Stability of Spectral Types for Jacobi Matrices Under Decaying Random Perturbations

0.32 MB·English

by  Jonathan Breuer

STABILITY OF SPECTRAL TYPES FOR JACOBI
MATRICES UNDER DECAYING RANDOM
7 PERTURBATIONS
0
0
2 JONATHAN BREUER1,2 AND YORAM LAST1,3

n
a
J Abstract. We study stability of spectral types for semi-infinite

0 self-adjointtridiagonalmatricesunder randomdecayingperturba-
1 tions. We show that absolutely continuous spectrum associated
with bounded eigenfunctions is stable under Hilbert-Schmidt ran-
]
P dom perturbations. We also obtain some results for singular spec-
S tral types.
.
h
t
a
m
1. Introduction
[

1 In this paper we study semi-infinite Jacobi matrices of the form
v
9 b(1) a(1) 0 0 ...
7
a(1) b(2) a(2) 0 ...
2
1 J ({a(n)}∞n=1,{b(n)}∞n=1) =  0 a(2) b(3) a(3) ...  (1.1)
0
7  ... ... ... ... ... 
0  
 
h/ with
at b(n) R, a(n) > 0,
m ∈
as operators on ℓ2(Z = 1,2,... ). We shall assume throughout that
: + { }
v J ( a(n) , b(n) ) is self-adjoint. For this to be true, Σ∞ a(n)−1 =
i { } { } n=1 ∞
X suffices [1]. In fact, we need a somewhat stronger restriction on the
r growth of a(n) (see (1.7) below).
a { }
Such operators are a natural generalization of discrete Schr¨odinger
operators on the half line. In particular, the discrete Laplacian on
ℓ2(Z ) can be described with the help of the constant sequences 1
+
a◦(n) , 0 b◦(n) , where a◦(n) = 1 and b◦(n) = 0 for all n Z≡,
+
{ } ≡ { } ∈
so that
∆ = J (1,0).
Date: December 4, 2006.
1 Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel.
2 E-mail: [email protected].
3
E-mail: [email protected].
1
2 J. BREUER ANDY. LAST
From the fact that the vector
1
0
δ1  0 
≡
.
 .. 
 
 
isacyclic vector forJ ( a(n) , b(n) ), itfollows([20])thatthereexists
{ } { }
a measure µ, which coincides with the spectral measure of the vector
δ , so that J ( a(n) , b(n) ) is unitarily equivalent to the operator of
1
multiplication{by th}e{param}eter on L2(R,dµ). µ decomposes as
µ = µ +µ +µ ,
ac sc pp
where µ is the part of µ that is absolutely continuous with respect
ac
to the Lebesgue measure, µ is a continuous measure that is singular
sc
with respect to the Lebesgue measure, and µ is a pure point measure.
pp
We want to investigate the stability of certain continuity properties
of µ under a decaying random perturbation of J ( a(n) , b(n) ). The
{ } { }
first part of the paper deals with the stability of the essential support
of the absolutely continuous spectrum. In the second part, we restrict
the discussion to the case a(n) = 1 (the discrete Schr¨odinger case)
{ }
and deal with the more delicate singular spectral types. In both cases,
a principal tool in the analysis is the connection between properties of
the spectral measure and the behavior at infinity of solutions of the
difference equation
a(n)ϕ(n+1)+a(n 1)ϕ(n 1)+b(n)ϕ(n) = Eϕ(n) (1.2)
− −
forfixed E R andn 1(we set a(0) = 1). Such a difference equation
∈ ≥
can be regarded as an initial value problem, which makes it natural to
introduce the single-step transfer matrices:
E−b(n) a(n−1)
SE(n) = a(n) − a(n) , n 1, (1.3)
1 0 ≥
(cid:18) (cid:19)
that satisfy
ϕ(n+1) ϕ(n)
= SE(n)
ϕ(n) ϕ(n 1)
(cid:18) (cid:19) (cid:18) − (cid:19)
for any ϕ(n) ∞ that solves (1.2). Thus, if we denote
{ }n=0
ϕ(n+1)
ϕ~(n) =
ϕ(n)
(cid:18) (cid:19)
and TE(n) SE(n) ... SE(1), then
≡ · ·
ϕ~(n) = TE(n)ϕ~(0). (1.4)
STABILITY OF SPECTRAL TYPES 3
The essential support of an absolutely continuous measure ν on R
is the equivalence class Σ (ν) of sets A R such that ν is supported
ac
⊆
on A and that the restriction of Lebesgue measure to A is absolutely
continuous w.r.t. ν. We shall use Σ ( a(n) , b(n) ) to denote the
ac
{ } { }
essential support of µ and refer to it as the essential support of the
ac
absolutely continuous spectrum of J ( a(n) , b(n) ).
{ } { }
Over the past decade, there has been a significant amount of work
done(see, e.g., [2,3,4,5,11,13,14,22]), inthe areaofone-dimensional
Schr¨odinger operators, towards determining conditions on a perturbing
˜
potential b(n) ensuring that
{ }
Σ (1, b(n) ) = Σ (1, b(n)+˜b(n) ). (1.5)
ac ac
{ } { }
That such an equality exists for any ˜b(n) ℓ1 is a well known result
{ } ∈
from scattering theory [21, Chapter XI.3]. For general b(n) , this is
{ }
the best there is at present, in terms of sheer ℓp properties of the per-
turbation. For b(n) = 0, however, it has been proven by Deift-Killip
{ }
[5] that (1.5) holds for ˜b(n) merely in ℓ2. This result has been later
{ }
extended by Killip [11] to include any periodic b(n) . For arbitrary
{ }
background potentials b(n) , it has been conjectured by Kiselev-Last-
{ }
Simon [16] that an ℓ2 perturbation does not change the essential sup-
port of the absolutely continuous spectrum. For a perturbation of the
off-diagonal entries as well as the diagonal entries, Killip-Simon [12]
have shown that if a˜(n) , ˜b(n) ℓ2, then
{ } { } ∈
Σ (1,0) = Σ (1+ a˜(n) , ˜b(n) ). (1.6)
ac ac
{ } { }
Our first result deals with the preservation of Σ ( a(n) , b(n) ) for
ac
{ } { }
general b(n) and a(n) obeying
{ } { }
L
1
limsup a(n)−1 > 0 (1.7)
L
L→∞
n=1
X
under a random decaying perturbation of both the diagonal and off-
diagonal entries. For a measurable set B R, Σ B denotes the
ac
⊆ ∩
equivalence class of sets A B such that A Σ .
ac
∩ ∈
Theorem 1.1. Let J ( a(n) , b(n) ) be a Jacobi matrix such that
{ } { }
a(n) obeys (1.7), and let a˜ (n) : Ω R and ˜b (n) : Ω R (n 1)
ω ω
{ } → → ≥
be two sequences of independent random variables with zero mean, de-
fined over a probability space (Ω, ,P). Assume that there exists a
F
δ > 0, for which
a(n)
δ−1 > > δ (1.8)
a(n)+a˜ (n)
ω
4 J. BREUER ANDY. LAST
for every n and ω Ω. Let J = J ( a(n) , b(n) ) and
0
∈ { } { }
˜
J = J( a(n)+a˜ (n) , b(n)+b (n) ).
ω ω ω
{ } { }
Then, for a.e. ω,
Σ (J ) Γ = Σ (J ) Γ, (1.9)
ac 0 ac ω
∩ ∩
where Γ is the set of all E R for which
∈
∞
a˜ (n)4 1/2 + ˜b (n)2 (a(n)+1)tE(n) 4 < , (1.10)
ω ω
∞
Xn=1(cid:16)(cid:10) (cid:11) D E(cid:17)(cid:0) (cid:1)
where we denote f f dP(ω) for any measurable function f
h ωi ≡ Ω ω ω
of ω and tE(n) TE(n) is the norm of the n’th transfer matrix
≡k Rk
corresponding to J .
0
We note that Kaluzhny-Last [10] recently studied Jacobi matrices of
theformJ( a(n)+a˜ (n) , b(n)+˜b (n) ), where a(n) 1and b(n)
ω ω
{ } { } { }− { }
˜
are decaying sequences of bounded variation and a˜ (n) , b (n) are
ω ω
{ } { }
as in Theorem 1.1 and obey
∞
a˜ (n)2 + ˜b (n)2 < .
ω ω
∞
Xn=1(cid:16)(cid:10) (cid:11) D E(cid:17)
They show that, with probability one, such operators have purely ab-
solutely continuous spectrum on ( 2,2) and moreover, this purity of
−
the absolutely continuous spectrum is stable under changing any fi-
nite number of entries in the Jacobi matrices. Since the unperturbed
J( a(n) , b(n) ) is known (see, e.g., [24]) in this case to have purely
{ } { }
absolutely continuous spectrum on ( 2,2) with tE(n) ∞ being a
− { }n=1
bounded sequence for every E ( 2,2), we see that a part of their
∈ −
result, namely, the fact that Σ (J ) = Σ (J ), can be recovered as a
ac 0 ac ω
special case of Theorem 1.1.
To further elucidate Theorem 1.1, consider the case a(n) = 1,
a˜ (n) = 0. TheconditiondefiningΓtranslatesintoanℓ2 typecondition
ω
on the perturbation when one studies energies for which the transfer
matricesarebounded: Fora givenbackground potential b(n) , denote
{ }
Γ Γ ( b(n) ) = E R tE(n) is bounded .
0 0
≡ { } { ∈ | }
Then it follows from the theory of subordinacy [8] (also see [24]) that
there exists a set A Σ (1, b(n) ) for which Γ A. From Theo-
ac 0
∈ { } ⊆
rem 1.1, it follows that
Corollary 1.2. Assume that
∞
˜b (n)2 < .
ω
∞
Xn=1D E
STABILITY OF SPECTRAL TYPES 5
Then, for a.e. ω,
Σ (1, b(n) ) Γ = Σ (1, b(n)+˜b (n) ) Γ .
ac 0 ac ω 0
{ } ∩ { } ∩
Corollary 1.2 constitutes some progress towards a random version of
the above mentioned conjecture of Kiselev-Last-Simon [16]. Whether
actually Γ ( b(n) ) Σ ( b(n) ) for any b(n) is a long standing
0 ac
{ } ∈ { } { }
open problem. For some related work, see Maslov-Molchanov-Gordon
[19].
The questionofstability ofsingular spectral typeshasreceived much
less attention than the one concerning Σ . One of the reasons for this
ac
is the fact that singular spectral types are not stable even under rank
one perturbations (see [7]). One may, however, bypass this problem
by using an idea of Del-Rio-Simon-Stolz [6] to consider the union of
spectralsupports over thedifferent boundaryconditions. This provides
a unified approach for the different spectral types, in that spectral
stabilityisobtainedforanycompactlysupportedperturbation(see[6]).
Kiselev-Last-Simon [16] have modified and extended this approach, via
the theory of subordinacy, to deal with the classification of spectral
types according to the singularity/continuity of the spectral measure
w.r.t. α-dimensional Hausdorff measures. In our definitions, we follow
their general methodology.
While it is possible, using the methods developed below, to deal
with the general Jacobi case, we restrict the discussion to the case
of diagonal perturbations of discrete Schr¨odinger operators. We take
this approach in order to avoid technical difficulties which may obscure
the main argument. Thus, for fixed E R, we shall be looking at
∈
properties of solutions of the equations
ϕ(n+1)+ϕ(n 1)+b(n)ϕ(n) = Eϕ(n) (1.11)
−
for n 2,
≥
ϕ(2)+(b(1) tan(θ))ϕ(1) = Eϕ(1) (1.12)
−
for π < θ < π. Such sequences are obviously eigenvectors (not nec-
−2 2
essarily in ℓ2) of the infinite matrix
b(1) tan(θ) 1 0 0 ...
−
1 b(2) 1 0 ...
Hθ =  0 1 b(3) 1 ... . (1.13)
 ... ... ... ... ... 
 
 
We denote by ϕE (n) the solution to (1.11), (1.12), normalized by
1,θ
ϕE (1) = cos(θ). (1.14)
1,θ
6 J. BREUER ANDY. LAST
We also include the case θ = π/2, for which (1.12) and (1.14) are
−
replaced by ϕE (1) = 0, ϕE (2) = 1. We shall use the notation
1,−π/2 1,−π/2
ϕE ϕE . (1.15)
2,θ ≡ 1,θ−π/2
Remark. One may define ϕE by referring only to (1.11) (for n 1)
1,θ ≥
and using ϕE (0) = sin(θ), ϕE (1) = cos(θ). This way ϕE is more
1,θ − 1,θ 1,θ
naturally defined on [ π, π), without anything special for θ = π/2.
−2 2 −
A basic object in the theory of subordinacy is the L’th norm (for
L > 0) of a function f : Z C,
+
→
⌊L⌋
f 2 f(n) 2 +(L L ) f( L +1) 2, (1.16)
k kL≡ | | −⌊ ⌋ | ⌊ ⌋ |
n=1
X
where denotes integer part. For a given E R, θ [ π, π), ϕE
⌊·⌋ ∈ ∈ −2 2 1,θ
is called subordinate if
ϕE
lim k 1,θ kL = 0. (1.17)
L→∞ k ϕE2,θ kL
It is clear that a subordinate solution does not necessarily exist for
every E, but whenever it does, it is unique. We denote the θ for which
ϕE is subordinate, if it exists, by θ(E). One may decompose R into
1,θ
three disjoint sets:
Σ E R θ(E) exists and ϕE ℓ2
pp ≡ { ∈ | 1,θ(E) ∈ }
Σ E R θ(E) exists and ϕE ℓ2
sc ≡ { ∈ | 1,θ(E) 6∈ }
R (Σ Σ )
pp sc
\ ∪
What makes the discussion of stability of singular spectral types inter-
esting is the fact (see, e.g., [16]) that these three sets have the following
spectral interpretation:
Σ = σ˜ (H ), where σ˜ (H ) is the set of eigenvalues of H .
pp θ pp θ pp θ θ
• ∪
For any θ, µ ( ) = µ (Σ ) and any other set A with this
θ,sc θ sc
• · ∩·
property equals Σ up to a set of Lebesgue measure zero.
sc
Σ R (Σ Σ ).
ac pp sc
• ∋ \ ∪
The above sets areclearly independent of θ andstable under compactly
supported perturbations.
The Jitomirskaya-Last extension of subordinacy theory [9] makes it
possible to investigate the stability of Hausdorff-dimensional properties
of the spectral measure. It follows from their analysis that for any
α (0,1], there exist sets Σ R and Σ R such that for any θ,
αc αs
∈ ⊆ ⊆
µ = µ (Σ ), µ = µ (Σ ) (1.18)
θ,αc θ αc θ,αs θ αs
∩· ∩·
STABILITY OF SPECTRAL TYPES 7
where µ is the part of µ that is continuous with respect to the α-
θ,αc θ
dimensional Hausdorff measure, and µ is the part which is singular
θ,αs
with respect to it. (For the study of decompositions of a measure w.r.t.
dimensional Hausdorff measures andforthesignificance of thisanalysis
to quantum mechanics, see, for example, [17] and references therein.)
For any α (0,1], Σ Σ Σ , and for any E Σ , whether
αs sc pp sc
∈ ⊆ ∪ ∈
E Σ or not, depends on the decay of the subordinate solution at
αs
∈
infinity:
E Σ
αs
∈
if and only if
ϕE
liminf k 1,θ(E) kL = 0 (1.19)
L→∞ ϕE β˜(α)
k 2,θ(E) kL
where β˜(α) = α (see [9]).
2−α
The discussion above motivates the following definition of [16]: Let
E Σ . Define
sc
∈
ln ϕE
β(E) = liminf k 1,θ(E) kL. (1.20)
L→∞ ln k ϕE2,θ(E) kL
For any E with β(E) > 0, we also define
1 β(E)
η(E) = − . (1.21)
β(E)
Again, it is clear that the sets Σ and Σ and the parameter β(E)
αs αc
(where it is defined) are stable under compactly supported perturba-
tions. To obtain more, one needs a regularity condition on the energy:
Following Kiselev-Last-Simon [16], we shall call an energy E regular if
for some θ and all ε > 0, we have
k ϕE1,θ kL< CεL12+ε.
Since almost every energy is regular both with respect to each µ (see
θ
[1]) and (by spectral averaging—see Theorem 1.8 in [23]) with respect
to Lebesgue measure, the demand that energies be regular is not a
severe restriction.
Let
Λ = E Σ E is regular and β(E) > 0 . (1.22)
0 sc
{ ∈ | }
For deterministic perturbations, Kiselev-Last-Simon [16] have shown
that, for any 0 < α < 1,
Λ˜ Σ (1, b(n) ) Σ 1, b(n)+˜b(n) ,
αc αc
∩ { } ⊆ { }
(cid:16) (cid:17)
Λ˜ Σ (1, b(n) ) Σ 1, b(n)+˜b(n) ,
αs αs
∩ { } ⊆ { }
(cid:16) (cid:17)
8 J. BREUER ANDY. LAST
where b(n) is any background potential, and
{ }
Λ˜ = E Λ ˜b(n) < Cn−γ for some γ > η(E)+1 . (1.23)
0
{ ∈ | | | }
For random potentials we show
Theorem 1.3. Let ˜b (n) be a sequence of independent real-valued
ω
{ }
random variables with zero mean on Ω. For any E Λ , η˜ > 0 and
0
∈
n 1, let
≥
rE(n) ϕE (n) 4n2η˜+ ϕE (n) 4, (1.24)
η˜ ≡ | 1,θ(E) | | 2,θ(E) |
and let
∞
Λ = E Λ rE(n) ˜b (n)2 < for some η˜> η(E) .
∈ 0 η˜ ω ∞
( (cid:12) )
(cid:12) Xn=1(cid:16) D E(cid:17)
(cid:12) (1.25)
(cid:12)
Then, for any 0 <(cid:12) α < 1 and any fixed measure ν on R, for a.e. ω,
Λ Σαc(1, b(n) ) Σαc(1, b(n)+˜bω(n) ),
∩ { } ⊆ { }
Λ Σαs(1, b(n) ) Σαs(1, b(n)+˜bω(n) ), (1.26)
∩ { } ⊆ { }
where the inclusion is up to a set of ν-measure zero.
Theorems 1.1 and 1.3 have the common feature of the appearance
of the 4th power of the norms of the transfer matrices (in 1.3, see the
definition of rE(n)). The reason for this is that our basic tool is a ran-
η˜
dom variation of parameters, where the perturbing potential is coupled
to the square of the transfer matrices and thus, when estimating the
variance of the perturbation, the 4th power enters the picture. For ex-
amples where the pointwise behavior of the solutions to (1.2) is known,
this does not constitute a problem. One such example is the class of
bounded sparse potentials studied by Zlatoˇs [25]. For this class, one
has stability of Σ and Σ under random perturbations decaying like
αc αs
n−γ for γ > η(E)+ 1 (compare with γ > η(E)+1 in (1.23)).
2
In light of these remarks, the general question of the pointwise be-
havior of thesolutions of (1.2) isone thatarises naturallyinconnection
with the results presented here. The more famous question of whether
or not there is almost-everywhere boundedness of solutions with re-
spect to the absolutely continuous part of the spectral measure is only
one facet of this general problem.
The rest of this paper is organized as follows. Section 2 covers some
preliminaries—especially a useful characterization of Σ due to Last-
ac
Simon [18] and a variation on a classic theorem concerning the almost
everywhere convergence of random series with convergent variances. In
Section3weintroducethemainideabehindouranalysis. Weformulate
and prove two different (but similar) lemmas which are central to the
STABILITY OF SPECTRAL TYPES 9
proofsofour two main theorems. These theorems areproved inSection
4. Section5hastheworkedoutapplicationofTheorem1.3totheabove
mentioned sparse potentials of Zlatoˇs [25].
ThisresearchwassupportedinpartbyTheIsraelScienceFoundation
(Grant No. 188/02) and by Grant No. 2002068 from the United States-
Israel Binational Science Foundation (BSF), Jerusalem, Israel.
2. Preliminaries
As explained in the introduction, we want to exploit the connection
between spectral properties of the operator J ( a(n) , b(n) ) and the
{ } { }
asymptotic properties of the solutions to the corresponding difference
equation. That is, we want to compare the asymptotic properties of
the solutions to the difference equation corresponding to the basic op-
erator, with those of the solutions to the equation corresponding to
the perturbed one. In the singular continuous case we will ‘equate’
the behavior at infinity of the perturbed and unperturbed solutions (in
a sense to be precisely defined in Section 4). For the absolutely con-
tinuous case, however, we need a little less. We rely on the following
characterization of Σ due to Last-Simon [18]:
ac
Proposition 2.1 (Last-Simon [18]). Let J ( a(n) , b(n) ) be a self-
{ } { }
adjoint Jacobi matrix such that a(n) satisfies (1.7), and let TE(n) be
{ }
the corresponding transfer matrices defined by (1.3)-(1.4). Let N ∞
{ j}j=1
be a sequence for which
1 Nj 1
lim > 0
j→∞ Nj a(n)
n=1
X
and let Σ Σ ( a(n) , b(n) ). Then
ac ac
≡ { } { }
1 Nj
E R liminf TE(n) 2< Σ .
ac
 ∈ (cid:12) j→∞ Nj k k ∞ ∈
 (cid:12) Xn=1 
(cid:12)
Remark. This is act(cid:12)ually a slight generalization of Theorem 1.1 of [18]
 (cid:12) 
to the general Jacobi case. Its proof is essentially the same as their
proof.
Thefollowingarevariantsofamartingaleinequalityandconvergence
theorem which play a crucial role in the proofs of Lemmas 3.1 and 3.2.
Lemma 2.2. Let (Ω, ,P) be a probability space and let x (n) be a
ω
F { }
sequence of independent random variables such that
x (n)dP(ω) x (n) = 0
ω ω
≡ h i
ZΩ
10 J. BREUER ANDY. LAST
for all n. Let
z (n) = x (n)f (x (n+1),x (n+2),...)
ω ω n ω ω
where the f are real-valued, measurable functions on R∞.
n
Then, for any N < N and r 0,
1 2
≥
N2 (z (j))2
P ω max z (n)+...+z (N ) > r j=N1h ω i.
ω ω 2
(cid:18)(cid:26) (cid:12) N1≤n≤N2| | (cid:27)(cid:19) ≤ P r2
(cid:12) (2.1)
(cid:12)
(cid:12)
Proof. Obviously, we may assume that (z (n))2 < for all n, since
ω
∞
otherwise there is nothing to prove. Denote
(cid:10) (cid:11)
n−1 N2
Y (n) = z (j), Q (n) = z (j),
ω ω ω ω
jX=N1 Xj=n
and let
A = ω Ω Q (j) > r; Q (j +1) ,..., Q (N ) r .
j ω ω ω 2
{ ∈ | | | | | | | ≤ }
Then, if i < j,
z (i)Q (j)χ = x (i) f (x (i+1),...)Q (j)χ = 0
ω ω j ω i ω ω j
h i h ih i
where
χ = χ = the characteristic function of A ,
j Aj j
and thus,
χ Y (j)Q (j) = 0
j ω ω
h i
so that
χ Q (j)2 χ (Y (j)+Q (j))2 .
j ω j ω ω
≤
Therefore
(cid:10) (cid:11) (cid:10) (cid:11)
r2 χ χ Q (j)2 χ (Y (j)+Q (j))2
j j ω j ω ω
h i ≤ ≤
and
(cid:10) (cid:11) (cid:10) (cid:11)
N2 N2 N2
r2 χ χ Q (j)2 χ (Y (j)+Q (j))2
j j ω j ω ω
h i ≤ ≤
jX=N1 jX=N1(cid:10) (cid:11) jX=N1(cid:10) (cid:11)
2 2
N2 N2 N2
= χ z (l) z (j)
j ω ω
≤
* ! + * ! +
jX=N1 lX=N1 jX=N1
N2
= z (j)2
ω
* +
jX=N1
where in the last equality we use
z (i)z (j) = 0 for i = j.
ω ω
h i 6