A FLOQUET OPERATOR WITH PURE POINT SPECTRUM AND ENERGY INSTABILITY

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1 A FLOQUET OPERATOR WITH PURE POINT SPECTRUM AND ENERGY INSTABILITY Abstract. An example of Floquet operator with purely point spectrum and energy instability is presented. In the unperturbed energy eigenbasis its eigenfunctions are exponentially localized.. Introduction It is not immediate whether a self-adjoint operator H with purely point spectrum implies absence of transport under the time evolution Ut) = e iht ; in fact, it is currently known examples of Schrödinger operators with such kind of spectrum and transport. In case of tight-binding models on l 2 IN) the transport is usually probed by the moments of order m > of the position operator Xe k = ke k, that is, ) X m = k IN k m e k, e k, where e k j) = δ kj Kronecker delta) is the canonical basis of l 2 IN). Analogous definition applies for l 2 ZZ) and even higher dimensional spaces. Then, by definition, transport at ψ, also called dynamical instability or dynamical delocalization, occurs if for some m the function 2) t ψt), X m ψt), ψt) := Ut)ψ, is unbounded. If for all m > the corresponding functions are bounded, one has dynamical stability, also called dynamical localization. The first rigorous example of a Schrödinger operator with purely point spectrum and dynamical instability has appeared in [7], Appendix 2, what the authors have called A Pathological Example; in this case the tight binding Schrödinger operator h on l 2 IN) with a Dirichlet condition at n = was with potential hu)n) = un + ) + un ) + vn)un) 3) vn) = 3 cosπαn + θ) + λδ n, 99 Mathematics Subject Classification. 8Q B,47B99). CRdeO was partially supported by CNPq Brazil). MSS was supported by CAPES Brazil).

2 2 that is, rank one perturbations of an instance of the almost Mathieu operator. An irrational number α was constructed so that for a.e. θ [, 2π) and a.e. λ [, ] the corresponding self-adjoint operator h has purely point spectrum with dynamical instability at e throughout, the term a.e. without specification means with respect to the Lebesgue measure under consideration). More precisely, it was shown that for all ɛ > lim sup t t 2 ɛ ψt), X2 ψt) =, ψ) = e. Compare with the absence of ballistic motion for point spectrum Hamiltonians [6] lim t t 2 ψt), X2 ψt) =. Additional examples of this behavior are known, even for random potentials, but with a strong local correlations [], as for the random dimer model in the Schrödinger case; there is also an adaptation [6] for the random Bernoulli Dirac operator with no correlation in the potential, although for the massless case. The time evolution of a quantum system with time-dependent Hamiltonian is given by a strongly continuous family of unitary operators Ut, r) the propagator). For an initial condition ψ at t =, its time evolution is given by Ut, )ψ. If the Hamiltonian is time-periodic with period T, then Ut + T, r + T ) = Ut, r), t, r, and we have the Floquet operator U F := UT, ) defined as the evolution generated by the Hamiltonian over a period. 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 consequence of the famous RAGE theorem, firstly proved for the autonomous case [5] and then for time-periodic Hamiltonians [8]. In principle, a Floquet operator with purely point spectrum would imply stability, but one should be alerted by the above mentioned pathological examples in the autonomous case. In this work we give an example of a Floquet operator with purely point spectrum and energy instability, which can be considered the partner concept of dynamical instability in case of autonomous systems. We shall consider a particular choice in the family of Floquet operators studied in [3]; such operators describe the quantum dynamics of certain interesting physical models see [, 3] and references therein), and display a band structure with respect to an orthogonal basis {ϕ k } of l 2 IN) or l 2 ZZ), constructed as eigenfunctions of an unperturbed energy operator. There are some conceptual differences with respect to the autonomous case mentioned before, since

3 PURE POINT AND ENERGY INSTABILITY 3 now the momentum X m is defined in the energy space 4) X m = k k m ϕ k, ϕ k, instead of the physical space IN. Thus, if for all m > the functions 5) n ψn), X m ψn), ψn) := U n F ψ, n, are bounded we say there is energy stability or energy localization, while if at least one of them is unbounded we say the system presents energy instability or energy delocalization; the latter reflects a kind of resonance. Our construction is a fusion of the Floquet operator studied in [3], now under suitable additional rank one perturbations, and the arguments presented in [7] for model 3). For suitable values of parameters we shall get the following properties:. The resulting unitary operator U λ β, θ) + after the rank one perturbation; see Eq. )) still belongs to the family of Floquet operators considered in [3]. 2. U λ β, θ) + has purely point spectrum with exponentially localized eigenfunctions. 3. The time evolution along the Floquet operator U λ β, θ) + of the initial condition ϕ presents energy instability. U λ β, θ) + will be obtained as a rank one perturbation of the almost periodic class of operators studied in the Section 7 of [3] we describe them ahead). In order to prove purely point spectrum, we borrow an argument from [9] that was used to prove localization for random unitary operators, and it combines spectral averaging and positivity of the Lyapunov exponent with polynomial boundedness of generalized eigenfunctions. In order to get dynamical instability, although we adapt ideas of [7], we underline that some results needed completely different proofs and they are not entirely trivial. It is worth mentioning that in [9] a form of dynamical stability was obtained for discrete evolution along some Floquet operators CMV matrices) related to random Verblunsky coefficients. This paper is organized as follows. In Section 2 we present the model of Floquet operator we shall consider, some preliminary results and the main result is stated in Theorem 2. In Section 3 we shall prove that our Floquet operator is pure point. Section 4 is devoted to the proof of dynamical instability. 2. The Floquet Operator We briefly recall the construction of the Floquet operator introduced in [3] based on the physical model discussed in []. The separable Hilbert space is l 2 ZZ) and {ϕ k } k ZZ denote its canonical basis. Consider the set of 2 2

4 4 matrices defined for any k ZZ by S k = e iθ k re iα k ite iγ k ) ite iγ k re iα k parameterized by the phases α k, γ k, θ k in the torus T and the real parameters t, r, the reflection and transition coefficients, respectively, linked by r 2 + t 2 =. Then, let P j be the orthogonal projection onto the span of ϕ j, ϕ j+ in l 2 ZZ), and let U e, U o be two 2 2 block diagonal unitary operators on l 2 ZZ) defined by U e = k ZZ P 2k S 2k P 2k and U o = k ZZ P 2k+ S 2k+ P 2k+. The matrix representation of U e in the canonical basis is... S 2 U e = S, S 2... and similarly for U o, with S 2k+ in place of S 2k. The Floquet operator U is defined by such that, for any k ZZ, U = U o U e, Uϕ 2k = irte iθ 2k+θ 2k ) e iα 2k γ 2k ) ϕ 2k +r 2 e iθ 2k+θ 2k ) e iα 2k α 2k ) ϕ 2k +irte iθ 2k+θ 2k+ ) e iγ 2k+α 2k+ ) ϕ 2k+ 6) t 2 e iθ 2k+θ 2k+ ) e iγ 2k+γ 2k+ ) ϕ 2k+2 Uϕ 2k+ = t 2 e iθ 2k+θ 2k ) e iγ 2k+γ 2k ) ϕ 2k +irte iθ 2k+θ 2k ) e iγ 2k+α 2k ) ϕ 2k +r 2 e iθ 2k+θ 2k+ ) e iα 2k α 2k+ ) ϕ 2k+ +irte iθ 2k+θ 2k+ ) e iα 2k γ 2k+ ) ϕ 2k+2 The extreme cases where rt = are spectrally trivial; in case t =, r =, U is pure point and if t =, r =, U is purely absolutely continuous Proposition 3. in [3]). From now on we suppose < r, t <. For the eigenvalue equation Uψ = e ie ψ ψ = k ZZ c k ϕ k, c k, E C,

5 PURE POINT AND ENERGY INSTABILITY 5 one gets the following relation between coefficients ) ) c2k c2k 2 = T c k E), 2k+ c 2k where the matrix T k E) has elements T k E) = e ie+γ 2k +γ 2k 2 +θ 2k +θ 2k 2 ), T k E) 2 = i r ) e ie+γ 2k α 2k 2 +θ 2k +θ 2k 2 ) e iγ 2k α 2k ), t T k E) 2 = i r e iθ 2k 2 θ 2k +γ 2k +γ 2k +γ 2k 2 +α 2k ) t 7) e ie+θ 2k 2+θ 2k +γ 2k +γ 2k +γ 2k 2 +α 2k ) ), T k E) 22 = t 2 eie+θ 2k+θ 2k γ 2k γ 2k ) and + r2 t 2 e iγ 2k+γ 2k ) e iθ 2k θ 2k 2 +α 2k 2 α 2k ) + e iα 2k α 2k ) ) r2 t 2 e ie+θ 2k 2+θ 2k +γ 2k +γ 2k +α 2k α 2k 2 ) det T k E) = e iθ 2k 2 θ 2k +γ 2k +2γ 2k +γ 2k 2 ). Given coefficients c, c ), for any k IN one has ) ) c2k c = T c k E)... T 2 E)T E), 2k+ c c 2k c 2k+ ) = T k+ E)... T E) T E) c c ). In the physical setting [], the natural Hilbert space is l 2 IN ), with IN the set of positive integers, and the definition according with [3] of the Floquet operator, denoted by U +, is 8) U + ϕ = re iθ +θ ) e iα ϕ + ite iθ +θ ) e iγ ϕ 2, U + ϕ k = Uϕ k, k > with Uϕ k as in 6). In this case the eigenvalue equation is U + ψ = e ie ψ with ψ = k= c kϕ k. Then starting from c 2, c 3, we have ) ) c2k c2 = T c k E)... T 2 E), k = 2, 3,... 2k+ c 3 where the transfer matrices T k E) are given by 7), along with the additional one ) ) c2 a E) = c c, 3 a 2 E)

6 6 where a E) = i t e ie+γ +θ +θ ) re iγ α ) ) a 2 E) = t 2 eie+θ 2+θ γ 2 γ ) + r t 2 e iγ 2+γ ) e iθ 2 θ α ) + re iα 2 α ) ) r t 2 e ie+θ +θ +γ 2 +γ +α 2 ) For further details and generalizations of this class of unitary operators, we refer the reader to [3,, 2, 9]. In particular, when the phases are i.i.d. random variables, it was proved to hold in the unitary case typical results obtained for discrete one-dimensional random Schrödinger operators. For example, the availability of a transfer matrix formalism to express generalized eigenvectors allows to introduce a Lyapunov exponent, to prove a unitary version of Oseledec s Theorem and of Ishii-Pastur Theorem and get absence of absolutely continuous spectrum in some cases). Our main interest is on the almost periodic example U U{θ k }, {α k }, {γ k }), where the phases α k are taken as constants, α k = α k ZZ, while the γ k s are arbitrary and can be replaced by ) k+ α see Lemma 3.2 in [3]). The almost periodicity due to the phases θ k defined by θ k = 2πβk + θ, where β IR, and θ [, 2π). We denote U above by U = Uβ, θ) and then for any k ZZ see 6)) Uβ, θ)ϕ 2k = irte i2πβ4k )+2θ) ϕ 2k +r 2 e i2πβ4k )+2θ) ϕ 2k +irte i2πβ4k+)+2θ) ϕ 2k+ 9) t 2 e i2πβ4k+)+2θ) ϕ 2k+2 Uβ, θ)ϕ 2k+ = t 2 e i2πβ4k )+2θ) ϕ 2k +itre i2πβ4k )+2θ) ϕ 2k +r 2 e i2πβ4k+)+2θ) ϕ 2k+ +itre i2πβ4k+)+2θ) ϕ 2k+2 Let Uβ, θ) + be the corresponding operator on l 2 IN ) defined by 8). The following result was proved in [3]. Theorem. i) For β rational and each θ [, 2π), Uβ, θ) is purely absolutely continuous, σ sc Uβ, θ) + ) =, σ ac Uβ, θ) + ) = σ ac Uβ, θ)) and the point spectrum of Uβ, θ) + consists of finitely many simple eigenvalues in the resolvent set of Uβ, θ). ii) Let Tk θ E) be the transfer matrices at E T corresponding to Uβ, θ).

7 PURE POINT AND ENERGY INSTABILITY 7 For β irrational, the Lyapunov exponent γe) satisfies, for almost all θ, γ θ E) = lim N ln N k= T k θe) ln N t 2 >, and so σ ac Uβ, θ)) =. The same is true for Uβ, θ) +. Finally, we introduce our study model. We consider a rank one perturbation of Uβ, θ) +, λ [, 2π) see also [4]) ) U λ β, θ) + := Uβ, θ) + e iλpϕ = Uβ, θ) + ) I d + e iλ )P ϕ, where P ϕ ) = ϕ, ϕ. We observe that Uβ, θ) + U + {θ k } k=, {α k} k=, {γ k} k= ) and U λ β, θ) + U { + θ ) k } k=, { α k} k=, { γ k} k= where θ = θ λ and θ k = θ k, α k = α k, γ k = γ k for k. Hence, the perturbed operator U λ β, θ) + also belongs to the family of Floquet operators studied in [3]. Note also that the Lyapunov exponent is independent on the parameter λ. We can now state our main result: Theorem 2. i) For β irrational, U λ β, θ) + has only point spectrum for a.e. θ, λ [, 2π), and in the basis {ϕ k } its eigenfunctions decay exponentially. ii) β can be chosen irrational so that lim sup n X U λ β, θ) + ) n ϕ 2 F n) =, for all θ [, 2π) and any λ [ π 6, π 2 ], where F n) = n2 moment of order m = given by 4). ln2+n) and X is the Remarks.. Joining up i) and ii) of the theorem above we proved that for some β irrational, for a.e. θ [, 2π) and λ [ π 6, π 2 ], U λβ, θ) + has pure point spectrum and the function n U λ β, θ) +) n ϕ, X 2 U λ β, θ) +) n ϕ is unbounded. That is, we have pure point spectrum and dynamical instability. 2. One can modify the proof to replace the logarithm function fn) = ln2 + n) for any monotone sequence f with lim n fn) =. 3. Pure Point Spectrum In this section we prove part i) of Theorem 2. We need a preliminary lemma.

8 8 Lemma. For any β and θ, the vector ϕ is cyclic for Uβ, θ) +. Proof. Fix β and θ. We indicate that any vector ϕ k, k IN can be written as a linear combination of the vectors Uβ, θ) + ) n ϕ, n ZZ. Since Uβ, θ) + ϕ = re i2πβ+2θ) e iα ϕ + ite i2πβ+2θ) e iα ϕ 2 then ) ϕ 2 = i t ei2πβ+2θ) e iα Uβ, θ) + ϕ + ir t ϕ. Now 2) Uβ, θ) + ) ϕ = a ϕ + a 2 ϕ 2 + a 3 ϕ 3, where a, a 2 and a 3 are nonzero complex numbers. Thus, using ) and 2), suitable linear combination of Uβ, θ) + ) ϕ, ϕ and Uβ, θ) + ϕ yields ϕ 3. Since Uβ, θ) + ϕ 2 = b ϕ + b 2 ϕ 2 + b 3 ϕ 3 + b 4 ϕ 4 we obtain that ϕ 4 can be written as a linear combination desired. Due to the structure of Uβ, θ) +, the process can be iterated to obtain any ϕ k. We are in conditions to prove pure point spectrum for our model. Proof. Theorem 2i)) Fix β irrational and let denote the Lebesgue measure on [, 2π). By Theorem ii), for any E [, 2π) there exists ΩE) [, 2π) with ΩE) = such that Thus, by Fubini s Theorem, = γ θ E) >, ΩE) de 2π and for θ in a set of measure one = = θ ΩE). χ ΩE) θ) de 2π =, χ ΩE) θ) dθ ) de 2π 2π χ ΩE) θ) de 2π ) dθ 2π that is, θ ΩE) for almost all E [, 2π). Then we get the existence of Ω [, 2π) with Ω = such that for any θ Ω there exists A θ [, 2π) with A θ = and γ θ E) >, E A c θ := [, 2π) \ A θ. Let µ k θ,λ be the spectral measures associated with U λ β, θ) + = e ie df θ,λ E) and respectively vectors ϕ k, so that for k IN and all Borel sets Λ [, 2π) µ k θ,λ Λ) = ϕ k, F θ,λ Λ)ϕ k.

9 PURE POINT AND ENERGY INSTABILITY 9 Now, for rank one perturbations of unitary operators there is a spectral averaging formula as for rank one perturbations of self-adjoint operators see [7, 2] for the self-adjoint case and [2, 4] for the unitary case). Thus, for any f L [, 2π)) one has 3) dλ fe)dµ θ,λ E) = fe) de 2π. Then, applying 3) with f the characteristic function of A θ we obtain = A θ = = dλ χ Aθ E) de 2π χ Aθ E)dµ θ,λ E) = µ θ,λ A θ)dλ, and so µ θ,λ A θ) = for almost all λ. Therefore, for each θ Ω, there is J θ [, 2π) with Jθ c = such that 4) µ θ,λ A θ) =, λ J θ. By Lemma and 4), it follows that F θ,λ A θ ) = for all θ Ω and λ J θ. Moreover, let S θ,λ denote the set of E [, 2π) so that the equation U λ β, θ) + ψ = e ie ψ has a nontrivial polynomially bounded solution. It is known that F θ,λ [, 2π) \ S θ,λ ) = see [3, 9]). Thus we conclude that S θ,λ A c θ is a support for F θ,λ ) see remark bellow) for all θ Ω and λ J θ. Now, if E S θ,λ A c θ then U λβ, θ) + ψ = e ie ψ has a nontrivial polynomially bounded solution ψ and γ θ E) >. By construction γ θ,λ E) = γ θ E) where γ θ,λ E) is the Lyapunov exponent associated with U λ β, θ) +. Thus, by Oseledec s Theorem, every solution which is polynomially bounded necessarily has to decay exponentially, so ψ is in l 2 IN ) and is an eigenfunction of U λ β, θ) +. Hence, we conclude that each E S θ,λ A c θ is an eigenvalue of U λ β, θ) + with corresponding eigenfunction decaying exponentially. As l 2 IN ) is separable, it follows that S θ,λ A c θ is countable and then F θ,λ ) has countable support for all θ Ω and λ J θ. Thus U λ β, θ) + has purely point spectrum for a.e. θ, λ [, 2π). Remark. We say that a Borel set S supports the spectral projection F ) if F [, 2π) \ S) =.

10 4. Energy Instability In this section we present the proof of Theorem 2ii). The initial strategy is that of Appendix 2 of [7], and Lemmas 2 and 3 ahead are similar to Lemmas B. and B.2 in [7]. However, some important technical issues needed quite different arguments. To begin with we shall discuss a series of preliminary lemmas, adapted to the unitary case from the self-adjoint setting. 4.. Preliminary Lemmas. Let P n a denote the projection onto those vectors supported by {n : n a}, that is, for ψ l 2 IN ) {, if n < a P n a ψ)n) = ψn), if n a, and similarly for P n<a. Let fn) be a monotone increasing sequence with fn) as n. Lemma 2. If there exists T m, T m IN for all m, such that 5) then T m + 2T P n f) m Uλ β, θ) +) j ϕ 2 lim sup X U λ β, θ) +) j ϕ 2 fj)5 j j 2 =. ft m ) 2, Proof. By hypothesis, for each m IN, there must be some j m [T m, 2T m ] such that P n Uλ β, θ) +) j m ϕ 2 f) ft m ) 2 and then X U λ β, θ) +) j m ϕ 2 = n 2 Uλ β, θ) +) j m ϕ n) 2 n IN T m P ft m ) n Uλ β, θ) +) ) j m ϕ n) f) n IN 2 Therefore T 2 m ft m ) 4. fj m ) 5 X U λ β, θ) +) j m ϕ 2 j 2 m and the lemma is proved. j m ) 2 ) fjm ) 4 fj m ) ft m ) 4 fj m)

11 PURE POINT AND ENERGY INSTABILITY In order to prove Theorem 2ii) we want to apply the above lemma with fn) = lnn + 2)) /5. By keeping this goal in mind, the estimate in relation 5) is crucial as well as the following lemmas. Lemma 3. Let ξ be a unit vector, P a projection, and U a unitary operator. If ξ = η + ψ with η, ψ =, then 6) T + I d P )U j ξ 2 ψ 2 3 T + P U j ψ 2 /2. Proof. Denote D := 2T T + I d P )U j ξ 2. Then D = = T + T + = η 2 T + + ψ 2 T + = A + B, P U j ξ 2 ) ψ 2 + η 2 P U j η + ψ) 2 ) P U j η 2 P U j ψ 2 + 2Re P U j η, P U j ψ )) with A = η 2 T + 2T P U j η 2 and B = ψ 2 T + 2T P U j ψ 2 + 2Re P U j η, P U j ψ )).

12 2 Clearly, 2T T + P U j η 2 η 2, and the same is true with η replaced by ψ. Hence A and B = ψ 2 T + ψ 2 T + ψ 2 T + 2 T + ψ 2 3 T + P U j ψ 2 2 T + P U j ψ 2 2 T + P U j ψ 2) 2 P U j η 2) 2 T + The result follows immediately. P U j ψ 2) 2. Re P U j η, P U j ψ ) P U j η P U j ψ P U j ψ 2) 2 The following lemma is an adaptation to the discrete setup of a classical estimate found in Lemma 4.5, page 543 of [3]. Lemma 4. Let U = e it de U t) be the spectral decomposition of a unitary operator U on the Hilbert space H. Let ξ H be an absolutely continuous vector for U, i.e., the spectral measure µ ξ, associated to U and ξ, is absolutely continuous with respect to Lebesgue measure, and denote by g = dµ ξ dx L [, 2π)) the corresponding Radon-Nikodym derivative. Define ξ U = g /2. Then, for any η H, one has U j ξ, η 2 2π ξ 2 U η 2. j ZZ If it is clear the unitary operator in question, then will be used to indicate U. Proof. If ξ = then the result is clear. Suppose ξ < and take η H. Denote by P ac the spectral projection onto the absolutely continuous subspace H ac of U, η = P ac η and g = dµη dλ ; then µ ξ,η is absolutely continuous and its Radon-Nikodym derivative h is estimate by hx) g g) 2 x) = g 2 x) g 2 x) ξ g 2 x). Hence h L 2 [, 2π)) with L 2 norm estimated by ) 2 h 2 ξ gx)dx = ξ dµ η ) 2

13 Since U j ξ, η = PURE POINT AND ENERGY INSTABILITY 3 = ξ η ξ η. e ijt dµ ξ,η t) = e ijt ht)dt = 2πFh)j), it follows that U j ξ, η 2 = 2π Fh)j) 2 = 2π h 2 2 2π ξ 2 η 2, j ZZ j ZZ which is precisely the stated result Cauchy and Borel Transforms. Given a probability measure µ on D = {z C : z = }, its Cauchy F µ z) and Borel R µ z) transforms are, respectively, for z C with z, and R µ is related to F µ by 7) F µ z) = R µ z) = D D e it + z e it z dµt) e it z dµt). F µ z) = 2zR µ z) +. Moreover, F µ has the following properties [8]: 8) lim r F µ re iθ ) exists for a.e. θ, and if one decomposes the measure in its absolutely continuous and singular parts then dµθ) = ωθ) dθ 2π + dµ sθ), ωθ) = lim r Re F µ re iθ ). θ is a pure point of µ if and only if lim r r)re F µ re iθ ). dµ s is supported on {θ : lim r F µ re iθ ) = }. Now, let U be a unitary operator on a separable Hilbert space H and φ a cyclic vector for U. Consider the rank one perturbation of U U λ = Ue iλp φ = UI d + e iλ )P φ ), where P φ ) = φ, φ and λ [, 2π). Denote by dµ λ the spectral measure associated with U λ and φ, F λ = F µλ and R λ = R µλ. We have the following relations between R λ and R, F λ and F :

14 4 Lemma 5. For z 9) and 2) In particular, for λ π, 2) where y = 22) R λ z) = R z) e iλ + ze iλ )R z) F λ z) = eiλ ) + e iλ + )F z) e iλ + ) + e iλ )F z) sin λ +cos λ, and for λ = π Re F λ z) = + y2 )Re F z) + iyf z) 2, Re F λ z) = Re F z) F z) 2. Proof. Relation 9) was got in [4]. For checking 2) we use relations 7) and 9). In fact, F λ z) = 2zR λ z) + = R z) 2z e iλ + ze iλ )R z) + = eiλ + ze iλ )R z) + 2zR z) e iλ + ze iλ )R z) = eiλ + ze iλ + )R z) e iλ + ze iλ )R z) = 2eiλ + 2ze iλ R z) + 2zR z) 2e iλ + 2ze iλ R z) 2zR z) = eiλ + e iλ + 2e iλ zr z) + + 2zR z) e iλ + + e iλ + 2e iλ zr z) 2zR z) = eiλ ) + e iλ + ) + 2zR z)) e iλ + ) + e iλ ) + 2zR z)) = eiλ ) + e iλ + )F z) e iλ + ) + e iλ )F z). Now, for λ π we have e iλ + and then F λ z) = e iλ e iλ + + F z) ) + e iλ F e iλ + z) = iy + F z) + iyf z) iyf z) iyf z) = iy + F z) iy F z) 2 + y 2 F z) + iyf z) 2,

15 where eiλ = iy and y = sin e iλ + PURE POINT AND ENERGY INSTABILITY 5 λ +cos λ. So, for λ π, Re F λ z) = + y2 )Re F z) + iyf z) 2 and 2) is obtained. For λ = π we have F λ z) = F z) and 22) follows. Lemma 6. Fix a rational number β. Then there exist C > and C 2 <, and for each θ [, 2π) and λ [ π 6, π 2 ] a decomposition so that ϕ = η θ,λ + ψ θ,λ 23) 24) 25) η θ,λ, ψ θ,λ =, ψ θ,λ C, ψ θ,λ Uλ β,θ) + C 2 the notation U was introduced in Lemma 4). Proof. We break the proof in some steps. Step. By Theorem, since β is rational, σ sc Uβ, θ) + ) =, σ ac Uβ, θ) + ) = σ ac Uβ, θ)) and the point spectrum of Uβ, θ) + consists of finitely many simple eigenvalues in the resolvent set of Uβ, θ). Denote by µ θ,λ the spectral measure associated to U λ β, θ) + and the cyclic vector) ϕ, and by µ θ the spectral measure associated to Uβ, θ) + and ϕ i.e., the case λ = ). Write dµ θ,λ E) = f θ,λ E) de 2π + dµθ,λ s E), dµ θ E) = f θ E) de 2π + dµθ pe). Step 2. Relation between f θ,λ and f θ : By Lemma 5, for λ π one has where y = sin λ +cos λ for almost all E. Re F µθ,λ z) = + y2 )Re F µθ z) + iyf µθ z) 2, and then f θ,λ E) = + y 2 )f θ E) + iy lim r F µθ re ie ) 2, Step 3. Relation between f θ and f : By 9) and 8) one gets 26) Uβ, θ) + = e i2θ Uβ, ) +

16 6 and using this relation it found that Uβ, θ) + ) j = e ij2θ Uβ, ) +) j for all j ZZ. Thus, by the spectral theorem, for any j ZZ, Hence 27) for almost all E. and e ije f θ E) de 2π = f θ E) = f E 2θ) e ije f E 2θ) de 2π. Step 4. Lower and upper bounds for f θ,λ : We have If we denote lim F µθ re ie ) = f θ E) + i lim Im F µθ re ie ) r r lim Im F µθ re ie ) = lim r r + lim r g θ E) = lim Im r e it + re ie Im e it re ie e it + re ie Im e it re ie e it + re ie e it re ie ) f θ t) dt 2π, ) f θ t) dt 2π ) dµ θ pt). then by 27) we obtain g θ E) = g E 2θ) for almost all E. On the other hand, by 26) we have that E is an eigenvalue of Uβ, θ) + if and only if E 2θ is an eigenvalue of Uβ, ) +. Let {Ej θ}n j= be the set of eigenvalues of Uβ, θ) + recall that n < ) and dµ θ p = n j= κθ j δ Ej θ δ E is the Dirac measure at E). Then e it + re ie ) lim Im r e it re ie dµ θ 2r sine t) pt) = lim r + r 2 2r cose t) dµθ pt) n 2r sine Ej θ = lim )κθ j r + r 2 2r cose Ej θ) = j= n 2 sine 2θ Ej )κθ j e ie j e ie 2θ) 2. Since f L [, 2π)), by a result of [4] Theorem.6 in Chapter III), the function g is of weak L type, i.e., g is measurable and there exits a constant C > such that for all T > the Lebesgue measure j= 28) {E : g E) T } C T.

17 Pick S > such that Ω S := PURE POINT AND ENERGY INSTABILITY 7 { } E : S f E) S satisfies Ω S > and dist Ω S, {Ej }n j= ) = L >. Then choose T sufficiently large such that A := Ω S {E : g E) T } satisfies A > ; by 28) this is possible. For θ [, 2π) put I θ := {E [, 2π) : E 2θ A}; thus I θ = A >. Then, for all θ [, 2π), λ [, π 2 ] equivalently y [, ]) and almost all E I θ one has + iy lim F µ θ re ie ) r + y f θ E) + g θ E) n 2 sine 2θ E + ) j )κθ j j= e ie j e ie 2θ) 2 n 2 κ θ j + f E 2θ) + g E 2θ) + + S + T + 2 L 2. j= L 2 So, for all θ [, 2π), λ [, π 2 ] and almost all E I θ f θ,λ E) = + y 2 )f θ E) + iy lim r F µθ re ie )) 2 f E 2θ) + S + T + 2/L 2 ) 2 S + S + T + 2/L 2 ) 2. In order to get un upper bound, note that + iy lim F µ θ re ie ) r yf θe), [ ] and so, for all θ [, 2π), λ [ π 6, π 2 ] equivalently y 2+, ) and almost 3 all E I θ f θ,λ E) = + y 2 )f θ E) + iy lim r F µθ re ie ) 2 + y2 )f θ E) y 2 f θ E) 2 = + y2 ) y 2 f E 2θ) ) 2 S.

18 8 Summing up, for all θ [, 2π), λ [ π 6, π 2 ] and almost all E I θ, we have proved that 29) S + S + T + 2/L 2 ) 2 f θ,λe) ) 2 S. Step 5. Conclusion: For λ [ π 6, π 2 ] and θ [, 2π) let ψ θ,λ = P θ,λ I θ ϕ, η θ,λ = I d P θ,λ I θ )ϕ, where P θ,λ I θ is the spectral projection of U λ β, θ) + ) onto I θ. Then for each θ [, 2π) and λ [ π 6, π 2 ] we have the decomposition ϕ = ψ θ,λ + η θ,λ that satisfies 23). By the construction in Step 4, we have that A = I is in the absolutely continuous spectrum of Uβ, ) +, so by 26) and the definition of I θ it follows that I θ is in the absolutely continuous spectrum of Uβ, θ) + ; thus using Birman-Krein s theorem on invariance of absolutely continuous spectrum under trace class perturbations, we conclude that I θ belongs to the absolutely continuous spectrum of U λ β, θ) + for all λ. Therefore by 29) ψ θ,λ 2 = ψ θ,λ, ψ θ,λ = P θ,λ ϕ, P θ,λ I θ ϕ and 24) holds with also I θ = ϕ, P θ,λ I θ ϕ = = f θ,λ E) de I θ 2π χ Iθ E)dµ θ,λ A 2πS + S + T + 2/L 2 ) 2 A ) /2 C = > ; 2πS + S + T + 2/L 2 ) 2 ψ θ,λ 2 U λ β,θ) = P θ,λ + I θ ϕ 2 = χ U λ β,θ) + I θ f θ,λ ) 2 S and 25) holds with C 2 = ) 2 S) /2 <. The lemma is proved Variation of β. The next lemma gives an estimate of the dependence of the dynamics on β. Its proof strongly uses the structure of U λ β, θ) +. Lemma 7. Let β, β IR. Then, for n, U λ β, θ) +) n ϕ U λ β, θ) +) n ϕ 2 4 n 2n 2 n)2π β β.

19 Proof. It is an induction. We have Thus PURE POINT AND ENERGY INSTABILITY 9 U λ β, θ) + ϕ j = Uβ, θ) + I d + e iλ )P ϕ )ϕ j { Uβ, θ) = + ϕ j if j > Uβ, θ) + ϕ + e iλ )Uβ, θ) + ϕ if j = { Uβ, θ) = + ϕ j if j > e iλ Uβ, θ) + ϕ if j = U λ β, θ) + ϕ = e iλ Uβ, θ) + ϕ = a e i2πβ) ϕ + a 2 e i2πβ) ϕ 2 where a = re iλ e iα+2θ) and a 2 = ite iλ e iα+2θ). Since 3) e ix e ix 2 x x and a j, j =, 2, then U λ β, θ) + ϕ U λ β, θ) + ϕ 2 e i2πβ) e i2πβ ) and the lemma is proved for n =. Now Uλ β, θ) +) 2 ϕ = U λ β, θ) + U λ β, θ) + ϕ 4 2 2πβ 2πβ = 2 4 2π β β = U λ β, θ) + a e i2πβ) ϕ + a 2 e i2πβ) ϕ 2 ) = e iλ a e i2πβ) Uβ, θ) + ϕ + a 2 e i2πβ) Uβ, θ) + ϕ 2 = e iλ a e i2πβ) b e i2πβ) ϕ + b 2 e i2πβ) ϕ 2 ) +a 2 e i2πβ) c e i3.2πβ)) ϕ + c 2 e i3.2πβ)) ϕ 2 +c 3 e i5.2πβ)) ϕ 3 + c 4 e i5.2πβ)) ϕ 4 ) Since a j <, b j <, c j < and there are < 4 4 terms in the expansion of U λ β, θ) + ) 2 ϕ and the largest exponent which provides the largest contribution by 3)) is obtained from the product of the exponentials e i2πβ) e i2+3)2πβ) = e i+2+3)2πβ), we obtain U λ β, θ) +) 2 ϕ U λ β, θ) +) 2 ϕ )2π β β = )2π β β, and the lemma is proved for n = 2. In a similar way by the structure of U λ β, θ) + we conclude that U λ β, θ) + ) 3 ϕ has at most terms where the largest exponent is in e i+2+3)2πβ e i4+5)2πβ) = e i )2πβ)

20 2 and so U λ β, θ) +) 3 ϕ U λ β, θ) +) 3 ϕ )2π β β = )2π β β. Inductively one finds that U λ β, θ) + ) n ϕ has at the most 4 n terms, and according to 3) the largest contribution comes from the product and then e i n 3)2πβ e i2n 2)+2n ))2πβ) = e i n )2πβ) U λ β, θ) +) n ϕ U λ β, θ) +) n ϕ 2 4 n n )2π β β ; since 2n 2 n = n, the result follows Proof of Theorem 2ii). Finally, using this preparatory set of results, we finish the proof of our main result. Let fn) = ln2+ n )) 5. Sequences β m, T m, m will be built inductively, starting with β =, so that i) β m+ β m = 2 κm! for some κ m IN; ii) T m+ 2 m P n f) U λ β, θ) + ) j ϕ 2 ft m) 2 λ [ π 6, π 2 ] and β with β β m m ; iii) β m+ β k < k for k =, 2,..., m. for all θ [, 2π), If i), ii) and iii) are satisfied then we conclude by i) that β = lim β m is irrational, by iii) that β β m m and then by ii) that T m + 2T P n f) m Uλ β, θ) +) j ϕ 2 for θ [, 2π) and λ [ π 6, π 2 ]. So by Lemma 2 ft m ) 2 lim sup X U λ β, θ) +) n ϕ 2 fn)5 n n 2 = for β = β and the result is proved. Then we shall construct β m, T m, m such that i), ii) and iii) hold. Start with β =. Given β,..., β m, T,..., T m and,..., m we shall show how to choose T m, m and β m+. Given β m, let ϕ = η +ψ be the decomposition given by Lemma 6 and let C, C 2 be the corresponding constants. Choose T m 2T m and T 2) so that 3) C 2 3 2πC 2 2fT m ) + T m ) 2 2f ).

21 PURE POINT AND ENERGY INSTABILITY 2 This is possible since C and C 2 are fixed given β m ) and fn). Note that 32) T + in fact P n< T ft ) Uλ β, θ) +) j ψ 2 2π T + # { n : n < T } ψ 2 ; ft ) = = T + T + T + T + P n< T ft ) n< T ft ) n< T ft ) n< T ft ) j= Uλ β, θ) +) j ψ 2 = Uλ β, θ) +) ) j ψ n) 2 Uλ β, θ) +) ) j ψ n) 2 ϕ n, U λ β, θ) +) j ψ 2, then by Lemma 4 T + P n< T ft ) and 32) follows. By Lemma 3 and 32) T m + 2T ψ 2 3 Uλ β, θ) +) j ψ 2 T + P n f) m T m + 2π ψ 2 3 2T n< T ft ) Uλ β m, θ) +) j ϕ 2 P n< f) m { T m + # n : n < T m ft m ) 2π { C 2 3 T m + # n : n < T m ft m ) = C 2 3 { 2πC 2 T m + # n : n < 2π ψ 2, Uλ β m, θ) +) j ψ 2 ) 2 } ψ 2) 2 } C 2 2 ) 2 T m ft m ) }) 2.

22 22 { Since # n : n < ft m) } T m + 2 ft m) + it follows that 2T P n f) m C 2 3 2πC 2 T m + C πC 2 ft m ) + Uλ β m, θ) +) j ϕ 2 2 )) ft m ) + 2 T m ) 2 for θ [, 2π) and λ [ π 6, π 2 ]. Thus by 3), we obtain T m + 2T P n f) m Uλ β m, θ) +) j ϕ 2 2 ft m ) for θ [, 2π) and λ [ π 6, π 2 ]. So, by Lemma 7, for β IR, θ [, 2π) and λ [ π 6, π 2 ] = T m + T m + T m + 2T +P n f) T m + P n f) m 2T P n f) m 2T P n f) m Uλ β, θ) +) j ϕ 2 Uλ β, θ) +) j ϕ ) 2 Uλ β m, θ) +) j ϕ Uλ β, θ) +) j ϕ U λ β m, θ) +) j ϕ ) ) 2 2T m P n f) Uλ β m, θ) +) j ϕ Uλ β, θ) +) j Uλ β m, θ) +) ) ) 2 j ϕ T m + 2T m P n f) 4 j+ 2j 2 j)π β β m ) Uλ β m, θ) +) j ϕ 2 ) 2 2 ft m ) 2T ) 2. 4 j+ 2j 2 j)π β β m ) T m + m

23 Taking PURE POINT AND ENERGY INSTABILITY 23 m = we obtain that, if β β m < m, T m + 2T T m + ft m ) 2T m m 4 j+ 2j 2 j)π P n f) m Finally, pick β m+ rational so that Uλ β, θ) +) j ϕ 2 β n β m+ < n n =,..., m, ft m ) 2. and β m+ = β m + 2 κm! for some κ m IN. This finishes the proof of Theorem 2ii). Remark. For the operator U λ β, θ) := Uβ, θ)i d +e iλ )P ϕ ) on l 2 ZZ) we can similarly prove an analogous result. The proof of dynamical instability for some irrational β is essentially unchanged except for Lemma 6 which is simplified since Uβ, θ) is purely absolutely continuous for β rational. On the other hand, about pure point spectrum, the main difference in this case is that ϕ might not be cyclic, an thus, we don t get pure point spectrum for U λ β, θ) for a.e. θ and λ as obtained on l 2 IN ), but we get that ϕ is in the point spectral subspace corresponding to U λ β, θ) for a.e. θ and λ. References [] Blatter G., Browne D.: Zener Tunneling and Localization in small Conducting Rings. Phys. Rev. B 37, ) [2] Bourget O.: Singular Continuous Floquet Operator for Periodic Quantum Systems. J. Math. Anal. Appl. 3, ) [3] Bourget O., Howland J. S., Joye A.: Spectral Analysis of Unitary Band Matrices. Commun. Math. Phys. 234, ) [4] Combescure M.: Spectral Properties of a Periodically Kicked Quantum Hamiltonian. J. Stat. Phys. 59, ) [5] Cycon H. L., Froese R. G., Kirsch W., Simon B.: Schrödinger Operators. Berlin: Springer-Verlag, 987 [6] de Oliveira C. R., Prado R. A.: Spectral and Localization Properties for the One- Dimensional Bernoulli Discrete Dirac Operator. J. Math. Phys. 46, ) [7] del Rio R., Jitomirskaya S., Last Y., Simon B.: Operators with Singular Continuous Spectrum IV: Hausdorff Dimensions, Rank One Perturbations and Localization. J. d Analyse Math. 69, ) [8] Enss V., Veselic K.: Bound States and Propagating States for Time-Dependent Hamiltonians. Ann. Inst. H. Poincaré Sect. A 39, ) [9] Hamza E., Joye A., Stolz G.: Localization for Random Unitary Operators. Lett. Math. Phys. 75, ) [] Jitomirskaya S., Schulz-Baldes H., Stolz G.: Delocalization in Random Polymer Models. Commun. Math. Phys. 233, ) [] Joye A.: Density of States and Thouless Formula for Random Unitary Band Matrices. Ann. Henri Poincaré 5, )

24 24 [2] Joye A.: Fractional Moment Estimates for Random Unitary Band Matrices. Lett. Math. Phys. 72, ) [3] Kato T.: Perturbation Theory for Linear Operators Second Edition. Berlin: Springer-Verlag, 98 [4] Katznelson Y.: An Introduction to Harmonic Analysis. New York: John Wiley, 968 [5] Reed M., Simon B.: Methods of Modern Mathematical Physics III Scattering Theory. New York: Acad. Press 979 [6] Simon B.: Absence of Ballistic Motion. Commun. Math. Phys. 34, ) [7] Simon B.: Spectral Analysis of rank one Perturbations and Applications. CRM Lecture Notes Vol. 8, ) J. Feldman, R. Froese, L. Rosen, eds.) [8] Simon B.: Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle. J. Comput. Appl. Math. 7, ) [9] Simon B.: Aizenman s Theorem for Orthogonal Polynomials On the Unit Circle. Constr. Approx. 23, ) [2] Simon B., Wolf T.: Singular Continuous Spectrum under rank one Perturbations and Localization for Random Hamiltonians. Commun. Pure Appl. Math. 39, ) Departamento de Matemática UFSCar, São Carlos, SP, Brazil address: oliveira@dm.ufscar.br Departamento de Matemática UFSCar, São Carlos, SP, Brazil address: mariza@dm.ufscar.br