Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 902506, 30 pages
doi:10.1155/2009/902506
Research Article
Quantum Energy Expectation in Periodic
Time-Dependent Hamiltonians via Green Functions
César R. de Oliveira1, 2 and Mariza S. Simsen3
1
Departamento de Matemática, Universidade Federal de São Carlos, 13560-970 São Carlos, SP, Brazil
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2
3
Instituto de Ciências Exatas, Universidade Federal de Itajubá, 37500-000 Itajubá, MG, Brazil
2
Correspondence should be addressed to César R. de Oliveira, oliveira@dm.ufscar.br
Received 10 November 2008; Revised 29 January 2009; Accepted 21 February 2009
Recommended by Edson Denis Leonel
Let UF be the Floquet operator of a time periodic Hamiltonian Ht. For each positive and discrete
observable A which we call a probe energy, we derive a formula for the Laplace time average of
its expectation value up to time T in terms of its eigenvalues and Green functions at the circle of
radius e1/T . Some simple applications are provided which support its usefulness.
Copyright q 2009 C. R. de Oliveira and M. S. Simsen. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Consider a general periodically driven quantum hamiltonian system
Ht H0 V t
1.1
with period τ acting in a separable Hilbert space H, and let UF denote its Floquet operator, so
that if ξ is the initial state at time zero of the system, then UFm ξ is this state at time mτ. Typically, the unperturbed hamiltonian H0 is assumed to have purely point spectrum so that the
same is true for e−iτH0 . What happens when H0 is perturbed by V t? A natural physical question is if the expectation values of the unperturbed energy H0 remain bounded when V t / 0.
This question is formulated based on many physical models, in particular on the Fermi
accelerator in which a particle can acquire unbounded energy from collisions with a heavy
periodically moving wall. Here quantum mechanics is considered and, more precisely, if
sup UFm ξ, H0 UFm ξ m∈N
is finite or not, where ξ ∈ dom H0 ⊂ H, the domain of H0 .
1.2
2
Mathematical Problems in Engineering
Motivated by models with hamiltonians as above Ht H0 V t, one is suggested to
probe quantum instability through the behavior of an “abstract energy operator” which we
call a probe operator and will be represented by a positive, unbounded, self-adjoint operator
A : dom A ⊂ H → H and with discrete spectrum,
Aϕn λn ϕn ,
1.3
0 ≤ λn < λn1 , such that if UFm ξ ∈ dom A for all m ∈ N, then, for each m, the expectation value
EξA m UFm ξ, AUFm ξ is finite. It is convenient to write EξA m ∞ if UFm ξ does not belong
to dom A.
We say the system is A-dynamically stable when EξA m is a bounded function of time
m, and A-dynamically unstable otherwise usually we say just (un)stable. If the function EξA m
is not bounded, one can ask about its asymptotic behavior, that is, how does EξA m behave
as m goes to infinity? Usually this is a very difficult question and sometimes the temporal
average of EξA m is considered, as we will do in this work.
Quantum systems governed by a time periodic hamiltonian have their dynamical
stability often characterized in terms of the spectral properties of the corresponding Floquet
operator. As in the autonomous case, the presence of continuous spectrum is a signature of
unstable quantum systems; this is a rigorous consequence of the famous RAGE theorem 1,
firstly proved for the autonomous case and then for time-periodic hamiltonians 2, 3. At first
sight a Floquet operator with purely point spectrum would imply stability, but one should
be alerted by examples with purely point spectrum and dynamically unstable 4–6 in the
autonomous case and, recently, also a time-periodic model with energy instability 7 was
found.
Dynamical stability of time-dependent systems was studied, for example, in references
2, 8–19. In 14 it was proved that the applicability of the KAM method gives a uniform
bound at the expectation value of the energy for a class of time-periodic hamiltonians
considered in 20.
For hamiltonians Ht H0 V t, not necessarily periodic, with H0 a positive selfadjoint operator whose spectrum consists of separated bands {σj }∞
j1 such that σj ⊂ λj , Λj ,
upper bounds of the type
Ut, 0ψ, H0 Ut, 0ψ ≤ cte t1α/nα
1.4
were obtained in 10 if the gaps λj1 − Λj grow like j α , with α > 0, and if V t is strongly
Cn with n ≥ 1 α/2α 1. The proof is based on adiabatic techniques that require smooth
time dependence and therefore do not apply to kicked systems. In 11, 13 upper bounds
complementary to those of 10 described above are obtained.
In 2, 8, 9, 15 stability results are obtained through topological properties of the orbits
ξt Ut, 0ξ for ξ ∈ H, while in 16–19 lower bounds for averages of the type
T
m
1
UF ξ, H0 UFm ξ ≥ CT γ
T m1
1.5
Mathematical Problems in Engineering
3
are obtained for periodic hamiltonians Ht H0 V t through dimensional properties of
the spectral measure μξ associated with UF and ξ the exponent γ depends on the measure
μξ .
In this work we study instability of periodic time-dependent systems. As for tightbinding models see 21 and references therein we consider the Laplace-like average of
UFm ξ, AUFm ξ, that is,
∞
2
e−2m/T UFm ξ, AUFm ξ,
T m0
1.6
where A is a probe energy, ξ is an element of dom A, and UF is the Floquet operator. The
main technical reason for working with this expression for the time average is that it can
be written in terms of see Theorem 2.3 the eigenvalues of A, that is, Aϕj λj ϕj , and the
matrix elements ϕj , Rz UF ξ of the resolvent operator Rz UF UF − z1−1 with z e−iE e1/T with respect to the orthonormal basis {ϕj } of the Hilbert space here 1 denotes the
identity operator. Lemma 2.2 relates the long run of Laplace-like average with the usual
Cesàro average. In Section 2 we shall prove this abstract results and present some applications
in Section 3.
Since our main results are for temporal Laplace averages of expectation values
of probe energies see Section 2, in practice we will think of instability in terms
of unboundedness of such averages. Note that unbounded Laplace averages imply
unboundedness of expectation values of probe energies themselves.
2. Average Energy and Green Functions
Consider a time-dependent hamiltonian Ht with Ht τ Ht for all t ∈ R, acting in
the separable Hilbert space H. Suppose the existence of the propagators Ut, s, so that the
Floquet operator UF Uτ, 0 is at our disposal. Let A be a probe energy and λj , ϕj as in the
introduction. Also, {ϕj }∞
j1 is an orthonormal basis of H.
The main interest is in the study of the expectation values, herein defined by
EξA m :
⎧
⎨ UFm ξ, AUFm ξ ,
if UFm ξ ∈ dom A,
⎩∞,
if UFm ξ ∈ H \ dom A,
2.1
as function of time m ∈ N. Another quantity of interest is the time dependence of the moment
of this probe energy, which takes values in 0, ∞ and is defined by
MξA m :
∞
2
λj ϕj , UFm ξ .
j1
Our first remark is the equivalence of both concepts under certain circumstances.
2.2
4
Mathematical Problems in Engineering
Proposition 2.1. If UFm ξ ∈ dom A for all m, then
EξA m MξA m,
∀m.
2.3
This holds, in particular, if dom A is invariant under the time evolution UFm and ξ ∈ dom A.
Proof. Since dom A ⊂ dom A1/2 1, one has UFm ξ ∈ dom A1/2 , for all m, and so
MξA m ∞ 2
1/2
λj ϕj , UFm ξ j1
∞ 2
1/2
A ϕj , UFm ξ j1
∞ 2
ϕj , A1/2 UFm ξ j1
2.4
2
A1/2 UFm ξ
A1/2 UFm ξ, A1/2 UFm ξ
UFm ξ, AUFm ξ
EξA m,
which is the stated result.
We introduce the temporal Laplace average of EξA see also the appendix by the
following function of T > 0, which also takes values in 0, ∞:
LA
ξ T :
∞
2
e−2m/T EξA m.
T m0
2.5
Under certain conditions, the next result shows that the upper β and lower β− growth
exponents of this average, that is, roughly they are the best exponents so that for large T
there exist 0 ≤ c1 ≤ c2 < ∞ with
−
β
c 1 T β ≤ LA
ξ T ≤ c2 T ,
2.6
and the corresponding exponents for the temporal Cesàro average
CξA T T
1
EA m
T m0 ξ
2.7
Mathematical Problems in Engineering
5
are closely related; this follows at once by Lemma 2.2, which perhaps could be improved
to get equality also between lower exponents. Note that, although not indicated, these
exponents depend on the initial condition ξ.
n
Lemma 2.2. If hm∞
m0 is a nonnegative sequence and hm ≤ Cm for some C > 0 and n ≥ 0,
−
−
then βe βd and βe ≤ βd , where
βe lim sup
log
T
m0 hm
log
T
m0 hm
,
βe− lim inf
,
T →∞
log T
log T
−2m/T
−2m/T
log ∞
hm
log ∞
hm
m0 e
m0 e
−
,
βd lim inf
.
βd lim sup
T →∞
log T
log T
T →∞
T →∞
2.8
Proof. Note that for 0 ≤ m ≤ T we have e−2 ≤ e−2m/T ≤ 1, and so
T
T
hm ≤
m0
∞
e2 e−2m/T hm ≤ e2
m0
e−2m/T hm.
2.9
m0
Hence βe± ≤ βd± .
On the other hand, for each > 0, denoting by x the smallest integer larger or equal
to x, one has
∞
e
−2m/T
hm m0
1
T
e−2m/T hm
mT 1 1
m0
≤
∞
e−2m/T hm 1
T
∞
hm C
m
m0
T 1
2.10
e−2m/T mn .
1
Now, for T large enough nT/2 < T 1 ≤ T 1 . Thus
∞
e−2m/T mn ≤
mT 1 1
∞
T 1 e−2t/T tn dt.
2.11
Therefore, for each > 0 and T large enough
∞
e
−2m/T
hm ≤
m0
1
T
m0
≤
1
T
m0
hm C
∞
T 1
e−2t/T tn dt
−2T T n .
hm Ce
2.12
6
Mathematical Problems in Engineering
Since e−2T T n → 0 as T → ∞, it follows that
βd lim sup
log
∞
m0 e
log
T 1 m0
hm
log T
T →∞
lim sup
hm
log T
T →∞
≤ lim sup
−2m/T
hm log T 1
log T
log T 1 log
T →∞
T 1 m0
T 1
≤ lim sup
log
T →∞
m0 hm
logT 1 1 lim sup
log
T 1 m0
2.13
log T 11
log T
hm
log T 1 T →∞
≤ 1 βe .
As > 0 was arbitrary, βd ≤ βe .
ξ
Recall that the Green functions Gz j associated with the operators A, UF at ξ ∈ H
and z ∈ C, |z| / 1, are defined by the matrices elements of the resolvent operator Rz UF −1
UF − z1 along the orthonormal basis {ϕj }∞
j1 , that is,
ξ Gz j : ϕj , Rz UF ξ.
2.14
ξ
1 that resolvent operator is bounded.
Note that Gz j is always well defined since for |z| /
T . It presents a
Theorem 2.3 is the main reason for considering the temporal averages LA
ξ
formula that translates the Laplace average of wavepackets at time T into an integral of the
Green functions over “energies” in the circle of radius e1/T in the complex plane centered at
the origin. As T grows, the integration region approaches the unit circle where the spectrum
of UF lives and Rz UF takes singular values, so that hopefully A-instability can be
quantitatively detected.
Theorem 2.3. Assume that UFm ξ ∈ dom A for all m ≥ 0. Then
LA
ξ T ∞
1
1
λj
πe−2/T T j1
2π ξ 2
Gz j dE,
z e−iE1/T .
2.15
0
Before the proof of this theorem, we underline that this formula, that is, the expression
on the right-hand side of 2.15, is a sum of positive terms and so it is well defined for all
ξ ∈ H if we let it take values in 0, ∞; hence, in principle it can happen that this formula is
Mathematical Problems in Engineering
7
T ∞. The general case, that
finite even for vectors UFm ξ not in the domain of A, where LA
ξ
is, ∀ξ ∈ H, can then be gathered in the following inequality:
LA
ξ T ≥
∞
1
1
λj
πe−2/T T j1
2π ξ 2
Gz j dE,
z e−iE1/T ,
2.16
0
so that lower bound estimates for this formula always imply lower bound estimates for the
Laplace average.
Proof of Theorem 2.3. First note that, by hypothesis, UFm ξ ∈ dom A1/2 for each m ∈ N. Denote
by μj the spectral measure of UF associated with the pair ϕj , ξ and by F the Fourier
transform F : L2 0, 2π → l2 Z. By the spectral theorem for unitary operators
ϕj , UF ξ 2π
e−iE dμj E .
2.17
0
For each j let aj aj mm∈Z be the sequence
j
a m ⎧
⎨0,
⎩e−m/T 2π e−iE m dμ E ,
j
0
if m < 0,
if m ≥ 0.
2.18
Since aj ∈ l1 Z∩l2 Z and F is a unitary operator, it follows that aj l2 Z F−1 aj L2 0,2π
and also
∞
1 F−1 aj E √
eiEm aj m
2π m−∞
2π
∞
1 iEm −m/T
√
e e
e−iE m dμj E
2π m0
0
2π ∞
1
imE−E i/T
dμj E
e
√
2π 0 m0
2π
1
1
√
dμj E
i/T iE−E
2π 0 1 − e
2π
dμj E 1
√
2π 0 eiEi/T e−iEi/T − e−iE
2π
dμj E 1
−√
2π eiEi/T 0 e−iE − e−iEi/T 1
−√
ϕj , Rz UF ξ
2π eiEi/T 1
ξ −√
Gz j ,
iE
−1/T
2π e e
2.19
8
Mathematical Problems in Engineering
with z e−iE1/T . Therefore
−1 j 2
F a E 1
ξ 2
j
G
,
z
2πe−2/T
2.20
and so
−1 j 2
F a 2
L 0,2π
1
2πe−2/T
2π ξ 2
Gz j dE.
2.21
0
From such relation it follows that
LA
ξ T ∞
2
T
m0
e−2m/T MξA m
∞
∞
2
2 −2m/T e
λj
| ϕj , UFm ξ |
T
j1 m0
2
2π
∞ 2
−m/T
−iE m
λj
e
dμj E e
T m0
0
j1
∞
2 2
λj aj 2
l Z
T
j1
∞
∞
j1
λj
2.22
2 −1 j 2
F a 2
L 0,2π
T
∞
1
1
λj
πe−2/T T j1
2π ξ 2
Gz j dE,
0
which is exactly the stated result.
Theorem 2.3 clearly remains true if the eigenvalues λj of A have finite multiplicity. In
this case, for each λj consider the corresponding orthonormal eigenvectors ϕj1 , . . . , ϕjk , and
one obtains
LA
ξ T with z as before.
∞
1
1
λj
πe−2/T T j1
2π
k
ϕj , Rz UF ξ2 dE ,
n
n1
0
2.23
Mathematical Problems in Engineering
9
In case the initial condition is ξ ϕ1 , put ηz : Rz UF ϕ1 . Thus, UF − zηz ϕ1 and
so UF ηz zηz ϕ1 . Hence
ϕj , UF ηz zϕj , ηz δj,1
2.24
ϕ Gz j : Gz 1 j ,
2.25
⎧ z
⎪1
⎨ z ϕ1 , UF η − 1 , if j 1,
Gz j ⎪
⎩ 1 ϕj , UF ηz ,
if j > 1.
z
2.26
and by denoting
one concludes.
Lemma 2.4.
In Section 3 we discuss some Floquet operators that are known in literature and
analyze their Green functions through the equation
UF − z1ηz ϕ1 .
2.27
3. Applications
This section is devoted to some applications of the formula obtained in Theorem 2.3. In
general it is not trivial to get expressions and/or bounds for the Green functions of Floquet
operators, and so one of the main goals of the applications that follow is to illustrate how to
approach the method we have just proposed.
3.1. Time-Independent Hamiltonians
As a first example and illustration of the formula proposed in Theorem 2.3, we consider the
special case of autonomous hamiltonians. In this case Ht H0 for all t, and we assume
that H0 is a positive, unbounded, self-adjoint operator and with simple discrete spectrum,
H0 ϕj χj ϕj , so that {ϕj }∞
j1 is an orthonormal basis of H and 0 ≤ χ1 < χ2 < χ3 < · · ·
q
with χj → ∞. For q > 0 we can consider H0 as our abstract energy operator A, so that its
q
eigenvalues are λj χj since A and H0 have the same eigenfunctions, we are justified in
using the notation ϕj for the eigenfunctions of H0 . We take UF e−iH0 time t 1 and for
ξ∈H
ϕj , ξ
ξ .
Gz j ϕj , Rz H0 ξ Rz H0 ϕj , ξ −iχ
e j −z
3.1
10
Mathematical Problems in Engineering
q
Since dom H0 is invariant under the time evolution e−itH0 , then for z e−iE e1/T and ξ ∈
q
dom H0 we have
H
q
q
Lξ T : Lξ 0 T ∞
1
1
q
χ
πe−2/T T j1 j
∞
1
1
q
χj
−2/T
T
πe
j1
1
πe−2/T
2π ξ 2
Gz j dE
0
2
2π ϕj , ξ
3.2
dE
2
|e−iχj − z|
∞
2 2π
1
dE
q
χj ϕj , ξ
.
2
−iχ
T j1
j
0 |e
− z|
0
2π
Thus we need to calculate the integral Ij : 0 dE/|e−iχj −z|2 . Let γ be the closed path
in C given by γE eiE with 0 ≤ E ≤ 2π, αj e1/T eiχj and βj e−1/T eiχj , then
Ij 2π
e
0
2π
dE
e−iχj − e−iE e1/T eiχj − eiE e1/T
0
2π
0
−
dE
− z eiχj − z
−iχj
e2/T
e−1/T e−iχj
2π
1
e2/T
dE
− e−iE e−1/T eiχj − eiE
0
3.3
dE
e−iE e−1/T e−iχj eiE − αj eiE − βj
1
1 2π
ieiE dE
e1/T e−iχj i 0 eiE − αj eiE − βj
dw
i
.
−iχj
1/T
w − βj
e e
γ w − αj
−
As |αj | > 1 and |βj | < 1, βj is the unique pole in the interior of γ. Thus, by using residues,
Ij and Ij is independent of χj .
i
e1/T e−iχj
1
2π
2/T
e
−1
βj − αj
2πi 3.4
Mathematical Problems in Engineering
11
Therefore by 3.2 it follows that
q
Lξ T ∞
2 2π
1
1
q
χj ϕj , ξ 2/T
−2/T
T
πe
e
−1
j1
2
e−2/T
∞
2
1
1
q
χj ϕj , ξ
2/T
T e
− 1 j1
3.5
2
1
q/2 2
H0 ξ .
−2/T
T
1−e
Since 1 − e−2/T 2/T O1/T 2 , for large T it is found that
q
q/2 2
Lξ T ≈ H0 ξ ,
3.6
q
with for ξ ∈ dom H0 q
lim L T T →∞ ξ
q
ξ, H0 ξ.
3.7
Then we conclude that the function
q
N m −→ e−iH0 m ξ, H0 e−iH0 m ξ
3.8
q
is bounded for ξ ∈ dom H0 , which is of course an expected result see Proposition 2.1.
3.2. Lower-Bounded Green Functions
As a first theoretical application we get dynamical instability from some lower bounds of the
Green functions. See 21 for a similar result in the one-dimensional discrete Schrödinger
operators context; there, a relation to transfer matrices allows interesting applications to
nontrivial models, what is not available in the unitary setting yet and it constitutes of
an important open problem. As before, λj denote the increasing sequence of positive
eigenvalues of the abstract energy operator A, the ones we use to probe instability.
Let · denotes the integer part of a real number, and | · | indicates Lebesgue measure.
Theorem 3.1. Suppose that there exist K > 0 and α > 0 such that for each 2N > 0 large enough
there exists a nonempty Borel set JN ⊂ S1 such that
K
ξ
Gz j ≥ α ,
N
N ≤ j ≤ 2N
3.9
12
Mathematical Problems in Engineering
holds for all z e−iE1/T with E ∈ JT N {E ∈ S1 : ∃ E ∈ JN; |E − E | ≤ 1/T} (the
1/T-neighborhood of JN). Let δ > 0, then for T large enough such that NT T δ , one has
δ1−2α−2
.
LA
ξ T ≥ cte λT δ T
3.10
δγ−2α1−2
LA
.
ξ T ≥ cte T
3.11
Moreover, if λj ≥ cte j γ , γ ≥ 0, then
Proof. By the formula in Theorem 2.3, or its more general version 2.16,
LA
ξ T ≥
≥
∞
1
1
λj
−2/T
T j1
πe
2π ξ 2
Gz j dE
0
2π 2NT cte ξ 2
λj
Gz j dE
T jNT 0
2NT
cte
ξ 2
λNT ≥
Gz j dE
T
jNT JT N
2NT
K2
cte
λNT ≥
|J N|
2α T
T
jNT NT K2
cte
|JT N|λNT T
NT 2α−1
1
cte
|JT N|λT δ 2α−1
δ
T
T 3.12
≥ cte λT δ T δ1−2α−2 ;
and we have used that |JT N| ≥ 1/T . If λj ≥ cte j γ , then
δγ δ1−2α−2
LA
cte T δγ−2α1−2 .
ξ T ≥ cte T T
3.13
The proof is complete.
The above theorem becomes appealing when the exponent of T is greater than zero and
instability is obtained, for instance, when δγ − 2α 1 > 2 in case λj ≥ cte j γ . However, up to
now we have not yet been able to find explicit estimates in models of interest; in any event,
we think that the future applications will be useful, and so we point out some speculations.
First, note that it applies even if the set JN is a single point! Nevertheless, we expect that
Theorem 3.1 will be applied to models whose Floquet operators have some kind of “fractal
spectrum” usually singular continuous or uniformly Hölder continuous spectral measures,
and, somehow, α should be related to dimensional properties of those spectra; indeed, this
Mathematical Problems in Engineering
13
was our first motivation for the derivation of this result, and, in our opinion, such applications
are among the most interesting open problems left here.
3.3. Rank-One Kicked Perturbations
Now consider
Ht H0 κPφ
δt − n2π,
3.14
n
with H0 as in Section 3.1, with eigenvectors {ϕj }∞
j1 and χj the corresponding eigenvalues;
Pφ · φ, ·φ where κ ∈ R and φ is a normalized cyclic vector for H0 , in the sense that
φ 1 and the closed subspace spanned by {H0m φ : m ∈ N} equals H. Let
φ
3.15
bj ϕj .
j
In this case see 22–24
UF U0 1 αPφ ,
3.16
q
q
with U0 e−i2πH0 and α e−i2πκ − 1. Note that φ ∈ dom H0 , ∀q > 0, and so for ξ ∈ dom H0 ,
UF ξ U0 ξ αφ, ξ U0 φ
q
3.17
q
also belongs to dom H0 ; a simple iteration process shows that UFm ξ ∈ dom H0 for all m ≥ 0,
and we are justified in using the formula in Theorem 2.3 to estimate Laplace averages.
1, it follows that ηz belongs to the
We are interested in ηz Rz UF ϕ1 . As |z| /
Hilbert space, and so one can write
ηz ∞
aj ϕj .
3.18
j1
Note that aj Gz j, and we have
UF ηz − zηz ϕ1 .
3.19
14
Mathematical Problems in Engineering
By the relation
UF ηz U0 ηz αU0 Pφ ηz
∞
aj U0 ϕj αU0 φ, ηz φ
j1
∞
∞
aj e−i2πχj ϕj αφ, ηz bj e−i2πχj ϕj
j1
3.20
j1
∞ aj αφ, ηz bj e−i2πχj ϕj ,
j1
and 3.19 it follows that
∞ ∞
aj αφ, ηz bj e−i2πχj ϕj − z aj ϕj ϕ1 ,
j1
3.21
j1
that is,
∞ aj e−i2πχj − z αφ, ηz bj e−i2πχj ϕj ϕ1 ,
3.22
j1
and we get the equations
a1 e−i2πχ1 − z αφ, ηz b1 e−i2πχ1 1,
aj e
−i2πχj
− z αφ, ηz bj e−i2πχj 0
3.23
for j > 1.
Thus
a1 1 − αφ, ηz b1 e−i2πχ1
,
e−i2πχ1 − z
aj −
αφ, ηz bj e−i2πχj
,
e−i2πχj − z
3.24
j > 1.
For the trivial case α 0 or, equivalently, κ ∈ Z, one has
a1 1
e−i2πχ1
aj 0,
−z
,
j > 1,
3.25
Mathematical Problems in Engineering
15
q
and ηz ϕ1 /e−i2πχ1 − z. In this case the analysis of Lϕ1 T is reduced to
2π
|a1 |2 dE 2π
0
dE
|e−i2πχ1 − z|
0
q
2
2π
−1
e2/T
3.26
q
as calculated in Section 3.1. Thus Lϕ1 T ≈ H0 ϕ1 for large T , as expected.
Returning to the general case α /
0, note that
φ, ηz ∞
bj aj
j1
b1
1 − αφ, ηz b1 e−i2πχ1
e−i2πχ1 − z
b1
e−i2πχ1
∞
−αφ, ηz bj e−i2πχj
bj
e−i2πχj − z
j2
3.27
∞ α|b |2 e−i2πχj
j
.
− φ, ηz −i2πχj
−z
e
−z
j1
So
⎡
b1
⎣1 e−i2πχ1 − z
φ, ηz ∞ α|b |2 e−i2πχj
j
j1
e−i2πχj − z
⎤−1
⎦ .
3.28
By denoting
τz 1 ∞ α|b |2 e−i2πχj
j
j1
e−i2πχj − z
3.29
,
by 3.24 we finally obtain the relations
a1 1
e−i2πχ1 − z
−
α|b1 |2 e−i2πχ1 τz−1
e−i2πχ1 − z
αbj b1 e−i2πχj τz−1
aj − −i2πχ
−i2πχ
,
j
1 − z
e
−z
e
2
,
3.30
j > 1.
16
Mathematical Problems in Engineering
3.3.1. A Harmonic Oscillator
Now we present an application of the above relations to a kicked harmonic oscillator with
H
q
q
natural frequency equals to 1; we will write Lξ Lξ 0 .
Proposition 3.2. Let H0 be a harmonic oscillator hamiltonian with appropriate parameters so that its
eigenvalues are integers j, j ≥ 1, and UF U0 1 αPφ as aforementioned. Then for any κ ∈ R and
cyclic vector φ for H0 , there exists C > 0 so that, for T large enough,
q
Lϕ1 T ≤ C,
3.31
q
where ϕ1 is the harmonic oscillator ground state. Hence one has H0 -dynamical stability.
q
Proof. We use the above notation; note that ϕ1 ∈ dom H0 , ∀q > 0 and Theorem 2.3 can be
applied. In this case we have
τz 1 ∞ α|b |2
j
j1
1
1−z
α φ2 1 − z α ,
1−z
1−z
3.32
and so
a1 α|b1 |2
1
−
,
1 − z 1 − ze−i2πκ − z
aj −
Now we evaluate Ij :
2π
0
2π
0
αbj b1
1 − ze−i2πκ − z
,
3.33
j > 1.
|aj |2 dE. For j > 1 and γE eiE , 0 ≤ E ≤ 2π,
2
2π αbj b1
|aj | dE dE
−i2πκ − z −
ze
1
0
2
2
2
|α| |bj | |b1 |
2
2π
0
|α|2 |bj |2 |b1 |
dE
1 − e−iE e1/T e−i2πκ − e−iE e1/T 2
2
ie2/T e−i2πκ
γ
w − β1
w dw
,
w − β2 w − β3 w − β4
3.34
Mathematical Problems in Engineering
17
where β1 e1/T , β2 e−1/T , β3 e1/T ei2πκ , and β4 e−1/T ei2πκ ; only β2 and β4 are poles in the
interior of γ. By residue, for j > 1,
Ij 2π|α|2 |bj |2 |b1 |
2
e2/T e−i2πκ
β2
β4
β2 − β1 β2 − β3 β2 − β4
β4 − β1 β4 − β2 β4 − β3
2
2
2πα|bj |2 |b1 |
2πα|bj |2 |b1 | ei2πκ
2/T
−
−i2πκ
−1 e
− e2/T
e
e2/T − 1 ei2πκ − e2/T
2 2 2παbj b1 1
ei2πκ
,
−
2/T
e−i2πκ − e2/T ei2πκ − e2/T
−1
e
3.35
and for j 1
2
2π α|b1 |2
1
I1 −
dE
−i2πκ
1
−
z
−
z
−
ze
1
0
2π
2π
dE
dE
2
− α|b1 |
z
−
z1
−
1
zei2πκ − z
−
z1
−
1
0
0
2π
dE
− α|b1 |2
−i2πκ − z
0 1 − z1 − ze
2π
dE
4
2
,
|α| |b1 |
−i2πκ − zei2πκ − z
ze
−
z1
−
1
0
3.36
and evaluating the integrals we obtain
2π|b1 |2
2π|b1 |2
2π
2πα|b1 |2
−
−
−
e2/T − 1
e2/T − 1
ei2πκ − e2/T
e2/T − 1 e−i2πκ − e2/T
1
ei2πκ
2πα|b1 |4
,
−
2/T
e−i2πκ − e2/T ei2πκ − e2/T
−1
e
I1 3.37
and after inserting this in the expression of the average energy we get
q
Lϕ1 T 2
1 − e−2/T T
α|b1 |2
1 − |b1 | − −i2πκ
− e2/T
e
2
2|b1 |2
e
ei2πκ − e2/T T
1
ei2πκ
2α|b1 |2
q
φ, H0 φ.
−
1 − e−2/T T e−i2πκ − e2/T ei2πκ − e2/T
−
−2/T
3.38
18
Mathematical Problems in Engineering
Therefore, for large T there is a constant Cκ, b1 > 0 so that
1
q
q
.
Lϕ1 T ≤ Cκ, b1 1 φ, H0 φ T
3.39
This completes the proof.
For harmonic oscillators with eigenvalues ωj, ω /
1, the evaluations of the resulting
integrals are more intricate and were not carried out.
3.4. Kicked Perturbations by a V in L2 S1 3.4.1. Kicked Linear Rotor
Consider
Ht ωp V x
δt − n2π,
3.40
n∈Z
where p −id/dx, ω ∈ R, and V ∈ L2 S1 . The Hilbert space is L2 S1 ; this model was
considered in 12, 25, 26 and references therein. The Floquet operator is
UF UV e−i2πωp e−iV x .
3.41
√
Denote ϕj x eijx / 2π, 0 ≤ x < 2π, and j ∈ Z to be the eigenvectors of p2
whose eigenvalues are the square of integers j 2 ; all eigenvalues have multiplicity 2 the
corresponding eigenvectors are ϕj and ϕ−j , except for the null eigenvalue which is simple.
Consider the case ω 1; then
UF − z−1 ϕ0 x √
2π
1
e−iV x
3.42
,
−z
and so
1
ϕ Gz 0 j ϕj , Rz UF ϕ0 2π
Denote Ij :
2π
0
2π
0
e−ijx
dx.
−z
3.43
e−iV x
ϕ
|Gz 0 j|2 dE. It follows that
2
2π 2π
e−ijx
dx
Ij dE
2
−iV
x
−z 2π 0
0 e
1
1
2π2
2π 2π
0
0
e−ijx eijy
2π
0
dE
−iV x
− z eiV y − z
e
3.44
dxdy.
Mathematical Problems in Engineering
For x, y ∈ S1 fixed denote Ixy :
one has
Ixy 2π
0
2π
e−iV x
0
dE/e−iV x − zeiV y − z. If γE eiE , 0 ≤ E ≤ 2π,
−
e−iE e1/T
dE
2π
0
19
e−iE e−iV x
1
1
− 1/T −iV x
i
e e
w−
γ
dE
eiV x e1/T
−
eiE
eiV y − eiE e1/T
e1/T e−1/T eiV y − eiE
eiV x e1/T
3.45
dw
,
w − e−1/T eiV y
and by residues
Ixy −
e1/T e−iV x
2π
e−1/T eiV y
−
eiV x e1/T
e2/T
2π
.
− e−iV x eiV y
3.46
Hence
2π 2π
2π
e−ijx eijy 2/T
dxdy
− e−iV x eiV y
e
2π2 0 0
2π
eijy dy
1 2π −ijx
e
2/T
dx
2π 0
− e−iV x eiV y
e
0
2π
eijy dy
1 2π e−ijx
2/T iV x
dx.
2π 0 e−iV x
− eiV y
e e
0
1
Ij 3.47
The analytical evaluation of these integrals is not a simple task. As an illustration,
consider the particular potential V x x; since by Cauchy’s integral formula
2π
eijy dy
1
−
iy
2/T
ix
i
e e −e
0
wj−1 dw
0,
w − e2/T eix
γ
if j ≥ 1
3.48
and by residue theorem
2π
0
eijy dy
1
−
iy
2/T
ix
i
e e −e
γ
w1−j
dw
2π
,
1−j
2/T
ix
w−e e
e2/T eix if j ≤ 0,
3.49
it is found that
Ij 0 if j ≥ 1,
2π
2π
1 2π e−ijx
1
2π
Ij dx
dx 2/T 1−j ,
−ix
1−j
2/T
1−j
2π 0 e
e
e
0
e2/T eix if j ≤ 0.
3.50
20
Mathematical Problems in Engineering
Therefore, by 2.16 it follows that for any q > 0
∞
∞
1
2
1
j 2q I−j j 2q e−2/T j ,
−2/T
T j1
T j1
πe
p2q
Lϕ0 T ≥
3.51
and we conclude that see the appendix
p2q
3.52
Lϕ0 m ≥ cte m2q
and also that the sequence m → UFm ϕ0 , p2q UFm ϕ0 is unbounded. This behavior is expected
since the spectrum of UF is absolutely continuous in this case 25, but here we got the
result explicitly without passing through spectral arguments, although in a rather involved
way; indeed, a much simpler derivation is possible by direct calculating UFm ϕ0 and the
corresponding expectation values.
For V x kx with integer k ≥ 2, similar results are obtained, that is
Ij ⎧
⎪
⎨0,
⎪
⎩
if j lk, l ≥ 1,
2π
,
e2/T 1 − l
3.53
if j lk, l ≤ 0
and so
p2q
Lϕ0 T ≥
∞
2k2q l2q e−2/T l .
T l1
3.54
Therefore, we have the following lower bound for the Laplace average:
p2q
Lϕ0 m ≥ C k, q m2q
3.55
see appendix. The same is valid if V x kx with k denoting any negative integer number.
3.4.2. Power-Kicked Systems
p2q
Due to the difficulty in evaluating the integrals in 3.47, in order to estimate Lϕ0 T in some
situations we take an alternative way.
Consider the Kicked models in L2 S1 with Floquet operator
UF UV e−i2πωfp e−iV x ,
3.56
corresponding to the hamiltonian
Ht ωf p V x δt − 2πn,
n∈Z
3.57
Mathematical Problems in Engineering
21
with p, V, ϕj as before and fp pN for some N ∈ N. Let F : L2 S1 → l2 Z be the Fourier
transform. Then FUV F−1 : l2 Z → l2 Z and
FUV F−1 Fe−i2πωfp e−iV x F−1 Fe−i2πωfp F−1 Fe−iV x F−1 ,
3.58
where Fe−i2πωfp F−1 is represented by a diagonal matrix D whose elements are
Dm, n e−i2πωfn δmn ,
3.59
and Fe−iV x F−1 is represented by a matrix W whose elements are
Wm, n Fρ m − n ρm
! − n,
3.60
√
where ρx 1/ 2πe−iV x . Denote B DW; so
! − n,
Bm, n e−i2πωfn ρm
3.61
UV F−1 BF.
3.62
UV ηz − zηz ϕ0 ,
3.63
BFηz − zFηz Fϕ0 .
3.64
BFηz n − zFηz n Fϕ0 n,
3.65
Put ηz Rz UV ϕ0 ; then
and using 3.62 we obtain
Thus, for each n ∈ Z,
so that
e−i2πωfn
ϕ ϕ
ρ! n − j Gz 0 j − zGz 0 n δn0 .
3.66
j∈Z
Tridiagonal Case
In order to deal with the above equations, we try to simplify them by supposing that V is
such that ρm
! − n 0 if |m − n| > 1. Then, for each n ∈ Z fixed 3.66 becomes
e−i2πωfn
ϕ ϕ
ρ! n − j Gz 0 j − zGz 0 n δn0 ,
|n−j |≤1
3.67
22
Mathematical Problems in Engineering
and F−1 UV F B is tridiagonal and has the structure
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
B⎜
⎜
⎜
⎜
⎜
⎜
⎝
..
⎞
.
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎠
g−1ρ0
!
g−1ρ−1
!
ρ1
!
ρ0
!
g1ρ1
!
ρ−1
!
g1ρ0
!
g1ρ−1
!
g2ρ1
!
g2ρ0
!
..
3.68
.
where gn e−i2πωfn .
Now, a tridiagonal unitary operator U on l2 Z is either unitarily equivalent to a
bilateral shift operator or an infinite direct sum of 2 × 2 and 1 × 1 unitary matrices, as
shown in 27, Lemma 3.1. For proving this result it was only used that U is unitary and
Uek αk ek−1 βk ek γk ek1 , where {ek } is the canonical basis of l2 Z, that is,
⎛
⎞
..
⎜ . αk−1
⎜
⎜
βk−1 αk
⎜
⎜
⎜
U⎜
γk−1 βk αk1
⎜
⎜
γk βk1
⎜
⎝
..
γk1
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎠
3.69
.
It then follows that for all k ∈ Z
|αk |2 |βk |2 |γk |2 1,
γk−1 βk−1 βk αk 0,
3.70
αk γk 0.
Applying these relations to B F−1 UV F we obtain the following.
i If ρ−1
!
0, then ρ1
!
ρ0
!
0 and |ρ−1|
!
1.
/
ii If ρ1
! /
0, then ρ−1
!
ρ0
!
0 and |ρ1|
!
1.
ii If ρ0
! /
0, then ρ1
!
ρ−1
!
0 and |ρ0|
!
1.
The next step is to investigate these cases. If ρ0
! /
0, it reduces to the autonomous case
Ht H0 previously considered.
The cases ρ−1
!
0 and ρ1
! /
0 are similar, so we only discuss that ρ1
! /
0. For n ∈ Z
/
fixed, 3.67 takes the form
ϕ
ϕ
0
0
!
e−i2πωfn ρ1G
z n − 1 − zGz n δn0 ,
3.71
Mathematical Problems in Engineering
ϕ
23
ϕ
ϕ
and so we can write Gz 0 n in terms of Gz 0 0 and Gz 0 −1 for all n ∈ Z. More precisely
ϕ
Gz 0 n ϕ
Gz 0 −n
e−i2πωfn···f1 ρ1
! n ϕ0
Gz 0,
zn
zn−1
e−i2πωf−n1···f−1 ρ1
! n−1
ϕ
n ≥ 1,
3.72
ϕ
Gz 0 −1,
n ≥ 2;
ϕ
0
0
−iE 1/T
e
and
moreover, for n 0 in 3.71 we obtain ρ1G
!
z −1 − zGz 0 1, and so for z e
T >1
ϕ
ϕ
1 ≤ Gz 0 −1 |z|Gz 0 0
ϕ
ϕ
Gz 0 −1 e1/T Gz 0 0
3.73
ϕ
ϕ
≤ e Gz 0 −1 Gz 0 0 ,
and there exists d > 0 so that
2 ϕ0
ϕ0 Gz −1 Gz 0 ≥ d > 0.
3.74
Therefore, by 2.16, for T > 1 one has
p2q
Lϕ0 T ∞
1
1
n2q
≥
πe−2/T T n1
∞
1
1
n2q
πe−2/T T n1
2π
2π 2
ϕ0 2
ϕ0
dE
n
Gz
Gz −n dE
0
e
0
2π 1
ϕ0 2
0
dE
G
z
2n/T
e
0
2n−1/T
2π 2
ϕ0
Gz −1 dE
3.75
0
≥
∞
1
1
n2q e−2n/T
−2/T
T n1
πe
≥d
2π 2
ϕ0 2 ϕ0
Gz 0 Gz −1 dE
0
∞
2
n 12q e−2n/T ,
T n0
so that, by the discussion at the end of the appendix,
p2q
Lϕ0 m ≥ Cm 12q ,
and UVm ϕ0 , p2q UVm ϕ0 is unbounded. Hence we have instability.
3.76
24
Mathematical Problems in Engineering
Pentadiagonal Case
Suppose now that V is such that ρm−n
!
0 if |m−n| > 2. Then for each n ∈ Z fixed, equation
3.66 becomes
e−i2πωfn
ϕ ϕ
ρ! n − j Gz 0 j − zGz 0 n δn0 ,
3.77
|n−j|≤2
and F−1 UV F is pentadiagonal and has a structure similar to the corresponding operator in
the previous case, just adding the elements whose distance to the diagonal is 2. The elements
!
and those in the new lower diagonal are
in the new upper diagonal are e−i2πωfn ρ−2,
!
e−i2πωfn ρ2.
For not repeating the tridiagonal case we suppose that either ρ2
!
or ρ−2
!
is different
from zero. If U is a pentadiagonal unitary operator in l2 Z, that is, Uek ζk ek−2 αk ek−1 βk ek γk ek1 θk ek2 , one gets the matrix representation
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
U⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
..
⎞
.
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
βk−2 αk−1 ζk
γk−2 βk−1 αk ζk1
θk−2 γk−1 βk αk1 ζk2
θk−1 γk βk1 αk2
θk γk1 βk2
..
3.78
.
From this we obtain the following relations, for each k ∈ Z,
2 2
|ζk |2 |αk |2 βk γk |θk |2 1,
ζk αk−1 αk βk−1 βk γk−1 γk θk−1 0,
βk−1 θk−1 αk γk ζk1 βk1 0,
3.79
αk−1 θk−1 ζk γk 0,
ζk θk 0.
Suppose that ρ2
! /
0. The case ρ−2
!
0 is similar. Then by the above relations we obtain
/
ρ−2
!
ρ−1
!
ρ0
!
ρ1
!
0, and so 3.77 becomes
ϕ
ϕ
0
0
e−i2πωfn ρ2G
!
z n − 2 − zGz n δn0 .
3.80
Mathematical Problems in Engineering
ϕ
25
ϕ
0
0
For n 0 one gets ρ2G
!
z −2 − zGz 0 1 and analogously to the previous case
2 ϕ0
ϕ0 2
Gz −2 Gz 0 ≥ d > 0,
3.81
with z e−iE e1/T and T > 1. Since for n ≥ 1
ϕ
Gz 0 2n ! n ϕ0
e−i2πωf2nf2n−2···f2 ρ2
Gz 0,
zn
ϕ
Gz 0 −2n
3.82
ϕ
zn−1 Gz 0 −2
ρ2
! n−1 e−i2πωf−2n−1···f−2
,
we obtain
p2q
Lϕ0 T ∞
1
1
2n2q
≥
πe−2/T T n1
∞
1
1
2n2q
πe−2/T T n1
2π
2π 2
2
ϕ0
ϕ0
2n
dE
Gz
Gz −2n dE
0
e
0
2π 1
ϕ0 2
0
dE
G
z
2n/T
e
0
2n−1/T
2π 2
ϕ0
Gz −2 dE
3.83
0
≥
∞
1
1
2n2q e−2n/T
−2/T
T n1
πe
≥d
2π 2
ϕ0 2 ϕ0
Gz 0 Gz −2 dE
0
∞
2
2n 12q e−2n/T ,
m n0
hence
p2q
Lϕ0 T ≥ C2m 12q ,
3.84
and UVm ϕ0 , p2q UVm ϕ0 is unbounded.
N-Diagonal Case
If V satisfies ρm
! − n 0 for |m − n| > N, we suppose that either ρN
!
or ρ−N
!
is different
!
−
from zero. In case ρN
!
0, by unitarity and the structure of F−1 UV F we obtain that ρN
/
1 · · · ρ0
!
ρ−1
!
· · · ρ−N
!
0, thus 3.66 becomes, for each n ∈ Z,
ϕ
ϕ
0
0
!
e−i2πωfn ρNG
z n − N − zGz n δn0 ,
3.85
26
Mathematical Problems in Engineering
and so
2 ϕ0
ϕ0 2
Gz −N Gz 0 ≥ d > 0,
3.86
with z e−iE e1/T , T > 1. Moreover, for n ≥ 1
ϕ
Gz 0 nN n
!
e−i2πωfnNfn−1N···fN ρN
ϕ
Gz 0 0,
zn
ϕ
Gz 0 −nN
ϕ
zn−1 Gz 0 −N
n−1 −i2πωf−Nn−1···f−N
ρN
!
e
3.87
.
Similarly to the previous cases we conclude that
p2q
Lϕ0 T ≥ d
∞
2
Nn 12q e−2n/T .
T n0
3.88
Therefore we can state the following result.
Theorem 3.3. For Kicked systems in L2 S1 with
UV e−i2πωfp e−iV x
3.89
as in 3.56, one can obtain that FUV F−1 : l2 Z → l2 Z is represented by the matrix B with
! − n, where ρx 2π−1/2 e−iV x . If V satisfies ρm
! − n 0
elements Bm, n e−i2πωfn ρm
∗
!
or ρ−N
!
is different from zero, then V x ±Nx θ, for
for |m − n| > N ∈ N and either ρN
some θ ∈ R, and FUV F−1 is unitarily equivalent to T N (the Nth power of T ) where T is the bilateral
shift. Furthermore,
p2q
Lϕ0 T ≥ d
∞
2
Nn 12q e−2n/T .
T n0
3.90
!
0
Proof. It is enough to prove that FUV F−1 is unitarily equivalent to T N . Suppose that ρN
/
the case for ρ−N
!
0 is similar; then by the above discussion we obtain
/
Bm, n ⎧
⎨0,
if m /
n N,
⎩e−i2πωfn ρN,
!
if m n N,
3.91
2
!
!
1,
that is, Ben e−i2πωfn ρNe
nN where {en } is the canonical basis of l Z. Since |ρN|
−iθ
write ρN
!
e . Let W be the unitary operator defined by
Wen eiϑn en ,
n ∈ Z,
3.92
Mathematical Problems in Engineering
27
where ϑn are elements in 0, 2π. If ϑn satisfies for all n ∈ Z
ϑnN − ϑn 2πωfn θ,
3.93
it follows that W −1 BW T N . Equation 3.93 is satisfied taking, for example, ϑ0 ϑ1 · · · ϑN−1 0 and another ϑn obeying 3.93.
Although Theorem 3.3 gives a nice illustration of the potential applications of our
expression for the Laplace average, since it is one of the few instances that such average can
be explicitly estimated from below, again it can be derived by more direct methods and one
can also conclude 25 that the spectrum of the corresponding Floquet operators is absolutely
continuous.
4. Conclusions
Although most of our applications of Theorem 2.3 give expected results sometimes known
results that can be derived in simpler ways, we believe that that formula is interesting and
has a potential to be applied to more sophisticated models as the Fermi accelerator. The
difficulty is to get expressions or estimates for the Green functions, since calculating the
resolvent of an operator is not always an easy task; sometimes we have the expressions for
resolvent operators e.g., for kicked systems, but the resulting integrals can be too involved.
We have not tried any numerical approach to formula 2.15, which might be useful for some
specific models.
In the case of one-dimensional discrete Schrödinger operators, where the hamiltonian
is HV : l2 Z → l2 Z defined by
HV ξn ξn 1 ξn − 1 V nξn,
4.1
V a bounded sequence, a similar formula can be handled in some cases by relating the
resolvent REi/T HV to transfer matrices. Then adequate upper bounds of such transfer
matrices, on some set of energies E, result in lower estimates for the corresponding Green
functions, and then transport properties are obtained for interesting models see 21 and
references therein.
In 27 a class of Floquet operators displaying a pentadiagonal structure was
introduced; for these models there is a transfer matrix formalism. However, such transfer
matrices are too complicated, and analytical estimates seem far from trivial.
Anyway, the technique here is quite general; it asks no particular regularity of the
time-dependence and can be virtually applied to any time-periodic system as soon as the
time evolution is well posed. As already said, the chief difficulty is related to suitable
bounds of matrix elements of the resolvents of unitary Floquet operators, a task harder
than we initially envisaged. Herein we put forward for consideration the challenge of getting
additional applications for the formula 2.15 deduced for the Laplace averages, including an
application of Theorem 3.1 to physical models. It is also worth mentioning the question left
open in Lemma 2.2, that is, is it true that βe− βd− ?
28
Mathematical Problems in Engineering
Appendix
Laplace Transform of Sequences
Let a an n∈N be a sequence of positive real numbers. The Laplace transform of a, denoted
by fa , is the function defined by
fa s ∞
e−sn an,
A.1
n0
for s in a subset of R. It will also be denoted by fa s La.
We say that the Laplace transform of a an exists if the series in A.1 converges for
2
some s. For example, if an en , then the sum in A.1 diverges for all s ∈ R.
Examples
1 For the constant sequence an 1 it follows that
fa s ∞
e−sn n0
1
,
1 − e−s A.2
for s > 0. By using Taylor expansion, for small s one finds that 1/1 − e−s ≈ 1/s.
n
2 Since ∞
n0 z 1/1 − z, for z ∈ C, |z| < 1, it follows that
∞
n kn k − 1 · · · n 1zn n0
k!
1 − zk1
,
A.3
for k 1, 2, 3, . . . , and z as above. Thus, the Laplace transform of ak n n kn k −
1 · · · n 1 is
fak s ∞
e−sn ak n n0
k!
1 − e−s k1
,
s > 0.
A.4
For small s, fak s ≈ k!/sk1 .
A sequence of complex numbers a an is said to be exponential of order σ0 real if
there exists M > 0 so that |an| ≤ Meσ0 n , ∀n. That is, an does not increase faster than eσ0 n
as n → ∞. If a an is exponential of order σ0 > 0, then
fa s ∞
n0
is convergent for any s > σ0 .
e−sn an
A.5
Mathematical Problems in Engineering
29
Let V denote the set of positive sequences of exponential order σ0 . The Laplace
transform L satisfies
Lca cLa,
La b La Lb,
A.6
where c is a positive number, and a and b are sequences in V. Moreover, if a ∈ V and La 0,
−sn
then ∞
an 0 and so an 0 for all n, that is, a 0. Thus L is injective on V.
n0 e
The Laplace average 2.5 is related to the Laplace transform of EξA n by
LA
ξ T ∞
2
2
2
.
e−2n/T EξA n fEξA
T n0
T
T
A.7
1
2 1
2
1
≈
T 1 − e−2/T
T 2/T
A.8
If an 1 for all n, then
2
2
fa
T
T
for T large enough. If an n kn k − 1 · · · n 1 ≈ nk , then
2
2
fa
T
T
2
k!
k!
2
T
≈
k!
k1
k1
T 1 − e−2/T T 2/T 2
k
A.9
for large T . Hence, if EξA T grows like T k , then the same law holds for its average Laplace
transform. We have a restricted converse, that is, if LA
n grows with a positive power of n
ξ
then, by Lemma 2.2, its Cesàro average is unbounded with a rather similar behavior at large
times and so is EξA n. These properties are repeatedly used in the text.
One should be aware that there are special situations of unbounded positive sequences
an with bounded average Laplace transforms so that βe βd 0; an explicit example is
∈ {k2 : k ∈ N}. The same phenomenon is well known for Cesàro
an2 n and an 0 for n /
averages, and, by Lemma 2.2, such phenomena are connected.
Acknowledgments
The authors thank the anonymous referees for careful readings and suggestions. C. R. de
Oliveira thanks the warm hospitality of PIMS Vancouver and the support by CAPES
Brazil.
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