DYNAMICAL LOWER BOUNDS FOR 1D DIRAC OPERATORS. 1. Introduction We consider discrete, resp. continuous, Dirac operators

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1 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA Abstract. Qantm dynamical lower bonds for continos and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then sch reslts are applied to varios models, inclding the Bernolli-Dirac one and, in contrast to the discrete case, critical energies are also fond for the continos Dirac case with positive mass.. Introdction We consider discrete, resp. continos, Dirac operators mc 2 cd Dm, c := D m, c V I 2 = cd mc 2 V I 2, with Dirichlet bondary conditions, acting on l 2 N, C 2, resp. L 2 [,, C 2, where c > represents the speed of light, m the mass of a particle, I 2 is the 2 2 identity matrix and V is a bonded real potential. In the discrete case D is the finite difference operator defined by Dϕn = ϕn ϕn, with adjoint D ϕn = ϕn ϕn, and in the continos case D = D = i d dx. Model in the continos case is well known in relativistic qantm mechanics [, 3], and the discrete version was introdced and stdied in [6, 7]. The goal of this paper is to establish lower bonds on the dynamics associated to Dm, c throgh the behavior of the corresponding transfer matrices. To this end we will consider the time averaged q-th moments A ψ of the position operator [ ] ϕ x ϕ x X x = ϕ x ϕ x acting in l 2 N, C 2, resp. L 2 [,, C 2, defined by T > A ψ m, T, q := 2 e 2t/T X q/2 e itdm,c 2 ψ 2 dt, T 99 Mathematics Sbject Classification. 8Q. addresses: rap@dm.fscar.br RAP, oliveira@dm.fscar.br CRdeO. Fax: RAP was spported by FAPESP Brazil. CRdeO was partially spported by CNPq Brazil.

2 2 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA with initial state ψ = δ in l2 N, C 2, resp. ψ = f in L 2 [,, C 2, where δ is an element of the canonical basis of l2 N, C 2 and f is an element of L 2 [,, C 2 with compact spport which satisfies a sitable technical condition. To investigate the polynomial behavior in time T of A ψ m, T, q, one sally considers the lower growth exponents 3 β log A ψ m, T, q ψ m, q := lim inf T log T In the Schrödinger setting, dynamical lower bonds was fond for random polymer models [] and for random palindrome models [2], de to existence of critical energies []. For discrete Schrödinger operators in l 2 N and l 2 Z, in [5] a general method was developed which allows one to derive dynamical lower bonds from pper bonds on the growth of norms of transfer matrices. Damanik, Lenz and Stolz [4] have presented an extension of this method to continos Schrödinger operators in L 2 [, and L 2 R, with application to the continos Bernolli-Anderson model. In this paper we adapt the above mentioned methods to the Dirac model for both discrete and continos cases. One important conseqence of Theorem ahead is the following: sppose that there is an energy E R sch that the transfer matrices Φ m E, x, y defined in Section 2 satisfies Φ m E, x, y CN α for all N large enogh, α, C > and x, y N, then it follows that. A ψ m, T, q CT q 4α α, for ψ as in 2 and C >. We then apply sch reslt to the continos Bernolli-Dirac model, the discrete Dirac model with zero mass m = and any two-valed potential, the The-Morse Dirac model and discrete Dirac model with Strmian potentials. There are some reasons jstifying the adaptation of known reslts in the Schrödinger setting to the Dirac one. First of all, althogh expected, it is not immediately clear nor trivial which and how sch adaptations work. Second, althogh we have fond the abstract reslts have similar statements, in applications sally different conditions on the potentials appear in case of Dirac operators see, e.g., Theorem 3. Third, and this was or main motivation for considering dynamical lower bonds for model, is that for the continos Bernolli-Dirac model it is possible to constrct examples see Sbsection 3. which have critical energies for m = and also for m >, in contrast with the discrete case which have critical energies only for m = [6, 7]. Forth, with respect to transfer matrices, the discrete Dirac operator has some kind of bilt-in dimerization [7] implying transport which motivates the stdy of the corresponding continos case. Finally, we have fond that the pper and lower components of some initial conditions in the Dirac setting prodce interferences so that the techniqe in the Schrödinger

3 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 3 case does not apply so leaving an interesting open problem; see the remark at the end of Sbsection 3.. We anticipate that the presence of critical energies in continm Bernolli- Dirac models prodces dynamical lower bonds in the sense that almost srely β f m, q q 2, for all q >, for any mass m and sitable initial conditions f. Another method to obtain dynamical lower bonds from pper bonds on transfer matrices was lately developed in [9], with application to Schrödinger operators with random decaying potentials and sparse potentials. Their method is sitable for models that admit pper bonds on transfer matrix norms for large sets of energies i.e., sets with positive Lebesge measre, while with the method sed here based on [4, 5] it is possible to get dynamical bonds for models with large or small e.g., finite sets of sch energies. An approach for qasi-ballistic dynamics for discrete Schrödinger as well Dirac operators with potentials along some dynamical systems have recently been obtained in [8]. This paper is organized as follows: In Section 2 the reslt abot dynamical lower bonds Theorem for the Dirac model is presented, whose proof appears in Section 4. In Section 3 applications of Theorem are discssed, inclding the continos Bernolli-Dirac model. 2. Dynamical Bonds In this section we will present reslts abot dynamical lower bonds for the operators Dm, c defined by in both the discrete and continos cases. For a given operator Dm, c on l 2 N, C 2, resp. L 2 [,, C 2, the transfer matrices Φ m E, x, y between sites y and x are defined as N Φ m E, x, y = x D x N N x D, resp. x D x x N x D, x where N N = and D D = denote the soltions of eqation N Dm, c = E, E R, satisfying N y N = y resp. N y =, D D y D y, D y = =.,

4 4 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA It follows from the definitions that if = eigenvale eqation Dm, c = E, then x y = Φ x m E, x, y y resp. x x y = Φ m E, x, y y is a soltion of the Note that in the discrete case, the matrix Φ m E, x, y, x > y, can be written as with T m E, V k =., Φ m E, x, y = T m E, V x T m E, V y, m2 c 4 E V k 2 c 2 mc 2 E V k c mc 2 E V k c. We denote by δ n ± the elements of the canonical position basis of l 2 N, C 2, for which all entries are except the nth one, which is given by and for the sperscript indices and, respectively. In the continos case, consider the measrable locally bonded vectorvaled fnctions w E, v E defined by w E x = N D x D N x D x N x and v E x = N D x D N x. D x N x g For g =, with g g, g measrable and locally bonded fnctions, f and f = L f 2 [,, C 2 of compact spport, define [g, f] := g t f t g t f t dt. Note that in case all involved fnctions are sqare integrable [, ] coincides with their inner prodct.

5 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 5 For fixed parameters m and c, let H E be the set of the vectors f = f L f 2 [,, C 2 with compact spport, which satisfies one of the following conditions: i f, f = and [, f] = tf tdt for some soltion = of Dm, c = E; ii f =, f and [, f] = tf tdt for some soltion = of Dm, c = E; iii f, f and [w E, f] or [v E, f] or both. For α, m, C > and N > define the set { } P m α, C, N = E R : Φ m E, x, y CN α for all x, y N. Now we are in position to state the main reslt abot dynamical lower bonds. Theorem. Let Dm, c be the operator defined by. Sppose E R is sch that there exist C > and α with E P m α, C, N for all sfficiently large N. i discrete case Let AN be a niformly bonded seqence of sbset of P m α, C, N containing E and µ m the spectral measre for Dm, c associated to δ. Then, there exists C > sch that for T > large enogh A δ m, T, q C B 2 T µ m B T T q 3α α, where B j T, j =, 2, is the j/t neighborhood of AT α. ii continos case Let AN be a sbset of P m α, C, N containing E sch that diaman as N. Then, for every f H E there exists C > sch that for T > large enogh A f m, T, q C B T T q 3α α. Remarks.. Theorem can be adapted to the operator Dm, c on l 2 Z, C 2 and L 2 R, C 2, and always with similar statements. 2. The dynamical lower bonds obtained in Theorem are stable nder sitable power-decaying pertrbations of the potential V as in [5], becase the power-law bonds of the transfer matrices keep nchanged. The proof of Theorem will be given in Section 4. As in [4, 5], Theorem have the following immediate conseqences.

6 6 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA Corollary. Let A be a nonempty bonded sbset of P m α, C, N for some C >, α and for all N large enogh, sch that µ m A >. Then β δ m, q q 3α α. Proof. Take AN = A for every N. Since µ m B T µ m A >, by Theorem i there exists C > sch that for T > large enogh Hence the reslt follows. A δ m, T, q C T q 3α α. Corollary 2. Sppose there is an energy E R sch that Φ m E, x, y CN α for all N large enogh and x, y N. Then, β q 4α ψ m, q α, for every ψ = f H E in the continos case and ψ = δ case. in the discrete Proof. Take AN = {E } for every N. Then B T = [ E T, E T and by Theorem there exists C > sch that for T large enogh A ψ m, T, q C T T q 3α q 4α α = C T for ψ as in the hypothesis. Hence the reslt follows. α, ] 3. Applications This section is devoted to applications of Theorem and its corollaries. 3.. The continos Bernolli-Dirac model. Let g and g be two realvaled potentials with spport in [, ]. Consider the family of Dirac operators in L 2 [,, C 2, 4 D ω m, c := D m, c V ω I 2, ω Ω := {, } N, with potential V ω x = n g ω n x n, where ω n {, } are i.i.d. Bernolli random variables with common probability measre µ satisfying µ{} = p, µ{} = p, for some < p <, and prodct measre P = n µ ω n on Ω. Let T m j E be the transfer matrix for D ω m, c with potential V j x = n g jx n, j =,, at energy E from to.

7 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 7 Definition []. E R is a critical energy for D ω m, c if the matrices T m j E, j =,, are elliptic i.e., trace T m j E < 2 or eqal to ±I 2, and commte. If E is a critical energy for D ω m, c, it follows from Definition that there exists a real invertible matrix Q sch that Q T m j E Q cosηj sinη = j, for j =,. sinη j cosη j Adapting the argments sed in [, 4] for the Bernolli-Dirac model 4, we obtain the following details omitted. Lemma. Assme that η η is not an integer mltiple of π. Let λ > be arbitrary. Then there are b > and C < sch that for every N N, there exists a set Ω N λ Ω with P Ω N λ Ce bn λ and Φ ω me, x, y C for all ω Ω\Ω N λ, x, y N and E [E N λ /2, E N λ /2 ]. We can now state or main reslt for model 4. Theorem 2. Assme that η η is not an integer mltiple of π. For every f H E one has β f m, q q 2, ω P a.s.. Proof. De to Lemma, for each λ >, P Ω N λ is smmable over N. Ths, by Lemma and a Borel-Cantelli argment, there exists < C < sch that Φ ω me, x, y C for all N, x, y N, for almost every ω and E AN := [E N λ /2, E N λ /2 ]. Note that B T AT = 2T λ /2. Applying Theorem ii with α =, it follows that almost srely β f m, q q 2 λ for every f H E. Taking λ = n and sing a contable intersection of fll measre sets, we obtain the reslt. It is possible to show, by applying similar argments of [4, ] for model 4, that if E is a critical energy for D ω m, c, then for every f H E one has β f m, q q, for every ω. Recently, we have established see [7] the same lower bonds obtained above for the discrete Bernolli-Dirac model with zero mass m =, de to existence of critical energies. Now we will present a continos Bernolli- Dirac model defined by 4 that have critical energies for both m = and m > note that for the latter case critical energies are absent in the discrete case. As a conseqence we will obtain lower bonds by Theorem 2.

8 8 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA In fact, consider the Bernolli-Dirac model 4 with g = and g = λχ [,], λ >. By solving the eqation D m, c = E one finds the following soltions for E 2 > m 2 c 4 : N N = N and D D = D, with N x = cosξ E x, N x = imc2 E cξ E sinξ E x, D x = icξ E mc 2 E sinξ E x, D x = cosξ E x, E where ξ E = 2 m 2 c 4, and they satisfy c N = and D =. Ths, the transfer matrices are cos ξ E T m E = ic ξ E mc 2 E sin ξ E imc 2 E c ξ E sin ξ E cos ξ E for E 2 > m 2 c 4 and T m E = T m E λ for E λ 2 > m 2 c 4. If E = ± m 2 c 4 n 2 π 2 c 2 for n N and m, then T m E = ±I 2. Moreover, taking < λ < m 2 c 4 n 2 π 2 c 2 mc 2 or λ > m 2 c 4 n 2 π 2 c 2 mc 2 this implies E λ 2 > m 2 c 4, it follows that trace T m E < 2 i.e., T m E is elliptic. On the other hand, if E = λ ± m 2 c 4 n 2 π 2 c 2 for n N, m and λ as above, we have T m E = ±I 2 and trace T m E < 2. Ths, for sch vales of λ we have the following set of critical energies: { ± m 2 c 4 n 2 π 2 c 2, λ ± } m 2 c 4 n 2 π 2 c 2 : n N, m. For sch energies the condition reqired in Theorem 2 holds, that is, η η kπ, k Z. Corollary 3. Let D ω m, c be defined by 4 with g = and g = λχ [,], λ >. If λ < m 2 c 4 n 2 π 2 c 2 mc 2 or λ > m 2 c 4 n 2 π 2 c 2 mc 2, then β f m, q q 2, ω P a.s., f for all masses m and any f = L f 2 [, ], C 2 satisfying one of the following conditions: i f L 2 [, ] and f =.

9 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 9 ii f = and f L 2 [, ]. iii f, f and [ ] inπc [w E, f] = mc 2 f x sinnπx f x cosnπx dx m 2 c 4 n 2 π 2 c 2 or [ mc 2 ] m [v E, f] = f x cosnπx i 2 c 4 n 2 π 2 c 2 f x sinnπx dx. nπc Note that in this case the above conditions on f depends on m. Proof. We consider two cases:. ω =, that is, V ω x = on [, ]. 2. ω =, that is, V ω x = λ on [, ]. If ω =, then applying Theorem 2 for the critical energies we obtain E = ± m 2 c 4 n 2 π 2 c 2, n N, m, β f m, q q 2, ω P a.s., for all mass vales m and for any f H E with spp f [, ]. Note that for sch energies cosnπx N x = i mc2 m 2 c 4 n 2 π 2 c 2 sinnπx nπc and D x = inπc mc 2 m 2 c 4 n 2 π 2 c 2 sinnπx cosnπx are fndamental soltions of D m, c = E. By definition we have the D vectors w E x = x N D and v E x = x x N. x For any f L 2 [, ], f, there is at least one n N sch that f t cosnπxdt or f t sinnπxdt similarly for f L 2 [, ]. This is valid becase {} {cos2kπx, sin2kπx : k N} form a basis of L 2 [, ]. Therefore, by sing the definition of the set H E the reqired reslt is obtained. If ω =, then we conclde the reslt in the same way, bt now based on the critical energies E = λ ± m 2 c 4 n 2 π 2 c 2, n N and m.

10 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA Remark. Note that Corollary 3iii does not assre β f m, q q 2 for f any f = L f 2 [, ], C 2, de to some kind of qantm interference. For instance, for any integer n, ñ, by taking f x = mc2 m 2 c 4 n 2 π 2 c 2 sinñπx and f x = cosñπx, inπc one obtains [w E, f] = and [v E, f] =. In the corresponding Schrödinger model [4] one has β f q q 2 for any f L2 [, ], f The discrete massless Dirac model with two-valed potentials. Consider the discrete Dirac operator D, c defined by. The following reslt holds. Theorem 3. Let V : N {a, b} R be a potential for D, c. i If a b < 2c, then for every q >, β, q q. δ ii If a b = 2c, then for every q >, β, q q 5 δ 2. Proof. We shall find pper bonds for the transfer matrices Φ E, x, y for a sitable energy E. Let E = a. Then a b2 a b c 2 c T E, a = I 2 and T E, b = a b c. This implies that Φ E, x, y = T E, b n b, where n b is the nmber of times that b occrs in the prodct. If a b < 2c, then T E, b is elliptic trace T E, b < 2 and hence Φ E, x, y CE, x, y N. Ths, by Corollary 2 with α =, we obtain β δ, q q, q >. On the other hand, if a b = 2c, then T E, b is parabolic trace T E, b = d 2 and hence T E, b can be written as with d. Becase nb d = nb d C d n b, it follows that Φ E, x, y CE n b CE x y, x, y N.

11 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS Therefore, by Corollary 2 with α =, we obtain β δ, q q 5 2, q > The The-Morse Dirac model. This model is defined as in by D ω m, c := D m, c V ω I 2, acting on l 2 N, C 2 or L 2 [,, C 2, where V ω is generated by the The- Morse sbstittion on the alphabet {a, b} given by Sa = ab, Sb = ba. For more details see [4, 5]. Let Ω TM be the associated sbshift. Since the bondedness of the transfer matrices in this case depends only on the strctre of the potential and it is independent on the explicit form of these matrices, by adapting a similar model [4, 5] in the Schrödinger setting we obtain the following reslt details omitted. Lemma 2. There are E R and C > sch that for every ω Ω TM and every m, Φ ω me, x, y C, x, y N or x, y [,. Ths, by Corollary 2 with α =, it follows that β ω,ψ m, q q, for every ω Ω TM, q >, m and for every ψ = f H E in the continos case and ψ = δ in the discrete case. This shold be compared with Theorem The discrete Dirac model with Strmian Potentials. We discss dynamical lower bonds for the model D λ,ω,θ m, c := D m, c V λ,ω,θ I 2 defined by on l 2 N, C 2, whose potential is given by V λ,ω,θ x = λχ [ ω, xω θ mod, where λ is the copling constant, ω, irrational is the rotation nmber and θ [, is the phase. For more details on this potential in the corresponding Schrödinger case see [3, ]. Since the bondedness of the transfer matrices in this case depends only on the strctre of the potential, again a direct adaptation of reslts in the Schrödinger setting shows that

12 2 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA Lemma 3. Sppose ω is a nmber of bonded density. For every λ, there are a constant C > and α = αλ, ω > sch that for every θ and every E σd λ,ω,θ we have for every x, y N and any m. Φ ω m,λ,θ E, x, y C x y α, Therefore, by Corollary with A = σd λ,ω,θ so µ m A =, it is fond that for every λ, θ, the operator D λ,ω,θ satisfies β m, q q 3α δ α, for every q > and any m. 4. Proof of Dynamical Bonds In this section the proof of Theorem will be presented. We first gather some preliminary reslts that we will sed in the proof. For the operator Dm, c, m, on l 2 N, C 2, we introdce the twocomponents Green s fnction so that 5 G m z, n G mz, n δ n, Dm, c z δ = δn, Dm, c z δ G Dm, c z m z, n G mz, n = δ n. By sing transfer matrices, one obtains for n, G m z, n G G = Φ mz, n m z, n, m z, 6 G mz,, z C\R,. Lemma 4. Let Dm, c be the operator. For z = E i/t T > and m, one has i A δ m, T, q = n q G πt m z, n 2 G mz, n 2 de, in the discrete case and ii A f m, T, q = πt n N x q R R Dm, c z fx for every f L 2 [,, C 2, in the continos case. 2 de dx, Proof. The identity i follows by Lemma 3.2 in [2] adapted for the operator Dm, c on l 2 N, C 2, and the identity ii follows by Lemma 2.3 in [4] applied to Dm, c in L 2 [,, C 2.

13 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 3 Lemma 5. Let E R, N >, m and consider L m N := Φm E, x, y. sp x,y N Then, there is < C < sch that for every δ C and x, y N, one has Φm E δ, x, y [ ] δ δ Lm N exp c c C L m N x y. Proof. We consider the discrete case with x, y N, x > y the continos case is similar. An indctive argment shows that, for δ C and m, we can write the identity x Φ m E δ, x, y = Φ m E, x, y δ Φ m E δ, x, j B δ E, j Φ m E, j, y, with B δ E, j = δ c 2 j=y 2 c E V j c By iteration, sing the hypothesis and the above identity, we obtain Φ m E δ, x, y [ L m N δ ] δ x y c c C L m N [ ] δ δ L m N exp c c C L m Nx y, for some < C < and for y < x N.. The following reslt will be important for the proof of Theorem in the continos case; it is based on Lemmas 2.6 and 2.7 of [4]. Lemma 6. Let Dm, c be the operator defined by on L 2 [,, C 2. For z C\R, define m f,z = Dm, c z f. Sppose E R and f = f L f 2 [,, C 2 with spp f [, s] are sch that 7 Then f / H E. lim inf δ { m f,z s : z C, z E δ } =. Proof. By 7 there exists a seqence z n C with z n E and m f,z n s for n. Since m f,z n = for all n and by continity, the inhomogeneos eqation 8 Dm, c E = f f

14 4 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA v has a soltion v = with v = vs =. v Let Y t be the fndamental matrix of the homogeneos eqation at x = s, i.e., Y t = vn t v D t, v N t v D t v where v N N = v v N and v D D = v D are soltions of the homogeneos eqation which satisfy v N s = and v D s =. By writing eqation 8 as x = x i c i c mc2 V x E i c mc2 V x E f x, f x we have the variation of parameters formla v x x = Y x Y t v x i c s f t f t dt. x x Replacing Y t in the above eqation and considering x =, we obtain 9 = v = i c [w E, f] and = v = i c [v E, f], where and f = and Now, w E t = v N vd t v D v D vn t, t v N t v E t = v N vd t v D v D vn t t v N t, with f f, f. f t := v N vd t v D v D vn t t v N t 2 t := v N vd t v D v D vn t t v N t

15 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 5 are soltions of eqation Dm, c and 2 = = E satisfying =. Ths,, 2 form a fndamental system of soltions of Dm, c = E and it follows from 9 that [ ] [ ] i f, = = i,, i =, 2. f f Therefore, if f =, f, resp. f = s for every soltion = f / H E. tf tdt =, resp. s f, f, one has tf tdt =, of Dm, c = E. Hence, we conclde that Proof. Theorem i By Lemma 4, we have for T >, A δ m, T, q = n q G πt m E i/t, n 2 G me i/t, n 2 de. n N R Define NT := T α. By hypothesis, L m NT := sp Φ m E, n, k C NT α, E ANT. n,k NT By Lemma 5, we obtain for every E B 2 T and n NT, Φ m E i/t, n, B NT α, with B = C e 3 c 3 c CC. For every E B 2 T and T sfficiently large, it follows from 6 and the above estimate that G m E i/t, n 2 G me i/t, n 2 Observe that n NT 2 NT n= NT 2 G m E i/t, n 2 G me i/t, n 2 B 2 4 NT 2α G me i/t, 2 2 G me i/t, 2 G me i/t, 2 G me i/t, 2. G me i/t, = δ, Dm, c E i/t δ = F m E i/t,

16 6 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA where F m z is the Borel transform of the spectral measre corresponding to the pair Dm, c, δ. Using eqation 5 one shows that G me i/t, 2 2 G me i/t, 2 G me i/t, 2 a > for some niform constant a. Therefore, it follows from that for T sfficiently large, G πt m E i/t, n 2 G me i/t, n 2 de R n NT 2 B NT 2α T B NT 2α T B 2 T B 2 T Im 2 F m E i/t de 2 ImF me i/t for some constant B >. In the last step it was sed that Im 2 F m z 2 ImF m z. For any set S R, denote by S ɛ the ɛ-neighborhood of S. It was shown in [5, 2] that S ɛ I mf m E i/t de π 2 µm S. de, Ths, taking S = B T we conclde that for T large enogh, A δ πt m, T, q NT q 2 C NT q 2α T C T q 3α α R n NT 2 B 2 T B2 T µ m B T. G m E i/t, n 2 G me i/t, n 2 de ImF m E i/t de ii As in Lemma 6 we write m f,z = Dm, c z f. Let s > with spp f [, s] and define NT := T α. By Lemma 4, we have for T >, A f m, T, q = x q m f,ei/t πt x 2 de dx R n n q m f,ei/t 2πT x 2 de dx n=s n R n n q m f,ei/t 2πT x 2 dx de. n=s B T n

17 DYNAMICAL LOWER BOUNDS FOR D DIRAC OPERATORS 7 Using the fact that m f,ei/t is a soltion of Dm, c = E i T on [n, n ] and the transfer matrices satisfy Φ m = Φ m, we obtain from that A f m, T, q n q 2πT n=s B T n n Φ m E i/t, x, s 2 m f,ei/t s 2 dx de. By hypothesis and Lemma 5, it follows that for T large enogh, A f m, T, q NT NT q C NT 2α m f,ei/t πt 2 s 2 de B T πt n= NT 2 NT q B T C NT 2α inf 2 distz,b T T m f,z s 2, for some constant C >. For every f H E with spp f [, s], Lemma 6 implies that there exists κ > and δ > satisfying inf { m f,z s 2 : z C, z E δ } κ. By hypothesis, diaman as N and E AN for all N. Hence, { inf m f,z s 2 : distz, B T } κ > T for T sfficiently large. Therefore, for T large enogh we obtain A f m, T, q C T NT q 2α B T = C T q 3α α B T. The proof is complete. References [] Bjorken, S. D., Drell, J. D.: Relativistic qantm mechanics. McGraw-Hill, New York 965 [2] Carvalho, T. O., de Oliveira, C. R.: Critical energies in random palindrome models. J. Math. Phys. 44, [3] Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional qasicrystals, II. The Lyapnov exponent. Lett. Math. Phys. 5, [4] Damanik, D., Lenz, D., Stolz, G.: Lower transport bonds for one-dimensional continm Schrödinger operators. Math. Ann. 336, [5] Damanik, D., Sütő, A., Tcheremchantsev, S.: Power-law bonds on transfer matrices and qantm dynamics in one dimension II. J. Fnct. Anal. 26, [6] de Oliveira, C. R., Prado, R. A.: Dynamical delocalization for the D Bernolli discrete Dirac operator. J. Phys. A: Math. Gen. 38, L5 L9 25

18 8 ROBERTO A. PRADO AND CÉSAR R. DE OLIVEIRA [7] de Oliveira, C. R., Prado, R. A.: Spectral and localization properties for the onedimensional Bernolli discrete Dirac operator. J. Math. Phys. 46, pp 25 [8] de Oliveira, C. R., Prado, R. A.: Qantm Hamiltonians with qasi-ballistic dynamics and point spectrm. J. Differential Eqations 235, [9] Germinet, F., Kiselev, A., Tcheremchantsev, S.: Transfer matrices and transport for D Schrödinger operators. Ann. Inst. Forier 54, [] Iochm, B., Raymond, L., Testard, D.: Resistance of one-dimensional qasicrystals. Physica A 87, [] Jitomirskaya, S., Schlz-Baldes, H., Stolz, G.: Delocalization in random polymer models. Commn. Math. Phys. 233, [2] Killip, R., Kiselev, A., Last, Y.: Dynamical pper bonds on wavepacket spreading. Am. J. Math. 25, [3] Thaller, B.: The Dirac eqation. Springer-Verlag, Berlin 99 Departamento de Matemática UFSCar, São Carlos, SP, Brazil, address: rap@dm.fscar.br Departamento de Matemática UFSCar, São Carlos, SP, Brazil address: oliveira@dm.fscar.br