# Quantum Energy Expectation in Periodic Time-Dependent Hamiltonians via Green Functions

Authors: César R. de Oliveira; Mariza S. Simsen
Journal: Mathematical Problems in Engineering
 (2009)

---

## Abstract

LetUF be the Floquet operator of a time periodic
Hamiltonian H(t).
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.


---

## Body

## 1. Introduction

Consider a general periodically driven quantum hamiltonian system(1.1)H(t)=H0+V(t)
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 UFmξ 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
(1.2)supm∈ℕ|〈UFmξ,H0UFmξ〉|
is finite or not, where ξ∈domH0⊂ℋ, the domain of
H0.Motivated by models with hamiltonians as aboveH(t)=H0+V(t), one is suggested to probe quantum (in)stability 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:domA⊂ℋ→ℋ and with discrete spectrum,
(1.3)Aφn=λnφn,0≤λn<λn+1, such that if UFmξ∈domA for all m∈ℕ, then, for each m, the expectation value EξA(m)=〈UFmξ,AUFmξ〉 is finite. It is convenient to write EξA(m)=+∞ if UFmξ does not belong to domA.We say the system isA-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 hamiltoniansH(t)=H0+V(t), not necessarily periodic, with H0 a positive self-adjoint operator whose spectrum consists of separated bands {σj}j=1∞ such that σj⊂[λj,Λj], upper bounds of the type
(1.4)〈U(t,0)ψ,H0U(t,0)ψ〉≤ctet(1+α)/nα
were obtained in [10] if the gaps λj+1-Λj grow like jα, 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 ξ(t)=U(t,0)ξ for ξ∈ℋ, while in [16–19] lower bounds for averages of the type
(1.5)1T∑m=1T‍〈UFmξ,H0UFmξ〉≥CTγ
are obtained for periodic hamiltonians H(t)=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 (in)stability of periodic time-dependent systems. As for tight-binding models (see [21] and references therein) we consider the Laplace-like average of 〈UFmξ,AUFmξ〉, that is,
(1.6)2T∑m=0∞‍e-2m/T〈UFmξ,AUFmξ〉,
where A is a probe energy, ξ is an element of domA, 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=λjφj, and the matrix elements 〈φj,Rz(UF)ξ〉 of the resolvent operator Rz(UF)=(UF−z1)−1 (with z=e-iEe1/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 Section2), in practice we will think of (in)stability in terms of (un)boundedness 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 hamiltonianH(t) with H(t+τ)=H(t) for all t∈ℝ, acting in the separable Hilbert space ℋ. Suppose the existence of the propagators U(t,s), so that the Floquet operator UF=U(τ,0) is at our disposal. Let A be a probe energy and λj,φj as in the introduction. Also, {φj}j=1∞ is an orthonormal basis of ℋ.The main interest is in the study of the expectation values, herein defined by(2.1)EξA(m):={〈UFmξ,AUFmξ〉,ifUFmξ∈domA,+∞,ifUFmξ∈ℋ∖domA,
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
(2.2)MξA(m):=∑j=1∞‍λj|〈φj,UFmξ〉|2.
Our first remark is the equivalence of both concepts (under certain  circumstances).Proposition 2.1.
IfUFmξ∈domA for all m, then
(2.3)EξA(m)=MξA(m),∀m.
This holds, in particular, if domA is invariant under the time evolution UFm and ξ∈domA.Proof.
SincedomA⊂domA1/2 [1], one has UFmξ∈domA1/2, for all m, and so
(2.4)MξA(m)=∑j=1∞‍|〈λj1/2φj,UFmξ〉|2=∑j=1∞‍|〈A1/2φj,UFmξ〉|2=∑j=1∞‍|〈φj,A1/2UFmξ〉|2=∥A1/2UFmξ∥2=〈A1/2UFmξ,A1/2UFmξ〉=〈UFmξ,AUFmξ〉=EξA(m),
which is the stated result.We introduce the temporal Laplace average ofEξA (see also the appendix) by the following function of T>0, which also takes values in [0,+∞]:
(2.5)LξA(T):=2T∑m=0∞‍e-2m/TEξA(m).
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
(2.6)c1Tβ-≤LξA(T)≤c2Tβ+,
and the corresponding exponents for the temporal Cesàro average
(2.7)CξA(T)=1T∑m=0T‍EξA(m)
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.Lemma 2.2.
If(h(m))m=0∞ is a nonnegative sequence  and h(m)≤Cmn for some C>0 and n≥0, then βe+=βd+ and βe-≤βd-, where
(2.8)βe+=lim supT→∞log(∑m=0Th(m))logT,βe-=lim infT→∞log(∑m=0Th(m))logT,βd+=lim supT→∞log(∑m=0∞e-2m/Th(m))logT,βd-=lim infT→∞log(∑m=0∞e-2m/Th(m))logT.Proof.
Note that for0≤m≤T we have e-2≤e-2m/T≤1, and so
(2.9)∑m=0T‍h(m)≤∑m=0T‍e2e-2m/Th(m)≤e2∑m=0∞‍e-2m/Th(m).
Hence βe±≤βd±.
On the other hand, for eachϵ>0, denoting by ⌈x⌉ the smallest integer larger or equal to x, one has
(2.10)∑m=0∞‍e-2m/Th(m)=∑m=0⌈T1+ϵ⌉‍e-2m/Th(m)+∑m=⌈T1+ϵ⌉+1∞‍e-2m/Th(m)≤∑m=0⌈T1+ϵ⌉‍h(m)+C∑m=⌈T1+ϵ⌉+1∞‍e-2m/Tmn.
Now, for T large enough  nT/2<T1+ϵ≤⌈T1+ϵ⌉. Thus
(2.11)∑m=⌈T1+ϵ⌉+1∞‍e-2m/Tmn≤∫⌈T1+ϵ⌉∞‍e-2t/Ttndt.
Therefore, for each ϵ>0 and T large enough
(2.12)∑m=0∞‍e-2m/Th(m)≤∑m=0⌈T1+ϵ⌉‍h(m)+C∫⌈T1+ϵ⌉∞‍e-2t/Ttndt≤∑m=0⌈T1+ϵ⌉‍h(m)+C̃e-2TϵTn.
Since e-2TϵTn→0 as T→∞, it follows that
(2.13)βd+=lim supT→∞log∑m=0∞e-2m/Th(m)logT≤lim supT→∞log∑m=0⌈T1+ϵ⌉h(m)logT=lim supT→∞log∑m=0⌈T1+ϵ⌉h(m)log⌈T1+ϵ⌉log⌈T1+ϵ⌉logT≤lim supT→∞log∑m=0⌈T1+ϵ⌉h(m)log⌈T1+ϵ⌉log(T+1)1+ϵlogT=(1+ϵ)lim supT→∞log∑m=0⌈T1+ϵ⌉h(m)log⌈T1+ϵ⌉≤(1+ϵ)βe+.
As ϵ>0 was arbitrary, βd+≤βe+.Recall that the Green functionsGzξ(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)=(UF-z1)-1 along the orthonormal basis {φj}j=1∞, that is,
(2.14)Gzξ(j):=〈φj,Rz(UF)ξ〉.
Note that Gzξ(j) is always well defined since for |z|≠1 that resolvent operator is bounded. Theorem 2.3 is the main reason for considering the temporal averages LξA(T). It presents a 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-(in)stability can be quantitatively detected.Theorem 2.3.
Assume thatUFmξ∈domA  for all m≥0. Then
(2.15)LξA(T)=1πe-2/T1T∑j=1∞‍λj∫02π‍|Gzξ(j)|2dE,z=e-iE+1/T.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 finite even for vectors UFmξ not in the domain of A, where LξA(T)=+∞. The general case, that is, ∀ξ∈ℋ, can then be gathered in the following inequality:
(2.16)LξA(T)≥1πe-2/T1T∑j=1∞‍λj∫02π‍|Gzξ(j)|2dE,z=e-iE+1/T,
so that lower bound estimates for this formula always imply lower bound estimates for the Laplace average.Proof of Theorem2.3.
First note that, by  hypothesis,UFmξ∈domA1/2 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
(2.17)〈φj,UFξ〉=∫02π‍e-iE′dμj(E′).
For each j let a(j)=(a(j)(m))m∈ℤ be the sequence
(2.18)a(j)(m)={0,ifm<0,e-m/T∫02π‍e-iE′mdμj(E′),ifm≥0.
Since a(j)∈l1(ℤ)∩l2(ℤ) and ℱ is a unitary operator, it follows that ∥a(j)∥l2(ℤ)=∥ℱ-1a(j)∥L2[0,2π] and also
(2.19)(ℱ-1a(j))(E)=12π∑m=-∞∞‍eiEma(j)(m)=12π∑m=0∞‍eiEme-m/T∫02π‍e-iE′mdμj(E′)=12π∫02π‍(∑m=0∞‍eim(E-E′)+i/T)dμj(E′)=12π∫02π‍11-ei(E-E′+i/T)dμj(E′)=12π∫02π‍dμj(E′)ei(E+i/T)(e-i(E+i/T)-e-iE′)=-12πei(E+i/T)∫02π‍dμj(E′)e-iE′-e-i(E+i/T)=-12πei(E+i/T)〈φj,Rz(UF)ξ〉=-12πeiEe-1/TGzξ(j),
with z=e-iE+1/T. Therefore
(2.20)|ℱ-1a(j)|2(E)=12πe-2/T|Gzξ(j)|2,
and so
(2.21)∥ℱ-1a(j)∥L2[0,2π]2=12πe-2/T∫02π‍|Gzξ(j)|2dE.
From such relation it follows that
(2.22)LξA(T)=∑m=0∞‍2Te-2m/TMξA(m)=∑j=1∞‍λj∑m=0∞‍2Te-2m/T|〈φj,UFmξ〉|2=∑j=1∞‍λj2T∑m=0∞‍|e-m/T∫02π‍e-iE′mdμj(E′)|2=∑j=1∞‍λj2T∥a(j)∥l2(ℤ)2=∑j=1∞‍λj2T∥ℱ-1a(j)∥L2[0,2π]2=1πe-2/T1T∑j=1∞‍λj∫02π‍|Gzξ(j)|2dE,
which is exactly the stated result.Theorem2.3 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
(2.23)LξA(T)=1πe-2/T1T∑j=1∞‍λj(∑n=1k‍∫02π‍|〈φjn,Rz(UF)ξ〉|2dE),
with z as  before.In case the initial condition isξ=φ1, put η(z):=Rz(UF)φ1. Thus, (UF-z)η(z)=φ1 and so UFη(z)=zη(z)+φ1. Hence
(2.24)〈φj,UFη(z)〉=z〈φj,η(z)〉+δj,1
and by denoting
(2.25)Gz(j):=Gzφ1(j),
one concludes.Lemma 2.4.
(2.26)Gz(j)={1z(〈φ1,UFη(z)〉-1),ifj=1,1z〈φj,UFη(z)〉,ifj>1.In Section3 we discuss some Floquet operators that are known in literature and analyze their Green functions through the equation
(2.27)(UF-z1)η(z)=φ1.


## 3. Applications

This section is devoted to some applications of the formula obtained in Theorem2.3. 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 Theorem2.3, we consider the special case of autonomous hamiltonians. In this case H(t)=H0 for all t, and we assume that H0 is a positive, unbounded, self-adjoint operator and with simple discrete spectrum, H0φj=χjφj, so that {φj}j=1∞ is an orthonormal basis of ℋ and 0≤χ1<χ2<χ3<⋯ with χj→∞. For q>0 we can consider H0q as our abstract energy operator A, so that its eigenvalues are λj=χjq (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(3.1)Gzξ(j)=〈φj,Rz(H0)ξ〉=〈Rz¯(H0)φj,ξ〉=〈φj,ξ〉e-iχj-z.
Since domH0q is invariant under the time evolution e-itH0, then for z=e-iEe1/T and ξ∈domH0q  we have
(3.2)Lξq(T):=LξH0q(T)=1πe-2/T1T∑j=1∞‍χjq∫02π‍|Gzξ(j)|2dE=1πe-2/T1T∑j=1∞‍χjq∫02π‍|〈φj,ξ〉|2|e-iχj-z|2dE=1πe-2/T1T∑j=1∞‍χjq|〈φj,ξ〉|2∫02π‍dE|e-iχj-z|2.Thus we need to calculate the integralIj:=∫02π‍(dE/|e-iχj-z|2). Let γ be the closed path in ℂ given by γ(E)=eiE with 0≤E≤2π, αj=e1/Teiχj and βj=e-1/Teiχj, then
(3.3)Ij=∫02π‍dE(e-iχj-z)(eiχj-z¯)=∫02π‍dE(e-iχj-e-iEe1/T)(eiχj-eiEe1/T)=∫02π‍dEe2/T(e-1/Te-iχj-e-iE)(e-1/Teiχj-eiE)=-1e2/T∫02π‍dEe-iEe-1/Te-iχj(eiE-αj)(eiE-βj)=-1e1/Te-iχj1i∫02π‍ieiEdE(eiE-αj)(eiE-βj)=ie1/Te-iχj∫γ‍dw(w-αj)(w-βj).
As |αj|>1 and |βj|<1, βj is the unique pole in the interior of γ. Thus, by using residues,
(3.4)Ij=ie1/Te-iχj2πi1(βj-αj)=2πe2/T-1
and Ij is independent of χj.Therefore by (3.2) it follows that
(3.5)Lξq(T)=1πe-2/T1T∑j=1∞‍χjq|〈φj,ξ〉|22πe2/T-1=2e-2/T1T1(e2/T-1)∑j=1∞‍χjq|〈φj,ξ〉|2=2(1-e-2/T)1T∥H0q/2ξ∥2.
Since (1-e-2/T)=2/T+𝒪(1/T2), for large T it is found that
(3.6)Lξq(T)≈∥H0q/2ξ∥2,
with (for ξ∈domH0q)
(3.7)limT→∞Lξq(T)=〈ξ,H0qξ〉.
Then we conclude that the function
(3.8)ℕ∋m↦〈e-iH0mξ,H0qe-iH0mξ〉
is bounded for ξ∈domH0q, 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 (in)stability.Let[·] denotes the integer part of a real number, and |·| indicates Lebesgue measure.Theorem 3.1.
Suppose that there existK>0 and α>0 such that for each 2N>0 large enough there exists a nonempty Borel set J(N)⊂S1 such that
(3.9)|Gzξ(j)|≥KNα,N≤j≤2N
holds for all z=e-iE+1/T with E∈JT(N)={E′′∈S1:∃E′∈J(N);|E′′-E′|≤1/T} (the (1/T)-neighborhood of J(N)). Let δ>0, then for T large enough such that N(T)=[Tδ], one has
(3.10)LξA(T)≥cteλ[Tδ]Tδ(1-2α)-2.
Moreover, if λj≥ctejγ, γ≥0, then
(3.11)LξA(T)≥cteTδ(γ-2α+1)-2.Proof.
By the formula in Theorem2.3, or its more general version (2.16),
(3.12)LξA(T)≥1πe-2/T1T∑j=1∞‍λj∫02π‍|Gzξ(j)|2dE≥cteT∑j=N(T)2N(T)‍λj∫02π‍|Gzξ(j)|2dE≥cteTλN(T)∑j=N(T)2N(T)‍∫JT(N)‍|Gzξ(j)|2dE≥cteTλN(T)∑j=N(T)2N(T)‍K2N(T)2α|JT(N)|=cteT|JT(N)|λN(T)K2N(T)2α-1=cteT|JT(N)|λ[Tδ]1[Tδ]2α-1≥cteλ[Tδ]Tδ(1-2α)-2;
and we have used that |JT(N)|≥1/T. If λj≥ctejγ, then
(3.13)LξA(T)≥cteTδγTδ(1-2α)-2=cteTδ(γ-2α+1)-2.
The proof is complete.The above theorem becomes appealing when the exponent ofT is greater than zero and instability is obtained, for instance, when δ(γ-2α+1)>2 in case λj≥ctejγ. 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 J(N) 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 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(3.14)H(t)=H0+κPϕ∑n‍δ(t-n2π),
with H0 as in Section 3.1, with eigenvectors {φj}j=1∞ 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 {H0mϕ:m∈ℕ} equals ℋ. Let
(3.15)ϕ=∑j‍bjφj.
In this case (see [22–24])
(3.16)UF=U0(1+αPϕ),
with U0=e-i2πH0 and α=(e-i2πκ-1). Note that ϕ∈domH0q, ∀q>0, and so for ξ∈domH0q,
(3.17)UFξ=U0ξ+α〈ϕ,ξ〉U0ϕ
also belongs to domH0q; a simple iteration process shows that UFmξ∈domH0q for all m≥0, and we are justified in using the formula in Theorem 2.3 to estimate Laplace averages.We are interested inη(z)=Rz(UF)φ1. As |z|≠1, it follows that η(z) belongs to the Hilbert space, and so one can write
(3.18)η(z)=∑j=1∞‍ajφj.
Note that aj=Gz(j), and we have
(3.19)UFη(z)-zη(z)=φ1.
By the relation
(3.20)UFη(z)=U0η(z)+αU0Pϕη(z)=∑j=1∞‍ajU0φj+αU0〈ϕ,η(z)〉ϕ=∑j=1∞‍aje-i2πχjφj+α〈ϕ,η(z)〉∑j=1∞‍bje-i2πχjφj=∑j=1∞‍(aj+α〈ϕ,η(z)〉bj)e-i2πχjφj,
and (3.19) it follows that
(3.21)∑j=1∞‍(aj+α〈ϕ,η(z)〉bj)e-i2πχjφj-z∑j=1∞‍ajφj=φ1,
that is,
(3.22)∑j=1∞‍[aj(e-i2πχj-z)+α〈ϕ,η(z)〉bje-i2πχj]φj=φ1,
and we get the equations
(3.23)a1(e-i2πχ1-z)+α〈ϕ,η(z)〉b1e-i2πχ1=1,aj(e-i2πχj-z)+α〈ϕ,η(z)〉bje-i2πχj=0forj>1.
Thus
(3.24)a1=1-α〈ϕ,η(z)〉b1e-i2πχ1e-i2πχ1-z,aj=-α〈ϕ,η(z)〉bje-i2πχje-i2πχj-z,j>1.For the trivial caseα=0 or, equivalently, κ∈ℤ, one has
(3.25)a1=1e-i2πχ1-z,aj=0,j>1,
and η(z)=φ1/(e-i2πχ1-z). In this case the analysis of Lφ1q(T) is reduced to
(3.26)∫02π‍|a1|2dE=∫02π‍dE|e-i2πχ1-z|2=2πe2/T-1
as calculated in Section 3.1. Thus Lφ1q(T)≈∥H0qφ1∥ for large T, as expected.Returning to the general caseα≠0, note that
(3.27)〈ϕ,η(z)〉=∑j=1∞‍b¯jaj=b¯1(1-α〈ϕ,η(z)〉b1e-i2πχ1e-i2πχ1-z)+∑j=2∞‍b¯j(-α)〈ϕ,η(z)〉bje-i2πχje-i2πχj-z=b¯1e-i2πχ1-z-〈ϕ,η(z)〉∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z.
So
(3.28)〈ϕ,η(z)〉=b¯1(e-i2πχ1-z)[1+∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z]-1.
By denoting
(3.29)τ(z)=1+∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z,
by (3.24)   we finally obtain the relations
(3.30)a1=1e-i2πχ1-z-α|b1|2e-i2πχ1τ(z)-1(e-i2πχ1-z)2,aj=-αbjb¯1e-i2πχjτ(z)-1(e-i2πχ1-z)(e-i2πχj-z),j>1.


#### 3.3.1. A Harmonic Oscillator

Now we present an application of the above relations to a kicked harmonic oscillator with natural frequency equals to 1; we will writeLξq=LξH0q.Proposition 3.2.
LetH0 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,
(3.31)Lφ1q(T)≤C,
where φ1 is the harmonic oscillator ground state. Hence one has  H0q-dynamical stability.Proof.
We use the above notation; note thatφ1∈domH0q, ∀q>0 and Theorem 2.3 can be applied. In this case we have
(3.32)τ(z)=1+∑j=1∞‍α|bj|21-z=1+α1-z∥ϕ∥2=1-z+α1-z,
and so
(3.33)a1=11-z-α|b1|2(1-z)(e-i2πκ-z),aj=-αbjb¯1(1-z)(e-i2πκ-z),j>1.
Now we evaluate Ij:=∫02π‍|aj|2dE. For j>1 and γ(E)=eiE, 0≤E≤2π,
(3.34)∫02π‍|aj|2dE=∫02π‍|αbjb¯1(1-z)(e-i2πκ-z)|2dE=|α|2|bj|2|b¯1|2∫02π‍dE|(1-e-iEe1/T)(e-i2πκ-e-iEe1/T)|2=|α|2|bj|2|b¯1|2ie2/Te-i2πκ∫γ‍wdw(w-β1)(w-β2)(w-β3)(w-β4),
where β1=e1/T, β2=e-1/T, β3=e1/Tei2πκ, and β4=e-1/Tei2πκ; only β2 and β4 are poles in the interior of γ. By residue, for j>1,
(3.35)Ij=2π|α|2|bj|2|b¯1|2e2/Te-i2πκ(β2(β2-β1)(β2-β3)(β2-β4)+β4(β4-β1)(β4-β2)(β4-β3))=2πα|bj|2|b1¯|2(e2/T-1)(e-i2πκ-e2/T)-2πα|bj|2|b1¯|2ei2πκ(e2/T-1)(ei2πκ-e2/T)=2πα|bj|2|b¯1|2(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and for j=1(3.36)I1=∫02π‍|11-z-α|b1|2(1-z)(e-i2πκ-z)|2dE=∫02π‍dE(1-z)(1-z¯)-α¯|b1|2∫02π‍dE(1-z)(1-z¯)(ei2πκ-z¯)-α|b1|2∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)+|α|2|b1|4∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)(ei2πκ-z¯),
and evaluating the integrals we obtain
(3.37)I1=2π(e2/T-1)-2π|b1|2(e2/T-1)-2π|b1|2(ei2πκ-e2/T)-2πα|b1|2(e2/T-1)(e-i2πκ-e2/T)+2πα|b1|4(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and after inserting this in the expression of the average energy we get
(3.38)Lφ1q(T)=2(1-e-2/T)T(1-|b1|2-α|b1|2(e-i2πκ-e2/T))-2|b1|2e-2/T(ei2πκ-e2/T)T+2α|b1|2(1-e-2/T)T(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T)〈ϕ,H0qϕ〉.
Therefore, for large T there is a constant C(κ,b1)>0 so that
(3.39)Lφ1q(T)≤C(κ,b1)(1+〈ϕ,H0qϕ〉+1T).
This completes the proof.For harmonic oscillators with eigenvaluesωj, ω≠1, the evaluations of the resulting integrals are more intricate and were not carried out.


### 3.4. Kicked Perturbations by aV in L2(S1)

#### 3.4.1. Kicked Linear Rotor

Consider(3.40)H(t)=ωp+V(x)∑n∈ℤ‍δ(t-n2π),
where p=-i(d/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
(3.41)UF=UV=e-i2πωpe-iV(x).
Denote  φj(x)=eijx/2π, 0≤x<2π, and j∈ℤ  to be the eigenvectors of p2 whose eigenvalues are the square of integers j2; all eigenvalues have multiplicity 2 (the corresponding eigenvectors are φj and φ-j), except for the null eigenvalue which is simple.Consider the caseω=1; then
(3.42)((UF-z)-1φ0)(x)=12π(e-iV(x)-z),
and so
(3.43)Gzφ0(j)=〈φj,Rz(UF)φ0〉=12π∫02π‍e-ijxe-iV(x)-zdx.
Denote Ij:=∫02π‍|Gzφ0(j)|2dE. It follows that
(3.44)Ij=1(2π)2∫02π‍|∫02π‍e-ijxe-iV(x)-zdx|2dE=1(2π)2∫02π‍∫02π‍e-ijxeijy(∫02π‍dE(e-iV(x)-z)(eiV(y)-z¯))dxdy.
For x,y∈S1 fixed denote Ixy:=∫02π‍(dE/(e-iV(x)-z)(eiV(y)-z¯)). If γ(E)=eiE, 0≤E≤2π, one has
(3.45)Ixy=∫02π‍dE(e-iV(x)-e-iEe1/T)(eiV(y)-eiEe1/T)=∫02π‍dEe-iEe-iV(x)(eiE-eiV(x)e1/T)e1/T(e-1/TeiV(y)-eiE)=-1e1/Te-iV(x)1i∫γ‍dw(w-eiV(x)e1/T)(w-e-1/TeiV(y)),
and by residues
(3.46)Ixy=-2πe1/Te-iV(x)(e-1/TeiV(y)-eiV(x)e1/T)=2π(e2/T-e-iV(x)eiV(y)).
Hence
(3.47)Ij=1(2π)2∫02π‍∫02π‍e-ijxeijy2π(e2/T-e-iV(x)eiV(y))dxdy=12π∫02π‍e-ijx(∫02π‍eijydy(e2/T-e-iV(x)eiV(y)))dx=12π∫02π‍e-ijxe-iV(x)(∫02π‍eijydy(e2/TeiV(x)-eiV(y)))dx.The analytical evaluation of these integrals is not a simple task. As an illustration, consider the particular potentialV(x)=x; since by Cauchy's integral formula
(3.48)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍wj-1dw(w-e2/Teix)=0,ifj≥1
and by residue theorem
(3.49)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍dww1-j(w-e2/Teix)=2π(e2/Teix)1-j,ifj≤0,
it is found that
(3.50)Ij=0ifj≥1,Ij=12π∫02π‍e-ijxe-ix2π(e2/Teix)1-jdx=1e(2/T)(1-j)∫02π‍dx=2πe(2/T)(1-j),ifj≤0.
Therefore, by (2.16) it follows that for any q>0(3.51)Lφ0p2q(T)≥1πe-2/T1T∑j=1∞‍j2qI-j=2T∑j=1∞‍j2qe-(2/T)j,
and we conclude that (see the appendix)
(3.52)Lφ0p2q(m)≥ctem2q
and also that the sequence m↦〈UFmφ0,p2qUFmφ0〉 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 UFmφ0 and the corresponding expectation values.ForV(x)=kx with integer k≥2, similar results are obtained, that is
(3.53)Ij={0,ifj=lk,l≥1,2πe2/T(1-l),ifj=lk,l≤0
and so
(3.54)Lφ0p2q(T)≥2k2qT∑l=1∞‍l2qe-(2/T)l.
Therefore, we have the following lower bound for the Laplace average:
(3.55)Lφ0p2q(m)≥C(k,q)m2q
(see appendix). The same is valid if V(x)=kx with k denoting any negative integer number.


#### 3.4.2. Power-Kicked Systems

Due to the difficulty in evaluating the integrals in (3.47), in order to estimate Lφ0p2q(T) in some situations we take an alternative way.Consider the Kicked models inL2(S1) with Floquet operator
(3.56)UF=UV=e-i2πωf(p)e-iV(x),
corresponding to the hamiltonian
(3.57)H(t)=ωf(p)+V(x)∑n∈ℤ‍δ(t-2πn),
with p,V,φj as before and f(p)=pN for some N∈ℕ. Let ℱ:L2(S1)→l2(ℤ) be the Fourier transform. Then ℱUVℱ-1:l2(ℤ)→l2(ℤ) and
(3.58)ℱUVℱ-1=ℱe-i2πωf(p)e-iV(x)ℱ-1=ℱe-i2πωf(p)ℱ-1ℱe-iV(x)ℱ-1,
where ℱe-i2πωf(p)ℱ-1 is represented by a diagonal matrix D whose elements are
(3.59)D(m,n)=e-i2πωf(n)δmn,
and ℱe-iV(x)ℱ-1 is represented by a matrix W whose elements are
(3.60)W(m,n)=(ℱρ)(m-n)=ρ̂(m-n),
where ρ(x)=(1/2π)e-iV(x). Denote B=DW; so
(3.61)B(m,n)=e-i2πωf(n)ρ̂(m-n),(3.62)UV=ℱ-1Bℱ.
Put η(z)=Rz(UV)φ0; then
(3.63)UVη(z)-zη(z)=φ0,
and using (3.62) we obtain
(3.64)Bℱη(z)-zℱη(z)=ℱφ0.
Thus, for each n∈ℤ,
(3.65)(Bℱη(z))(n)-(zℱη(z))(n)=(ℱφ0)(n),
so that
(3.66)e-i2πωf(n)∑j∈ℤ‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0.Tridiagonal Case
In order to deal with the above equations, we try to simplify them by supposing thatV is such that ρ̂(m-n)=0 if |m-n|>1. Then, for each n∈ℤ fixed (3.66) becomes
(3.67)e-i2πωf(n)∑|n-j|≤1‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ=B is tridiagonal and has the structure
(3.68)B=(⋱g(-1)ρ̂(0)g(-1)ρ̂(-1)ρ̂(1)ρ̂(0)ρ̂(-1)g(1)ρ̂(1)g(1)ρ̂(0)g(1)ρ̂(-1)g(2)ρ̂(1)g(2)ρ̂(0)⋱),
where g(n)=e-i2πωf(n).
Now, a tridiagonal unitary operatorU 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=αkek-1+βkek+γkek+1, where {ek} is the canonical basis of l2(ℤ), that is,
(3.69)U=(⋱αk-1βk-1αkγk-1βkαk+1γkβk+1γk+1⋱).
It then follows that for all k∈ℤ(3.70)|αk|2+|βk|2+|γk|2=1,γk-1βk-1¯+βkαk¯=0,αkγk¯=0.
Applying these relations to B=ℱ-1UVℱ we obtain the following.
(i)
Ifρ̂(-1)≠0, then ρ̂(1)=ρ̂(0)=0 and |ρ̂(-1)|=1.(ii)
Ifρ̂(1)≠0, then ρ̂(-1)=ρ̂(0)=0 and |ρ̂(1)|=1.(ii)
Ifρ̂(0)≠0, then ρ̂(1)=ρ̂(-1)=0 and |ρ̂(0)|=1.
The next step is to investigate these cases. Ifρ̂(0)≠0, it reduces to the autonomous case H(t)=H0 previously considered.
The casesρ̂(-1)≠0 and ρ̂(1)≠0 are similar, so we only discuss that ρ̂(1)≠0. For n∈ℤ fixed,   (3.67) takes the form
(3.71)e-i2πωf(n)ρ̂(1)Gzφ0(n-1)-zGzφ0(n)=δn0,
and so we can write Gzφ0(n) in terms of Gzφ0(0) and Gzφ0(-1) for all n∈ℤ. More precisely
(3.72)Gzφ0(n)=e-i2πω(f(n)+⋯+f(1))ρ̂(1)nznGzφ0(0),n≥1,Gzφ0(-n)=zn-1e-i2πω(f(-n+1)+⋯+f(-1))ρ̂(1)n-1Gzφ0(-1),n≥2;
moreover, for n=0 in (3.71) we obtain ρ̂(1)Gzφ0(-1)-zGzφ0(0)=1, and so for z=e-iEe1/T and T>1(3.73)1≤|Gzφ0(-1)|+|z||Gzφ0(0)|=|Gzφ0(-1)|+e1/T|Gzφ0(0)|≤e(|Gzφ0(-1)|+|Gzφ0(0)|),
and there exists d>0 so that
(3.74)|Gzφ0(-1)|2+|Gzφ0(0)|≥d>0.
Therefore, by (2.16), for T>1 one has
(3.75)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍n2q(∫02π‍|Gzφ0(n)|2dE+∫02π‍|Gzφ0(-n)|2dE)=1πe-2/T1T∑n=1∞‍n2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-1)|2dE)≥1πe-2/T1T∑n=1∞‍n2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-1)|2)dE≥d2T∑n=0∞‍(n+1)2qe-2n/T,
so that, by the discussion at the end of the appendix,
(3.76)Lφ0p2q(m)≥C(m+1)2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded. Hence we have instability.Pentadiagonal Case
Suppose now thatV is such that ρ̂(m-n)=0 if |m-n|>2. Then for each n∈ℤ fixed, equation (3.66) becomes
(3.77)e-i2πωf(n)∑|n-j|≤2‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ 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 in the new upper diagonal are e-i2πωf(n)ρ̂(-2), and those in the new lower diagonal are e-i2πωf(n)ρ̂(2).
For not repeating the tridiagonal case we suppose that eitherρ̂(2) or ρ̂(-2) is different from zero. If U is a pentadiagonal unitary operator in l2(ℤ), that is, Uek=ζkek-2+αkek-1+βkek+γkek+1+θkek+2, one gets the matrix representation
(3.78)U=(⋱βk-2αk-1ζkγk-2βk-1αkζk+1θk-2γk-1βkαk+1ζk+2θk-1γkβk+1αk+2θkγk+1βk+2⋱).
From this we obtain the following relations, for each k∈ℤ,
(3.79)|ζk|2+|αk|2+|βk|2+|γk|2+|θk|2=1,ζk¯αk-1+αk¯βk-1+βk¯γk-1+γk¯θk-1=0,βk-1¯θk-1+αk¯γk+ζk+1¯βk+1=0,αk-1¯θk-1+ζk¯γk=0,ζk¯θk=0.
Suppose that ρ̂(2)≠0. The case ρ̂(-2)≠0 is similar. Then by the above relations we obtain ρ̂(-2)=ρ̂(-1)=ρ̂(0)=ρ̂(1)=0, and so (3.77) becomes
(3.80)e-i2πωf(n)ρ̂(2)Gzφ0(n-2)-zGzφ0(n)=δn0.
For n=0 one gets ρ̂(2)Gzφ0(-2)-zGzφ0(0)=1 and analogously to the previous case
(3.81)|Gzφ0(-2)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T and T>1. Since for n≥1(3.82)Gzφ0(2n)=e-i2πω(f(2n)+f(2n-2)+⋯+f(2))ρ̂(2)nznGzφ0(0),Gzφ0(-2n)=zn-1Gzφ0(-2)ρ̂(2)n-1e-i2πω(f(-2(n-1))+⋯+f(-2)),
we obtain
(3.83)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍(2n)2q(∫02π‍|Gzφ0(2n)|2dE+∫02π‍|Gzφ0(-2n)|2dE)=1πe-2/T1T∑n=1∞‍(2n)2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-2)|2dE)≥1πe-2/T1T∑n=1∞‍(2n)2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-2)|2)dE≥d2m∑n=0∞‍(2(n+1))2qe-2n/T,
hence
(3.84)Lφ0p2q(T)≥C(2(m+1))2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded.N-Diagonal Case
IfV satisfies ρ̂(m-n)=0 for |m-n|>N, we suppose that either ρ̂(N) or ρ̂(-N) is different from zero. In case ρ̂(N)≠0, by unitarity and the structure of ℱ-1UVℱ we obtain that ρ̂(N-1)=⋯ρ̂(0)=ρ̂(-1)=⋯=ρ̂(-N)=0, thus (3.66) becomes, for each n∈ℤ,
(3.85)e-i2πωf(n)ρ̂(N)Gzφ0(n-N)-zGzφ0(n)=δn0,
and so
(3.86)|Gzφ0(-N)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T, T>1. Moreover, for n≥1(3.87)Gzφ0(nN)=e-i2πω(f(nN)+f((n-1)N)+⋯+f(N))ρ̂(N)nznGzφ0(0),Gzφ0(-nN)=zn-1Gzφ0(-N)ρ̂(N)n-1e-i2πω(f(-N(n-1))+⋯+f(-N)).
Similarly to the previous cases we conclude that
(3.88)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Therefore we can state the following result.Theorem 3.3.
For Kicked systems in L2(S1) with
(3.89)UV=e-i2πωf(p)e-iV(x)
as in (3.56),  one can obtain that ℱUVℱ-1:l2(ℤ)→l2(ℤ) is represented by the matrix B with elements B(m,n)=e-i2πωf(n)ρ̂(m-n), where ρ(x)=(2π)-1/2e-iV(x). If V satisfies ρ̂(m-n)=0 for |m-n|>N∈ℕ* and either ρ̂(N) or ρ̂(-N) is different from zero, then V(x)=±Nx+θ, for some θ∈ℝ, and ℱUVℱ-1 is unitarily equivalent to TN (the Nth power of T) where T is the bilateral shift. Furthermore,
(3.90)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Proof.
It is enough to prove thatℱUVℱ-1 is unitarily equivalent to TN. Suppose that ρ̂(N)≠0 (the case for ρ̂(-N)≠0 is similar); then by the above discussion we obtain
(3.91)B(m,n)={0,ifm≠n+N,e-i2πωf(n)ρ̂(N),ifm=n+N,
that is, Ben=e-i2πωf(n)ρ̂(N)en+N where {en} is the canonical basis of l2(ℤ). Since |ρ̂(N)|=1, write ρ̂(N)=e-iθ. Let W be the unitary operator defined by
(3.92)Wen=eiϑnen,n∈ℤ,
where ϑn are elements in [0,2π). If ϑn satisfies for all n∈ℤ(3.93)ϑn+N-ϑn=2πωf(n)+θ,
it follows that W-1BW=TN. Equation (3.93) is satisfied taking, for example, ϑ0=ϑ1=⋯=ϑN-1=0 and   another ϑn obeying (3.93).Although Theorem3.3 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.


## 3.1. Time-Independent Hamiltonians

As a first example and illustration of the formula proposed in Theorem2.3, we consider the special case of autonomous hamiltonians. In this case H(t)=H0 for all t, and we assume that H0 is a positive, unbounded, self-adjoint operator and with simple discrete spectrum, H0φj=χjφj, so that {φj}j=1∞ is an orthonormal basis of ℋ and 0≤χ1<χ2<χ3<⋯ with χj→∞. For q>0 we can consider H0q as our abstract energy operator A, so that its eigenvalues are λj=χjq (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(3.1)Gzξ(j)=〈φj,Rz(H0)ξ〉=〈Rz¯(H0)φj,ξ〉=〈φj,ξ〉e-iχj-z.
Since domH0q is invariant under the time evolution e-itH0, then for z=e-iEe1/T and ξ∈domH0q  we have
(3.2)Lξq(T):=LξH0q(T)=1πe-2/T1T∑j=1∞‍χjq∫02π‍|Gzξ(j)|2dE=1πe-2/T1T∑j=1∞‍χjq∫02π‍|〈φj,ξ〉|2|e-iχj-z|2dE=1πe-2/T1T∑j=1∞‍χjq|〈φj,ξ〉|2∫02π‍dE|e-iχj-z|2.Thus we need to calculate the integralIj:=∫02π‍(dE/|e-iχj-z|2). Let γ be the closed path in ℂ given by γ(E)=eiE with 0≤E≤2π, αj=e1/Teiχj and βj=e-1/Teiχj, then
(3.3)Ij=∫02π‍dE(e-iχj-z)(eiχj-z¯)=∫02π‍dE(e-iχj-e-iEe1/T)(eiχj-eiEe1/T)=∫02π‍dEe2/T(e-1/Te-iχj-e-iE)(e-1/Teiχj-eiE)=-1e2/T∫02π‍dEe-iEe-1/Te-iχj(eiE-αj)(eiE-βj)=-1e1/Te-iχj1i∫02π‍ieiEdE(eiE-αj)(eiE-βj)=ie1/Te-iχj∫γ‍dw(w-αj)(w-βj).
As |αj|>1 and |βj|<1, βj is the unique pole in the interior of γ. Thus, by using residues,
(3.4)Ij=ie1/Te-iχj2πi1(βj-αj)=2πe2/T-1
and Ij is independent of χj.Therefore by (3.2) it follows that
(3.5)Lξq(T)=1πe-2/T1T∑j=1∞‍χjq|〈φj,ξ〉|22πe2/T-1=2e-2/T1T1(e2/T-1)∑j=1∞‍χjq|〈φj,ξ〉|2=2(1-e-2/T)1T∥H0q/2ξ∥2.
Since (1-e-2/T)=2/T+𝒪(1/T2), for large T it is found that
(3.6)Lξq(T)≈∥H0q/2ξ∥2,
with (for ξ∈domH0q)
(3.7)limT→∞Lξq(T)=〈ξ,H0qξ〉.
Then we conclude that the function
(3.8)ℕ∋m↦〈e-iH0mξ,H0qe-iH0mξ〉
is bounded for ξ∈domH0q, 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 (in)stability.Let[·] denotes the integer part of a real number, and |·| indicates Lebesgue measure.Theorem 3.1.
Suppose that there existK>0 and α>0 such that for each 2N>0 large enough there exists a nonempty Borel set J(N)⊂S1 such that
(3.9)|Gzξ(j)|≥KNα,N≤j≤2N
holds for all z=e-iE+1/T with E∈JT(N)={E′′∈S1:∃E′∈J(N);|E′′-E′|≤1/T} (the (1/T)-neighborhood of J(N)). Let δ>0, then for T large enough such that N(T)=[Tδ], one has
(3.10)LξA(T)≥cteλ[Tδ]Tδ(1-2α)-2.
Moreover, if λj≥ctejγ, γ≥0, then
(3.11)LξA(T)≥cteTδ(γ-2α+1)-2.Proof.
By the formula in Theorem2.3, or its more general version (2.16),
(3.12)LξA(T)≥1πe-2/T1T∑j=1∞‍λj∫02π‍|Gzξ(j)|2dE≥cteT∑j=N(T)2N(T)‍λj∫02π‍|Gzξ(j)|2dE≥cteTλN(T)∑j=N(T)2N(T)‍∫JT(N)‍|Gzξ(j)|2dE≥cteTλN(T)∑j=N(T)2N(T)‍K2N(T)2α|JT(N)|=cteT|JT(N)|λN(T)K2N(T)2α-1=cteT|JT(N)|λ[Tδ]1[Tδ]2α-1≥cteλ[Tδ]Tδ(1-2α)-2;
and we have used that |JT(N)|≥1/T. If λj≥ctejγ, then
(3.13)LξA(T)≥cteTδγTδ(1-2α)-2=cteTδ(γ-2α+1)-2.
The proof is complete.The above theorem becomes appealing when the exponent ofT is greater than zero and instability is obtained, for instance, when δ(γ-2α+1)>2 in case λj≥ctejγ. 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 J(N) 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 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(3.14)H(t)=H0+κPϕ∑n‍δ(t-n2π),
with H0 as in Section 3.1, with eigenvectors {φj}j=1∞ 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 {H0mϕ:m∈ℕ} equals ℋ. Let
(3.15)ϕ=∑j‍bjφj.
In this case (see [22–24])
(3.16)UF=U0(1+αPϕ),
with U0=e-i2πH0 and α=(e-i2πκ-1). Note that ϕ∈domH0q, ∀q>0, and so for ξ∈domH0q,
(3.17)UFξ=U0ξ+α〈ϕ,ξ〉U0ϕ
also belongs to domH0q; a simple iteration process shows that UFmξ∈domH0q for all m≥0, and we are justified in using the formula in Theorem 2.3 to estimate Laplace averages.We are interested inη(z)=Rz(UF)φ1. As |z|≠1, it follows that η(z) belongs to the Hilbert space, and so one can write
(3.18)η(z)=∑j=1∞‍ajφj.
Note that aj=Gz(j), and we have
(3.19)UFη(z)-zη(z)=φ1.
By the relation
(3.20)UFη(z)=U0η(z)+αU0Pϕη(z)=∑j=1∞‍ajU0φj+αU0〈ϕ,η(z)〉ϕ=∑j=1∞‍aje-i2πχjφj+α〈ϕ,η(z)〉∑j=1∞‍bje-i2πχjφj=∑j=1∞‍(aj+α〈ϕ,η(z)〉bj)e-i2πχjφj,
and (3.19) it follows that
(3.21)∑j=1∞‍(aj+α〈ϕ,η(z)〉bj)e-i2πχjφj-z∑j=1∞‍ajφj=φ1,
that is,
(3.22)∑j=1∞‍[aj(e-i2πχj-z)+α〈ϕ,η(z)〉bje-i2πχj]φj=φ1,
and we get the equations
(3.23)a1(e-i2πχ1-z)+α〈ϕ,η(z)〉b1e-i2πχ1=1,aj(e-i2πχj-z)+α〈ϕ,η(z)〉bje-i2πχj=0forj>1.
Thus
(3.24)a1=1-α〈ϕ,η(z)〉b1e-i2πχ1e-i2πχ1-z,aj=-α〈ϕ,η(z)〉bje-i2πχje-i2πχj-z,j>1.For the trivial caseα=0 or, equivalently, κ∈ℤ, one has
(3.25)a1=1e-i2πχ1-z,aj=0,j>1,
and η(z)=φ1/(e-i2πχ1-z). In this case the analysis of Lφ1q(T) is reduced to
(3.26)∫02π‍|a1|2dE=∫02π‍dE|e-i2πχ1-z|2=2πe2/T-1
as calculated in Section 3.1. Thus Lφ1q(T)≈∥H0qφ1∥ for large T, as expected.Returning to the general caseα≠0, note that
(3.27)〈ϕ,η(z)〉=∑j=1∞‍b¯jaj=b¯1(1-α〈ϕ,η(z)〉b1e-i2πχ1e-i2πχ1-z)+∑j=2∞‍b¯j(-α)〈ϕ,η(z)〉bje-i2πχje-i2πχj-z=b¯1e-i2πχ1-z-〈ϕ,η(z)〉∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z.
So
(3.28)〈ϕ,η(z)〉=b¯1(e-i2πχ1-z)[1+∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z]-1.
By denoting
(3.29)τ(z)=1+∑j=1∞‍α|bj|2e-i2πχje-i2πχj-z,
by (3.24)   we finally obtain the relations
(3.30)a1=1e-i2πχ1-z-α|b1|2e-i2πχ1τ(z)-1(e-i2πχ1-z)2,aj=-αbjb¯1e-i2πχjτ(z)-1(e-i2πχ1-z)(e-i2πχj-z),j>1.


### 3.3.1. A Harmonic Oscillator

Now we present an application of the above relations to a kicked harmonic oscillator with natural frequency equals to 1; we will writeLξq=LξH0q.Proposition 3.2.
LetH0 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,
(3.31)Lφ1q(T)≤C,
where φ1 is the harmonic oscillator ground state. Hence one has  H0q-dynamical stability.Proof.
We use the above notation; note thatφ1∈domH0q, ∀q>0 and Theorem 2.3 can be applied. In this case we have
(3.32)τ(z)=1+∑j=1∞‍α|bj|21-z=1+α1-z∥ϕ∥2=1-z+α1-z,
and so
(3.33)a1=11-z-α|b1|2(1-z)(e-i2πκ-z),aj=-αbjb¯1(1-z)(e-i2πκ-z),j>1.
Now we evaluate Ij:=∫02π‍|aj|2dE. For j>1 and γ(E)=eiE, 0≤E≤2π,
(3.34)∫02π‍|aj|2dE=∫02π‍|αbjb¯1(1-z)(e-i2πκ-z)|2dE=|α|2|bj|2|b¯1|2∫02π‍dE|(1-e-iEe1/T)(e-i2πκ-e-iEe1/T)|2=|α|2|bj|2|b¯1|2ie2/Te-i2πκ∫γ‍wdw(w-β1)(w-β2)(w-β3)(w-β4),
where β1=e1/T, β2=e-1/T, β3=e1/Tei2πκ, and β4=e-1/Tei2πκ; only β2 and β4 are poles in the interior of γ. By residue, for j>1,
(3.35)Ij=2π|α|2|bj|2|b¯1|2e2/Te-i2πκ(β2(β2-β1)(β2-β3)(β2-β4)+β4(β4-β1)(β4-β2)(β4-β3))=2πα|bj|2|b1¯|2(e2/T-1)(e-i2πκ-e2/T)-2πα|bj|2|b1¯|2ei2πκ(e2/T-1)(ei2πκ-e2/T)=2πα|bj|2|b¯1|2(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and for j=1(3.36)I1=∫02π‍|11-z-α|b1|2(1-z)(e-i2πκ-z)|2dE=∫02π‍dE(1-z)(1-z¯)-α¯|b1|2∫02π‍dE(1-z)(1-z¯)(ei2πκ-z¯)-α|b1|2∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)+|α|2|b1|4∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)(ei2πκ-z¯),
and evaluating the integrals we obtain
(3.37)I1=2π(e2/T-1)-2π|b1|2(e2/T-1)-2π|b1|2(ei2πκ-e2/T)-2πα|b1|2(e2/T-1)(e-i2πκ-e2/T)+2πα|b1|4(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and after inserting this in the expression of the average energy we get
(3.38)Lφ1q(T)=2(1-e-2/T)T(1-|b1|2-α|b1|2(e-i2πκ-e2/T))-2|b1|2e-2/T(ei2πκ-e2/T)T+2α|b1|2(1-e-2/T)T(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T)〈ϕ,H0qϕ〉.
Therefore, for large T there is a constant C(κ,b1)>0 so that
(3.39)Lφ1q(T)≤C(κ,b1)(1+〈ϕ,H0qϕ〉+1T).
This completes the proof.For harmonic oscillators with eigenvaluesωj, ω≠1, the evaluations of the resulting integrals are more intricate and were not carried out.


## 3.3.1. A Harmonic Oscillator

Now we present an application of the above relations to a kicked harmonic oscillator with natural frequency equals to 1; we will writeLξq=LξH0q.Proposition 3.2.
LetH0 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,
(3.31)Lφ1q(T)≤C,
where φ1 is the harmonic oscillator ground state. Hence one has  H0q-dynamical stability.Proof.
We use the above notation; note thatφ1∈domH0q, ∀q>0 and Theorem 2.3 can be applied. In this case we have
(3.32)τ(z)=1+∑j=1∞‍α|bj|21-z=1+α1-z∥ϕ∥2=1-z+α1-z,
and so
(3.33)a1=11-z-α|b1|2(1-z)(e-i2πκ-z),aj=-αbjb¯1(1-z)(e-i2πκ-z),j>1.
Now we evaluate Ij:=∫02π‍|aj|2dE. For j>1 and γ(E)=eiE, 0≤E≤2π,
(3.34)∫02π‍|aj|2dE=∫02π‍|αbjb¯1(1-z)(e-i2πκ-z)|2dE=|α|2|bj|2|b¯1|2∫02π‍dE|(1-e-iEe1/T)(e-i2πκ-e-iEe1/T)|2=|α|2|bj|2|b¯1|2ie2/Te-i2πκ∫γ‍wdw(w-β1)(w-β2)(w-β3)(w-β4),
where β1=e1/T, β2=e-1/T, β3=e1/Tei2πκ, and β4=e-1/Tei2πκ; only β2 and β4 are poles in the interior of γ. By residue, for j>1,
(3.35)Ij=2π|α|2|bj|2|b¯1|2e2/Te-i2πκ(β2(β2-β1)(β2-β3)(β2-β4)+β4(β4-β1)(β4-β2)(β4-β3))=2πα|bj|2|b1¯|2(e2/T-1)(e-i2πκ-e2/T)-2πα|bj|2|b1¯|2ei2πκ(e2/T-1)(ei2πκ-e2/T)=2πα|bj|2|b¯1|2(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and for j=1(3.36)I1=∫02π‍|11-z-α|b1|2(1-z)(e-i2πκ-z)|2dE=∫02π‍dE(1-z)(1-z¯)-α¯|b1|2∫02π‍dE(1-z)(1-z¯)(ei2πκ-z¯)-α|b1|2∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)+|α|2|b1|4∫02π‍dE(1-z)(1-z¯)(e-i2πκ-z)(ei2πκ-z¯),
and evaluating the integrals we obtain
(3.37)I1=2π(e2/T-1)-2π|b1|2(e2/T-1)-2π|b1|2(ei2πκ-e2/T)-2πα|b1|2(e2/T-1)(e-i2πκ-e2/T)+2πα|b1|4(e2/T-1)(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T),
and after inserting this in the expression of the average energy we get
(3.38)Lφ1q(T)=2(1-e-2/T)T(1-|b1|2-α|b1|2(e-i2πκ-e2/T))-2|b1|2e-2/T(ei2πκ-e2/T)T+2α|b1|2(1-e-2/T)T(1e-i2πκ-e2/T-ei2πκei2πκ-e2/T)〈ϕ,H0qϕ〉.
Therefore, for large T there is a constant C(κ,b1)>0 so that
(3.39)Lφ1q(T)≤C(κ,b1)(1+〈ϕ,H0qϕ〉+1T).
This completes the proof.For harmonic oscillators with eigenvaluesωj, ω≠1, the evaluations of the resulting integrals are more intricate and were not carried out.


## 3.4. Kicked Perturbations by aV in L2(S1)

### 3.4.1. Kicked Linear Rotor

Consider(3.40)H(t)=ωp+V(x)∑n∈ℤ‍δ(t-n2π),
where p=-i(d/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
(3.41)UF=UV=e-i2πωpe-iV(x).
Denote  φj(x)=eijx/2π, 0≤x<2π, and j∈ℤ  to be the eigenvectors of p2 whose eigenvalues are the square of integers j2; all eigenvalues have multiplicity 2 (the corresponding eigenvectors are φj and φ-j), except for the null eigenvalue which is simple.Consider the caseω=1; then
(3.42)((UF-z)-1φ0)(x)=12π(e-iV(x)-z),
and so
(3.43)Gzφ0(j)=〈φj,Rz(UF)φ0〉=12π∫02π‍e-ijxe-iV(x)-zdx.
Denote Ij:=∫02π‍|Gzφ0(j)|2dE. It follows that
(3.44)Ij=1(2π)2∫02π‍|∫02π‍e-ijxe-iV(x)-zdx|2dE=1(2π)2∫02π‍∫02π‍e-ijxeijy(∫02π‍dE(e-iV(x)-z)(eiV(y)-z¯))dxdy.
For x,y∈S1 fixed denote Ixy:=∫02π‍(dE/(e-iV(x)-z)(eiV(y)-z¯)). If γ(E)=eiE, 0≤E≤2π, one has
(3.45)Ixy=∫02π‍dE(e-iV(x)-e-iEe1/T)(eiV(y)-eiEe1/T)=∫02π‍dEe-iEe-iV(x)(eiE-eiV(x)e1/T)e1/T(e-1/TeiV(y)-eiE)=-1e1/Te-iV(x)1i∫γ‍dw(w-eiV(x)e1/T)(w-e-1/TeiV(y)),
and by residues
(3.46)Ixy=-2πe1/Te-iV(x)(e-1/TeiV(y)-eiV(x)e1/T)=2π(e2/T-e-iV(x)eiV(y)).
Hence
(3.47)Ij=1(2π)2∫02π‍∫02π‍e-ijxeijy2π(e2/T-e-iV(x)eiV(y))dxdy=12π∫02π‍e-ijx(∫02π‍eijydy(e2/T-e-iV(x)eiV(y)))dx=12π∫02π‍e-ijxe-iV(x)(∫02π‍eijydy(e2/TeiV(x)-eiV(y)))dx.The analytical evaluation of these integrals is not a simple task. As an illustration, consider the particular potentialV(x)=x; since by Cauchy's integral formula
(3.48)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍wj-1dw(w-e2/Teix)=0,ifj≥1
and by residue theorem
(3.49)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍dww1-j(w-e2/Teix)=2π(e2/Teix)1-j,ifj≤0,
it is found that
(3.50)Ij=0ifj≥1,Ij=12π∫02π‍e-ijxe-ix2π(e2/Teix)1-jdx=1e(2/T)(1-j)∫02π‍dx=2πe(2/T)(1-j),ifj≤0.
Therefore, by (2.16) it follows that for any q>0(3.51)Lφ0p2q(T)≥1πe-2/T1T∑j=1∞‍j2qI-j=2T∑j=1∞‍j2qe-(2/T)j,
and we conclude that (see the appendix)
(3.52)Lφ0p2q(m)≥ctem2q
and also that the sequence m↦〈UFmφ0,p2qUFmφ0〉 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 UFmφ0 and the corresponding expectation values.ForV(x)=kx with integer k≥2, similar results are obtained, that is
(3.53)Ij={0,ifj=lk,l≥1,2πe2/T(1-l),ifj=lk,l≤0
and so
(3.54)Lφ0p2q(T)≥2k2qT∑l=1∞‍l2qe-(2/T)l.
Therefore, we have the following lower bound for the Laplace average:
(3.55)Lφ0p2q(m)≥C(k,q)m2q
(see appendix). The same is valid if V(x)=kx with k denoting any negative integer number.


### 3.4.2. Power-Kicked Systems

Due to the difficulty in evaluating the integrals in (3.47), in order to estimate Lφ0p2q(T) in some situations we take an alternative way.Consider the Kicked models inL2(S1) with Floquet operator
(3.56)UF=UV=e-i2πωf(p)e-iV(x),
corresponding to the hamiltonian
(3.57)H(t)=ωf(p)+V(x)∑n∈ℤ‍δ(t-2πn),
with p,V,φj as before and f(p)=pN for some N∈ℕ. Let ℱ:L2(S1)→l2(ℤ) be the Fourier transform. Then ℱUVℱ-1:l2(ℤ)→l2(ℤ) and
(3.58)ℱUVℱ-1=ℱe-i2πωf(p)e-iV(x)ℱ-1=ℱe-i2πωf(p)ℱ-1ℱe-iV(x)ℱ-1,
where ℱe-i2πωf(p)ℱ-1 is represented by a diagonal matrix D whose elements are
(3.59)D(m,n)=e-i2πωf(n)δmn,
and ℱe-iV(x)ℱ-1 is represented by a matrix W whose elements are
(3.60)W(m,n)=(ℱρ)(m-n)=ρ̂(m-n),
where ρ(x)=(1/2π)e-iV(x). Denote B=DW; so
(3.61)B(m,n)=e-i2πωf(n)ρ̂(m-n),(3.62)UV=ℱ-1Bℱ.
Put η(z)=Rz(UV)φ0; then
(3.63)UVη(z)-zη(z)=φ0,
and using (3.62) we obtain
(3.64)Bℱη(z)-zℱη(z)=ℱφ0.
Thus, for each n∈ℤ,
(3.65)(Bℱη(z))(n)-(zℱη(z))(n)=(ℱφ0)(n),
so that
(3.66)e-i2πωf(n)∑j∈ℤ‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0.Tridiagonal Case
In order to deal with the above equations, we try to simplify them by supposing thatV is such that ρ̂(m-n)=0 if |m-n|>1. Then, for each n∈ℤ fixed (3.66) becomes
(3.67)e-i2πωf(n)∑|n-j|≤1‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ=B is tridiagonal and has the structure
(3.68)B=(⋱g(-1)ρ̂(0)g(-1)ρ̂(-1)ρ̂(1)ρ̂(0)ρ̂(-1)g(1)ρ̂(1)g(1)ρ̂(0)g(1)ρ̂(-1)g(2)ρ̂(1)g(2)ρ̂(0)⋱),
where g(n)=e-i2πωf(n).
Now, a tridiagonal unitary operatorU 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=αkek-1+βkek+γkek+1, where {ek} is the canonical basis of l2(ℤ), that is,
(3.69)U=(⋱αk-1βk-1αkγk-1βkαk+1γkβk+1γk+1⋱).
It then follows that for all k∈ℤ(3.70)|αk|2+|βk|2+|γk|2=1,γk-1βk-1¯+βkαk¯=0,αkγk¯=0.
Applying these relations to B=ℱ-1UVℱ we obtain the following.
(i)
Ifρ̂(-1)≠0, then ρ̂(1)=ρ̂(0)=0 and |ρ̂(-1)|=1.(ii)
Ifρ̂(1)≠0, then ρ̂(-1)=ρ̂(0)=0 and |ρ̂(1)|=1.(ii)
Ifρ̂(0)≠0, then ρ̂(1)=ρ̂(-1)=0 and |ρ̂(0)|=1.
The next step is to investigate these cases. Ifρ̂(0)≠0, it reduces to the autonomous case H(t)=H0 previously considered.
The casesρ̂(-1)≠0 and ρ̂(1)≠0 are similar, so we only discuss that ρ̂(1)≠0. For n∈ℤ fixed,   (3.67) takes the form
(3.71)e-i2πωf(n)ρ̂(1)Gzφ0(n-1)-zGzφ0(n)=δn0,
and so we can write Gzφ0(n) in terms of Gzφ0(0) and Gzφ0(-1) for all n∈ℤ. More precisely
(3.72)Gzφ0(n)=e-i2πω(f(n)+⋯+f(1))ρ̂(1)nznGzφ0(0),n≥1,Gzφ0(-n)=zn-1e-i2πω(f(-n+1)+⋯+f(-1))ρ̂(1)n-1Gzφ0(-1),n≥2;
moreover, for n=0 in (3.71) we obtain ρ̂(1)Gzφ0(-1)-zGzφ0(0)=1, and so for z=e-iEe1/T and T>1(3.73)1≤|Gzφ0(-1)|+|z||Gzφ0(0)|=|Gzφ0(-1)|+e1/T|Gzφ0(0)|≤e(|Gzφ0(-1)|+|Gzφ0(0)|),
and there exists d>0 so that
(3.74)|Gzφ0(-1)|2+|Gzφ0(0)|≥d>0.
Therefore, by (2.16), for T>1 one has
(3.75)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍n2q(∫02π‍|Gzφ0(n)|2dE+∫02π‍|Gzφ0(-n)|2dE)=1πe-2/T1T∑n=1∞‍n2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-1)|2dE)≥1πe-2/T1T∑n=1∞‍n2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-1)|2)dE≥d2T∑n=0∞‍(n+1)2qe-2n/T,
so that, by the discussion at the end of the appendix,
(3.76)Lφ0p2q(m)≥C(m+1)2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded. Hence we have instability.Pentadiagonal Case
Suppose now thatV is such that ρ̂(m-n)=0 if |m-n|>2. Then for each n∈ℤ fixed, equation (3.66) becomes
(3.77)e-i2πωf(n)∑|n-j|≤2‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ 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 in the new upper diagonal are e-i2πωf(n)ρ̂(-2), and those in the new lower diagonal are e-i2πωf(n)ρ̂(2).
For not repeating the tridiagonal case we suppose that eitherρ̂(2) or ρ̂(-2) is different from zero. If U is a pentadiagonal unitary operator in l2(ℤ), that is, Uek=ζkek-2+αkek-1+βkek+γkek+1+θkek+2, one gets the matrix representation
(3.78)U=(⋱βk-2αk-1ζkγk-2βk-1αkζk+1θk-2γk-1βkαk+1ζk+2θk-1γkβk+1αk+2θkγk+1βk+2⋱).
From this we obtain the following relations, for each k∈ℤ,
(3.79)|ζk|2+|αk|2+|βk|2+|γk|2+|θk|2=1,ζk¯αk-1+αk¯βk-1+βk¯γk-1+γk¯θk-1=0,βk-1¯θk-1+αk¯γk+ζk+1¯βk+1=0,αk-1¯θk-1+ζk¯γk=0,ζk¯θk=0.
Suppose that ρ̂(2)≠0. The case ρ̂(-2)≠0 is similar. Then by the above relations we obtain ρ̂(-2)=ρ̂(-1)=ρ̂(0)=ρ̂(1)=0, and so (3.77) becomes
(3.80)e-i2πωf(n)ρ̂(2)Gzφ0(n-2)-zGzφ0(n)=δn0.
For n=0 one gets ρ̂(2)Gzφ0(-2)-zGzφ0(0)=1 and analogously to the previous case
(3.81)|Gzφ0(-2)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T and T>1. Since for n≥1(3.82)Gzφ0(2n)=e-i2πω(f(2n)+f(2n-2)+⋯+f(2))ρ̂(2)nznGzφ0(0),Gzφ0(-2n)=zn-1Gzφ0(-2)ρ̂(2)n-1e-i2πω(f(-2(n-1))+⋯+f(-2)),
we obtain
(3.83)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍(2n)2q(∫02π‍|Gzφ0(2n)|2dE+∫02π‍|Gzφ0(-2n)|2dE)=1πe-2/T1T∑n=1∞‍(2n)2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-2)|2dE)≥1πe-2/T1T∑n=1∞‍(2n)2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-2)|2)dE≥d2m∑n=0∞‍(2(n+1))2qe-2n/T,
hence
(3.84)Lφ0p2q(T)≥C(2(m+1))2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded.N-Diagonal Case
IfV satisfies ρ̂(m-n)=0 for |m-n|>N, we suppose that either ρ̂(N) or ρ̂(-N) is different from zero. In case ρ̂(N)≠0, by unitarity and the structure of ℱ-1UVℱ we obtain that ρ̂(N-1)=⋯ρ̂(0)=ρ̂(-1)=⋯=ρ̂(-N)=0, thus (3.66) becomes, for each n∈ℤ,
(3.85)e-i2πωf(n)ρ̂(N)Gzφ0(n-N)-zGzφ0(n)=δn0,
and so
(3.86)|Gzφ0(-N)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T, T>1. Moreover, for n≥1(3.87)Gzφ0(nN)=e-i2πω(f(nN)+f((n-1)N)+⋯+f(N))ρ̂(N)nznGzφ0(0),Gzφ0(-nN)=zn-1Gzφ0(-N)ρ̂(N)n-1e-i2πω(f(-N(n-1))+⋯+f(-N)).
Similarly to the previous cases we conclude that
(3.88)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Therefore we can state the following result.Theorem 3.3.
For Kicked systems in L2(S1) with
(3.89)UV=e-i2πωf(p)e-iV(x)
as in (3.56),  one can obtain that ℱUVℱ-1:l2(ℤ)→l2(ℤ) is represented by the matrix B with elements B(m,n)=e-i2πωf(n)ρ̂(m-n), where ρ(x)=(2π)-1/2e-iV(x). If V satisfies ρ̂(m-n)=0 for |m-n|>N∈ℕ* and either ρ̂(N) or ρ̂(-N) is different from zero, then V(x)=±Nx+θ, for some θ∈ℝ, and ℱUVℱ-1 is unitarily equivalent to TN (the Nth power of T) where T is the bilateral shift. Furthermore,
(3.90)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Proof.
It is enough to prove thatℱUVℱ-1 is unitarily equivalent to TN. Suppose that ρ̂(N)≠0 (the case for ρ̂(-N)≠0 is similar); then by the above discussion we obtain
(3.91)B(m,n)={0,ifm≠n+N,e-i2πωf(n)ρ̂(N),ifm=n+N,
that is, Ben=e-i2πωf(n)ρ̂(N)en+N where {en} is the canonical basis of l2(ℤ). Since |ρ̂(N)|=1, write ρ̂(N)=e-iθ. Let W be the unitary operator defined by
(3.92)Wen=eiϑnen,n∈ℤ,
where ϑn are elements in [0,2π). If ϑn satisfies for all n∈ℤ(3.93)ϑn+N-ϑn=2πωf(n)+θ,
it follows that W-1BW=TN. Equation (3.93) is satisfied taking, for example, ϑ0=ϑ1=⋯=ϑN-1=0 and   another ϑn obeying (3.93).Although Theorem3.3 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.


## 3.4.1. Kicked Linear Rotor

Consider(3.40)H(t)=ωp+V(x)∑n∈ℤ‍δ(t-n2π),
where p=-i(d/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
(3.41)UF=UV=e-i2πωpe-iV(x).
Denote  φj(x)=eijx/2π, 0≤x<2π, and j∈ℤ  to be the eigenvectors of p2 whose eigenvalues are the square of integers j2; all eigenvalues have multiplicity 2 (the corresponding eigenvectors are φj and φ-j), except for the null eigenvalue which is simple.Consider the caseω=1; then
(3.42)((UF-z)-1φ0)(x)=12π(e-iV(x)-z),
and so
(3.43)Gzφ0(j)=〈φj,Rz(UF)φ0〉=12π∫02π‍e-ijxe-iV(x)-zdx.
Denote Ij:=∫02π‍|Gzφ0(j)|2dE. It follows that
(3.44)Ij=1(2π)2∫02π‍|∫02π‍e-ijxe-iV(x)-zdx|2dE=1(2π)2∫02π‍∫02π‍e-ijxeijy(∫02π‍dE(e-iV(x)-z)(eiV(y)-z¯))dxdy.
For x,y∈S1 fixed denote Ixy:=∫02π‍(dE/(e-iV(x)-z)(eiV(y)-z¯)). If γ(E)=eiE, 0≤E≤2π, one has
(3.45)Ixy=∫02π‍dE(e-iV(x)-e-iEe1/T)(eiV(y)-eiEe1/T)=∫02π‍dEe-iEe-iV(x)(eiE-eiV(x)e1/T)e1/T(e-1/TeiV(y)-eiE)=-1e1/Te-iV(x)1i∫γ‍dw(w-eiV(x)e1/T)(w-e-1/TeiV(y)),
and by residues
(3.46)Ixy=-2πe1/Te-iV(x)(e-1/TeiV(y)-eiV(x)e1/T)=2π(e2/T-e-iV(x)eiV(y)).
Hence
(3.47)Ij=1(2π)2∫02π‍∫02π‍e-ijxeijy2π(e2/T-e-iV(x)eiV(y))dxdy=12π∫02π‍e-ijx(∫02π‍eijydy(e2/T-e-iV(x)eiV(y)))dx=12π∫02π‍e-ijxe-iV(x)(∫02π‍eijydy(e2/TeiV(x)-eiV(y)))dx.The analytical evaluation of these integrals is not a simple task. As an illustration, consider the particular potentialV(x)=x; since by Cauchy's integral formula
(3.48)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍wj-1dw(w-e2/Teix)=0,ifj≥1
and by residue theorem
(3.49)∫02π‍eijydy(e2/Teix-eiy)=-1i∫γ‍dww1-j(w-e2/Teix)=2π(e2/Teix)1-j,ifj≤0,
it is found that
(3.50)Ij=0ifj≥1,Ij=12π∫02π‍e-ijxe-ix2π(e2/Teix)1-jdx=1e(2/T)(1-j)∫02π‍dx=2πe(2/T)(1-j),ifj≤0.
Therefore, by (2.16) it follows that for any q>0(3.51)Lφ0p2q(T)≥1πe-2/T1T∑j=1∞‍j2qI-j=2T∑j=1∞‍j2qe-(2/T)j,
and we conclude that (see the appendix)
(3.52)Lφ0p2q(m)≥ctem2q
and also that the sequence m↦〈UFmφ0,p2qUFmφ0〉 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 UFmφ0 and the corresponding expectation values.ForV(x)=kx with integer k≥2, similar results are obtained, that is
(3.53)Ij={0,ifj=lk,l≥1,2πe2/T(1-l),ifj=lk,l≤0
and so
(3.54)Lφ0p2q(T)≥2k2qT∑l=1∞‍l2qe-(2/T)l.
Therefore, we have the following lower bound for the Laplace average:
(3.55)Lφ0p2q(m)≥C(k,q)m2q
(see appendix). The same is valid if V(x)=kx with k denoting any negative integer number.


## 3.4.2. Power-Kicked Systems

Due to the difficulty in evaluating the integrals in (3.47), in order to estimate Lφ0p2q(T) in some situations we take an alternative way.Consider the Kicked models inL2(S1) with Floquet operator
(3.56)UF=UV=e-i2πωf(p)e-iV(x),
corresponding to the hamiltonian
(3.57)H(t)=ωf(p)+V(x)∑n∈ℤ‍δ(t-2πn),
with p,V,φj as before and f(p)=pN for some N∈ℕ. Let ℱ:L2(S1)→l2(ℤ) be the Fourier transform. Then ℱUVℱ-1:l2(ℤ)→l2(ℤ) and
(3.58)ℱUVℱ-1=ℱe-i2πωf(p)e-iV(x)ℱ-1=ℱe-i2πωf(p)ℱ-1ℱe-iV(x)ℱ-1,
where ℱe-i2πωf(p)ℱ-1 is represented by a diagonal matrix D whose elements are
(3.59)D(m,n)=e-i2πωf(n)δmn,
and ℱe-iV(x)ℱ-1 is represented by a matrix W whose elements are
(3.60)W(m,n)=(ℱρ)(m-n)=ρ̂(m-n),
where ρ(x)=(1/2π)e-iV(x). Denote B=DW; so
(3.61)B(m,n)=e-i2πωf(n)ρ̂(m-n),(3.62)UV=ℱ-1Bℱ.
Put η(z)=Rz(UV)φ0; then
(3.63)UVη(z)-zη(z)=φ0,
and using (3.62) we obtain
(3.64)Bℱη(z)-zℱη(z)=ℱφ0.
Thus, for each n∈ℤ,
(3.65)(Bℱη(z))(n)-(zℱη(z))(n)=(ℱφ0)(n),
so that
(3.66)e-i2πωf(n)∑j∈ℤ‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0.Tridiagonal Case
In order to deal with the above equations, we try to simplify them by supposing thatV is such that ρ̂(m-n)=0 if |m-n|>1. Then, for each n∈ℤ fixed (3.66) becomes
(3.67)e-i2πωf(n)∑|n-j|≤1‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ=B is tridiagonal and has the structure
(3.68)B=(⋱g(-1)ρ̂(0)g(-1)ρ̂(-1)ρ̂(1)ρ̂(0)ρ̂(-1)g(1)ρ̂(1)g(1)ρ̂(0)g(1)ρ̂(-1)g(2)ρ̂(1)g(2)ρ̂(0)⋱),
where g(n)=e-i2πωf(n).
Now, a tridiagonal unitary operatorU 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=αkek-1+βkek+γkek+1, where {ek} is the canonical basis of l2(ℤ), that is,
(3.69)U=(⋱αk-1βk-1αkγk-1βkαk+1γkβk+1γk+1⋱).
It then follows that for all k∈ℤ(3.70)|αk|2+|βk|2+|γk|2=1,γk-1βk-1¯+βkαk¯=0,αkγk¯=0.
Applying these relations to B=ℱ-1UVℱ we obtain the following.
(i)
Ifρ̂(-1)≠0, then ρ̂(1)=ρ̂(0)=0 and |ρ̂(-1)|=1.(ii)
Ifρ̂(1)≠0, then ρ̂(-1)=ρ̂(0)=0 and |ρ̂(1)|=1.(ii)
Ifρ̂(0)≠0, then ρ̂(1)=ρ̂(-1)=0 and |ρ̂(0)|=1.
The next step is to investigate these cases. Ifρ̂(0)≠0, it reduces to the autonomous case H(t)=H0 previously considered.
The casesρ̂(-1)≠0 and ρ̂(1)≠0 are similar, so we only discuss that ρ̂(1)≠0. For n∈ℤ fixed,   (3.67) takes the form
(3.71)e-i2πωf(n)ρ̂(1)Gzφ0(n-1)-zGzφ0(n)=δn0,
and so we can write Gzφ0(n) in terms of Gzφ0(0) and Gzφ0(-1) for all n∈ℤ. More precisely
(3.72)Gzφ0(n)=e-i2πω(f(n)+⋯+f(1))ρ̂(1)nznGzφ0(0),n≥1,Gzφ0(-n)=zn-1e-i2πω(f(-n+1)+⋯+f(-1))ρ̂(1)n-1Gzφ0(-1),n≥2;
moreover, for n=0 in (3.71) we obtain ρ̂(1)Gzφ0(-1)-zGzφ0(0)=1, and so for z=e-iEe1/T and T>1(3.73)1≤|Gzφ0(-1)|+|z||Gzφ0(0)|=|Gzφ0(-1)|+e1/T|Gzφ0(0)|≤e(|Gzφ0(-1)|+|Gzφ0(0)|),
and there exists d>0 so that
(3.74)|Gzφ0(-1)|2+|Gzφ0(0)|≥d>0.
Therefore, by (2.16), for T>1 one has
(3.75)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍n2q(∫02π‍|Gzφ0(n)|2dE+∫02π‍|Gzφ0(-n)|2dE)=1πe-2/T1T∑n=1∞‍n2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-1)|2dE)≥1πe-2/T1T∑n=1∞‍n2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-1)|2)dE≥d2T∑n=0∞‍(n+1)2qe-2n/T,
so that, by the discussion at the end of the appendix,
(3.76)Lφ0p2q(m)≥C(m+1)2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded. Hence we have instability.Pentadiagonal Case
Suppose now thatV is such that ρ̂(m-n)=0 if |m-n|>2. Then for each n∈ℤ fixed, equation (3.66) becomes
(3.77)e-i2πωf(n)∑|n-j|≤2‍ρ̂(n-j)Gzφ0(j)-zGzφ0(n)=δn0,
and ℱ-1UVℱ 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 in the new upper diagonal are e-i2πωf(n)ρ̂(-2), and those in the new lower diagonal are e-i2πωf(n)ρ̂(2).
For not repeating the tridiagonal case we suppose that eitherρ̂(2) or ρ̂(-2) is different from zero. If U is a pentadiagonal unitary operator in l2(ℤ), that is, Uek=ζkek-2+αkek-1+βkek+γkek+1+θkek+2, one gets the matrix representation
(3.78)U=(⋱βk-2αk-1ζkγk-2βk-1αkζk+1θk-2γk-1βkαk+1ζk+2θk-1γkβk+1αk+2θkγk+1βk+2⋱).
From this we obtain the following relations, for each k∈ℤ,
(3.79)|ζk|2+|αk|2+|βk|2+|γk|2+|θk|2=1,ζk¯αk-1+αk¯βk-1+βk¯γk-1+γk¯θk-1=0,βk-1¯θk-1+αk¯γk+ζk+1¯βk+1=0,αk-1¯θk-1+ζk¯γk=0,ζk¯θk=0.
Suppose that ρ̂(2)≠0. The case ρ̂(-2)≠0 is similar. Then by the above relations we obtain ρ̂(-2)=ρ̂(-1)=ρ̂(0)=ρ̂(1)=0, and so (3.77) becomes
(3.80)e-i2πωf(n)ρ̂(2)Gzφ0(n-2)-zGzφ0(n)=δn0.
For n=0 one gets ρ̂(2)Gzφ0(-2)-zGzφ0(0)=1 and analogously to the previous case
(3.81)|Gzφ0(-2)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T and T>1. Since for n≥1(3.82)Gzφ0(2n)=e-i2πω(f(2n)+f(2n-2)+⋯+f(2))ρ̂(2)nznGzφ0(0),Gzφ0(-2n)=zn-1Gzφ0(-2)ρ̂(2)n-1e-i2πω(f(-2(n-1))+⋯+f(-2)),
we obtain
(3.83)Lφ0p2q(T)≥1πe-2/T1T∑n=1∞‍(2n)2q(∫02π‍|Gzφ0(2n)|2dE+∫02π‍|Gzφ0(-2n)|2dE)=1πe-2/T1T∑n=1∞‍(2n)2q(1e2n/T∫02π‍|Gzφ0(0)|2dE+e2(n-1)/T∫02π‍|Gzφ0(-2)|2dE)≥1πe-2/T1T∑n=1∞‍(2n)2qe-2n/T∫02π‍(|Gzφ0(0)|2+|Gzφ0(-2)|2)dE≥d2m∑n=0∞‍(2(n+1))2qe-2n/T,
hence
(3.84)Lφ0p2q(T)≥C(2(m+1))2q,
and 〈UVmφ0,p2qUVmφ0〉 is unbounded.N-Diagonal Case
IfV satisfies ρ̂(m-n)=0 for |m-n|>N, we suppose that either ρ̂(N) or ρ̂(-N) is different from zero. In case ρ̂(N)≠0, by unitarity and the structure of ℱ-1UVℱ we obtain that ρ̂(N-1)=⋯ρ̂(0)=ρ̂(-1)=⋯=ρ̂(-N)=0, thus (3.66) becomes, for each n∈ℤ,
(3.85)e-i2πωf(n)ρ̂(N)Gzφ0(n-N)-zGzφ0(n)=δn0,
and so
(3.86)|Gzφ0(-N)|2+|Gzφ0(0)|2≥d>0,
with z=e-iEe1/T, T>1. Moreover, for n≥1(3.87)Gzφ0(nN)=e-i2πω(f(nN)+f((n-1)N)+⋯+f(N))ρ̂(N)nznGzφ0(0),Gzφ0(-nN)=zn-1Gzφ0(-N)ρ̂(N)n-1e-i2πω(f(-N(n-1))+⋯+f(-N)).
Similarly to the previous cases we conclude that
(3.88)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Therefore we can state the following result.Theorem 3.3.
For Kicked systems in L2(S1) with
(3.89)UV=e-i2πωf(p)e-iV(x)
as in (3.56),  one can obtain that ℱUVℱ-1:l2(ℤ)→l2(ℤ) is represented by the matrix B with elements B(m,n)=e-i2πωf(n)ρ̂(m-n), where ρ(x)=(2π)-1/2e-iV(x). If V satisfies ρ̂(m-n)=0 for |m-n|>N∈ℕ* and either ρ̂(N) or ρ̂(-N) is different from zero, then V(x)=±Nx+θ, for some θ∈ℝ, and ℱUVℱ-1 is unitarily equivalent to TN (the Nth power of T) where T is the bilateral shift. Furthermore,
(3.90)Lφ0p2q(T)≥d2T∑n=0∞‍(N(n+1))2qe-2n/T.Proof.
It is enough to prove thatℱUVℱ-1 is unitarily equivalent to TN. Suppose that ρ̂(N)≠0 (the case for ρ̂(-N)≠0 is similar); then by the above discussion we obtain
(3.91)B(m,n)={0,ifm≠n+N,e-i2πωf(n)ρ̂(N),ifm=n+N,
that is, Ben=e-i2πωf(n)ρ̂(N)en+N where {en} is the canonical basis of l2(ℤ). Since |ρ̂(N)|=1, write ρ̂(N)=e-iθ. Let W be the unitary operator defined by
(3.92)Wen=eiϑnen,n∈ℤ,
where ϑn are elements in [0,2π). If ϑn satisfies for all n∈ℤ(3.93)ϑn+N-ϑn=2πωf(n)+θ,
it follows that W-1BW=TN. Equation (3.93) is satisfied taking, for example, ϑ0=ϑ1=⋯=ϑN-1=0 and   another ϑn obeying (3.93).Although Theorem3.3 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 Theorem2.3 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 isHV:l2(ℤ)→l2(ℤ) defined by
(4.1)(HVξ)(n)=ξ(n+1)+ξ(n-1)+V(n)ξ(n),V a bounded sequence, a similar formula can be handled in some cases by relating the resolvent RE+i/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-?



---

*Source: 902506-2009-06-21.xml*