The Conductivity Measure for the Anderson Model
We study the ac-conductivity in linear response theory for the Anderson tight-binding model. We de ne the electrical ac-conductivity and calculate the linear-response current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended s...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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irk-123456789-1064982016-09-30T03:02:50Z The Conductivity Measure for the Anderson Model Klein, A. Müller, P. We study the ac-conductivity in linear response theory for the Anderson tight-binding model. We de ne the electrical ac-conductivity and calculate the linear-response current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended states are believed to exist. 2008 Article The Conductivity Measure for the Anderson Model / A. Klein, P. Müller // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 128-150. — Бібліогр.: 29 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106498 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study the ac-conductivity in linear response theory for the Anderson tight-binding model. We de ne the electrical ac-conductivity and calculate the linear-response current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended states are believed to exist. |
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Klein, A. Müller, P. |
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Klein, A. Müller, P. The Conductivity Measure for the Anderson Model Журнал математической физики, анализа, геометрии |
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Klein, A. Müller, P. |
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Klein, A. |
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The Conductivity Measure for the Anderson Model |
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The Conductivity Measure for the Anderson Model |
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The Conductivity Measure for the Anderson Model |
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The Conductivity Measure for the Anderson Model |
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The Conductivity Measure for the Anderson Model |
| title_sort |
conductivity measure for the anderson model |
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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/123456789/106498 |
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The Conductivity Measure for the Anderson Model / A. Klein, P. Müller // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 128-150. — Бібліогр.: 29 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT kleina theconductivitymeasurefortheandersonmodel AT mullerp theconductivitymeasurefortheandersonmodel AT kleina conductivitymeasurefortheandersonmodel AT mullerp conductivitymeasurefortheandersonmodel |
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2025-07-07T18:34:10Z |
| last_indexed |
2025-07-07T18:34:10Z |
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1837014189551386624 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 1, pp. 128�150
The Conductivity Measure for the Anderson Model
Abel Klein
�
Department of Mathematics, University of California
Irvine, CA 92697-3875, USA
E-mail:aklein@math.uci.edu
Peter M�uller
Institut f�ur Theoretische Physik
Friedrich-Hund-Platz 1, Georg-August-Universit�at, 37077 G�ottingen, Germany
E-mail:peter.mueller@physik.uni-goe.de
Received September 22, 2007
We study the ac-conductivity in linear response theory for the Anderson
tight-binding model. We de�ne the electrical ac-conductivity and calculate
the linear-response current at zero temperature for arbitrary Fermi energy.
In particular, the Fermi energy may lie in a spectral region where extended
states are believed to exist.
Key words: Anderson model, random Schr�odinger operator, conductivity,
linear response theory.
Mathematics Subject Classi�cation 2000: 82B44 (primary); 47B80, 60H25
(secondary).
Dedicated to Leonid A. Pastur on the occasion of his 70th birthday
1. Introduction
In this article we study the ac-conductivity in linear response theory for
the Anderson tight-binding model. We de�ne the electrical ac-conductivity and
calculate the linear-response current at temperature T = 0 for arbitrary Fermi
energy �.
At temperature T = 0, if the Fermi energy � is either in the region of lo-
calization or outside the spectrum of the random Schr�odinger operator, this was
already done in [KlLM] by a careful mathematical analysis of the ac-conductivity
in linear response theory, following the approach of [BoGKS], and the introduc-
tion of a new concept, the conductivity measure. This approach can be easily
extended to the nonzero temperature case, T > 0, with � (here the chemical
�Supported in part by NSF Grant DMS-0457474.
c
A. Klein and P. M�uller, 2008
The Conductivity Measure for the Anderson Model
potential) arbitrary. The conductivity measure �T
� (d�), with � the frequency of
the applied electric �eld, is a �nite positive even Borel measure on the real line.
If �T
� (d�) was known to be an absolutely continuous measure, the in-phase or
active conductivity Re �T� (�) would then be well-de�ned as its density. The con-
ductivity measure �T
� (d�) is thus an analogous concept to the density of states
measure N (dE), whose formal density is the density of states n(E). Given a
spatially homogeneous, time-dependent electric �eld E(t), the in-phase linear-
response current at time t, J in
lin(t;�; T;E), has a simple expression in terms of this
conductivity measure:
J in
lin(t;�; T;E) =
Z
R
�T
� (d�) ei�t bE(�): (1.1)
This procedure is conjectured to break down at T = 0 for, say, Fermi ener-
gies � in the region of extended states. In this case there has been no suitable
derivation of the in-phase linear-response current. In this paper we de�ne the
conductivity measure �0
�(d�) and the in-phase linear-response current for arbi-
trary Fermi energy �. We give an explicit expression for �0
�(d�), and justify the
de�nition by proving that
�0
�(d�) = lim
T#0
�T
� (d�) weakly for Lebesgue-a.e. � 2 R : (1.2)
The in-phase linear-response current is then de�ned by (1.1), and justi�ed by
J in
lin(t;�; 0;E) = lim
T#0
J in
lin(t;�; T;E) for Lebesgue-a.e. � 2 R : (1.3)
Acknowledgement. This paper originated from discussions with Leonid
A. Pastur, to whom this paper is dedicated on the occasion of his 70th birthday.
Pastur is a founding father of the theory of random Schr�odinger operators; of
particular relevance to this paper is his work on the electrical conductivity, e.g.,
[BeP, P1, P2, LGP, KP, P3, P4, KiLP]. The Authors also thank Olivier Lenoble
for many discussions.
2. De�nitions and Results
The Anderson tight-binding model is described by the random Schr�odinger op-
erator H, a measurable map ! 7! H! from a probability space (
;P) (with
expectation E ) to bounded selfadjoint operators on `2(Zd), given by
H! := ��+ V!: (2.1)
Here � is the centered discrete Laplacian,
(�')(x) := �
X
y2Zd; jx�yj=1
'(y) for ' 2 `2(Zd); (2.2)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 129
A. Klein and P. M�uller
and the random potential V consists of independent, identically distributed ran-
dom variables fV (x);x 2 Zdg on (
;P), such that the common single site proba-
bility distribution has a bounded density � with compact support.
The Anderson Hamiltonian H given by (2.1) is Zd-ergodic, and hence its spec-
trum, as well as its spectral components in the Lebesgue decomposition, are given
by nonrandom sets P-almost surely [KiM, CL, PF]. This nonrandom spectrum
will be denoted by S, with S{ , { = pp, ac, sc, denoting its nonrandom spectral
components.
We now outline the derivation of electrical ac-conductivities within linear re-
sponse theory for the Anderson model. We refer to [BoGKS] and [KlLM] for
mathematical details, generalizations and proofs.
At the reference time t = �1, the system is assumed to be in thermal equi-
librium at absolute temperature T > 0 and chemical potential � 2 R. On the
single-particle level, this equilibrium state is given by the random operator fT� (H),
where
fT� (E) :=
8<:
�
e
E��
T +1
��1
if T > 0
�
]�1;�](E) if T = 0
(2.3)
stands for the Fermi function. By �B we denote the indicator function of the set B.
A spatially homogeneous, time-dependent electric �eld E(t) is then introduced
adiabatically: Starting at time t = �1, we switch on the (adiabatic) electric
�eld E�(t) := e�t E(t) with � > 0, and then let � ! 0.
On account of isotropy we assume without restriction that the electric �eld
is pointing in the x1-direction: E(t) = E(t)bx1, where E(t) is the (real-valued)
amplitude of the electric �eld, and bx1 is the unit vector in the x1-direction. Our
precise requirements for the real-valued, time-dependent amplitude E(t) are stated
in the following assumption, which we assume valid from now on.
Assumption (E). The time-dependent amplitude E(t) of the electric �eld is
of the form
E(t) =
Z
R
d� ei �t bE(�); (2.4)
where bE 2 C(R) \ L1(R) with bE(�) = bE(��).
For each � > 0 this procedure results in a time-dependent random Hamiltonian
H!(�; t) := G(�; t)H!G(�; t)
�; with G(�; t) := eiX1
R t
�1
ds e�s E(s); (2.5)
where X1 stands for the operator of multiplication by the �rst coordinate of the
electron's position. H!(�; t) is, of course, gauge equivalent to H! + e�t E(t)X1.
130 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
At time t, the state of the system is described by the random operator %!(�; t),
the solution to the Liouville equation(
i@t%!(�; t) = [H!(�; t); %!(�; t)]
lim
t!�1
%!(�; t) = fT� (H!)
: (2.6)
The adiabatic electric �eld generates a time-dependent electric current.
Thanks to re�ection covariance in all but the �rst direction, the current is also
oriented along the �rst coordinate axis. Its amplitude is
J�(t;�; T; E) = �T �%!(�; t) _X1(t)
�
; (2.7)
where T is the trace per unit volume (see (A.14) and (A.15) in App. A) and _X1
is the �rst component of the velocity operator:
_X1 := i[H!;X1] = i[��;X1]: (2.8)
Note that we are using the Schr�odinger picture in (2.7). The time dependence of
the velocity operator _X1(t) := G(�; t) _X1G(�; t)
� there results from our particular
gauge. Finally, the adiabatic linear-response current is de�ned as
J�;lin(t;�; T; E) := d
d�
J�(t;�; T; �E)
��
�=0
: (2.9)
The detailed analysis in [BoGKS] shows that one can give a mathematical
meaning to the formal procedure leading to (2.9), for �xed temperature T > 0
and chemical potential � 2 R, if the corresponding thermal equilibrium random
operator fT� (H) satis�es the condition
E
�
X1 f
T
� (H!)Æ0
2 <1; (2.10)
where fÆaga2Zd is the canonical orthonormal basis in `2(Zd): Æa(x) = 1 if x = a
and Æa(x) = 0 otherwise. (This is the condition originally identi�ed in [BES].)
The derivation of a Kubo formula for the ac-conductivity [BES, SB, BoGKS]
requires normed spaces of measurable covariant operators. The required math-
ematical framework is described in App. A; here we will be somewhat informal.
K2 is the Hilbert space of measurable covariant operators A on `2(Zd), i.e., mea-
surable, covariant maps ! 7! A! from the probability space (
;P) to operators
on `2(Zd), with inner product
hhA;Bii := E
�hA!Æ0; B!Æ0i
= T fA�!B!g (2.11)
and norm jjjAjjj2 :=
phhA;Aii. Here T , given by T (A) := EfhÆ0 ; A!Æ0ig, is the
trace per unit volume. The Liouvillian L is the (bounded in the case of the
Anderson model) selfadjoint operator on K2 given by the commutator with H:
(LA)! := [H!; A!]: (2.12)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 131
A. Klein and P. M�uller
We also introduce operators HL and HR on K2 given by left and right multipli-
cation by H:
(HLA)! := H!A! and (HRA)! := A!H!: (2.13)
Note that HL and HR are commuting, bounded (for the Anderson Hamiltonian),
selfadjoint operators on K2, anti-unitarily equivalent (see (A.10)), and L = HL�
HR. It follows from the Wegner estimate for the Anderson Hamiltonian that in
this case the operators HL and HR have purely absolutely continuous spectrum
(see Lem. 1 in Sect. 3). For each T > 0 and � 2 R we consider the bounded
selfadjoint operator FT
� in K2 given by
FT
� := fT� (HL)� fT� (HR); i.e.;
�FT
� A
�
!
= [fT� (H!); A!]: (2.14)
In this setting the key condition (2.10) may be rewritten as
Y T
� := i[X1; f
T
� (H)] 2 K2: (2.15)
Note that condition (2.15) is always true for T > 0 with arbitrary � 2 R, since
in this case fT� (H) = g(H) for some g 2 S(Rd ) (cf. [BoGKS, Remark 5.2(iii)]).
We set
�0 :=
�
� 2 R; Y 0
� 2 K2
: (2.16)
For the same reason as when T > 0, we have � 2 �0 if either � =2 S or � is
the left edge of a spectral gap for H. Moreover, letting �cl denote the region of
complete localization, de�ned as the region of validity of the multiscale analysis,
or equivalently, of the fractional moment method, we have (cf. [AG, GK4])
�cl � �0: (2.17)
A precise de�nition of the region of complete localization is given in App. B.
Note that we included the complement of the spectrum S in �cl for convenience,
and that �cl is an open set by its de�nition. Note also that for � 2 �cl the
Fermi projection f0�(H) satis�es a much stronger condition than (2.10), namely
exponential decay of its kernel [AG, Th. 2] (see (B.2)). Conversely, fast enough
polynomial decay of the kernel of the Fermi projection for all energies in an interval
implies complete localization in the interval [GK4, Th. 3].
If Y T
� 2 K2, we proceed as in [KlLM], with a slight variation to include also
the case when T > 0. An inspection of the proof of [BoGKS, Th. 5.9] shows that
the adiabatic linear-response current (2.9) is well de�ned for every time t 2 R,
and given by
J�;lin(t;�; T; E) = T
8<:
tZ
�1
ds e�s E(s) _X1 e
� i(t�s)L Y T
�
9=; : (2.18)
132 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
It is convenient to rewrite (2.18) in terms of the conductivity measure �T
� ,
which we now introduce if either T > 0 or � 2 �0.
De�nition 1. If either T > 0 or � 2 �0, the ac-conductivity measure (x1-x1
component) at temperature T and chemical potential � is de�ned by
�T
� (B) := �hh _X1; �B(L)Y T
� ii for all Borel sets B � R : (2.19)
This de�nition is justi�ed by the following theorem, whose proof, as the proofs
of all other results in this section, is postponed to Sect. 3. M(R) will denote the
vector space of complex Borel measures on R, withM+(R) being the cone of �nite
positive Borel measures, and with M(e)
+ (R) the �nite positive even Borel mea-
sures. We recall that M(R) = C0(R)
� , where C0(R) denotes the Banach space
of complex-valued continuous functions on R vanishing at in�nity with the sup
norm. We will use two locally convex topologies on M(R). The �rst is the weak�
topology, de�ned by the linear functionals f� 2M(R) 7! �(g); g 2 C0(R)g. (By
�(g) :=
R
R
�(ds) g(s) we denote the integral of a function g with respect to a
measure �.) The second is the one de�ned by the similarly de�ned linear func-
tionals where g is any bounded measurable function on R. `Weak' will refer to
the weak� topology and `strong' to the other topology. We will write w-lim and
s-lim to denote the respective limits.
Theorem 1. (i) If either T > 0 or � 2 �0, the conductivity measure �T
�
is a �nite positive even Borel measure on the real line, i.e., �T
� 2 M(e)
+ (R), such
that
�T
� (R) = ��E�hÆbx1 + Æ�bx1 ; f
T
� (H)Æ0i
6
p
2�: (2.20)
(ii) For every � 2 �0 we have
�0
�(B) = �hhY 0
� ; �B(L) (�L)F0
�Y
0
� ii for all Borel sets B � R : (2.21)
(iii) The map ]0;1[3 T 7! �T
� 2 M(e)
+ (R) is strongly continuous for every
� 2 R.
(iv) For every � 2 �0 we have
s-lim
T#0
�T
� = �0
�: (2.22)
(v) If � 2 �cl
we also have limT#0 Y
T
� = Y 0
� in K2.
R e m a r k 1.
(i) Theorem 1(ii) shows that for T = 0 and � 2 �0 the conductivity measure
�0
� de�ned by (2.19) coincides with the one given in [KlLM, Def. 3.3].
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 133
A. Klein and P. M�uller
(ii) If the Fermi energy � is above or below the almost-sure spectrum S of H,
we have Y 0
� = 0, and hence also �0
� = 0. If ]a; b[ is a spectral gap, we clearly have
Y 0
� = Y 0
a , and hence �0
� = �0
a, for all � 2]a; b[. Moreover, it is shown in [KlLM,
Prop. 3.7] that the measure �0
� can be expressed in terms of a measure � on R2 ,
supported by the set S� given in [KlLM, Eq. (3.41)]. Since � depends on � only
through Y 0
� , we have � = a for all � 2]a; b[, and hence a is supported by the
set \
�2[a;b[
S� =
�
]�1; a]� [b;1[
[ �[b;1[�]�1; a]
: (2.23)
It then follows from [KlLM, Eq. (3.40)] that for all � 2 [a; b[ we have
�0
�([��; �]) = �0
a([��; �]) = 0 for all � 2]0; b� a[. (2.24)
(iii) If � 2 �0, as shown in [N, BoGKS], the direct-current conductivity van-
ishes at zero temperature:
�0�;dc := lim
�#0
DD
_X1;
1
iL+ �
Y 0
�
EE
= 0: (2.25)
(iv) For � 2 �cl, the region of complete localization, the Mott-type bound
lim sup
�#0
1
�
�0
�([0; �])
�2
�
log 1
�
�d+2
6 constant (2.26)
for the ac-conductivity measure was established in [KlLM].
We may now rewrite (2.18) in terms of the conductivity measure as follows.
If either T > 0 or � 2 �0, the same argument leading to [KlLM, Eq. (3.30) and
Th. 3.4] gives
J�;lin(t;�; T; E) = e�t
Z
R
d� ei �t �T� (�; �)
bE(�); (2.27)
where �T� (�; �) is the Stieltjes transform of the conductivity measure �T
� :
�T� (�; �) := � i
�
Z
R
�T
� (d�)
1
�+ � + i �
: (2.28)
The adiabatic in-phase linear-response current is now de�ned by
J in
�;lin(t;�; T; E) := e�t
Z
R
d� ei �t
�
Re�T� (�; �)
� bE(�): (2.29)
134 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
Turning o� the adiabatic switching, we obtain a simple expression for the in-
phase linear-response current in terms of the conductivity measure, as in [KlLM,
Cor. 3.5], given by
J in
lin(t;�; T; E) := lim
�#0
J in
�;lin(t;�; T; E) =
Z
R
�T
� (d�) ei �t bE(�): (2.30)
This gives a derivation of the in-phase linear-response current (1.1), and (2.30)
is valid as long as either T > 0 or � 2 �0. Moreover, it follows from (2.30) and
Theorem 1(iv) that
J in
lin(t;�; 0; E) = lim
T#0
J in
lin(t;�; T; E) for all � 2 �0: (2.31)
We have so far constructed the conductivity measure and the in-phase linear-
response current at T = 0 if � 2 �0. But what if, say, there is absolutely
continuous spectrum and � 2 Sac? In this case there is no reason to expect
� 2 �0. In view of Remark 1 (iii) we conjecture that � =2 �0 for most � 2 Sac.
In this article we show that the conductivity measure at zero temperature can
be constructed for arbitrary Fermi energy � in a physically sensible way as the
weak limit of the �nite-temperature conductivity measures as T # 0, with the
corresponding in-phase linear-response current given by (2.31).
To motivate our construction, we take T > 0 and decompose �T
� as
�T
� = �T
� (f0g) Æ0 +
�
�T
� � �T
� (f0g) Æ0
�
; (2.32)
where the Dirac measure Æ0 is the Borel measure on R concentrated at 0 with
total measure one. The details of this decomposition, presented in the following
theorem, will lead to a natural de�nition of �0
� for arbitrary �. We recall that
the Anderson model satis�es the Wegner estimate [W], and hence the density of
states measure N 2M+(R), de�ned by
N (B) := T (�B(H)) = EfhÆ0 ; �B(H!)Æ0ig for all Borel sets B � R ; (2.33)
supported by the spectrum S of H, is absolutely continuous with density n sat-
isfying knk1 6 k�k1.
We will use the following convention: If � 2M+(R) is absolutely continuous
and supported by the closed set F � R, we always assume that its density
is
also supported by F .
We set
Q0 := �
f0g(L) and Q? := I �Q0; (2.34)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 135
A. Klein and P. M�uller
the orthogonal projections onto the kernel of L in K2 and its orthogonal comple-
ment. Note that Q0 and Q? commute with HL and HR, and we have
g(HL)Q0 = g(HR)Q0 for all bounded Borel functions g: (2.35)
For each T > 0 and � 2 R, the bounded selfadjoint operator FT
� , de�ned in (2.14),
satis�es
Q0FT
� = FT
�Q0 = 0 and FT
� = FT
� Q? = Q?FT
� : (2.36)
We let L�1
? denote the pseudoinverse to L, that is,
L�1
? := g(L) with g(t) :=
1
t
if t 6= 0 and g(0) = 0: (2.37)
In particular,
L�1
? L = Q?: (2.38)
Moreover, we have �LFT
� > 0 and
�L�1
? FT
� = F T
� (HL;HR); (2.39)
where
F T
� (�1; �2) :=
8<:�
fT� (�1)�f
T
� (�2)
�1��2
=
���fT� (�1)�f
T
� (�2)
�1��2
��� if �1 6= �2
0 otherwise
: (2.40)
We write D(A) for the domain of an unbounded operator A in K2.
Theorem 2. (i) Let
(B) := �hh _X1;Q0 �B(HL) _X1ii for all Borel sets B � R : (2.41)
Then 2 M+(R) is absolutely continuous with respect to the density of states
measure N , and its density with respect to Lebesgue measure, , satis�es (E) 6
4�n(E) 6 4� k�k1 for Lebesgue-a.e. E 2 R. Moreover, we have supp �
R n �0 � R n �cl
.
(ii) For each T > 0 and � 2 R we have _X1 2 D
�
(�L�1
? FT
� )
1
2
�
. Setting
�T� (B) := �hh��L�1
? FT
�
� 1
2 _X1; �B(L)
��L�1
? FT
�
� 1
2 _X1ii (2.42)
for all Borel sets B � R, we have �T� 2M(e)
+ (R) with �T� (f0g) = 0.
(iii) If either T > 0 or � 2 �0, we have FT
�
_X1 2 D(L�1
? ) and
�T� (B) = �hh _X1; �B(L)
��L�1
? FT
�
�
_X1ii for all Borel sets B � R : (2.43)
136 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
(iv) For all T > 0 and � 2 R we have
�T
� (f0g) =
�
(�fT� )0
�
; (2.44)
�T
� (B n f0g) = �T� (B) for all Borel sets B � R ; (2.45)
yielding the following decomposition of the conductivity measure into mutually
singular measures:
�T
� =
�
(�fT� )0
�
Æ0 + �T� : (2.46)
(v) For all � 2 �0 we have
�0
� = �0
�: (2.47)
R e m a r k 2. On account of Theorem 2(i) we assume without loss of
generality that (�) = 0 for all � 2 �0.
R e m a r k 3. The measure �T� given in (2.42) can be expressed in terms
of the velocity-velocity correlation measure � 2M+(R
2), de�ned by (cf. [KlLM,
Eq. (3.46)])
�(C) := hh _X1; �C(HL;HR) _X1ii for all Borel sets C � R
2 : (2.48)
It follows from (2.39) that for each T > 0 and � 2 R the measure �T� can be
written as
�T� (B) = �
Z
R2
�(d�1d�2)F
T
� (�1; �2)�B(�1 � �2): (2.49)
We are thus led to the following de�nition.
De�nition 2. The ac-conductivity measure (x1-x1 component) at T = 0 and
� 2 R is the �nite positive even Borel measure �0
� on the real line given by
�0
� := (�)Æ0 + �0
�: (2.50)
The corresponding in-phase linear-response current is de�ned by
J in
lin(t;�; 0; E) :=
Z
R
�0
�(d�) ei �t bE(�): (2.51)
R e m a r k 4. In view of Theorem 2(v) and Remark 2, De�nition 2 agrees
with De�nition 1 on the common domain of de�nition, i.e., we have a unique
de�nition for �0
� for all � 2 R.
R e m a r k 5. In the absence of randomness, i.e., H = ��, we may still
carry out the above procedure and de�ne �0
� by (2.50) with as in (2.41) and
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 137
A. Klein and P. M�uller
�0
� as in (2.42) . In this case _X1 commutes with H, and hence Q0
_X1 = _X1. Thus
�0
� = 0 and, for a Borel set B � R,
(B) = �hh _X1; �B(��) _X1ii = �h�Æbx1 � Æ�bx1
�
; �B(��)
�
Æbx1 � Æ�bx1
�i: (2.52)
It follows that has a density given by a continuous function , the limit in
(3.38) holds for every �, and (recall �(��) = [�2d; 2d])
�0
� = (�)Æ0 with (�)
(
> 0 if � 2]� 2d; 2d[
= 0 otherwise
: (2.53)
Since the in-phase conductivity Re �0�(�) is formally the density of �0
�, (2.53) is
formally equivalent to the usual statement that for H = �� we have
Re�0�(�) = (�)Æ(�); (2.54)
with Æ(�) the formal Dirac delta function.
R e m a r k 6. The picture described in Remark 5 changes in the presence of
any amount of randomness. Let us introduce a disorder parameter in the Anderson
Hamiltonian by setting H
(�)
! := ��+�V!, where � 2 R is the disorder parameter.
Although the velocity operator _X1 does not depend on �, any amount of random-
ness (i.e., � 6= 0) implies Q(�)
0
_X1 6= _X1 since then [ _X1;H
(�)
! ] = �[ _X1; V!] 6= 0 for
a.e. !. In the region of complete localization we know (�)(�) = 0 by Theorem 2
(i), and hence the conductivity measure has no atom at 0 and we have (2.47). At
high disorder it is known that the region of complete localization (we include the
complement of the spectrum) is the whole real line, in which case we can conclude
that Q(�)
0
_X1 = 0, i.e., Q(�)
?
_X1 = _X1.
What happens if the Fermi energy � lies in a spectral region where extended
states are believed to exist is an open question. Common belief says that the
conductivity is nonzero in the region of extended states, but it is �nite for all
Fermi energies. The latter seems to rule out the existence of an atom of �0
� at
0 for all Fermi energies, which is equivalent to having Q(�)
0
_X1 = 0. That would
mean that any amount of disorder would have a very strong e�ect on the kernel
of the Liouvillian, since we would have Q(�)
?
_X1 = _X1 for all � 6= 0 although we
know that Q(0)
0
_X1 = _X1.
The justi�cation for Def. 2 is given in the following theorem.
Theorem 3. (i) For all T > 0 the map � 2 R 7! �T
� 2 M(e)
+ (R) is
strongly measurable, and for every T > 0 and � 2 R we have
�T
� =
�
(�fT0 )0 � �0
�
�
(�); that is;
�T
� (B) =
Z
R
dE (�fT� )0(E) �0
E(B) for all Borel sets B � R :
(2.55)
138 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
(ii) We have
�0
� =
(
s-limT#0�
T
� for all � 2 �0
w-limT#0 �
T
� for a.e. � 2 R n �0
: (2.56)
(iii) We have
J in
lin(t;�; 0; E) = lim
T#0
J in
lin(t;�; T; E)
(
for all � 2 �0
for a.e. � 2 R n �0
: (2.57)
3. Proofs
In this section we prove Ths. 1, 2 and 3. We refer to App. A for the mathe-
matical framework and basic notation.
We start with a consequence of the Wegner inequality [W].
Lemma 1. HL and HR have purely absolutely continuous spectrum.
P r o o f. In view of (A.10) it su�ces to prove that HL has purely absolutely
continuous spectrum. Given K2, let �A 2M+(R) be de�ned by
�A(B) := hhA;�B(HL)Aii for all Borel sets B � R : (3.1)
Since K1 is dense in K2, to prove the lemma it su�ces to show that �A is ab-
solutely continuous for all A 2 K1. In this case, using (A.6) and (2.33), we
get
�A(B) = jjj�B(H)Ajjj22 = jjjA��B(H)jjj22 6 jjjAjjj21 jjj�B(H)jjj22 = jjjAjjj21N (B): (3.2)
Since N is absolutely continuous, we conclude that �A is also absolutely continu-
ous.
Lemma 2. For all g 2 S(R) we have
Q0[X1; g(H)] = i g
0(HL)Q0
_X1: (3.3)
P r o o f. The lemma is proved by means of the Hel�er-Sj�ostrand formula for
smooth functions of selfadjoint operators (cf. [HS, App. B]). If g 2 S(R), then
for any selfadjoint operator K we have
g(K) =
Z
R2
d~g(z) (K � z)�1; (3.4)
g0(K) = �
Z
R2
d~g(z) (K � z)�2; (3.5)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 139
A. Klein and P. M�uller
where the integrals converge absolutely in operator norm. Here z = x + i y,
~g(z) is an almost analytic extension of g to the complex plane, and d~g(z) :=
1
2�@�z~g(z) dxdy with @�z = @x + i@y.
Thus, for g 2 S(R) we have, with R!(z) = (H! � z)�1, RL(z) = (HL � z)�1,
RR(z) = (HR � z)�1,
[X1; g(H)] =
Z
R2
d~g(z) [X1; R(z)] = � i
Z
R2
d~g(z)R(z) _X1R(z)
= � i
Z
R2
d~g(z)RL(z)RR(z) _X1: (3.6)
We recall [X1; g(H)]; [X1; R(z)] 2 K2, and the integrals converge absolutely in
operator norm in K2 (see [BoGKS, Prop. 2.4] and its proof). It follows, using
(2.35), that
Q0[X1; g(H)] = � i
Z
R2
d~g(z)RL(z)
2Q0
_X1 = i g
0(HL)Q0
_X1: (3.7)
The following lemma plays an important role in our analysis.
Lemma 3. (i) If either T > 0 or � 2 �0, we have
FT
�
_X1 = �LY T
� : (3.8)
In particular, we conclude that FT
�
_X1 2 D(L�1
? ).
(ii) Let T > 0. Then for all � 2 R we have
Y T
� = (�fT� )0(HL) Q0
_X1 �L�1
? FT
�
_X1: (3.9)
P r o o f. Let either T > 0 or � 2 �0, so Y
T
� 2 K2. Given ' 2 `2(Zd) with
compact support, we have
FT
�
_X1' = i
n
fT� (H)�L [H;X1]� [H;X1]�R f
T
� (H)
o
'
= � i
n
H �L [X1; f
T
� (H)]� [X1; f
T
�
�R (H)]H
o
'
= �(HL �HR)Y
T
� ' = �LY T
� '; (3.10)
since fT� (H)� 2 D(X1) for � 2 `2(Zd) with compact support by (2.10). Thus
(3.8) follows, and, in view of (2.36), we have FT
�
_X1 2 D(L�1
? ).
We now let T > 0, and note that (3.9) follows from (3.8) since Lem. 2 gives
Q0Y
T
� = (�fT� )0(HL)Q0
_X1: (3.11)
140 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
Lemma 4. The map ]0;1[3 T 7! Y T
� 2 K2 is norm continuous for every
� 2 R.
P r o o f. If g 2 S(R), it follows from [BoGKS, Prop. 2.4] and (A.7) that
jjj[X1; g(H)]jjj2 6 jjj[X1; g(H)]jjj1 6 C ffggg3 ; (3.12)
where C is a constant depending only on H and
ffggg3 :=
3X
r=0
Z
R
du jg(r)(u)j (1 + juj2) r�12 : (3.13)
The lemma follows in view of (2.15).
We are ready to prove Th. 1. Note that for all T > 0 and � 2 R we have
0 6
�FT
�
�2
6 1: (3.14)
Moreover, for all � 2 R the operator
�F0
�
�2
is an orthogonal projection in K2, and
hence �F0
�
�3
= F0
�: (3.15)
In addition, if � 2 �0 we have �F0
�
�2
Y 0
� = Y 0
� ; (3.16)
F0
�Y
T
� = FT
� Y
0
� for all T > 0: (3.17)
P r o o f o f T h e o r e m 1. Let � 2 �0 and �0
� be given by (2.19). Using
(3.16) and (3.8), we have
�0
�(B) = �hh _X1; �B(L)
�F0
�
�2
Y 0
� ii = �hhF0
�
_X1; �B(L)F0
�Y
0
� ii
= �hhY 0
� ; �B(L)(�L)F0
�Y
0
� ii; (3.18)
and hence coincides with [KlLM, Eq. (3.31)], a �nite positive even Borel measure
by [KlLM, Th. 3.4].
If T > 0 and � 2 R arbitrary, we use (3.9) to rewrite �T
� given by (2.19) as in
(2.46), where , given by (2.41), is clearly in M+(R), and �T� , given in (2.43), is
also seen to be in M+(R) by (2.39). We conclude that �T
� 2M+(R). The same
argument as in [KlLM, Proof of Th. 3.4] shows that the measure �T� , and hence
also �T
� , is even.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 141
A. Klein and P. M�uller
To prove (2.20), note that for either T > 0 or � 2 �0 it follows from (2.19),
the Cauchy�Schwarz inequality and jfT� j 6 1, that
�T
� (R) = ��E�hX2
1H!Æ0; f
T
� (H!)Æ0i
= ��E�hÆbx1 + Æ�bx1 ; f
T
� (H!)Æ0i
6
p
2 �
������fT� (H)
������
2
6
p
2 �
������fT� (H)
������
1
6
p
2�: (3.19)
We have thus proved parts (i) and (ii). Part (iii) is an immediate consequence
of Lem. 4. To prove (iv), given a bounded measurable function g and T > 0, we
write
�T
� (g) = �hh _X1; g(L)(F0
�)
2Y T
� ii+ �hh _X1; g(L)
�
1� (F0
�)
2
�
Y T
� ii: (3.20)
In view of (3.14), the same argument used to prove �T
� 2 M+(R) shows that
both terms on the right-hand side of (3.20) are integrals of g with respect to
�nite positive Borel measures on R. On account of (3.17) we have
hh _X1; g(L)(F0
�)
2Y T
� ii = hh _X1; g(L)F0
�FT
� Y
0
� ii = hhFT
�
_X1; g(L)F0
�Y
0
� ii: (3.21)
Using the Cauchy�Schwarz inequality, we get������(FT
� �F0
�)
_X1
������
2
6 2
������ _X1
������
1
������fT� (H)� f0�(H)
������
2
: (3.22)
Recalling (2.33), we have
������fT� (H)� f0�(H)
������2
2
=
Z
R
N (dE)
��fT� (E)� f0�(E)
��2; (3.23)
and hence
lim
T#0
������fT� (H)� f0�(H)
������
2
= 0 (3.24)
by dominated convergence. It follows that limT#0
������(FT
� � F0
�)
_X1
������
2
= 0. We
conclude, using (3.16), that
� lim
T#0
hh _X1; g(L)(F0
�)
2Y T
� ii = �hhF0
�
_X1; g(L)F0
�Y
0
� ii = �0
�(g): (3.25)
On the other hand, it follows from (2.20) that
lim
T#0
�T
� (R) = �0
�(R): (3.26)
Combining this with (3.25), where we set g = 1, we conclude that
lim
T#0
hh _X1;
�
1� (F0
�)
2
�
Y T
� ii = 0: (3.27)
142 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
Since hh _X1; �B(L)
�
1�(F0
�)
2
�
Y T
� ii is a positive measure, it converges to 0 strongly.
Part (iv) is proven.
It remains to prove part (v). Let � 2 �cl, so Y T
� 2 K2 for all T > 0. We need
to prove that
lim
T#0
������Y T
� � Y 0
�
������
2
= 0: (3.28)
Standard calculations give������Y T
� � Y 0
�
������2
2
= E
nD�
fT� (H)� f0�(H)
�
Æ0;X
2
1
�
fT� (H)� f0�(H)
�
Æ0
Eo
6
������fT� (H)� f0�(H)
������
2
�
E
n
X2
1
�
fT� (H)� f0�(H)
�
Æ0
2o� 1
2
:
(3.29)
In view of (3.24), the desired (3.28) follows if we prove that
lim sup
T#0
E
n
X2
1
�
fT� (H)� f0�(H)
�
Æ0
2o <1: (3.30)
To prove (3.30) we use that � 2 �cl, and hence there exists Æ > 0 such that
IÆ � �cl, where I� :=]� � �; � + �[ for � > 0. We pick functions gj 2 C1
c (R),
j = 1; 2, such that 0 6 gj 6 1, �S = (g1 + g2)�S, supp g1 � IÆ, supp g2 � R n I Æ
2
.
Letting gT� = fT� � f0�, we have
fT� (H)� f0�(H) = gT� (H) = gT� (H)g1(H) + gT� (H)g2(H): (3.31)
Since supp g1 � �cl and
��gT� �� 6 2 for all T > 0, standard estimates [A, AG,
GK1, GK4] give
sup
T>0
E
n
X2
1g
T
� (H)g1(H)Æ0
2o <1: (3.32)
On the other hand, explicit calculations show that
sup
T>0
�gT� �(k)�RnI Æ
2
1
<1 for all k = 0; 1; 2 : : : : (3.33)
Since supp g2 � R n I Æ
2
, a calculation using [GK2, Th. 2] shows that
sup
T>0
E
n
X2
1g
T
� (H)g2(H)Æ0
2o <1: (3.34)
The estimate (3.30) follows.
We now turn to Th. 2.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 143
A. Klein and P. M�uller
P r o o f o f T h e o r e m 2. Note that we already proved parts (iii) and
(iv) while proving Th. 1. To prove (v), note that it follows from (2.19), (3.16),
(2.38), (3.8), and (3.15) that for all Borel sets B � R we have
�0
�(B) = �hh _X1; �B(L)(F0
�)
2Y 0
� ii = �hh _X1; �B(L)(F0
�)
2L�1
? LY 0
� ii
= ��hh _X1; �B(L)(F0
�)
2L�1
? F0
�
_X1ii = �0
�(B): (3.35)
Now, we turn to part (i). Let be given by (2.41), it is clearly in M+(R).
Since
_X1Æ0 = � i
�
Æbx1 � Æ�bx1
�
; (3.36)
we have, for all Borel sets B � R, recalling (2.33),
1
�
(B) 6 hh _X1; �B(HL) _X1ii = E
�h�Æbx1 � Æ�bx1
�
; �B(H)
�
Æbx1 � Æ�bx1
�i
6 2N (B) + 2 E
�
�
B(H)Æbx1
�
B(H)�bx1
6 4N (B): (3.37)
It follows that is absolutely continuous with respect to the density of states
measure N , and that its density with respect to Lebesgue measure, , satis�es
(E) 6 4�n(E) for Lebesgue-a.e. E 2 R. Since the functions (�fT0 )0 form an
approximate identity as T # 0, it follows from the absolute continuity of and
the Lebesgue Di�erentiation Theorem (cf. [Gr, Cor. 2.1.17]) that
lim
T#0
�
(�fT� )0
�
= (�) for a.e. �: (3.38)
From parts (ii) and (iv) of Th. 1 and (2.44) (which is proved already) we conclude
that limT#0
�
(�fT� )0
�
= 0 for Lebesgue-almost all � 2 �0. Th. 2(i) is proven.
To �nish, we need to prove part (ii). Let � 2M+(R
2 ) be the velocity-velocity
correlation measure given in (2.48). As a consequence of (2.49), (2.46) and (2.20),
we have Z
R2
�(d�1d�2)F
T
� (�1; �2) 6
p
2 for all T > 0 and � 2 R : (3.39)
But for all � 2 R we have
lim
T#0
F T
� (�1; �2) = F 0
� (�1; �2) for �-a.e. (�1; �2) 2 R
2 ; (3.40)
where we used the fact that the two marginals of � are absolutely continuous, a
consequence of Lem. 1. (More is true: the two marginals are equal to the measure
� _X1
, and hence have a bounded density, cf. (3.2).) Using Fatou's Lemma and
(3.39), we conclude that for all � 2 R we haveZ
R2
�(d�1d�2)F
0
� (�1; �2) 6 lim inf
T#0
Z
R2
�(d�1d�2)F
T
� (�1; �2) 6
p
2: (3.41)
144 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
Theorem 2(ii) follows.
It remains to prove Th. 3.
P r o o f o f T h e o r e m 3. To prove part (i), we remark that measurability
in � follows from (2.46) and (2.49) if T > 0, respectively from Def. 2 and (2.49)
if T = 0. Now, De�nition 2, Theorem 2(iv) and (i) imply that it su�ces to prove
(2.55) with �T� substituted for �T
� , that is,
�T� (B) =
Z
R
dE (�fT� )0(E) �0
E(B) for all Borel sets B � R : (3.42)
But this follows from (2.49) using Fubini's Theorem plus the fact that
fT� (t) =
Z
R
ds (�fT� )0(s) f0s (t) for all t 2 R : (3.43)
Next we turn to part (ii). As in the proof of (3.38), it follows from (2.55)
and the Lebesgue Di�erentiation Theorem that for each Borel set B � R
we have limT#0 �
T
� (B) = �0
�(B) for Lebesgue-a.e. � 2 R (the exceptional set
depending on B!). Let fIngn2N denote an enumeration of the bounded intervals
with rational endpoints. It follows that for a.e. � we have limT#0 �
T
� (In) = �0
�(In)
for all n 2 N, and hence we have w-limT#0 �
T
� = �0
� for a.e. �. Part (ii) now
follows using Th. 1(iv) for � 2 �0.
Part (iii) is an immediate consequence of part (ii).
Appendix A. The Mathematical Framework for Linear
Response Theory
In this appendix we recall the mathematical framework for linear response
theory, following [BoGKS, Sect. 3] and [KlLM, Sect. 3] (see also [BES, SB]). We
restrict ourselves to the Anderson model. The Hamiltonian H!, given in (2.1), is
a measurable map from the probability space (
;P) to the bounded selfadjoint
operators on H = `2(Zd). The probability space (
;P) is equipped with an
ergodic group f�a; a 2 Zdg of measure preserving transformations, satisfying the
covariance relation
U(a)H!U(a)
� = H�a(!) for all a 2 Z
d; (A.1)
where U(a) denotes translation by a, i.e., U(a)Æb := Æb+a when applied to any
member of the canonical orthonormal basis fÆb; b 2 Z
dg for `2(Zd).
Let Hc = `2c(Z
d) be the (dense) subspace of �nite linear combinations of the
canonical basis vectors. By Kmc we denote the vector space of measurable covari-
ant operators A :
! Lin
�Hc;H), identifying measurable covariant operators
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 145
A. Klein and P. M�uller
that agree P-a.e.; all properties stated are assumed to hold for P-a.e. ! 2
. Here
Lin
�Hc;H) is the vector space of linear operators from Hc to H. Recall that A
is measurable if the functions ! ! h�;A!�i are measurable for all � 2 Hc, A is
covariant if
U(a)A!U(a)
� = A�a(!) for all a 2 Z
d: (A.2)
It follows (for H = `2(Zd)) that D(A�!) � Hc for A 2 Kmc, i.e., A is locally
bounded. Thus, the operator A
z
! := A�!
��
Hc
is well de�ned. Note that (JA)! :=
A
z
! de�nes a conjugation in Kmc.
We introduce norms on Kmc given by
jjjAjjj1 := k kA!k kL1(
;P);
jjjAjjjpp := E
�hÆ0; jA! jpÆ0i
; p = 1; 2;
(A.3)
and consider the normed spaces
Kp := fA 2 Kmc; jjjAjjjp <1g; p = 1; 2;1: (A.4)
It turns out that K1 is a Banach space and K2 is a Hilbert space with inner
product
hhA;Bii := E
�hA!Æ0; B!Æ0i
; (A.5)
and we have
hhA;Bii = hhBz; Azii: (A.6)
Since K1 is not complete, we introduce its (abstract) completion K1. The conju-
gation J is an isometry on each Kp, p = 1; 2;1. We also have
jjjAjjj1 6 jjjAjjj2 6 jjjAjjj1 and hence K1 � K2 � K1; (A.7)
and K1 is dense in Kp, p = 1; 2. Moreover, we have H;�; _X1 2 K1.
Given A 2 K1, we identify A! with its closure A!, a bounded operator in H.
We may then introduce a product in K1 by pointwise operator multiplication,
and K1 becomes a C�-algebra. (K1 is actually a von Neumann algebra [BoGKS,
Subsect. 3.5].) This C�-algebra acts by left and right multiplication in Kp, p =
1; 2. Given A 2 Kp, B 2 K1, left multiplication B �LA is simply de�ned by
(B �LA)! := B!A!. Right multiplication is more subtle, we set (A�RB)! :=
A
z�
! B! (see [BoGKS, Lem. 3.4] for a justi�cation), and note that (A�RB)z =
B�
�LA
z. Moreover, left and right multiplication commute:
B �LA�RC := B �L(A�RC) = (B �LA)�RC (A.8)
for A 2 Kp, B;C 2 K1. We refer to [BoGKS, Sect. 3] for an extensive set of
rules and properties which facilitate calculations in these spaces of measurable
covariant operators.
146 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
The Conductivity Measure for the Anderson Model
Since H 2 K1, we de�ne bounded commuting selfadjoint operators HL and
HR on K2 by
HLA := H �LA and HRA := A�RH; (A.9)
note that
HR = JHLJ : (A.10)
The Liouvillian is then de�ned by
L := HL �HR; (A.11)
and hence satis�es
L = �JLJ : (A.12)
Note that (cf. [BoGKS, argument below Eq. (5.91)])
kerL = fA 2 K2; A�
Lf(H) = f(H)�RA for all f 2 S(R)g : (A.13)
The trace per unit volume is given by
T (A) := E
�hÆ0; A!Æ0i
for A 2 K1; (A.14)
a well de�ned linear functional on K1 with jT (A)j 6 jjjAjjj1, and hence can be
extended to K1. Note that T is indeed the trace per unit volume:
T (A) = lim
L!1
1
j�Lj
tr f��LA!
��Lg for P-a.e. ! ; (A.15)
where �L denotes the cube of side L centered at 0 (see [BoGKS, Prop. 3.20]).
Moreover,
hhA;Bii = T fA�Bg for all A;B 2 K2: (A.16)
Appendix B. The Region of Complete Localization
There is a wealth of localization results for the Anderson model in arbitrary
dimension, based either on the multiscale analysis [FS, FMSS, DK], or on the
fractional moment method [AM, A]. The spectral region of applicability of both
methods turns out to be the same, and in fact it can be characterized by many
equivalent conditions [GK3, GK4]. For this reason we call it the region of complete
localization as in [GK4].
The most convenient de�nition for this paper is by the conclusions of [GK4,
Th. 3]. For convenience we include the complement of the spectrum in the region
of complete localization.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 147
A. Klein and P. M�uller
De�nition 3. The region of complete localization �cl
for the Anderson Hamil-
tonian H is the set of energies E 2 R for which there is an open interval I 3 E
and constants � > 0 and C <1 such that
E
(
sup
�2I
��hÆx; f0� (H!) Æ0i
��2) � C e�jxj
�
for all x 2 Z
d: (B.1)
R e m a r k 7. As remarked in the comments below [GK4, Th. 3], it su�ces to
require fast enough polynomial decay in (B.1); subexponential decay then follows.
R e m a r k 8. For the Anderson model, it follows from [A, AG] that we have
exponential decay in (B.1). More precisely, if E 2 �cl, there is an open interval
I 3 E and constants m > 0 and C <1 such that
E
(
sup
�2I
��hÆx; f0� (H!) Æ0i
��2) � C e�mjxj for all x 2 Z
d: (B.2)
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