectation we denote as E�
v(�)
.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 125
Werner Kirsch
With this notation (3.16) equals
X
n
2�Z
�2�
dt
X
�2�1
E
�
v(�)
� Z @'(Em(H
� �E + t)
@v(�)
�
�
v(�)
�
dv(�)
�
(3.17)
� jj � jj
X
n
2�Z
�2�
dt
X
�2�1
E
�
v(�)
� Z @'(Em(H
� �E + t)
@v(�)
dv(�)
�
: (3.18)
By the fundamental theorem of calculus we have
Z
@'
�
En(H
�)�E + t
�
@v(�)
dv(�) (3.19)
= '
�
En(H
�
v(�)=max)�E + t
�
� '
�
En(H
�
v(�)=min)�E + t
�
; (3.20)
where H�
v(�)=max
(resp. H�
v(�)=min
) denotes the operator H� with the potential
v(�) set to its maximal (resp. minimal) value, i.e. with v(�) = sup (supp(�))
or v(�) = inf (supp(�)). Note that we include the cases sup (supp(�)) = 1 and
inf (supp(�)) = �1.
Changing v(�) from its minimal to its maximal value is a (positive) perturba-
tion of rank at most M = K
j�j
j�1j
. Thus
En(H
�
v(�)=min) � En(H
�
v(�)=max) � En+M (H�
v(�)=min): (3.21)
To estimate (3.18) we use the following simple Lemma:
Lemma 3.2. Let ' be a nondecreasing function on R with 0 � ' � 1. If an
and bn are nondecreasing sequences satisfying an � bn � an+M for all n, thenX
n
�
'(bn)� '(an)
�
� M: (3.22)
Combining the above estimates we get
P
�
dist
�
�
�
H�
�
; E
�
< �
�
(3.23)
� jj � jj
X
n
2�Z
�2�
dt
X
�2�1
E
�
v(�)
� Z @'(Em(H
� �E + t)
@v(�)
dv(�)
�
(3.24)
� jj � jj
2�Z
�2�
dt
X
�2�1
E
�
v(�)
X
n
�
'(En(H
�
v(�)=max)� '(En(H
�
v(�)=min)
�
� jj � jj 4� j�j: (3.25)
126 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
A Wegner Estimate for Multi-Particle Random Hamiltonians
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Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 127
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