Scattering from Sparse Potentials on Graphs
We study the spectral structure of Schrodinger operators H = Δ+V for random potentials supported on sparse sets. In the past years examples of such operators whose spectra almost surely satisfy the following properties have been exhibited: Anderson localization holds outside spec(Δ), while the wave...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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| Цитувати: | Scattering from Sparse Potentials on Graphs / Ph. Poulin // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 151-170. — Бібліогр.: 29 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860186932521205760 |
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| author | Poulin, Ph. |
| author_facet | Poulin, Ph. |
| citation_txt | Scattering from Sparse Potentials on Graphs / Ph. Poulin // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 151-170. — Бібліогр.: 29 назв. — англ. |
| collection | DSpace DC |
| container_title | Журнал математической физики, анализа, геометрии |
| description | We study the spectral structure of Schrodinger operators H = Δ+V for random potentials supported on sparse sets. In the past years examples of such operators whose spectra almost surely satisfy the following properties have been exhibited: Anderson localization holds outside spec(Δ), while the wave operators Ω⁺(H, Δ) exist inside this last set. We continue this program by presenting sparseness conditions under which Ω⁺(Δ, H) also exist.
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| first_indexed | 2025-12-07T18:04:45Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 1, pp. 151�170
Scattering from Sparse Potentials on Graphs
Ph. Poulin
Department of Mathematical Sciences
Norwegian University of Science and Technology
7491 Trondheim, Norway
E-mail:poulin@math.ntnu.no
Received October 1, 2007
We study the spectral structure of Schr�odinger operators H = �+V for
random potentials supported on sparse sets. In the past years examples of
such operators whose spectra almost surely satisfy the following properties
have been exhibited: Anderson localization holds outside spec(�), while the
wave operators
�(H;�) exist inside this last set. We continue this program
by presenting sparseness conditions under which
�
(�; H) also exist.
Key words: random Schr�odinger operators, spectral analysis, scattering
theory.
Mathematics Subject Classi�cation 2000: 81Q10, 47B80.
1. Introduction
Since its introduction in 1958, there has been considerable interest in the
Anderson model [4], which describes potentials that are not completely known,
but are a�ected by a probability distribution. By focusing on almost sure results
(and hence by discarding pathological constructions), research on this model has
given a new insight into quantum physics. A random potential, V , lies on a lattice
Z
d. It is described by the following operator on l2(Zd):
V =
X
N2Zd
V (N)hÆN j �iÆN ;
where ÆN (M) is the Kronecker delta and fV (N)g
N2Zd
is a family of i.i.d. random
variables of law �.� The spectral structure of the random Hamiltonian
H = �+ �V
�Explicitly, the probability space is given by
= R
(Zd) equipped with its Borel �-algebra
and the probability measure P =
Q
Zd
�. The random variable V (N) then maps an element of
to its N -th coordinate.
c
Ph. Poulin, 2008
Ph. Poulin
has been investigated�where � is a positive number (the so-called disorder) and
� is the centered discrete Laplacian. It was proven by L. Pastur that the abso-
lutely continuous, essential, singular continuous and point spectra ofH are almost
surely constant [20]. Indeed, from the �rst days Anderson has conjectured that
H has the following spectral structure (almost surely): if � is small, spec(H)
is purely absolutely continuous (delocalization) except near its edges, where it is
pure point with exponentially decaying eigenfunctions (Anderson localization); on
the other hand, if � is large, Anderson localization occurs on the whole spec(H).
While the structure of the a.c. spectrum of H is still not completely understood,
the localization part of the above conjecture was proven by M. Aizenman and
S. Molchanov [3, 1]. In their works these authors developed a method for esti-
mating the sth-moment of the resolvent's matrix elements
R(M;N; z) = hÆM j (H � z)�1ÆN i
(in absolute value) for suitable �, s and z approaching the real line. This method,
which is used in the present paper, is based on the following decoupling lemmas �
which apply to a large class of probability measures including Gaussian, Cauchy,
and uniform distributions [1�3, 5, 11, 15]:�
Proposition 1. Suppose there exists an s 2 (0; 1) such that
ks = inf
�;�2C
R
R
jx� �jsjx� �j�s d�(x)R
R
jx� �j�s d�(x)
> 0:
Then, for any deterministic function F (M;N; z),
E jV (M)� F (M;N; z)jsjR(M;N; z)js > ksE jR(M;N; z)js :
Suppose instead there exists an s 2 (0; 1) such that
Ks = sup
�2C
R
R
jxjsjx� �j�s d�(x)R
R
jx� �j�s d�(x)
<1:
Then, E jV (M)jsjR(M;N; z)js 6 KsE jR(M;N; z)js :
In addition to the Anderson model, several researchers (M. Krishna et al.
[13, 14], W. Kirsch et al. [6, 12], S. Molchanov et al. [15�19]) have investigated
various sparse models, which describe random potentials lying on a set � subject
to various geometric constraints, having in common that the distance between
�In the sequel we use parentheses with E in the same way as with
P
. For instance, E Xs =
E (Xs), not (E X)s.
152 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
N 2 � and its closest neighbor in � tends to in�nity when jN j ! 1. In the
discrete case the following Hamiltonian on l2(Zd) has been investigated,
H = �+ V; V =
X
n2�
V (n)hÆn j �iÆn;
where fV (n)gn2� is a family of i.i.d. random variables.
Since such a model is not ergodic, Pastur's theorem fails for the singular
continuous and point spectra of H, but still holds for the essential and continuous
spectra. Indeed, the essential spectrum ofH has been completely characterized by
S. Molchanov and B. Vainberg under appropriate sparseness conditions [17, 19].
In addition, the spectral structure of H (for the above model or its continuous
analog) has been clari�ed in di�erent cases. Families of random Hamiltonians
with the following, almost sure properties have been exhibited: the spectrum of
H is (possibly dense) pure point outside spec(�), while the wave operators
�
E(H;�) = lim
t!�1
eitHe�it�1E(�) (strongly)
exist on the whole E = spec(�)�yielding that specac(H) = spec(�).
In order to complete this program we show that under suitable sparseness
conditions the above wave operators are almost surely complete, i.e.,
�
E(�;H)
also exist. We conclude this work by exhibiting a family of random operators
H = �+V with sparse potentials satisfying almost surely the following properties:
1o the spectrum of H is purely absolutely continuous on spec(�), 2o the wave
operators exist and are complete on spec(�), 3o the spectrum of H is (possibly
dense) pure point outside spec(�).
This work, based on a private communication with V. Jak�si�c, is an applica-
tion of a completeness criterion found in [9] � a paper of V. Jak�si�c and Y. Last
dedicated to L. Pastur.
Acknowledgements The author is grateful to Vojkan Jak�si�c for his sub-
stantial collaboration, his generous teaching (covering many results used in the
present paper), and for having made this invitation possible. The present article
is based on the second part of the doctoral dissertation of the author, who wants
to acknowledge his thesis' referees for instructive comments.
2. Abstract Results
2.1. The Model
At a higher level of generality the lattice Zd is replaced with a countable set
X endowed with a graph structure. We assume that this graph consists of �nitely
many connected components and that the degrees of the vertices are bounded. Let
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 153
Ph. Poulin
d(M;N) be the distance between M;N 2 X, that is, the length of the shortest
path connecting them in X (1 if M and N lie on two di�erent components).
The usual centered Laplacian is then replaced with the adjacency operator of X:
for ' 2 l2(X),
�'(N) =
X
d(M;N)=1
'(M):
Notice that � is a bounded selfadjoint operator on l2(X). The Euclidean distance
is replaced with a weight on the set X, that is, a function
: X �X ! [0;1)
satisfying all axioms of metric distance, except that
(M;N) = 0 does not neces-
sarily imply M = N .
For a �xed � � X, a family fV (n)gn2� of i.i.d. random variables is given.
Their law, �, is assumed to be absolutely continuous and to satisfy both hypothe-
ses of Prop. 1 for a �xed s 2 (0; 1). We study the following random Hamiltonian
on l2(X):
H = �+ V; V =
X
n2�
V (n)hÆn j �iÆn:
N o t a t i o n. In the sequel the connected components of the graph are
denoted by Xj . For 0 6 R 61, the R-fattening of � is de�ned as
�R = fN 2 X ; d(N;�) 6 Rg;
while the projection on l2(�R) is denoted by 1R. For the sake of clarity, we shall
use the following fonts: n varies in a certain subset of �, n varies in �, N varies
in a certain fattening of � and N in the whole X.
The abbreviation a.e. & a.s. stands for almost everywhere and almost surely,
where the former refers to the Lebesgue measure and the latter to the given
probability measure P. Here, the underlying probability space is given by
=
R
(Zd) equipped with its Borel �-algebra and the probability measure P =
Q
Zd
�.
2.2. Preliminaries
Our work is based on the following Jak�si�c�Last criterion of completeness [9],
whose conclusion trivially persists for disconnected graphs:�
Proposition 2. Suppose that the spectrum of H is purely a.c. on a given Borel
set E � R. Suppose also that 11 is �-smooth on E, that is,��
sup
0<"<1
e2E
k11(�� e� i")�111k <1:
�This last observation is deduced from elementary properties of the projections, Pj , onto
l
2(Xj), namely: PjPk = 0 if j 6= k;
P
Pj is the identity; Pj1R = 1RPj for any j and R;
f(T )Pj = f(TPj) = Pjf(T )Pj for any bounded Borel function f and T 2 f�; V; Hg.
�� See [25].
154 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
If for all n 2 � and almost all e 2 EX
M2�1
jIm hÆM j (H � e� i0)�1Ænij
2 <1;
then the wave operators
�
E(�;H) exist.
Since in this context the usual wave operators are
�
E(H;�), this last criterion
asserts their completeness, but without assuming their existence.
In order to prove localization we shall use the following Simon�Wol� theorem
[27]. It is easily seen that its conclusion is valid for disconnected graphs with
�nitely many components, except regarding simplicity of the eigenvalues � which
follows from a recent theorem of V. Jak�si�c and Y. Last [10].
Proposition 3. Let E � R be a Borel set. If with probability one
k(H � e� i0)�1Ænk <1
for all n 2 � and almost all e 2 E, then the spectrum of H on E is almost surely
pure point with simple eigenvalues.�
Suppose in addition that for almost all V 2
, almost all e 2 E, and all n 2 �
there exist constants K; k > 0 independent of M 2 X such that
jhÆn j (H � e� i0)�1ÆM ij 6 Ke�k
(n;M):
Then, the eigenfunctions are exponentially bounded in the following sense: for
such an eigenfunction (N) and an arbitrarily �xed site N0, there exists a coe�-
cient Const (depending on V , N0 and the associated eigenvalue) and a universal
exponent k > 0 such that
j (N)j 6 Const e�k
(N;N0)
for all N 2 X.
Given a selfadjoint operator T on l2(X), let Tj be its restriction to l2(Xj).
The essential support of the a.c. spectrum of Tj is given by
�(Tj) = fe 2 R ;
X
N2Xj
jIm hÆN j (Tj � e� i0)�1ÆN ij > 0g a.e.
Notice that �(Tj) is de�ned up to a set of Lebesgue measure zero; however, its
equivalence class is almost surely constant (by a variant of Pastur's theorem). We
de�ne
�(T ) = \j�(Tj):
The Jak�si�c�Last theorem [8] asserts:
�Recall that the spectrum of H on E is de�ned as spec(H�E(H)), where �E is the charac-
teristic function of E; it is not equal to spec(H)\E in general. Moreover, the above conclusion
includes the trivial case where H has no spectrum on E.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 155
Ph. Poulin
Proposition 4. Let E � R be a Borel set. If with probability one E � �(H)
(in the sense that E n �(H) has Lebesgue measure zero), then the spectrum of H
on E is purely a.c., almost surely.
2.3. Main Results
As mentioned in the previous section we shall determine the spectral structure
of H on a given interval [a; b] by using the Jak�si�c�Last and the Simon�Wol�
criteria (depending on the location of [a; b]). In both cases the matrix elements
of the resolvent of H have to be estimated. This will be done in one step, using
the Aizenman�Molchanov method.�
Consider the following quantity,
�(M;N) = sup
z2S
jhÆM j (�� z)�1ÆN ij;
where M; N 2 X and S = fa 6 Re z 6 b; 0 < Im z < 1g. In concrete models
�(M;N) decays whenM and N become distant. This motivates our choice in the
present abstract setting to make sparseness assumptions on �(M;N):
A s s u m p t i o n A. For all " > 0 there exists a �nite set F � � such thatP
n2�nfmg �(n;m)s < " for all m 2 � n F .
Given an R 2 [0;1],
A s s u m p t i o n B. supn2�
P
M2�R
�(n;M)s <1.
Let I = infn2�;z2S jhÆn j (�� z)�1Ænij. We also assume
A s s u m p t i o n C. I > 0.
Our chief lemma is:
Lemma 1. Suppose 0 6 R 6 1. Under Assumptions A, B and C, for all
n 2 �,
k1R(H � e� i0)�1Ænk <1 a.e. & a.s.
on [a; b]�
.
�Compared with the original Aizenman�Molchanov argument complications from two sources
arise: since we play with sparseness instead of the disorder, in order to control the norm of a
certain operator we remove a �nite number of sites and then put them back using the resolvent
identity repeatedly; moreover, deletion of these sites never prevents a remaining site to be close
to itself, so the diagonal elements have to be treated di�erently.
156 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
We deduce the following result inside spec(�):
Theorem 1. Suppose A, C, and supN2�1
P
M2�1
�(N;M )s < 1 for an in-
terval [a; b] � �(�). If
�
[a;b]
(H;�) exist a.s., then the spectrum of H on [a; b] is
purely a.c. and the wave operators are complete there, almost surely.
In order to derive Anderson localization outside spec(�) we make the following
assumptions on the weight:
A s s u m p t i o n D. For any k > 0, supN2X
P
M2X e�k
(N;M) <1:
A s s u m p t i o n E. For each L > 0 there exists a �nite set E � � such
that for all m 2 � n E, infn2�nfmg
(n;m) > L:
Given an R 2 [0;1],
A s s u m p t i o n F. There exist D; � such that �(n;M)s 6 De��
(n;M)
for all n 2 � and M 2 �R.
Our main lemma is:
Lemma 2. Suppose 0 6 R 6 1. Under Assumptions C, D, E, and F, there
exists a universal constant k > 0 such that the following holds a.e. & a.s. on
[a; b]�
: for all n 2 � there exists a K > 0 such that
jhÆn j (H � e� i0)�1ÆM ij 6 Ke�k
(n;M)
for all M 2 �R.
From Lemmas 1 and 2 we deduce:
Theorem 2. Suppose C, D, and E. Suppose in addition F holds with
R =1. Then, the spectrum of H on [a; b] is almost surely pure point with simple
eigenvalues and exponentially bounded eigenfunctions (in the sense of Prop. 3).
2.4. Proof of the First Lemma
In this section Assumption A is used in the following form: there exists a �nite
set F � � such that
sup
m2�nF
X
n2�nfmg
�(n;m)s <
I
sks
2Ks
: (1)
We also assume B for an arbitrary R 2 [0;1], and C.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 157
Ph. Poulin
Let bH = �+
P
n2�nF V (n)hÆn j �iÆn. We use the abbreviations
R0(N;M; z) = hÆN j (�� z)�1ÆM i;
R(N;M; z) = hÆN j (H � z)�1ÆM i;bR(N;M; z) = hÆN j ( bH � z)�1ÆM i;
where M , N 2 X and z 2 S. Since the spectral measure of ÆM and ÆN with
respect to H is real-valued [9], R(N;M; z) = R(M;N; z) for any z 2 S; similar
relations hold for R0 and bR.
In the sequel we use the Aizenman�Molchanov decoupling lemmas (Prop. 1)
in conjunction with the resolvent identity; this latter implies
bR(N;M; z) = R0(N;M; z) �
X
p2�nF
R0(N; p; z)V (p) bR(p;M; z) (2)
for all M;N 2 X. As a �rst instance, with the convention that p varies in � n F ,
Lemma 3. For all n; m 2 � n F and z 2 S,
E j bR(n;m; z)js 6
1
ksIs
�(n;m)s +
Ks
ksIs
X
p6=n
�(n; p)sE j bR(p;m; z)js:
P r o o f. By the equation (2),
bR(n;m; z)(1 +R0(n; n; z)V (n)) = R0(n;m; z)�
X
p6=n
R0(n; p; z)V (p) bR(p;m; z):
Using the triangle inequality for j�js, taking the expectation, and then apply-
ing the decoupling lemmas give ksjR0(n; n; z)j
s
E j bR(n;m; z)js 6 jR0(n;m; z)j
s +
Ks
P
p6=n jR0(n; p; z)j
s
E j bR(p;m; z)js; from which the result follows.
Let us �x m 2 � n F and z 2 S, n 2 � n F being thought as the only variable.
We de�ne the following vectors on l1(� n F):
X(n) = E j bR(n;m; z)js;
B(n) =
1
ksIs
�(n;m)s:
They are well de�ned, since kXk1 6 jIm zj�s and kBk1 < 1, the latter by
Assumption B (which also ensures kBk1 <1). Let us de�ne the operator
(A )(n) =
Ks
ksIs
X
p6=n
�(n; p)s (p);
158 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
which acts on both l1(� n F) and l1(� n F). By the equation (1),
kAk1 = kAk1 =
Ks
ksIs
sup
n
X
p6=n
�(n; p)s <
1
2
: (3)
In addition, the previous lemma gives (1�A)X 6 B (pointwise).
Lemma 4. supz2S supm2�nF
P
n2�nF E j
bR(n;m; z)js <1:
P r o o f. The relation (3) implies that (1�A)�1 =
P1
j=0A
j is well-de�ned
and satis�es k(1�A)�1k1 6 2: Observe that, since all matrix elements of A are
positive, those of (1�A)�1 are also positive, i.e., (1�A)�1 preserves pointwise
positivity. Therefore, by the previous lemma
X 6 (1�A)�1B (pointwise), (4)
so kXk1 6 2kBk1: In other words,
P
n E j
bR(n;m; z)js 6 2
ksIs
P
n �(n;m)s: Since
m and z are arbitrary, Assumption B yields the result.
Lemma 5. For all M; N 2 X and z 2 S,
E j bR(N;M; z)js 6 �(N;M)s +Ks
X
p2�nF
�(N; p)sE j bR(p;M; z)js:
P r o o f. The result is obtained by applying the triangle inequality for j�js
to (2), taking the expectation, and then using the decoupling lemma.
Lemma 6. supz2S supn2�
P
M2�R
E j bR(n;M; z)js <1:
P r o o f. Assumption B and Lemma 4 imply that C = supn2�
P
M2�R
�(n;M )s
and D = supz2S supm2�nF
P
n2�nF E j
bR(n;m; z)js are �nite. By the previous
lemma, for all N 2 �R, m 2 � n F and z 2 S,
E j bR(N;m; z)js 6 �(N;m)s +Ks
X
p2�nF
�(N; p)sE j bR(p;m; z)js;
and hence supz2S supm2�nF
P
N2�R
E j( bR(N;m; z)js 6 C +KsCD: By the same
lemma, for all n 2 �, M 2 �R and z 2 S
E j bR(n;M; z)js 6 �(n;M )s +Ks
X
p2�nF
�(n; p)sE j bR(p;M; z)js;
and hence
P
M2�R
E j bR(n;M; z)js 6 C + KsC(C +KsCD) uniformly in n 2 �
and z 2 S, as desired.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 159
Ph. Poulin
We want to deduce information about bR(n;M; e+ i0) for n 2 �, M 2 �R and
e 2 [a; b]; this last limit exists a.e. & a.s. on [a; b] �
(by classical Analysis and
Fubini's theorem).
Lemma 7. For all n 2 �,
X
M2�R
j bR(n;M; e+ i0)j2 < 1 a.e. & a.s. on
[a; b]�
.
P r o o f. For a �xed n 2 �,
bZ
a
E
X
M2�R
j bR(n;M; e+ i0)js de 6 (b� a) ess sup
a<e<b
X
M2�R
E j bR(n;M; e+ i0)js;
where ess sup denotes the essential supremum w.r.t. the Lebesgue measure. Hence,
by Fatou's lemma
bZ
a
E
X
M2�R
j bR(n;M; e+ i0)js de 6 (b� a) sup
z2S
X
M2�R
E j bR(n;M; z)js:
The result follows from the previous lemma and the triangle inequality for j�j
s
2 .
We are now ready to prove Lemma 1. Let n 2 �. By the resolvent identity,
for all M 2 �R
R(n;M; e+ i0) = bR(n;M; e+ i0)�
X
p2F
V (p) bR(p;M; e+ i0)R(n; p; e+ i0)
a.e. & a.s. on [a; b] �
. Consequently,
P
M2�R
jR(n;M; e+ i0)j2 is less than
or equal to A (
P
M2�R
j bR(n;M; e+ i0)j2 +M(e)
P
p2F jV (p)j
2jR(n; p; e+ i0)j2)
a.e. & a.s., whereM(e) = maxp2F
P
M2�R
j bR(p;M; e+ i0)j2 and A is the number
of elements of F plus one. Then the �niteness of F and the previous lemma
complete the proof.
2.5. Proof of the Second Lemma
Now we assume C, D, E, and F. Assumption D extends by induction:
Lemma 8. For any k and � such that 0 < � < k there exists a Ck;� > 0
satisfying X
P1;:::;Pl2X
e�k(
(N;P1)+
(P1;P2)+���+
(Pl;M))
6 C l
k;�e
��
(N;M) (5)
for every N;M 2 X and l 2 N.
160 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
P r o o f. There exists an s 2 (0; 1) such that � = sk. By Assumption D,
Bk0 = supN2X
P
M2X e�k
0
(N;M) < 1 for any k0 > 0. Let us show that Ck;� =
Btk then satis�es the desired property, where t = 1� s.
The triangle inequality for
implies that the left-hand side in (5) is bounded
above by
P
P1;:::;Pl
e�tk(
(N;P1)+���+
(Pl;M))e��
(N;M) for any �xed l > 0. It is thus
su�cient to show
P
P1;:::;Pl
e�tk(
(N;P1)+���+
(Pl;M)) 6 Bl
tk for any l > 0.
The result is trivial if l = 0. Suppose it holds for l � 1. Then,X
P1;:::;Pl
e�tk(
(N;P1)+���+
(Pl;M)) =
X
P1
e�tk
(N;P1)
X
P2;:::;Pl
e�tk(
(P1;P2)+���+
(Pl;M))
6 BtkB
l�1
tk = Bl
tk;
as desired.
As a �nal preliminary remark,
Lemma 9. All assumptions of the previous section are satis�ed.
P r o o f. Assumption B follows from Assumptions D and F. Assumption A
is satis�ed, since for any �nite E � � and n 2 � n E ,X
m2�nfng
�(m;n)s 6 (D sup
p2�
X
q2�nfpg
e�
�
2
(p;q)) sup
m2�nfng
e�
�
2
(n;m);
where the right-hand side goes to zero as E " X (by Assumptions D and E).
Finally, Assumption C is satis�ed by �at.
We are thus free to use the results and computations of the previous section.
Recall that F � � is a �nite set chosen in such a way that the relation (1) holds.
From now, by enlarging F if necessary, we also require�
e�
�
2
bd <
I
sks
KsC�
2
;
�
3
D
; (6)
where bd = infm2�nF infn2(�nF)nfmg
(n;m); this may be done by Assumption E.
Letm 2 �nF and z 2 S be �xed, n 2 �nF being thought as the only variable.
Then, with the notation of the previous section the inequation (4) applies, namely
X 6 (1�A)�1B (pointwise). Consequently,
Lemma 10. X 6 Const (1�A)�1Æm (pointwise).
�Here, �, D, and C �
2
;
�
3
refer to Assumption F and Lem. 8.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 161
Ph. Poulin
P r o o f. Observe that (AÆm)(n) = KsB(n)�
Ks
ksIs
�(m;m)sÆm(n); and hence
B = 1
Ks
AÆm + 1
ksIs
�(m;m)sÆm: By the inequation (4),
X 6
�
1
Ks
+
�(m;m)s
ksIs
�
(1�A)�1Æm (pointwise):
The result follows, with Const = 1
Ks
+ 1
ksIs
supp2�nF �(p; p)
s (which is �nite by
Assumption B).
Lemma 11. There exist universal constants Const and k such that
E j bR(n;m; z)js 6 Const e�k
(n;m)
for all n;m 2 � n F and z 2 S.
P r o o f. By the previous lemma,
E j bR(n;m; z)js 6 Const
1X
j=0
hÆn jA
jÆmi: (7)
Moreover,
Aj(n;m) =
�
Ks
ksIs
�j X
p
1
;��� ;p
j�1
2�nF
1n 6=p
1
�(n; p
1
)s : : : 1p
j�1
6=m�(p j�1;m)s;
where 1p6=q = 1 � Æp(q). By Assumption F, 1p6=q�(p; q)
s 6 De�
� bd
2 e�
�
2
(p;q) for
p; q 2 � n F . Hence, Lem. 8 implies
Aj(n;m) 6
0
@KsDe�
� bd
2
ksIs
1
Aj X
p
1
;��� ;p
j�1
e�
�
2
(n;p
1
) : : : e
�
�
2
(p
j�1
;m)
6
1
C�
2
;
�
3
0
B@KsC�
2
;�
3
De�
� bd
2
ksIs
1
CA
j
e�
�
3
(n;m):
By choice of F the equation (6) holds, so there exist constants Const and k
such that
P1
j=0A
j(n;m) 6 Const e�k
(n;m): The equation (7) then completes the
proof.
Lemma 12. There exist constants Const and k such that for each n 2 �,
M 2 �R and z 2 S,
E j bR(n;M; z)js 6 Const e�k
(n;M):
162 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
P r o o f. For N 2 �R and m 2 � n F , Lem. 5, Assumption F, and the
previous lemma yield
E j bR(N;m; z)js 6 �(N;m)s +Ks
X
p2�nF
�(N; p)sE j bR(p;m; z)js
6 Const e�k
(N;m) +Ks
X
p2�nF
Const e�k
(N;p)e�k
(p;m);
where Const and k denote generic constants. It follows from Lem. 8 that
E j bR(N;m; z)js 6 Const e�k
(N;m). Using this last inequation and Lem. 5 again,
a similar computation then gives the result.
Lemma 13. For all n 2 � and almost all (e; V ) 2 [a; b] �
there exist
constants, Const and k, the latter being universal, satisfying
j bR(n;M; e+ i0)j 6 Const e�k
(n;M)
for all M 2 �R.
P r o o f. Let n 2 � be �xed and M 2 �R. Recall that bR(n;M; e+ i0) exists
for almost all (e; V ) 2 [a; b] �
. Thus, the previous result and Fatou's lemma
yield
E
bZ
a
j bR(n;M; e+ i0)js de 6 Const e�k
(n;M):
Let AM = f(e; V ) 2 [a; b]�
; j bR(n;M; e+ i0)j > e�
k
2s
(n;M)g; where k refers to
the previous inequality. Then, denoting by d the Lebesgue measure,
X
M2�R
( d� dP)(AM ) 6
X
M2�R
E
bZ
a
e
k
2
(n;M)j bR(n;M; e+ i0)js de
6 Const
X
M2�R
e�
k
2
(n;M);
which is �nite by Assumption D. Hence, by Cantelli's lemma there exists a �nite
E � �R such that for all M 2 �R n E
j bR(n;M; e+ i0)j 6 e�
k
2s
(n;M) a.e. & a.s.;
where n 2 � is arbitrarily �xed. Since E is �nite, the result follows.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 163
Ph. Poulin
Lemma 14. Let E � � be �nite. For a given n 2 � and almost all (e; V ) 2
[a; b]�
there exist constants, K and k, the latter being universal, satisfying
j bR(q;M; e+ i0)j 6 Ke�k
(n;M)
for all M 2 �R and q 2 E.
P r o o f. Since E is �nite, the last lemma ensures for almost all (e; V )
the existence of constants satisfying j bR(q;M; e+ i0)j 6 Const e�k
(q;M) for all
M 2 �R and q 2 E . Since e�k
(q;M) 6 ek
(n;q)e�k
(n;M); the result follows, with
K = Const supq2E e
k
(n;q).
We are now ready to prove Lem. 2. By the resolvent identity, for all n 2 �,
M 2 �R, and almost all (e; V ) 2 [a; b]�
R(n;M; e+ i0) = bR(n;M; e+ i0)�
X
p2F
R(n; p; e+ i0)V (p) bR(p;M; e+ i0):
In particular, there exists a constant, namely, L = supp2F jR(n; p; e+ i0)V (p)j,
which depends on n, e, and V , but not on M , satisfying
jR(n;M; e+ i0)j 6 j bR(n;M; e+ i0)j+ L
X
p2F
j bR(p;M; e+ i0)j:
The result follows from the previous lemma applied to E = F [ fng.
2.6. Proofs of the Theorems
Lemma 15. Let 0 6 R 61. If
sup
N2�R
X
M2�R
�(N;M )s <1;
then 1R is �-smooth on [a; b].
P r o o f. The triangle inequality for j�js and the hypothesis yield
sup
N2�R
X
M2�R
jhÆN j (�� z)�1ÆM ij 6 Const
uniformly in z 2 S. Interpreting 1R(�� z)�11R as an operator on l2(�R), its l
1
and l1 norms are given by the above expression. Therefore, Schur's interpolation
theorem implies supz2S k1R(�� z)�11Rk <1; as desired.
164 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
P r o o f o f t h e f i r s t t h e o r e m. Since [a; b] � �(�), we also have
[a; b] � �(H) for all V such that
�
[a;b]
(H;�) exist, i.e., almost surely. Hence, by
Prop. 4 the spectrum of H is purely a.c. on [a; b]. Moreover, the previous lemma
(with R = 1) and the assumption of the theorem imply that 11 is �-smooth.
Lemma 1 (with R = 1) and Prop. 2 thus complete the proof.
P r o o f o f t h e s e c o n d t h e o r e m. Lemma 9 and the assumption
of the theorem imply Lems. 1 and 2 (both with R =1). The result then follows
from Prop. 3.
3. Models on Z
d
We now turn our attention to the case where X = Z
d (d > 2), and the
graph distance, d(M;N), is translational invariant. The graph (Zd;d) is then
determined by V = fN 2 Z
d ; d(N; 0) = 1g. We set
(M;N) = jN �M j.
Recall that the Fourier transform of 2 l2(Zd) is de�ned as
b (x) = (F )(x) = (2�)�
d
2
X
N2Zd
eiN �x (N);
where x 2 T
d. The symbol of� is b� = F�F�1: Thus, given a V � Z
d, the symbol
of the Laplacian associated with V is the multiplication by
�(x) =
X
V 2V
eiV �x =
X
V 2V
cos (V � x):
It follows from a change of variables that the spectrum of � is purely a.c. and
equal to [min�;#V], where #V denotes the cardinality of V.
The Green function of � is de�ned as G(M;N; z) = hÆM j (�� z)�1ÆN i for
M;N 2 Z
d and z 2 C + . Since (Zd;d) is translational invariant, G(M;N; z) =
G(0; N �M; z); this last is abbreviated by G(N �M; z). Hence, for any N 2 Z
d
and z 2 C + ,
G(N; z) = hbÆ0(x) j (b�� z)�1cÆN (x)i2
= (2�)�d
Z
Td
eiN �x
�(x)� z
dx: (8)
Recall that our sparseness assumptions are formulated in terms of
�(M;N) = sup
z2S
jG(N �M; z)j;
where [a; b] is a given interval and S = fa 6 Re z 6 b; 0 < Im z < 1g. Hence the
decay of G(N; z) when N !1 has to be a priori known. It is clear that G(N; z)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 165
Ph. Poulin
decays exponentially when [a; b] is outside the spectrum of �. Moreover, the case
where [a; b] � spec(�) has been studied in [24, 23] using material from [26, 28, 29]:
Proposition 5. Given a real-valued, analytic and periodic function �(x) on
T
d, let �(e) = fx 2 T
d ; �(x) = eg and let G(N; z) be de�ned by (8). Assume,
for (a0; b0) � Ran� and S 0 =
S
e2(a0;b0) �(e):
� r�(x) 6= 0 for all x 2 S 0;
� for all e 2 (a0; b0), �(e) admits at least � nonvanishing principal curvatures
at any point, where � > 1 is a �xed integer.
Then, for N = jN j! and [a; b] � (a0; b0), limz!e; z2C+ G(N; z) exists� and is
O(jN j�
�
2 ) uniformly in (e; !) 2 [a; b]� Sd�1. More generally,
G(N; z) = O(jN j�
�
2 log jN j)
uniformly in (z; !) 2 S � Sd�1, where S = fe+ iy ; a 6 e 6 b; 0 < y < 1g.
For example, in the case of the centered Laplacian, which is speci�ed by
V = f(�1; 0; : : : ; 0); (0;�1; : : : ; 0); : : : ; (0; 0; : : : ;�1)g
and whose spectrum is equal to [�2d; 2d], �(e) de�nes a regular surface for
e =2 f�2d;�2d + 4; : : : ; 2d � 4; 2dg, exempt of planarity if in addition e 6= 0.
Hence, letting E = f�2d;�2d + 4; : : : ; 2d � 4; 2dg[f0g, G(N; e+i0) = O(jN j�
1
2 )
uniformly on compact subsets of [�2d; 2d]nE. As an alternative, in order to avoid
convexity problems, S. Molchanov and B. Vainberg [17] have suggested to base
the discretization of the Laplacian on the diagonal neighbors
V = f(v(1); : : : ; v(d)) ; v(j) 2 f1;�1g for j = 1; : : : ; dg:
The resulting graph consists of 2d�1 connected components, and the spectrum of
its Laplacian is equal to
�
�2d; 2d
�
. Remarkably, �(e) de�nes a regular, strictly
convex surface for e =2 f�2d; 0; 2dg, as shown in [22]; hence, with E = f�2d; 0; 2dg,
G(N; e+ i0) = O(jN j�
d�1
2 ) uniformly on compact subsets of [�2d; 2d] n E.
Let us translate our abstract results to the present concrete models using the
previous proposition. Assumption A and the strengthened version of B assumed
in Th. 1 easily reduce to the following sparseness assumption:
A s s u m p t i o n G. There exists an � > 0 such that
P
m2�nfng jn �
mj�
�s
2
+� <1 for all n 2 �, and
lim
jnj!1
n2�
X
m2�nfng
jn�mj�
�s
2
+� = 0:
�Without constraints on the approach.
166 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1
Scattering from Sparse Potentials on Graphs
First consider the case where [a; b] � (a0; b0) � spec(�) for a given (a0; b0)
satisfying the hypotheses of the previous proposition. Since (Zd;d) is translational
invariant,
I = inf
z2S
jhÆ0 j (�� z)�1Æ0ij = inf
z2S
jG(0; z)j:
Moreover, by Th. 6.1 in [24]
lim
z!e
z2C+
ImG(0; z) = �
Z
�(e)
krx�(x)k
�1ds(x) > 0: (9)
Since in addition ImG(0; z) > 0 on S, the above implies C.
Let �j = Pj�Pj, where Pj denotes the projection onto l2(Xj). Observe that
for any z =2 R
hÆN j (�j � z)�1ÆN i =
�
G(0; z) if N 2 Xj
0 otherwise.
Hence, the equation (9) implies [a; b] � �(�).
Consider now the case where [a; b] is at a positive distance of spec(�). Then, I
is clearly positive, i.e., C holds. Assumption D is satis�ed for
(M;N) = jM�N j.
Moreover, Assumption F holds, since supz2S jG(N; z)j is exponentially decaying.
Finally, Assumption E yields our sparseness condition in this case, namely
A s s u m p t i o n H. lim
jnj!1
n2�
inf
m2�nfng
jn�mj =1:
Let � be a reunion of intervals (a0; b0) like above. We have proven:
Theorem 3. Suppose � satis�es G. If the wave operators
�
�(H;�) exist a.e.,
then they are complete (and the spectrum of H is purely a.c.) on �, almost
surely. Suppose instead � satis�es the weaker assumption H. Then, the spectrum
of H outside spec(�) is almost surely pure point with simple eigenvalues and
exponentially decaying eigenfunctions.
R e m a r k s.
1. In particular, the previous theorem holds for the standard Laplacian (with
� = 1) and the Molchanov�Vainberg Laplacian (with � = d � 1) on � =
spec(�) n E, where in both cases E is a �nite, deterministic set (described
after Prop. 5). By Proposition 4 (for instance), such an E does not contain
eigenvalues of H, almost surely. In both cases completeness (a.s.) of the
wave operators on the whole spec(�) follows.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 1 167
Ph. Poulin
2. Additional conditions may be imposed on the geometry of � in order to
assure the existence of the wave operators, including additional sparseness
conditions [19].
3. As mentioned in the introduction, by Pastur's theorem the essential spec-
trum of H is almost surely equal to a deterministic set, which was cha-
racterized by S. Molchanov and B. Vainberg [17, 19].� Using their result,
one may construct examples in which specess(H) = R. This is the case for
instance when the random potential at each site has a Cauchy or a normal
distribution. Then, the spectrum of H is dense pure point in R n spec(�).
4. Our study includes another approach, based on Fredholm analytic theory
and valid for bounded, deterministic potentials [23]. Under suitable sparse-
ness conditions both existence and completeness of the wave operators are
derived on spec(�) minus a set of Lebesgue measure zero � which disap-
pears in the random frame.
Example. Consider H = �+V , where � is the standard (or the Molchanov�
Vainberg) Laplacian. Suppose fV (n)gn2� is a family of i.i.d. random variables
lying on � = f(j4; 0; : : : ; 0) 2 Z
d ; j 2 Zg, whose common distribution is Cauchy
(alternatively, normal). Then, � is sparse in the sense of Th. 3 (with s su�-
ciently close to 1). Moreover, since � is included in the hyperplane Zd�1 � Z
d,
the existence of
�(H;�) follows from a deterministic result of V. Jak�si�c and
Y. Last [7].�� Hence, by Th. 3 (and the �rst remark following it), spec(H) is
purely a.c. on spec(�) and the wave operators are complete there (almost surely).
Moreover, by the same theorem (and the third remark following it), the spectrum
of H on Rnspec(�) is dense pure point with simple eigenvalues and exponentially
decaying eigenfunctions, almost surely.
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|
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| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Poulin, Ph. 2016-09-29T17:35:37Z 2016-09-29T17:35:37Z 2008 Scattering from Sparse Potentials on Graphs / Ph. Poulin // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 151-170. — Бібліогр.: 29 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106499 We study the spectral structure of Schrodinger operators H = Δ+V for random potentials supported on sparse sets. In the past years examples of such operators whose spectra almost surely satisfy the following properties have been exhibited: Anderson localization holds outside spec(Δ), while the wave operators Ω⁺(H, Δ) exist inside this last set. We continue this program by presenting sparseness conditions under which Ω⁺(Δ, H) also exist. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Scattering from Sparse Potentials on Graphs Article published earlier |
| spellingShingle | Scattering from Sparse Potentials on Graphs Poulin, Ph. |
| title | Scattering from Sparse Potentials on Graphs |
| title_full | Scattering from Sparse Potentials on Graphs |
| title_fullStr | Scattering from Sparse Potentials on Graphs |
| title_full_unstemmed | Scattering from Sparse Potentials on Graphs |
| title_short | Scattering from Sparse Potentials on Graphs |
| title_sort | scattering from sparse potentials on graphs |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106499 |
| work_keys_str_mv | AT poulinph scatteringfromsparsepotentialsongraphs |