ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaSingular Continuous Spectra andDiscrete Wave Packet DynamicsItalo Guarneri

Vienna, Preprint ESI 352 (1996) July 3, 1996Supported by Federal Ministry of Science and Research, AustriaAvailable via http://www.esi.ac.at

Singular Continuous Spectra and Discrete Wave Packet DynamicsItalo GuarneriCenter for Nonlinear StudiesUniversity of Milano at Comovia Lucini, 3 - 22100 Como, ItalyIstituto Nazionale di Fisica della MateriaSezione di Milano.AbstractAsymptotic estimates which relate the di�usion of wave packets on discretelattices to Hausdor� dimensions of the local density of states are discussed.PACS05.45.+b,02.30.-f,71.30.+h,71.55.JvI.- Introduction.The basic mathematical objects of this paper are a discrete unitary group fU tgt2Z in anin�nite dimensional separable Hilbert space H , and a complete Hilbert basis B = fengn2Z.The discrete-time evolution of vectors 2 H is obtained by repeated application of thegenerator U . Upon expanding vectors U t on the basis B, vectors in l2(Z) are obtained,which can be physically depicted as quantum wave packets propagating on a discrete one-dimensional lattice, the sites of which are in one-to-one correspondence to the basis vectors.According to conventional wisdom, long-time wave packet dynamics and spectral structureare qualitatively connected: wave packets will remain essentially con�ned within a �niteregion, or will propagate unboundedly, depending on the discrete or continuous nature of thespectrum. The intuitive idea, that the motion of wave packets will be the more "delocalized",the more "continuous" is the spectrum, turns out to have a validity all the way inbetweenthe extreme cases of discrete and absolutely continuous spectra. It has in fact been turned1

into rigorous estimates, which relate the asymptotical (in time) wave packet dynamics, to"spectral" dimensions of the Hausdor� type, which describe how continuous is the spectralmeasure (local density of states) associated with the vector . Estimates of this kind makethe subject of this paper.Physical motivation is provided by the increasingly frequent apparition of singular con-tinuous spectra in quantum mechanical models for electron dynamics in solids , where suchspectra may have a great in uence on quantum transport. This issue has been numericallyinvestigated on several quasi-periodic, incommensurate models. Fractal dimensions whichare somehow related to the singular continuous structure of the spectrum have been numer-ically obtained from a scaling analysis of the band spectra of periodic approximants1{4. Theresults obtained in this way suggest a connection between "band-scaling" fractal dimensionsand (anomalous) wave packet di�usion. Recourse to more sophisticated concepts of fractalanalysis5;6 appears now necessary, in order to further pursue this line of investigation; atheoretical approach has been proposed5.Further numerical works have investigated the multifractal structure of the spectral mea-sure itself, on di�erent models6{8. A comparison of such data with data from numericalsimulation of the quantum evolution in time7;8 con�rms a qualitative connection, but at thesame time indicates that no simple exact relation exists between di�usion exponents anddimensions of the local density of states.Rigorous results available to date concern the decay in time of the time-averaged probabil-ity of survival in the initial state3;9{11, and the algebraic growth of the "width" of wavepack-ets, as measured, e.g., by the moments of the associated probability distribution. In thelatter case, through successive generalizations the result has been obtained12{16 that theexponent of growth is bounded from below by the information dimension D1 of the spectralmeasure, independently of the choice of the basis B.No such general upper bound is to be expected. The very de�nition of moments de-pends on the labeling of the basis vectors, and so do the corresponding growth exponents,independently of the properties of spectral measures. Thus, although the growth exponents2

are in all cases subject to the above mentioned lower bound, they do actually depend on thechoice of a basis; therefore the 'spatial' structure of the operator U on the basis B has to betaken into account in order to get upper bounds on growth exponents. For instance, it willbe shown in sec.5 that a ballistic upper bound is valid as soon as the operator U satis�es acertain "analyticity" property with respect to the basis B.According to empirical evidence, the actual growth exponents can be signi�cantly di�er-ent from both the ballistic upper bound and theD1 lower bound. Getting precise asymptoticestimates on the long-time behaviour of wave packets is an open problem, that may not besolvable on the level of generality attained by the existing bounds.In this paper a di�erent dynamical characterization of singular continuous spectra isgiven, based on the growth with time of the dimension of the Hilbert subspace explored bythe trajectory of a state : By re�ning and extending results of ref.15, it will be shown thatthe corresponding growth exponents are related by upper and lower estimates to the fractaland Hausdor� dimensions, respectively, of the spectral measure. These results are presentedin sec.4 below. In secs.2,3 a survey of some basic de�nitions and of previous results, whichare needed for the elaborations in sec.4,6 is given.Finally, in this paper some steps are taken towards a multifractal analysis of the spatialstructure of the wave packet, following an idea of Evangelou and Katsanos17 . Such ananalysis is suggested by numerical evidence15;17 that growth exponents of moments do notscale in a simple way with the order of the moments (multiscaling), and so give rise toa nontrivial spectrum of growth exponents. This can be taken as an indication that wavepackets develop a sort of multifractal structure in space, that may in turn re ect an analogousstructure of eigenfunctions1;18. From this viewpoint, estimating growth exponents is but aspecial aspect of a more general problem: how is this structure related to the multifractalstructure of the spectral measures. Some results in this direction, which follow from generallower bound on growth exponents, are presented in sec.6.Multiscaling is also a justi�cation for an aspect of the present paper, which may appeardisturbing to the reader: the proliferation of di�erent quantities, all of which somehow3

measure the growth of wavepackets, but still may behave di�erently in time. Establishingprecise connections between these di�erent de�nitions would enable reducing their number;however, this as yet unsolved problem is closely related to the problem of upper bounds.Previous general remarks about the latter problem are valid in this case, too.II.- Spectral dimensions.In the following, the set f� 2 [0; 2�] : ei� 2 Spec(U)g will be called "spectrum". Thespectral measure of the vector will be supported in this set, and denoted � ; the label will be usually omitted. The spectral measure � is uniquely de�ned by( ;U t ) = 2�Z0 eit�d� (�)valid at all times t. The distribution of the spectral measure is also called (integrated)local density of states, at least in concrete cases in which is an eigenvector of a positionoperator. In this section we assume � ([0; 2�]) = k k2 = 1:By d�� (x); d+� (x) we will denote the lower and upper pointwise dimensions of � at thepoint x of the spectrum, de�ned byd�� (x) = lim inf�&0 log �(I�(x))log � d+� (x) = limsup�&0 log �(I�(x))log �where I�(x) is an interval of size � centered at x. If d�� (x) = d+� (x), then their common valuede�nes the local , or pointwise, dimension of the measure � at the point x: Upper and lowerglobal dimensions of � will be de�ned bydim+H(�) = �� ess sup d�� (x)dim�H(�) = � � ess inf d�� (x):Note that both de�nitions involve the lower local dimension only. The subscript H standsfor Hausdor�; the connection to Hausdor� dimensions is clari�ed by the following result,where dH(A) denotes the Hausdor� dimension of a set A.4

Theorem 0: dim+H(�) := inffdH(A) : �(A) = 1g;dim�H(�) = supf� : �(A) = 0 if dH(A) < �gProof. These equalities follow from general results of Rodgers and Taylor19, compactlyreviewed in refs.16;20. Let �> �-ess sup d�� (x); then � is supported by a set of dimension �20,so � � dim+H(�) by de�nition; hence, �-ess sup d�� (x) � dim+H(�): Conversely, let � <�-esssup d�� (x); and let S be any set supporting �: Then d�� (x) > � for all x in a set B � Sof positive measure. If �B is the characteristic function of B; and d�B := �Bd�, then themeasure �B is supported by B; and satis�es d��B (x) � d�� (x) > � for �B-almost all x. Any setof Hausdor� dimension less than � must have zero �B measure20, which entails dH(B) � �,and also dH(S) � �: This holds for any S supporting �, therefore dim+H(�) � �; it followsthat �-ess sup d�� (x) � dim+H(�):The proof of the second equality is similar. Let = �-ess inf d�� (x). Let us prove � dim�H(�). If = 0 this is obvious. If > 0, let � < ; then d�� (x) > � for �-almost allx, so � gives zero weight to any set of dimension less than �, which means � � dim�H(�).Conversely, if � > , then d�� (x) < � on a set S of positive measure. This set S is a subsetof the set T�, de�ned as T� = (x : limsup�&0 �(I�(x))�� = +1): (1)It is a known result20 that a set T 0� exists, with dH(T 0�) � �, and �(T 0� \ T�) = �(T�).Therefore, if S 0 = S \ T 0�, then �(S 0) = �(S) > 0, and dH(S 0) � dH(T 0�) � �. Hence,� > dim�H(�). 2If dim�H(�) = dim+H(�) = d (that is, if d�� (x) = d; �-a:e:); the measure is said to haveexact dimension d . This denomination is not universally agreed; for instance, a measure issometimes21 said to have exact dimension d if d�� (x) = d+� (x) = d; �-a:e:; that is, if the5

measure has a well de�ned constant local dimension �-a:e. A measure with this propertywill be called here an exactly scaling (ES) measure. ES measures also have exact dimensionin the sense used here, but the converse is not true - in ref.20 examples are given of measureswith exact dimension 0 which have d+� (x) = 1 �-a:e:. Although exceptional on mathematicalgrounds, in the physical literature the ES property is often assumed of "smooth multifractal"measures. On purely empirical grounds, and on the present level of numerical accuracy, thisassumption has not shown, so far, any serious inconsistency with numerical data, at least ina few test cases which are relevant to the subject of this paper, and which were mentionedin the Introduction. On the other hand, there are certain non generic features in thesecases, which enforce great caution in assuming that a likewise smooth structure of spectralmeasures will be typical, even within the class of quasi-periodic Schr�odinger operators.dim+H(�) is known as the Hausdor� dimension of the measure � . It is also sometimescalled the information dimension DI (�) of �22, although the latter name is more often givento a generalized fractal dimension, usually denoted D1(�) (which corresponds to q = 1 inthe set of generalized fractal (box-counting) dimensions Dq(�)). For "smooth" multifractalmeasures (as discussed, e.g., in23) DI(�) = D1(�); but this is not true in general, not evenof exactly dimensional measures.The fractal dimension of � will instead be de�ned asdimF (�) = sup0<�<1 infK fdF (K) : K compact; �(K) > 1� �g (2)where the fractal dimension dF (K) of a compact set K is de�ned by24dF (K) = limsup�&0 logNK(�)log 1�Here, NK(�) is the minimum number of closed intervals of size � � which are needed tocover K:Theorem 1. dim F (�) � �-ess sup d+� (x):6

Proof. Let d+� (x) < � for �-almost all x; we have to show that dimF (�) � � . Becauselim�0!0 sup�<�0 log �(I�(x))log � < � (3)for all x in a set of full measure, then, on the strength of the Egorov theorem, we can �nda set J� � (0; 2�); of measure �(J�) > 1 � �2; in which the monotone limit in the lhs of (3)is uniform, so there is a ���;� such that�(I�(x)) > �� 8x 2 J�; 8� < ���;� (4)Let us �x � so small that (4) holds. The family fI�(x)gx2J� is a covering of J� , from whichwe can extract a �nite or countable covering of the same set with no more than c overlaps,with c a �xed integer (this comes, e.g., of the Besicovitch covering lemma, as formulatedin ref.25). Denote I1; ::: the intervals in this covering, and let M be their number. Fromm�� � Pm1 �(Ij) � c�(Sm1 Ij) � c , which holds for any positive integer m < M , it followsthat M is �nite, M � c��� . Finally, choose a compact K � J� , with �(J� nK) < �2 . Forall su�ciently small �, NK(�) � M; so dF (K) � �; moreover, �(K) > 1 � �; so, from (2),we conclude dim F (�) � �:2Remark. Theorem 1 shows that the Hausdor� and the fractal dimension of ES measurescoincide.II.- Spreading of wave packets.Let B � fengn2Z an orthonormal set. For any positive integer time t letpn(t) := 1t t�1Xs=0 j n(s)j2 (5)where n(t) := (en; U t ) . Since time averages like the one in (5) appear frequently in thefollowing, the shorthand notation h�it2t1 will be used for time averages from time t1 to timet2 � 1: 7

Eqn.(5) de�nes a �nite measure Pt; on subsets A of Z, via Pt; (A) = Pk2A pk(t) . If B iscomplete, then the total mass of this measure is just k k2. A minimal �- support of the mea-sure Pt; will be any �nite family F� � Z such that (i) Pt; (F�) > (1 � �2)Pt; (Z), and (ii)Pt; (F 0) � (1 � �2)Pt; (Z) for any other �nite family of indices F 0 such that ](F 0) < ](F�),where ] denotes cardinality. The distribution ( 5) may have di�erent minimal ��supports,but they must have the same "size" ](F�), which will be denoted by n�( ; t;B). For simplic-ity's sake, in the following we shall often omit the complete list of parameters on which n�,and other quantities, depend, leaving understood those, whose speci�cation is not strictlynecessary.If � is a purely continuous measure, then it is a classical result that pk(t)! 0 as t!1,8k, so Pt; gives smaller and smaller weight to any �nite set A � Z. Thus, if the total massis constant, or at least bounded away from zero at all times, then n� diverges in the limitt ! 1. This is true, in particular, when B is complete. It will be shown below that, inthe latter case, n� diverges as soon as � has a continuous component. Our aim here isto describe asymptotic bounds on the growth of n�, in terms of spectral dimensions of � ,de�ned in sec.2.The asymptotic spreading of Pt; can also be quantitatively described by other quantities,e.g. by the moments m(�)(t), which are de�ned, for � > 0; bym(�)(t) := Xk2Z jkj� pk(t)The growth of n� as t ! 1 bounds the growth of (m(�))1=� from below, via a Chebyshev-like inequality. Instead, from the divergence of moments nothing can be inferred about thebehaviour of n�; examples are known, in which � is pure-point, n� is bounded in time, butmoments diverge algebraically.A description of the asymptotical growth in time of quantities which in one way or anothermeasure the "size" of wavepackets is provided by upper and lower growth exponents, which,for a given positive sequence c � fctg ; labelled by the discrete time t, will be de�ned anddenoted as follows, 8

��(c) = lim inft!1 log ctlog t �+(c) = lim supt!1 log ctlog t : (6)Previous results by the present author14;15 can be cast in the form of theorem 2 below. Theseresults can be signi�cantly strengthened, see remark 1 below, but the version used here isconvenient for the purposes of this paper.Theorem 2. Let k k = 1, 0 < � < 1, and suppose that B is complete. Then��(fn�( ; t)g) � dim�H(�):Remark 1. An immediate corollary is ��(nm(�)(t)o) � �dim�H(�), because the growth ofn�( ; t) bounds the growth of moments from below. Y. Last16 has strengthened this result,proving that m(�)(t) > const:t�dim+H(�) for any spectral measure � (Last's formulation issomewhat di�erent, as it is given in terms of Hausdor� decompositions of �). A similargeneralization of Thm.2 is also possible, yieldingsup� ��(fn�g) � dim+H(�) (7)This step will be explained later.Remark 2. The result is also true if B is not complete, still hkPB (t)k2it0 � �2 > 0,8t > 0, PB being projection onto the subspace spanned by B. In this case thm.2 holds for0 < � < �. The same is not true of the generalization mentioned in remark 1, though.Remark 3. In refs.14;15 theorem 2 was formulated for the case of exactly scaling measures�: The proof given there is also valid for the present version, without any modi�cation.Other minor di�erences require no special comment.Remark 4. Combes13;11 and Last16 generalize similar results to continuous unitary groups,generated by Schr�odinger operators in L2(Rn):The following form of thm.2 is proven in the same way, and will be used later, in theproof of proposition 1. 9

Theorem 2a. If B is an orthonormal (not necessarily complete) set, and there is asequence of times tk !1 such that, 8kDkPBU s k2Etk+1tk � �2 (8)then, for all su�ciently small positive � < �,lim infk!1 log n�( ; tk)log tk � dim�H(�) (9)IV.- E�ective dimensions.For 6= 0 consider �nite strings �s;t( ) = fU s ;U s+1 ; :::U s+t�1 g of the orbit of . If � has a continuous component, then , 8s; t, �s;t spans a subspace �s;t of H, ofdimension t. Nevertheless it may happen that the "e�ective dimension" of �s;t is signi�cantlysmaller, in the sense that some subspace of �s;t , of dimension � t; exists, such thatU s ;U s+1 ; :::U s+t�1 have but a small component outside it. This raises the question,how does the e�ective dimension of the subspace spanned by a string of length t increasewith t?This question will be formalized as follows. Given � 2 (0; 1); let ��(�s;t( )) be theminimum dimension of an orthogonal projector P in H; such that P?U j < � for s �j � s+ t� 1: In other words, ��(�s;t( )) is the minimum dimension of a subspace such thatthe string �s;t( ) lies within a distance � of it. Moreover, let ���(�s;t( )) be the minimumdimension of a subspace such that the same string lies within � of it in the average, that is,h P?U j 2is+ts < �2. The mentioned minimal subspaces can be assumed to be subspaces ofthe subspace spanned by �s;t( ). Moreover, since �s;t( ) = U s(�0;t( )), all the strings of agiven length are unitary images of one another, so ��(�s;t( )) and ���(�s;t( )) only dependon t, and will therefore be denoted ��( ; t) and ���( ; t). If � > k k, then ��( ; t) = 0:Immediate consequences of these de�nitions are,10

Proposition 0:(i) ���( ; t) � ��( ; t) � t, .(ii)��( ; t+ s) � ��( ; t) + ��( ; s);(iii) if 1 2 H1, 2 2 H2 where H1 ,H2 are mutually orthogonal subspaces, invariantunder U; then ��( 1 + 2; t) � max[��( 1; t); ��( 2; t)]: (10)Moreover, (ii-iii) also hold for ���.Proof. (i) is obvious. (ii) follows from ��(�0;t [ �t;s) � ��(�0;t) + ��(�t;s): As to (iii),note that �0;t( i) � Hi, and that vectors �1; :::�t exist, which span a subspace of dimension��( 1+ 2; t); and satisfy k�j � U j( 1 + 2)k < � for j = 0; :::t� 1: Now let projections Pi�jon Hi span a subspace Si of Hi; with i = 1; 2: On one hand, dim( Si) cannot be larger than��( 1 + 2; t); and on the other it cannot be less than ��( i; t); because kPi�j � U j ik < �for j = 0; :::t�1: Thus ��( 1+ 2; t) � ��( i; t). For ��� the proof proceeds in a similar way.2At given �; , both ��( ; t) and ���( ; t) are positive nondecreasing sequences. Theirgrowth exponents will be presently investigated.Proposition 1. If d�� (x) > �; ��a:e:; then, for su�ciently small positive �, ��(���) � �.Proof. Given a sequence of times tk !1, such that��(���) = limk!1 log ���( ; tk)log tklet us recursively construct a subsequence ntkjo , with k1 = 1; and11

kj+1 = minnk : tk > 2ljo ;where lj = Pjs=1 tks. Let us partition the orbit of into segments �j � f (s); lj � s < lj+1g ;the length of �j is thus tkj . According to the de�nition of ���( ; t) we can �nd a sequencef'tg of vectors such that, 8j, Dk's � U s k2Elj+1lj < �2 (11)(so that �j lies within an average distance � of the subspace spanned by f's; lj � s < lj+1g),and, moreover, the subspace spanned by f's; lj � s < lj+1g has exactly dimension ���( ; tkj):On orthonormalizing the sequence f'tg we obtain an orthonormal set B� with the propertythat, 8j DkPB�U s k2Elj+1lj > k k2 � �2 (12)Therefore we can use Thm.2a to the e�ect that, if � is small enough,lim infj!1 log n�( ;B�; lj)log lj � �: (13)On the other hand, (0); ::: (lj) lie within an average distance � of the subspace spannedby the �rst lj vectors 't. This subspace has dimension N(lj) � Pjs=1 ���( ; tks) , and is alsospanned by the �rst N(lj) vectors in B�: From the de�nition of a minimal �� support itfollows that n�( ;B�; lj) � N(lj) �Pjs=1 ���( ; tks); therefore, using Proposition 0,log 2n�( ;B�; lj)log 2lj � log 2Pjs=1 ���( ; tks)log 2lj� log 2Pj�1s=1 tkslog 2tkj + log 2���( ; tkj)log 2tkj� log 2lj�1lj�1 + log 2���( ; tkj)log 2tkjTaking the limit j !1 and using (13) the required result is obtained.212

One further step can be taken, in the spirit of Last's extension of thm.216. Recall thatdim+H(�) = �-ess sup d�� (x). For any small �, let A� = fx : d�� (x) > dim+H(�) � �g; then�(A�) > 0; and one can write = 1 + 2, the spectral decomposition of according tothe set A� and its complement. Then d� 1 = �A�d�, so d�� 1 (x) � d�� (x), and dim�H(� 1) �dim+H(�)� �. On account of Proposition 1, and of proposition 0(iii), ��(���) � dim+H(�)� �if � is smaller than some �0. The latter depends in general on �, so we can concludeProposition 2. For any 6= 0, and for su�ciently small �, ��(���( ; t)) � dim�H(�).Moreover, sup� ��(���( ; t)) � dimH +(�)Remark. This result includes all the versions of Thm.1 discussed in remarks (1-3), andinequality (7) in particular.Proposition 3. 8� > 0; �+(��) � dimF (�) (the fractal dimension of �).Proof. By spectral equivalence, we can work in the space L2([0; 2�] ; �); where (t) � eitx�� a:e: Choose � > 0 and de�ne, for all integer times t, �t = �=tp2: Then, if � > dim F (�);we can �nd a compact K with �(K) > 1 � �22 , and with a fractal dimension dF (K) < �:For all su�ciently large t, K can be covered by Nt closed intervals Ij of width � �t , withNt � ���t . We can assume that the intervals Ij have no more than 2 overlaps; therefore,using their endpoints we can de�ne a covering of K with a number Mt � 2Nt � 2���t ofdisjoint intervals of width � �t; which for simplicity will be denoted again by Ij, and whichsatisfy �([Ij) > 1 � �22 : In every interval Ij choose a point xj and, for 0 � s � t� 1; de�ne13

�s(x) � MtPj=1 eisxj�j(x) for x 2 S Ij�s(x) � 0 elsewherewhere �j(x) is the characteristic function of Ij: Thenk (s)� �sk2 = 2�R0 jeisx � �s(x)j2 d�(x)� s2�2t + �([0; 2�] n [Ij)< t2�2t + �22< �2Now �s (s = 0; :::t�1) belong by construction in the subspace spanned by f�jg1�j�Mt , which,for su�ciently large t; has dimension Mt � 2Nt � 2(p2t=�)�: Thus, this subspace containsvectors which are ��close to ; ::: (t� 1); therefore, at large times, ��( ; t) � 2(p2t=�)�:2Putting Theorem 1 and propositions 0(i),1,2,3 together, we getProposition 4. If � is exactly scaling, with dimension d; then, for su�ciently small� > 0 the limits limt!1 log ��(�; ; t)log t ; limt!1 log �(�; ; t)log t (14)exist, and have the same value d = dimH(�).Remark. One may speculate whether Proposition 2, too (hence, Theorem 2 itself) canbe proven by a "box counting" argument of the kind used in the proof of proposition 3.V.- A Ballistic Upper Bound.It is not possible to extract from the above results upper bounds for moments, and noteven for the growth of the size of minimal ��supports. Further assumptions are needed,14

which concern the "spatial" structure of the operator U on the basis B: Here a result inthis vein is presented. For convenience, we'll say that the spread is not faster than ballisticif hm(�)(t)i 1� does not increase faster than linearly with t,8� > 0. In this case, it is easilyseen that n�( ; t;B), too, does not increase faster than linearly.Proposition 5. For � � 0 letX� := ( 2 H s:t: k k� := supn2Z j(en; ) exp� jnjj <1)and suppose that U (X�) � X� for some � > 0: Then, if 2 X�, the spread is not fasterthan ballistic. Moreover, 8 2 H , n�( ; t;B) does not increase faster than linearly.Proof. Under the stated hypothesis , UeX� is an everywhere de�ned linear operatorin X�; which is a Banach space under the norm k:k� : Let f ng be a convergent sequencein the X��norm, such that fU ng also converges in that norm, and let 1; 01 be thecorresponding limits. Then n ! 1 and U n ! 01 in the weaker Hilbert norm, too;therefore U n ! U 1 in the Hilbert norm because U is unitary. It follows that 01 = U 1,so UeX� is a closed operator in X� . By the Closed Graph Theorem, it must be bounded.Therefore, if 2 X�; then 8t; nj n(t)j � U t � e��jnj � kUkt� k k� e��jnj (15)Using (15) for n � nt := ��1t log kUk� + ��1 log k k�, and j n(t)j � 1 for n < nt, theannounced bound on the growth of moments is immediately obtained. Finally, given any 2 H; there is a 0 2 X� with k � 0k < �2; at any time t; any �2� support for U t 0 isalso a ��support for U t . 2Remark 1. Under the hypotheses of proposition 5, the spread of wavepackets in thepresence of an absolutely continuous component is exactly ballistic.15

Remark 2. Proposition 5 applies in several cases of concrete interest, including tight-binding discrete models and quantum maps like the kicked rotor26 or the kicked -Harpermodel4.VI.- Dynamical Dimensions.In this section, we shall discuss a di�erent characterization of the structure of wavepack-ets, which still makes reference to a speci�c basis, but, unlike moments, does not dependon the labelling of the basis vectors, or sites. On formal grounds, the construction to bepresently described looks like a multifractal analysis of the measure Pt; , with the scalinglimit de�ned by t!1: It has been proposed, and numerically implemented, on the criticalHarper model, in17. In the following, k k = 1, and completeness of B, are always assumed.Let us de�ne the family of partition functionsZq( ; t) := Xk2Z pqk(t); (16)for all values of q such that the series converges at all times t. Such values include a half lineq > q0;with 0 � q0 � 1; for instance, under the assumptions of Proposition 5, Zq is �nite8t;8 q > 0. In the following q0 < 1 is assumed. Generalized entropies are de�ned, for q 6= 1,by Sq := 11 � q logZqand for q = 1 by S1 = Xk2Z�(pk(t)) (17)where �(x) = �x log x for 0< x � 1; �(0) = 1: S1 is the Shannon entropy of the distribution(5). With such de�nitions, Sq is a continuous function of q: Finally, de�ne the number ofstates 16

Nq(t) := eSq(t) = Z 11�qq (t) (18)(the 2nd equality for q 6= 1) which, at �xed t, is a nonincreasing function of q:The growth exponents ��(Nq) = D�q will be called here dynamical dimensions. Fromtheir very de�nition it follows thatProposition 6. D�q are nonincreasing functions of q, and (1�q)D�q are convex functionsof q: Therefore, D�q are continuous functions of q, except possibly at q = 1.Proposition 7. (i) D�q � dim+H(�) for q < 1, (ii) D�1 � dim�H(�):Proof. Given � > 0; q < 1; de�ne A�;q;t := nk 2 Z : pk(t) < � 11�qNq(t)�1oand B�;q;t := ZnA�;q;t . Then pq�1k > ��1Zq(t) if k 2 A�;q;t. FromZq(t) � Xk2A�;q;t pk(t)pq�1k (t) � ��1Zq(t)P ;t(A�;q;t)we obtain P ;t(B�;q;t) � 1� �. >From the de�nition of a minimal -support of P ;t it followsthat ](B�;q;t) � n�( ; t); therefore, Zq(t) � Pk2B�;q;t pqk(t)� n�( ; t)� q1�qNq(t)�q= n�( ; t)� q1�qZq(t)� q1�qwhich entails Nq(t) = Zq(t) 11�q � � q1�qn�( ; t). Since this holds for arbitrarily small �, part(i) of the thesis follows from ineq.(7).If q = 1; let A�;t := nk 2 Z : pk(t) < e�S1(t)=�o, and B�;t = Z n A�;t. If k 2 A�;t, thenlog 1pk(t) > S1(t)� , which implies P ;t(B�;t) � 1 � �, so ](B�;t) � n�( ; t). Now let �B�;t :=B�;t \ fk 2 Z : pk(t) < e�1g. Then ]( �B�;t) � n�( ; t)� 3. Consequently,17

S1(t) � Pk2 �B�;q;t�(pk(t))� (n�( ; t)� 3)��1S1(t)e�S1(t)�because �(x) is increasing in (0; e�1). Now, if ��(n�) = 0, then also dim�(�) = 0 bythm.2, so (ii) is obvious; if, instead, ��(n�) > 0, then n� > 3 eventually, so, from (),N1 � ���(n�( ; t)� 3)�, and �nally D�1 � ���(n�) � �dim�H(�) by Theorem 2. Since � < 1is arbitrary, (ii) is proved.2Remark 1. D�1 � dim+H(�) is false in general. If the "width" of the wavepacket onthe basis B is measured by means of the "informational" number of states N1, then thewavepacket does not necessarily spread as fast as its fastest component, contrary to whathappens with moments. The following elementary example illustrates this and other pecu-liarities. Let Uek = ek+1 for k 6= 0;�1, and Ue�1 = e1; Ue0 = e0: If = e0 cos� +e1 sin�;then d� = (cos 2�)�+(sin 2�)dx, with �, dx the Dirac and Lebesgue measures respectively;so dim+H(�) = 1. On the other hand, D�q = 0; sin 2�; 1 respectively for q > 1; q = 1; q < 1:This example shows that, at �xed dim+H(�); the value of D�1 is a�ected by the weight ofdi�erent spectral components , and suggests that a discontinuity of D�q at q = 1 may betypical of non-exactly dimensional spectral measures. In particular, such a discontinuityshould always appear, in case of coexistence of both a point and a continuous component ofthe spectral measure.Remark 2. Inequalities of "thermodynamic" type show that N1 cannot increase fasterthan (m(�)(t)) 1� :Remark 3. Like moments, Nq (q � 1) can, in principle, diverge as t ! 1, even in theabsence of a continuous spectrum.Remark 4. In case � has a point component (and only in that case), some at least ofthe pk(t) will not converge to 0 as t!1; then Zq will be bounded away from zero at largetimes, and Nq will not diverge in the limit t!1 if q > 1: Therefore the presence of a point18

component of � enforces D�q = 0 for q > 1:Finding upper bounds for dynamical dimensions appears di�cult in general, except inthe limit q ! 1, and in the particular case in which is one of the basis vectors, whichwithout loss of generality we may assume to be e0. Then letlim inft!1 log p0(t)log t = �c+ ; lim supt!1 log p0(t)log t = �c�Now c� can be exactly identi�ed with certain fractal dimensions of �. Rigorous reformu-lations of a result by Ketzmerick,Petschel, and Geisel3, due to Holschneider9, Barbaroux,Combes, and Motcho11, and Guerin and Holschneider10), show that c� = D�2 (�); the upperand lower correlation dimensions of the spectral measure (for the exact de�nition of whichthe reader is deferred to the quoted papers) . In the case when D+2 (�) = D�2 (�), theircommon value is just called correlation dimension and denoted D2(�)). Then, denoting D�1the q!1 limits of the monotonic functions D�q , we haveProposition 8. If 2 B, then D�1 � D�2 (�); and D+1 � D+2 (�):Proof. If q > 1; from Zq(t) � pq0(t) � t�(c++�)q, which is eventually true (in t) for anysmall �; we get D+q � qq�1(c+ + �): Analogously, from Zq(t) � pq0(t) � t�(c�+�)q, which isfrequently true, we get Nq(t) � tq(c�+�) frequently, that is, D�q � qq�1(c� + �):2The above results intuitively �t in a rough qualitative picture of the role of spectraldimensions. The behaviour of Zq(t) at q < 1 is mainly determined by that part of theprobability distribution, which decays faster. This part corresponds to the fastest componentof the wave packet, which, according to thm.2 (in the form of ineq.(7)), probes the "mostcontinuous" part of the spectrum; this is the sense of proposition 7. As q > 1 increases,instead, the slowly propagating part of the wave packet becomes more and more important.19

This very part also accounts for the tail in the decay of the probability at the initial site,which is determined by a less continuous region of the spectrum, and is in fact described bythe correlation dimension(s).The author wishes to thank the organizers of the semester on Mathematical Methods ofSolid State Physics, held in Wien at the Erwin Schr�odinger Institut for Mathematical Physics(ESI). The hospitality of ESI, where most of this work was done, is gratefully acknowledged.Discussions with G.Mantica and Y.Last, and their comments on a draft of this paper, arealso acknowledged.

20

REFERENCES1M.Kohmoto, Phys.Rev.Lett. 51 1198 (1983); C.Tang and M.Kohmoto, Phys.Rev. B 342041 (1986); H.Hiramoto and M.Kohmoto, Int. J. Mod. Phys.B 6 281 (1992) .2T.Geisel, R.Ketzmerick, and G.Petschel, Phys.Rev.Lett. 67, 3635 (1991).3R.Ketzmerick, G.Petschel and T.Geisel, Phys.Rev.Lett. 69, 695 (1992).4R.Lima, D.Shepelyansky, Phys.Rev.Lett. 67 1377 (1991); T.Geisel et al., loc.cit.;R.Artuso, G.Casati and D.L.Shepelyansky, Phys.Rev.Lett. 68 3826 (1992); R.Artuso,F.Borgonovi, I.Guarneri, L.Rebuzzini and G.Casati, Phys.Rev.Lett. 69 3302 (1992).5M.Wilkinson and E.J.Austin, Phys. Rev B 50 1420 (1994).6 J.X.Zhong, J.Bellissard and R.Mosseri, J.Phys.:Condens.Matter 7, 3507 (1995).7 I.Guarneri and M.diMeo, J.Phys. A: Math.Gen. 28 2717 (1995).8 I.Guarneri and G.Mantica, Phys.Rev.Lett. 73, 3379 (1994).9M.Holschneider, Comm.Math.Phys. 160 457 (1994).10C.A.Guerin and M.Holschneider, preprint CPT-95/P.3261, Marseille 1995.11 J.M.Barbaroux, J.M.Combes, and R.Montcho, preprint CPT-96/P.3303, Marseille 1996.12 I.Guarneri, Europhys.Lett. 10 95 (1989).13 J.M.Combes, in Di�erential equations, with applications to Mathematical Physics,W.F.Ames, E.M.Harrel and J.V. Herod eds., Boston, AP 1993.14 I.Guarneri, Europhys.Lett. 21 729 (1993).15 I.Guarneri, G.Mantica, Ann.Inst.Henri Poincar�e, 71 369 (1994).16Y.Last, "Quantum Dynamics and Decompositions of Singular Continuous Spectra",preprint, May 1995 (to appear in J.Funct.Anal.).21

17 S.N.Evangelou and D.E.Katsanos, J.Phys. A: Math.Gen. 26 L1243 (1993).18B.Huckenstein and L.Schweitzer, Phys. Rev. Lett. 72 713 (1994).19C.A.Rodgers, Hausdor� Measures, Cambridge University Press, 1970.20R.del Rio, S.Jitomirskaya, Y.Last and B.Simon, "Operators with Singular ContinuousSpectrum, IV(...)" preprint (submitted to J. d'An. Math.)1995.21Y.Pesin and H.Weiss, "A Multifractal Analysis of Gibbs Measures....", preprint, May 1995.22 J.P.Eckmann, D.Ruelle, Revs.Mod.Phys. 57,3,617 (1995) .23D.Bessis, J.S.Geronimo, and P.Moussa, J.Stat.Phys. 34, 75 (1984).24M.F.Barnsley, Fractals Everywhere, 2nd ed., Academic Press 1993, p.171.25R.S.Stricharz, J.Funct.Anal. 89, 154 (1990).26G.Casati,I.Guarneri, Comm.Math.Phys., 95 121 (1984). .

22