Expectation Values of Observables in Time ... - Semantic Scholar

Centre de Physique Theorique, CNRS Luminy, Case 907 F-13288 Marseille { Cedex 9

Expectation Values of Observables in Time-Dependent Quantum Mechanics J.M. Barbaroux and A. Joye , 1

1

Abstract

Let U (t) be the evolution operator of the Schrodinger equation generated by an hamiltonian of the form H0 (t) + W (t) where H0 (t) commutes for all t with a complete set of time independent projectors fPj g1 j =1 . Consider the P observable A = j Pj j , where j ' j  ,  > 0 for j large. Assuming that the "matrix elements" of W (t) behave as kPj W (t)Pk k ' 1=jj ? kjp , j 6= k for p > 0 large enough, we prove estimates on the expectation value hU (t)'jAU (t)'i  hAi' (t) for large times of the type hAi' (t)  ct , where  > 0 depends on p and . Typical applications concern the energy expectation hH0 i' (t) in case H0(t)  H0 or the expectation of the position operator hx2 i'(t) on the lattice where W (t) is the discrete laplacian or a variant of it and H0 (t) is a time dependent multiplicative potential.

Key-Words: Time-dependent hamiltonians, Schrodinger operator, quantum stability, quantum dynamics. May 1997 CPT-97/P.3479 anonymous ftp or gopher: cpt.univ-mrs.fr  1

Unite Propre de Recherche 7061 and PHYMAT, Universite de Toulon et du Var, B.P.132, F-83957 La Garde Cedex, France

1 Introduction This paper is concerned with estimates on the long time behaviour of expectation values of certain quantum observables. Such estimates pertain to the study of quantum stability and quantum diusion as well. Indeed, although our main result is quite general, the two paradigms we have in mind are estimates on the energy expectation for general time-dependent driven systems and estimates on the moments of the position operator for time-dependent driven systems de ned on the lattice. For more details on the origin of the problem, on quantum stability in general and on related results, the reader should consult [2], [14], [13], [5], [18], [17] and [15], for example. Let us start by describing our typical results for these cases. The rst type of systems are characterized by the self-adjoint hamiltonian H (t) = H + W (t) on some Hilbert space H which generates an evolution U (t), U (0) = 11 and the energy expectation is de ned by hH i'(t) = hU (t)'jH U (t)'i; where ' 2 H. Let us assume here for simplicity that the spectrum of the timeindependant part H is discrete (H ) = fj gj2N and that the eigenvalues behave according to j ' j , as j ! 1, for some  > 0. We denote by Pj the corresponding spectral projectors. We further assume that the perturbation W (t) is bounded with matrix elements characterized by kPj W (t)Pk k  w=jj ? kjp as jj ? kj ! 1; (1.1) for some time-independant constant w and for some p > 0 large enough. We show that there exists a diusion exponent  > 0 depending on  and p such that hH i'(t)  ct (1.2) for some nite constant c and for any initial condition ' belonging to some dense domain in H. More precisely, there exist real valued functions P () and (p) (see proposition 4.1 ) such that provided p > P () and 8" > 0,  = =(1 ? (p)) + "; where (p) = O(p? ) as p ! 1; (1.3) and P () is essentially linear in . In particular, when   1, p > 2 is enough. This estimate holds for any  > 0 if p large enough and regardless of the time dependence of W (t). Similar results were rst obtained for general systems by Nenciu in [17]. He made use of recently developped tools in quantum adiabatic theory which require increasing gaps between successive eigenvalues, that is  > 1, and dierentiability with respect to time of the perturbation W (t). However, nothing is required on the behaviour of the matrix elements of the perturbation. The diusion exponent obtained in [17] is of the form  =  ()=n, where n 2 N , the order of dierentiability of W (t), is large enough. It was then showed by Joye in [15] that estimates of the type (1.2) can be obtained for any  > 0 and without dierentiability of W (t), provided the coupling induced by W (t) with high energy levels is weak enough. Typically, if 0

0

0

0

0

0

2

1

0

1

kPj W (t)k ' j ? with 2 > 1+ , then  = =( ? 1=2). In terms of matrix elements, this weak coupling condition means that kPj W (t)Pk k ! 0 as j + k ! 1. The 2

2

present results complete those of [15] to the physically important situation where the strength of the couplings between neighbouring levels is independent of the energy. This is caracterized by condition (1.1) requiring decay of the matrix elements kPj W (t)Pk k to zero as jj ? kj ! 1 only. As a result, the diusion exponent (1.3) is bounded below by  > 0, uniformly in p, whereas the exponents obtained in [17] and [15] can be made arbitrarily small by adjusting the parameters of the perturbation W (t). Note however that our bounds are optimal in some cases, as discussed below. Finally, in the speci c case where H + W (t) is a time-dependent forced harmonic oscillator, the quantum dynamics can be solved explicitly, so that the asymptotic behavior of hH i'(t) can sometimes be derived. See for example [11], [3] and [4]. However, the perturbation W (t) is unbounded in this situation. Consider now the Hilbert space l (Zd) whose vectors we denote by u = fu(n)gn2Zd. Let H (t) be a time-dependent real-valued multiplicative operator, possibly unbounded, (potential) such that (HP (t)u)(n) = H (n; t)u(n), 8n 2 Zd. Let W be the discrete laplacian (Wu)(n) = j2Zd; jj?nj u(j ), 8nP2 N , and for xed m > 0, let jxjm be the moment of order m de ned by jxjm = n2Zd jnjmePn, where Pn is the projector on the site n and j je is the euclidian norm. Under some regularity assumptions, the time-dependent Schrodinger operator H (t) + W gives rise to an evolution U (t), with U (0) = 11. We show that for any m  0, the expectation value of jxjm, 0

0

2

0

0

0

=1

0

hjxjmi'(t) = hU (t)'j jxjm U (t)'i satis es the estimate hjxjmi'(t)  ctm ; (1.4) for some nite constant c and for any ' in the dense domain D(jxjm) 2 l (Z). Since this estimate holds in the free case as well, (1.4) is optimal. As a corollary, we note that it is impossible to accelerate a particle on the lattice. Such results are known in the time-independant case, see e.g. [19], [20]. Moreover, lower bounds are proven as well in case the potential is constant or periodic in time [9], [10], [1], [16]. However, such estimates are new for general time-dependent potentials. We also consider similar estimates when the discrete laplacian is replaced by a long range interaction W of the form X Wu(n) = !(n ? k)u(k); 8n 2 Z; (1.5) 2

k2Z

where ! is real valued and belongs to l (Z), we call long range laplacian. This allows comparison with a model of AC Stark eect studied recently by De Bievre and Forni [7] (see also Gallavotti [8], Bellissard [2]), for which they prove lower bounds on the expectation of jxj . Moreover, for some speci c choices of potential H (t), we are able to prove lower bounds on hjxj i'(t), some of which grow arbitrarily fast in time, although W de ned in (1.5) is bounded. 1

2

0

2

2

We actually deal with both situations in a uni ed manner as follows. We consider a self-adjoint hamiltonian H (t) = H (t) + W (t) such that H (t) commutes with a . We estimate the complete set of time independent orthogonal projectors fPj gj2NP expectation value of the positive observable A of the form A = j2N j Pj , where fj gj2N is a positive stricly increasing sequence such that j ! 1 as j ! 1. In order to estimate hAi'(t) = hU (t)'jAU (t)'i, the idea is to compare the evolution U (t) generated by H (t) + W (t) with the one generated by H (t) + Wqd(t), where Wqd(t) is obtained from W (t) by replacing by zeros all matrix elements kPj W (t)Pk k such that jj ? kj > q(j + k), where q : R ! R is a real valued (typically increasing) function. Wqd (t) can be considered as a generalized band matrix, whose width increases as we move along the main diagonal. This new evolution, we call Vq (t), diers from U (t) by a term of order Wqo(t) = W (t) ? Wqd(t), by Duhamel's formula. Provided kA Wqo(t)A? = k is uniformly bounded in time for some  1=2, hAi'(t) can be essentially estimated by hVq (t)'jAVq (t)'i. The control of this expectation value with respect to the evolution Vq (t), generated by the troncated perturbation Wq (t) is the main technical point of the paper. We can get ecient estimates on this expectation value provided we can show that the quantity X S (N; t) = j sup kPj Vq (t)Vq? (s)A? k  21 ; (1.6) st j>N 0

0

0

0

+

+

1 2

1

2

0

decays to zero as N ! 1, with a rate we control, as explained in the next section. By inserting the characteristics of the matrix elements of A and W (t) roughly described above, and by making the choice q(x) = x ,  2]0; 1[, we can achieve sucient control on S (N; t), by appropriate choice of the parameters  and . Note that we have a con ict between the parameters and , i.e. between and q. On the one hand, we wish to take q and as large as possible to make Vq closer and closer to U (t) and to take advantage of the decay induced by A? in (1.6). On the other hand, A WqoA? = is more and more likely to be bounded when is small and Wqo is sparser and sparser, i.e. q is small. In case W (t) = Wq (t) is a genuine band matrix, as is the case with the second application we deal with in section 5, this procedure is much easier. Actually, we found this example quite instructive and inspiring for the proof of the general case. 1 2

2 General Strategy

Let H (t) = H (t) + W (t), t 2 R be the generator of the Schrodinger equation de ned on a separable Hilbert space H. 0

+

Hypothesis H0:

H (t) is self-adjoint on a dense domain D for any t and W (t) is symmetric and relatively bounded with respect to H (t), with relative bound a < 1. There exists a set of t-independent complete orthogonal spectral projectors fPj g1 j 0

0

=1

such that

[H (t); Pj ]  0; 8j 2 N  ; t 2 R : 0

(2.7)

+

3

Let q : R ! R . We de ne X Wqd(t) = Pj W (t)Pk and Wqo(t) = W (t) ? Wqd(t); +

+

jj ?kjq(j +k)

using a strong sum. Note P that with our notations, W d(t) corresponds to the diagonal operator W d(t) = j Pj W (t)Pj . 0

Hypothesis H1: sup kWqd(t) ? W d (t)k < 1: 0

0

t2R+

We assume that the following evolution equations (where 0  dtd )

iU 0 (t)' = (H (t) + W (t))U (t)'; U (0) = 11 iVq0(t)' = (H (t) + Wqd(t))Vq (t)'; Vq (0) = 11 iV 0(t)' = (H (t) + W d(t))V (t)'; V (0) = 11; ' 2 D; 0

0

0

0

0

0

0

give rise to unitary operators which, together with there inverse, map D into D. Let j > 0, 8j 2 N  and let

A=

X j

j Pj de ned on D(A) = f' 2 Hj

X j

j kPj 'k < 1g 2

2

be the positive observable the expectation value of which we will consider.

Hypothesis H2:

The eigenvalues of A are positive and form a strictly increasing sequence

j > j > 0; 8j 2 N  ;

(2.8) and there exists two constants 0 < L?  L < 1 and  > 0 such that for any j large enough, 0 < L? j   j  L j  : (2.9) Let us describe the general strategy we adopt. Following [17, 15], we get from Duhamel's formula (in the strong sense), +1

+

+

Zt U (t) = Vq (t) ? iVq (t) Vq? (s)Wqo(s)U (s)ds  Vq (t) + Rq (t) where kRq (t)k  2. Let ' 2 D(A ) with  1=2 and consider hAi'(t)  hU (t)'jAU (t)'i. We have, due to the positivity of A hAi'(t)  2 (hVq (t)'jAVq (t)'i + hRq (t)'jARq (t)'i) : (2.10) 1

0

4

Then, with the notation Vq (t; s) = Vq (t)Vq? (s), kA = Rq (t)'k 1

1 2

2

X

Z t o  4N k'k + j Pj Vq (t; s)Wq (s)U (s)'ds

j>N ! X  4N k'k + t j sup kPj Vq (t; s)A? kkA Wqo(s)U (s)'k :

(2.11)

2

2

0

2

2

2

st

j>N

0

We work under the following hypothesis.

Hypothesis H3: There exists  1=2 such that sup kA Wqo(t)A? = k  kA WqoA? = k < 1: 1 2

1 2

t2R+

Under H3, we get from (2.10) and (2.11) hAi'(t)  2kA = Vq (t)'k + 8N k'k X +2kA WqoA? = k t sup kA = U (s)'k j sup kPj Vq (t; s)A? k 1 2

2

2

1 2

2

2

1 2

st

2

2

st

j>N

0

0

Hence, noting that kA = U (s)'k = hAi'(s) and writing hAi'(t) = sup st hAi'(s), we obtain hAi'(t)  2 sup kA = Vq (t)'k + 8N k'k (2.12) 1 2

st

2

0

1 2

2

2

0

+ 2kA WqoA? = k t hAi'(t) 1 2

2

2

X

j sup kPj Vq (t; s)A? k :

j>N 0st hAi'(t) provided

We can estimate the large t behaviour of behaviour of X S (N; t) = j sup kPj Vq (t; s)A? k : j>N = Vq (t)'k2

2

we control the large N

2

st

0

Indeed, kA is related to S (N; t) by the estimates (Vq (t) is unitary) X = kA Vq (t)'k  N k'k + j kPj Vq (t)'k 1 2

1 2

2

2

2

j>N

 N k'k + S (N; t)kA 'k : (2.13) Thus, provided we can show for some  1=2 that S (N; t) ! 0 as N ! 1, we get 2

2

from (2.12) and (2.13) under hypothesis H3   hAi'(t)  10N k'k + 2S (N; t) kA 'k + kA WqoA? = k t hAi'(t) : 2

2

1 2

2

2

Hence we require N (t) to be such that 2kA WqoA? = k t S (N (t); t)  K < 1 as t ! 1, so that we can write by means of a bootstrap argument hAi'(t)  hAi'(t)  c(N t k'k + S (N (t); t)kA 'k )  cN t (k'k + kA 'k ) ; for some nite constant c. Remark that the function q is not speci ed yet and that we don't make use of (2.9) at this point. 1 2

( )

( )

2

2

2

2

5

2

2

3 Main Result

In order to give explicit estimates on hAi'(t), we specify a little bit more our concern. Proposition 3.1 Assume H0, H1, H2 and H3 and let q be de ned by q:R !R (3.14) x 7! x for some  2]0; 1[: If (2 ? 1) > 1 + , there exists nite positive constants b (see (6.31)) and c (see (6.34)) such that if +

+

0

N (t) = [bt ? ] + 1; ( [ : ] being the integer part) ; we get for all t large enough, ? ? X S (N (t); t) = j sup kPj Vq (t; s)A? k  c t ? : (1

1

)2

st

j>N (t)

2

2

0

0

(2

(1

)2

1)

(3.15)

Corollary 3.1 Under the same hypotheses as above, and provided (2 ?1) > 1+, we get from (2.13)

sup kA = Vq (t; s)A? k  c (t st

1 2

2

1

0

?(2 ?1) (1?)2

2

+t



?)2 )

(1

for some nite constant c1 . Our results on hAi'(t) follow directly from the above considerations. Theorem 3.1 Assume the hypotheses of proposition If (2 ? 1) > 2 + 2(1 ? )2 , we get for any ' 2 D(A ):

hAi'(t)  c t 2



?)2

(1





kA 'k + k'k ; 2

2

for some nite constant c2 . The proofs of the above results are given in section 6

4 Applications In order to apply the above results, we need to determine which perturbations W (t) satisfy the hypotheses H3. Let us consider the typical situation where the perturbation W (t) is characterized by its matrix elements in the large Pj W (t)Pk in the xed subspaces Pj H, j 2 N  .

Hypothesis H4:

There exist positive constants w0 and p independent of t such that kPj W (t)Pk k  w0 p

jk ? j j

(4.16)

for jk ? j j large enough. The constraint on W (t) expressed through H3 can be expressed in terms of p above and  using estimates on kA WqoA?1=2 k.

6

Lemma 4.1 Under H2 and H4, for 0 <  < 1, ; p > 0,  1=2, one has  2p ? 2  > 1 kA WqoA? = k < 1 if  + (2 (4.17) p ? 1) ? 2  > 1 1 2

and

 kA WqoA? = k < 1 if =2 +p ?(p ?>1)1  Proof: Under assumption H4, one gets:

(4.18)

1 2

kA Wq A? k  ! 0

1 2

and

2

2 0

X

X

X  j (j ) j  jj ? k1j pk  ! 2 0

2

j 2N jk?j j(k+j )

2

2

j 2N

X

X 1 1 + p  p  kj j  jj ? kj k kj ?j  jj ? kj k By usual estimates (see e.g. [12], [15]), one obtains if 2p +  > 1: j + log j ) (j )  c( j p 1? + j1p + j p1  + log p j j  p?  for some nite constant c, which implies (4.17). To prove (4.18), one estimates the Schur norm of kA Wq A? k: (j ) 

2

+2

2 +

1

2

2

1 2

0

2

+

2

(2

n

1)+

kA Wq A? k  ! max sup j 0(j ); sup j ?  00 (j ) 0

where

1 2

0

j

j

2

o

X

1 p  jk?j j k j  jk ? j j k X k  00(j ) = p jk?j j k j  jk ? j j By using similar estimates as in [12], [15], one has, for some nite constant c, and if p ?  ? 1 > 0:  j+ 1 ; 0(j )  c p 1 ? + j1p + p1  + log j p j  p?  j j and    00(j )  c j p1?  + j p?1 ? + j j p? : which gives (4.18). 2 Gathering the conditions in the above lemma together with the bootstrap condition (2 ? 1) > 2 + 2(1 ? ) stated in Theorem 3.1, we get

0(j ) =

2

( + )

( + )

+2

1

+2

1

(

(

1)

2

7

1)+ 2

Proposition 4.1 Under the hypotheses H0, H1, H2 and H4, there exists > 1=2

such that







k'k + kA 'k ; for any parameter  2]0; 1[ and p < 1 such that p 8 < p > 2 and  > (p)  p ? p ? ; if   1 : p > 3 +  and  > (p)  p?p p ? ; if  > 1 hAi'(t)  c t 4

?)2

(1

2

2

2 +3

(2 +3)2 4

1+

where

2

(

40

(1+ )2

8

2

> > +  + ? ; if   1 = +  p? if  > 1

p?1)+?1 2

(2

Remarks:

1 2

1 2 (

1

(1

)2

1)

For  > 1, we have used (4.18). In this case, the values of p and  are not optimal and it is sometimes possible to improve them by doing more technical calculations using (4.17) and (4.18). If   1, which is the situation where the analysis in [17] does not apply, the conditions on p and  are optimal according to (4.17). In particular, one can take p close to 2 provided  is close to 1 and we can take  close to zero, provided p is large enough: (p) ! 1 when p & 2 and (p) = O(p? ) when p ! 1. Note also that in the limit p ! 1, W (t) becomes a band matrix. For such matrices, we derive the estimate hAi'  c t in section 5 below, which is known to be optimal in certain cases (see e.g. [1], [6]). It should be clear from the above computations that our results can be easily extended in order to accomodate more general asymptotic behaviours than these considered in (4.16) 1

5 Examples The above result is quite general since it holds under very weak assumptions on the behaviour of the matrix elements of W (t). We can get sharper estimates if we know more about the structure of the perturbation W (t), as can be seen on the following examples. Actually, the rst example also shows that there are cases where hAi'(t) can grow arbitrarily fast in time, eventhough W (t) is uniformly bounded.

5.1 Long range laplacian

Let H = l (Z), H (t) be the time-dependent multiplication operator (H (t)u)(n) = a(t)nu(n); 8n 2 Z; where a : R ! R and W (t) is a long range laplacian in l (Z) de ned as in (1.5). X (W (t)u)(n) = !(n ? k; t)u(k); 8n 2 Z; 2

0

0

+

2

k2Z

8

where ! : Z  R ! R is real valued, such that !(m; t) = !(?m; t) for all t  0 and belongs to l (Z) uniformly in t 2 R . By the Schur condition, we deduce that W (t) is uniformly bounded in t. We assume hypotheses H0 and H1 on the evolutions U (t) P and V (t) and we consider jxj = j2Z j jj ihj j. As is well known, see e.g. [8], [2], this system is explicitly soluble by Fourier series. Note that jxj is not invertible, but it's expectation value behaves like the one of the operator (jxj + 1) which satis es H2 with Pj = j ? j ih?j j + jj ihj j, j 2 N  and P = j0ih0j. +

1

+

2

2

0

2

2

0

Proposition 5.1 Let H (t), W (t) be as above. Let g(n; t) = w(n; t)ein P g^(x; t) = k2Z eikxg(k; t) 2 L [0; 2]. Then 0

0

( )

2

and

Z  Rt (U (t)' )(n) = '(n; t) = 1 e?inx e?i g x;s ds dx; where ' (n) = n; ; 2 2

0

0

and

R t a s ds

^(

)

0

0

 Z  Z t X 1 @x g^(x; s)ds dx = n hjxj i' (t) = 2 n2Z 2

2

2

0

0

0

2

0

Z t g(n; s)ds : 2

(5.19)

0

It is readily seen from (5.19) that if !(m; t)  ! jmj?p with p > 3=2, uniformly in t 2 R , then hjxj i' (t)  ct , for some nite constant c, independently of the function a(t). Moreover, this upper bound is optimal if a(t) = E + E cos(ft) ("alternating electric eld"), for speci c values of the parameters E ; E and f , as can be seen from [2], [8], [9] and [7]. Note that we recover this lower bound from our results in the limit p ! 1. However, if the decay of the matrix elements !(j ? k; t) of the uniformly bounded operator W (t) is not fast enough as jj ? kj ! 1, the expectation value hjxj i' (t) can grow arbitrarily fast, as shown in the example below. Proposition 5.2 Let H (t) and W (t) be as above with a(t) = (1 + t)?q , q > 1 and !(m; t) = !(m) = jmj?p, where 1 < p < 3=2. Then, there exists a constant c > 0 such that hx i' (t)  ct ? p q p? ; as t ! 1 : Remark: Since p > 1, the perturbation W (t) is bounded. However, the domain of x is not invariant under W alone so the electric eld helps stabilizing the system. The limit a(t) ! 0 as t ! 1 thus explains the behaviour of hx i' (t). Proof: Consider +

2

0

2

0

0

0

1

1

2

0

2

(3

2 ) +2

1

0

2

0

Z t Z t Z t Rs n g(n; s)ds = ing(n; s)ds = inein a u du ds !(n) : Integrating by parts twice yields Zt Rt Rs inein a u du ds = ein a u du =a(t) ? 1=a(0) + Rt ein a u du a0 (t)=(ina(t) ) ? a0(0)=(ina(0) ) ? Z t Rs 0  ein a u du a0 (s)=a(s) ds=in : 2

2

2

0

0

0

0

0

( )

0

( )

0

0

( )

0

3

( )

3

3

0

9

( )

2

0

Hence, with our choice of a(t),

Z t ing(n; s)ds = j!(n) j(1 + t) q n1 + O((1 + t)?q ) + O((1 + t)q? =n)o : There exists a constant K > 0 such that the last parentheses above is bigger than 1=2, provided t is large enough and jnj  K (1 + t)q? . Thus, for t large enough, we 2

2

1

2

0

1

get

Z t X Z t X n g(n; s)ds  n g(n; s)ds  n2Z jnjK t q? X 1 : (1 + t) q 2 jnjK t q? jnj p 2

2

2

2

0

1

(1+ )

0

2

(1+ )

2

1

It remains to use the estimate X 1  2Z 1 q? jnj p jnjK (1+t)q?1

K (1+t)

2

1

2

1 ]+1

[

x? pdx = 2p 2? 1 ([K (1 + t)q? ] + 1) ? p ; 1

2

2

to end the proof.

5.2 Time-dependent band matrix

The estimate we get for this second example is certainly not surprizing, but it has the merit of beeing optimal, which is useful for the sake of comparisons with general cases (see the remark below proposition 4.1). Moreover, the proof of it is relatively easy and contains some ideas we generalize to prove proposition 3.1 in section 6, so we give it here. W (t) is a band matrix with time-dependent width, i.e. W (t) is such that kW (t) ? W d(t)k  !, 8t 2 R , and Pj W (t)Pk = 0 if jj ? kj > q(t) where q(t) is a positive real-valued function. We further assume that H0 and H1 holds for the evolutions U (t) and V (t) and we consider a generic observable A satisfying H2. Proposition 5.3 Let H (t) and W (t) be as above and let q : R ! R . Denoting q(t)  sup st q(s), for all  1 there exist constants c < 1 and > 0 such that for any ' 2 D(A )   hAi'(t)  c (tq(t))k'k + (q(t) e? t + (tq(t)) ? ? )kA 'k : Furthermore, if there exists a K > 0 such that k > K implies Pk ' = 0, then   hAi'(t)  c (tq(t)) k'k + q(t) e? t kA'k : In case H ; W and q are independant of time, the second estimate can be found in [1]. +

0

0

+

0

+

0

2

+2

2

2

+2

(

1)

2

2

0

Proof:

Several constants appear in the proof, which we all denote by the same symbol c. Due to hypothesis H1, the unitary operator (t)  V ? (t)U (t) satis es for any '2D i 0 (t)' = V ? (t)(W (t) ? W d(t))V (t) (t)' f (t) (t)'; = W 0

0

1

0

0

10

1

f (t) is bounded by !. We can write, using Dyson's series where W

(t) =

1 X

n=0

n (t) with n (t) =

Zt 0

ds




Z sn?

1

1

f (s )


W f (sn) ; (t) = 11 ; dsnW 1

0

0

where, due to [Pj ; V (t)]  0, 8j 2 N  ; t 2 R , Pj n (t)Pk = 0 if jj ? kj > nq(t) ; and n  !t n ( !t ) kPj n (t)Pk k  n!  n : +

0

Moreover ( (t) is unitary and fj gj2N is increasing), hAi'(t) = hX

(t)'jA (t)'i kPj A = (t)'k = 1 2

j 2N

 N k'k + 2

2

X

j>N

j kPj (t)'k

2

for any N 2 N  . Let A(j; k) = fn 2 N j jj ? kj  q(t)ng, A (j ) = fk 2 N  j jj ? kj  atq(t)g and A (j ) = N  nA (j ), where a is determined below. X X j kPj (t)'k = j k + Pj (t)Pk 'k 1

2

1

2

2

k2A j k2A j 0 1 X X X  c @k j = Pj n (t)Pk 'k + k j = Pj (t)Pk 'k A k2A j n2A j;k k2A j 0 1 X X =  c@ j kPj n(t)Pk kkPk 'kA + k2A j n2A j;k 1 0 = X j A (5.20) c @  ? = kPj (t)kkPk 'k k2A j j 1( )

2( )

2

1( )

(

2

2

)

2

2( )

2

2

1( )

(

)

2

2

2( )

(

1) 2

By construction, if k 2 A (j ) and n 2 A(j; k), one has: j =  (k + q(t)n) =  2 = k = (q(t)n) = ; and there exists a (a) ! 1 as a ! 1 such that the rst term in (5.20) is bounded above by a constant times 1

2

2

2

2

2

0  !t n1 X X @ k = kPk 'k n = n A k2A j n2A j;k 0  ! n 1  X X q ( t ) k = kPk 'k n = a A  j  ? @ k2A j n2A j;k 0 1  X X q ( t ) k = kPk 'k  j  ? @ e? a n A k2A j n2A j;k q(t)  j ( ?1)

2

1( )

2

2

(

)

2

2

(

1)

1( )

2

(

)

2

2

(

1)

1( )

( )

(

11

)

0 1  ? a jj ?kj=q t X q ( t ) e k = kPk 'k 1 ? e? a A  j  ? @ k2A j 0  X = q ( t )  c N  ? e? a at @ k kPk 'ke? a jj?kj= ( )

2

(

1)

( )

( )

1( )

( )

(

2

2

1)

qt

(2 ( ))

( )

k2A1 (j )

1 A: 2

(where we used j > N ). Introducing f 2 l (Z) and g 2 l (Z) by  k = kP 'k if k  1 k f (k) = 0 otherwise g(k) = e? a jkj= q t ; we can estimate the last parenthesis above by kgkl Z kf kl Z  cq (t)kA 'k : The second term of (5.20) is bounded by a constant times 0 1 = X j @ A  ? = kPk 'k k2A j j 2

1

2

( )

(2 ( ))

2

1(

(

 N  ?

1)

1

 N  ? (

Hence,

0 X@ X

j>N k2A2 (j )

2

)

2

1) 2

1

(

2(

2

2

2( )

2

)

1)

0 X @ j ?atq t N ?atq t matq t 0 1  ? X X N @ (m + j )  kPm j 'k A (2atq(t) + 1)  (1 ? atq(t)=N ) ?atq t matq t j>N  ?  c (1 ? Natq(t)=N ) (tq(t)) kA 'k ; where we need atq(t) < N . Thus choosing N (t) = [2atq(t)] + 1, we get by gathering these estimates     hAi'(t)  c (tq(t))k'k + q(t) e? a at + (tq(t))  ? kA 'k : If there exists a K > 0 such that k > K implies Pk ' = 0, the set A (j ) is empty if N is so large that j > N (t)  atq(t) + K . Thus, with N (t) = [2atq(t)] + 1 again, we get in this case for enough,   a large hAi'(t)  c (tq(t)) k'k + q(t) e? a at kA 'k : 2 X

(1

2

)

2

( )

(1

+

( )

)

2

+

( )

(1

)

( )

2

2

2

+2

( )

2+ (1

2

)

2

2

+2

12

( )

2

5.3 Laplacian and time dependant potential on

l

2(Zd)

Let W be the discrete laplacian on l (Zd): (Wu)(n) = Pj2Zd; jj?nj u(j ) and H (t) be a time dependent multiplication operator de ned on a dense domain D for any t, which satis es hypothesis H1. Let j = jj ihj j ; j 2 Zd ; and X j ; k 2 N  ; Pk = 2

0

=1

j 2Zd; jj j=k?1

then for any m > 0, if X X m k Pk ; and jxjm = jj jmej ; A= k2N

j 2Zd

where j : je is the euclidian norm, one has, for all  2 l (Zd) hjxjm j  i  hA j  i  dm= h(jxj + 1)m j  i : Now, for all j; k 2 N  , X X l W  m Pj WPk = 2

2

l2Zd; jlj=j m2Zd; jmj=k

= 0 if jj ? kj > 1 : Obviously, for all j 2 N , [H (t); Pj ] = 0. Then, from the results of section 5.2 one obtains that for all ' 2 D(A ), where = max (2=m; 1), there exists c < 1 such that for all t  1, hjxjmi'(t)  c tm(k'k + k(jxj + 1) m'k ) : 0

2

2

6 Technicalities

Proof of Proposition 3.1:

From H1, one can easily prove that (V ? Vq )(t; s)  V ? (t)Vq (t; s)V (s) is the evoluf (t)  V ? (t)(Wqd ? W d )(t)V (t); tion operator associated to the bounded operator W thus one can write the norm convergent Dyson's series: 0

1

0

1

0

0

(V ?1 V 0

q )(t; s)

1 X

=

Zj=0t

j (t; s) ;

Z sj ?

1

0

0

where

f (s )


W f (sj ) j  1

j (t; s) = ds


dsj W s s

(t; s) = 11 : Moreover, if w  supt> k(Wqd ? W d)(t)k then n n kPj n (t; s)k  t nw!  e? n ; 80  s  t and 8n  at 1

1

1

0

0

0

13

(6.21)

for suitable non negative nite constants a and . For all  2 D(A ) one then obtains, using the unitarity of V (s) and [V (s); Pj ] = 0, X j sup kPj Vq (t; s) k 0

0

2

j>N (t)

s2[0;t]

X

j sup kPj (V ? Vq )(t; s)V ? (s) k s2 ;t j>N t X XX  j sup k Pj n (t; s)Pk V ? (s) k

=

[0 ]

( )

s2[0;t]

j>N (t)

1

0

0

1

2

1

0

k n

(6.22)

2

We now make use of the following lemma which is proven at the end of this section: Lemma 6.1 Under the assumptions H0, H1 and H2, there exists K < 1 such that for all s; t 2 R ,

Pk n (t; s)Pj = 0 if jk ? j j  K ? min(k; j )n ? (6.23) >From this, kPk V ?  k = kPk  k, (6.21), and the de nitions B (j; k)  fn 2 ? k ? j   ?  ?  N s:t: jk ? j j < K min(k; j ) n g, Ai  fk 2 N j j  ?  Katg, Aii  fk 2 ? N  j kj ? k ?  Katg and Aiii  N  ? fAi [ Aii g, one proves that the right hand side of (6.22) is bounded above by X n = X X ? n X X ? n 3 (j e kPk  k) + (j = e kPk  k) k2Ai n2B j;k k2Aii n2B j;k j>N t o X +(j = kPk  k) k2Aiii ? X n ? at X = ? jk?j ? ) e ( k kPk  ke  C k2Ai j>N t o

j ?k ? X = X = (6.24) j kPk  k) +e? at ( k kPk  ke? k ? ) + ( 1

1

(

)1

(1

1

0

1

1

1

1

1

1

(

)1

(1

)

)

1 2

1 2

2

( )

(

1 2

2

)

(

)

2

0

)1 (1 )

~ ( 2

1 2

~

2

( )

)1 (1 )

~ ( 2

1 2

~

1 2

2

k2Aiii

k2Aii

2

where ~ > 0, and C < 1. Indeed, for xed j 2 N  , if k 2 Ai [ Aii then for all n 2 B (j; k), one has n  at and if k 2 Ai and n 2 B (j; k) 0

j N (t)

1

2

2

2

2

where c is a nite constant. Similarly, for

 (j )  2

X

k2Aii

k= kPk  ke? 1 2

~ (j ?k)1? 2 k(1?)

one gets from the same lemma

?

 X  X e? kj? k ? k P ' k  (t)   (j )  C k  ? = j>N t j>N t klt j k  ck'k (6.28) where lt (j ) = supfx 2 R s:t: j  x + (Kat) = ? x g and c is a nite constant. Finally, X =  (j )  j kPk  k X

2

2

2

2

( )

( )

1

( )

~ ( 2

)1 (1 )

(

1 2)

2

2

1 (1

1 2

3

k2Aiii

is bounded above by X X j = kPk 'k  c k2Aiii k lt j ?j m Kat

j = kP 'k  j m ? j  (j + m)

1 2

( )

(

2

)1=(1

Since m  lt (j ) ? j , one has  1  j  ? = j (j + m) 1 + mj    j  ? l (jj )  t 2

(2

1

)

(1 2

)

)

+

)

(6.29)

(6.30) 15

On the other hand, by the de nition of lt (j ), one has j = lt (j ) + (Kat) which implies lt jj  , if ( )

1 2

? lt (j ) ,

1

1

(6.31) N (t) = [2 ? (Kat) ? ] + 1 : This result together with (6.29) and (6.30) gives for some nite constant c : X j  ? kPj m'k  (j )  c (6.32) 1

1

(1

1

)2

3

1

3

3

lt (j )?j m(Kat)1=(1?) j 

From (6.32) one obtains:  (t) 3

X

j>N (t)

 (j )  c 2

3

X

3

j

(2

)

+

?2 ) 



X

(1

kPj m'k

2

+

j>N (t) (Kat)1=(1?) j  X X (1?2 ) jmjX kPj+m'k2) 1 )( c3 j ( j>N jmj(Kat)1=(1?) j  jmj(Kat)1=(1?) j  )1=(1?) k'k2 c4 (NKat (t)(2 ?1)??1

 

 c (Kat)

(1?2 )+2 (1?)2

k'k

(6.33) where c ; c are a nite constants. Inequalities (6.27), (6.28) and (6.33) imply together with (6.24), for any  2 D(A? ) : 5

4

2

5

  Pj nPk  k  3 e? at  (t) +  (t) + 3  (t) k> n2N j>N t   ?   c e? at + t ? kA  k for some nite constant c independant of t and ', which proves (3.15). X

( )

j k

XX

2

~

1

2

3

0

6

Proof of lemma 6.1:

(1 2 )+2 (1 )2

~

(6.34)

2

6

For all t > 0 and for xed n, we rst calculate the values of j and k, 0 < j  k such that for all p 2 f1;


; ng and for all si 2 [0; t], (i 2 f1;


; pg) , one has Pk W~ (s )W~ (s )


W~ (sp)Pj = 0. For n = 1, consider the positive continuous concave function f (x) = j + q(j + x), where q(x) = x as in proposition 3.1. Let l (j ) be it's unique positive xed point; then for all k > l (j ), one has k ? j > q(k + j ) which implies that Pk W~ (s)Pj = 0, for all s 2 [0; t], since [V ; Pj ] = [V ; Pk ] = 0; thus Pk (t; s)Pj = 0. Now de ne f (x) = l (j ) + q(l (j ) + x) and let l (j ) be it's unique xed point; then for all s ; s 2 [0; t]: 1

2

1

+ 1

+ 1

1

+ 1

2

1

0

+ 1

2

X Pk W~ (s )W~ (s )Pj = Pk W~ (s )Pl W~ (s )Pj 1

2

l2N

1

2

If l > l (j ), Pl W~ (s )Pj = 0. If l  l (j ), then for all k > l (j ) one has k > l (j ) + q(k + l (j ))  l + q(k + l) ; and thus Pk W~ (s )Pl = 0. Finally if k > l (j ), Pk (t; s)Pj = 0. + 1

+ 1

2

+ 1

+ 2

+ 1

1

+ 2

16

2

0 + 2

By using the same arguments, one can easily construct an increasing sequence (ln (j ))n2N (l (j )  j ) of xed points of fn(x)  ln? (j ) + q(ln? (j ) + x) such that for any m 2 N , if k > lm (j ), then Pk m (t; s)Pj = 0. Note that lm(j ) does not depend on s and t. Similarly, there exists a decreasing sequence (possibly nite) of positive values (ln?(j )) solutions of the xed point equations + 0

+

+

+

1

+

1

+

ln?? (j ) ? q(ln?? + x) = x; l? (j ) = j : 1

1

0

Those xed points are such that Pk n (t; s)Pj = 0 as soon as k < ln?(j ). We now give an estimate of ln (j ). One has: ln (j ) ? ln? (j ) = (ln (j ) + ln? (j )) +  (l (j )) < 11 ?  n? +

+

+

+

1

+

1

1

(6.35)

Indeed, using the following estimate

ln (j ) ? ln? (j ) < (ln? (j )) (1 +  l ln (j(j)) ) n?    (ln? (j )) 1 +  +  (ln (j )l + l(nj?) (j )) +

+

+

1

+

+

1

+

1

+

+

1

1

+

n?1

which holds since (1 + x) < 1 + x for 0 <   1 and 0 < x < 1, we get   (ln (j ) + ln? (j )) < (ln? (j )) 1 ? (1l + (j ))? n? 1 +   1 ?  (ln? (j )) : +

+

+

1

1

+

+

1

1

1

By induction, suppose that for some xed m that for all 1  i  m ? 1,

li (j ) ? j < Kj i ? ; +

1

1

then (li (j )) < (Kj i ? + j )  2K  j i ? ; +

1

1

1

and

      lm (j ) ? j = lm (j ) ? lm? (j ) + lm? (j ) ? lm? (j ) +


+ l (j ) ? j +  l (j ) + l (j ) +


l (j ) + j   < 11 ? m?  m? mX ?  11 ?+  2K  j  i ? i   1 +   1 ?  2K  j  (1 ? )m ? +  : (6.36) +

+

+

+

+

1

+

1

1

+ 1

2

1

1

=0

1

17

+

1

2

+ 1

 ) ? , the last inequality in (6.36) gives If K  (2 11 + ? 1

1

lm (j ) ? j < Kj  m ? : +

1

(6.37)

1

n

o 

Then from (6.37) and (6.35), with n = 1 and K = max (2 ? ) ? ; ? , one obtains for all n 2 N  : ln (j ) ? j < Kj  n ? which proves (6.23) if k > j . With similar arguments on ln?(j ), one can prove for k and j such that k < j that +

1

1+ 1

1

1

1+ 1

1

Pk n(t; s)Pj = 0 if j ? k  Kkn ? : 1

1

Proof of lemma 6.2: Since (2 ? 1) > 1 + , there exists  2 R such that 2(1 ? ) > 1 and (1 ? 2 ) + 2(1 ? ) < ?1; then for some nite constant C depending only on , and for all t > 1:

XX

k

?

= ? ) e? ~2 jkk?(1jj?) 1

(1 2

j 2N k2N



kPk 'k

2

00 1 1 X B@ X  = ?  C () @ k jk ? j j? ? k ? kPk 'kA + kPj 'k CA k2N ;k6 j j 2N 0 1 X@ X  ? k'k  C () k jk ? j j?  ? k  ? A + k'k  j 2N 0 0k2N ;k6 j  ?  ? 1 1 X X  C~ ()k'k @ @ (jk(jjk+?1)j j + 1)  ? A + 1A ; j 2Z k2Z 2

(1 2

)

(1

)

(1

)

2

=

2

(1

2 )

2 (1

)

2

(1

)

2

=

(1

2

2 )+2

2 (1

(1

)

)

where C~ () = C ()2 ?  ? . By the choice of  one has f (k)  (jkj + 1) ?  ? 2 l (Z) and f (k)  (jkj + 1)?  ? 2 l (Z); then one gets (2

(1

2 )+2

(1

X X

j 2N k2N

)

1

k

1)+2 (1

2 (1

2

?

= ? ) e? ~2 jkk?(1jj?)

(1 2

)2

1



kPk 'k

2

)

1

1

 C~ () (kf  f kl 1

 C~ ()(kf kl 1

2

Z)

1(

k'k

Z) + 1)

1(

kf kl 2

Z)

1(

2

+ 1)k'k : 2

References [1] J.M. Barbaroux, J.M. Combes and R. Montcho: Remarks on the Relation between quantum Dynamics and fractal Spectra, to appear in J. Math. Anal. Appl. (1997). [2] J. Bellissard: Stability and Instability in Quantum Mechanics in "Trends and developments in the eighties", S. Albeverio, Ph. Blanchard Edts, World Scienti c, Singapore 1985. 18

[3] L. Bunimovich, H.R. Jauslin, J.L. Lebowitz, A. Pellegrinotti and P. Nielaba: Diusive Energy Growth in Classical and Quantum Driven Oscillators, J. Stat. Phys. 62 (1991) 793-817. [4] M. Combescure: Recurrent versus diusive dynamics for a kicked quantum oscillator, Ann. Inst. H. Poincare 57 (1992) 67-87. [5] M. Combescure: Recurrent versus diusive quantum behavior for timedependent hamiltonians, In "Operator Theory: Advances and Applications", Vol57, Birkhauser Verlag Basel 1992. [6] R. Del Rio, S. Jitomirskaya, Y. Last and B. Simon: Operators with singular continuous spectrum, IV. Hausdor dimensions, rank one perturbations and localization, to appear in J. d'An. Math. [7] S. Debievre and G. Forni: Transport Properties of kicked and quasi-periodic Hamiltonians, Preprint Universite de Lille (1997). [8] G. Gallavotti: Elements of classical mechanics, Springer Texts and Monographs in Physics (1983). [9] I. Guarneri: Singular Properties of Quantum Diusion on Discrete Lattices, Europhysics Letters 10 (2) (1989) 95-100. [10] I. Guarneri and G. Mantica: On the Asymptotic Properties of Quantum Dynamics in the Presence of a Fractal Spectrum, Ann. Inst. H. Poincare, sect. A61 (1994) 369-379. [11] G. Hagedorn, M. Loss and J. Slawny: Non Stochasticity of Time-dependent Quadratic Hamiltonians and Spectra of Cononical Transformations, J. Phys. A. 19 (1986) 521-531. [12] J. Howland: Floquet Operator with Singular Spectrum II, Ann. Inst. H. Poincare 49 (1989) 325-334. [13] J. Howland: Quantum Stability in "Schrodinger operators, the quantum mechanical many bodies problem", Ed. E. Balslev, Springer Lecture Notes in Physics 403 (1992) 100-122. [14] H.R. Jauslin and J.L. Lebowitz: Spectral and Stability Aspects of Quantum Chaos, Chaos 1 (1991) 114-137. [15] A. Joye: Upper Bounds for the Energy Expectation in Time-dependent Quantum Mechanics, J. Stat. Phys. 85 (1996) 575-606. [16] Y. Last: Quantum Dynamics and Decompositions of singular continuous Spectra, J. Funct. Anal. 142 (1996) 406-445. [17] G. Nenciu: Adiabatic Theory: Stability of Systems with Increasing Gaps, CPT95/P.3171 preprint (1995). 19

[18] C.R. de Oliveira: Some Remarks Concerning Stability for Nonstationary Quantum Systems, J.Stat.Phys. 78 (1995) 1055-1066. [19] C. Radin and B. Simon: Invariant Domains for the Time-dependent Schrodinger equation., J. Di. Equ. 29 (1978) 289-296. [20] B. Simon: Absence of Ballistic Motion, Commun. Math. Phys. 132 (1990) 209-212.

20