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DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1,

FRACTAL DIMENSIONS AND THE PHENOMENON OF INTERMITTENCY IN QUANTUM DYNAMICS JEAN-MARIE BARBAROUX, FRANÇOIS GERMINET, AND SERGUEI TCHEREMCHANTSEV

Abstract We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions Dµ±ψ 1/(1 + p/d) of the measure µψ (where d is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension. 1. Introduction A by now wide number of articles deals with the links between the quantum dynamics of wave packet solutions of the Schrödinger equation and the spectral properties of the associated Hamiltonian H . Actually, during the last decade an analysis originated by I. Guarneri in [15] and [16] and refined by others in [6], [8], [24], [18], and [27] established that the fractal properties of the spectral measures were relevant for the study of the spreading of wave packets. Consider a separable Hilbert space H , an orthonormal basis {en }n∈N , and a self-adjoint operator H on H . Let ψt = e−it H ψ be the solution of the Schrödinger equation ( t i ∂ψ ∂t = H ψt , ψt=0 = ψ. P With X = n nh· en i en being the position operator, we define the time-averaged moments of order p for ψ as Z Z

1 T 1 TX p

p/2 −it H 2 p hh|X | iiψ,T := ψ dt = |n| |hψt , en i|2 dt.

|X | e H T 0 T 0 n∈N (1.1) DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 29 December 1999. Revision received 28 September 2000. 2000 Mathematics Subject Classification. Primary 81Q10, 28A80; Secondary 35J10.

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162

BARBAROUX, GERMINET, AND TCHEREMCHANTSEV

In the specific case H = `2 (Zd ), more relevant from a physical point of view, {en } RT P p 2 is the canonical basis {δn }n∈Zd and hh|X | p iiψ,T = 1/T 0 n∈Zd |n| |ψt (n)| dt. It is now well known from a series of results (see [1], [7], [8], [15], [16], [27]) in the case H = `2 (Zd ) (extendable to H = L2 (Rd )) that if dim H (µψ ) is the Hausdorff dimension of the spectral measure µψ , then α − (ψ, p, d) := lim inf T →∞

loghh|X | p iiψ,T ≥ dim H (µψ ) p/d. log T

(1.2)

More recently, in [18] a lower bound has been proven for the upper oscillations of hh|X | p iiψ,T , namely, α + (ψ, p, d) := lim sup T →∞

loghh|X | p iiψ,T ≥ dim P (µψ ) p/d, log T

(1.3)

where dim P (µψ ) is the packing dimension of µψ . However, those results are certainly not optimal. Some 1-dimensional quantum systems with pure point spectrum can give rise to an almost ballistic motion (see [34]); that is, α + (ψ, 2, 1) = 2. Meanwhile, dim P (µψ ) = dim H (µψ ) = 0 for pure point measures. A similar phenomenon has been argued to hold for the random dimer model (see [13] [12]). In quasiperiodic models almost ballistic motion (α + (ψ, 2, 1) = 2) turns out to be a common phenomenon, actually a generic phenomenon (see [11]), even in presence of purely zero Hausdorff dimensionality of the spectral measures (see [27], [11]). These examples show how far we are from a complete understanding of “What determines the spreading of a wave packet" (see [23]). In this paper we go one step further in this undertanding and also supply (see Appendix B) a new enlightment concerning the main technique used in this field for the past ten years. We obtain new lower bounds for the growth exponents of hh|X | p iiψ,T , namely, (1.4) and (1.5). As in (1.2) and (1.3), these bounds rely only on the fractal properties of the spectral measure µψ . We point out right now that unlike the existing results, the bounds (1.4) and (1.5) we get can be nontrivial in the presence of zero dimensionality of the spectral measure (dim P (µψ ) = dim H (µψ ) = 0), even in the case of pure point spectrum (see Appendix D). Moreover, in Appendix D, Theorem D.1, we show that there is no hope to improve our result by only taking into account the fractal properties of the spectral measure encoded in its generalized fractal dimensions Dµ±ψ (q). Our result also provides a precise statement of a phenomenon discovered by recent numerical computations in some quantum models for quasicrystals (see [28], [29], [32], [38]); this phenomenon has been called intermittency. Namely, it has been suggested by physicists that the growth exponents α ± (ψ, p, d) should grow faster than linearly in p/d as proposed in (1.2) and (1.3); one should observe a more complex law for the behaviour of α ± (ψ, p, d) in the variable p/d: α ± (ψ, p, d) =

QUANTUM DYNAMICS AND MULTIFRACTAL DIMENSIONS

163

β ± (ψ, p/d) p/d, where β ± (ψ, p/d) are nondecreasing functions of p/d (and of course not smaller than dim H and dim P ). These recent numerical investigations emphasized, in this phenomenon of intermittency, the role of more refined fractal quantities, the so-called qth generalized fractal dimensions Dµ±ψ (q). The lower bounds (1.4) and (1.5) that we establish for hh|X | p iiψ,T actually appear to be intermittent lower bounds in the following sense: when Dµ± (q) is nonconstant for q ∈ (0, 1), these lower bounds grow faster than linearly in p/d. After Theorem 2.1 we discuss the application of our result to a Hamiltonian constructed with Julia matrices with self-similar spectra (see [5], [19]). For “real" Schrödinger operators, good candidates would be operators on `2 (N) or `2 (Z) with sparse or quasiperiodic (e.g., generated by substitution sequences) potential. But a careful analysis of the links between spectral properties and behaviour of the eigenfunctions would then be required, an analysis that goes, for instance, beyond the scope of [22]. Our main result (Theorem 2.1) holds for any self-adjoint operators H and for any initial state ψ such that the associated spectral measure dµψ satisfies (H)

Dµ±ψ (s) < +∞ for any s ∈ (0, 1).

The result reads then as follows: α (ψ, p, d) ≥ −

Dµ−ψ

and α + (ψ, p, d) ≥ Dµ+ψ

1 1 + p/d

1 1 + p/d

p/d

(1.4)

p/d,

(1.5)

where Dµ−ψ (q) and Dµ+ψ (q) are the lower and upper qth generalized fractal dimensions (see Definition 2.2). In Appendix C we discuss the validity of Hypothesis (H), which, we note, holds for any compactly supported measures. In particular, Theorem 2.1 applies to the examples where the intermittency phenomenon has been argued to hold. To achieve this we first derive a ( p/d)-dependent lower bound L ψ (T ) for hh|X | p iiψ,T (see Theorem 3.1). Then we establish via Theorem 4.2 the connection between L ψ (T ) and the generalized fractal dimensions, by adjusting for each single T the “thin" part of µψ that supplies the faster dynamical travel. Concerning the connection between L ψ (T ) and Dµ±ψ (q), we moreover obtain a kind of optimality, in the sense that this quantity L ψ (T ) that minors hh|X | p iiψ,T is shown to have its growth exponents exactly equal to Dµ±ψ 1/(1 + p/d) p/d. We point out that Dµ±ψ 1/(1 + p/d) are increasing functions of p/d and are, respectively, not smaller than dim H (µψ ) and dim P (µψ ) (for all p/d > 0). Therefore (1.4) and (1.5) do improve the bounds (1.2) and (1.3) above.

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These new bounds are a consequence of a double improvement of the approaches of Guarneri [15], [16], J.-M. Combes [8], and Y. Last [27] and of J.-M. Barbaroux and S. Tcheremchantsev [6]. The first improvement is due, after a decomposition ψ = φ + χ, φ ⊥ χ , to a better control of the key quantity Z 1 ∞ X |he−it H ψ, en i|2 h(t/T ) dt Bψ (T, N ) := T 0 |n|≤N Z ∞ X 2 ≤ Bχ (T, N ) + Re he−it H φ, en ihe−it H ψ, en ih(t/T ) dt, T 0 |n|≤N

(1.6) and, more particularly, of the last term (the crossed term). We stress right now that this better control of the crossed term is essential, meaning that using the former available estimates (see [27], [6]) does not lead to the right fractal dimensions Dµ±ψ 1/(1 + β) , as illustrated in Appendix B. R1 Here h(z) is some positive function in C0∞ ([0, 1]) such that 0 h(z) dz = 1. Afterwards, and this is Theorem 3.1, one is able to obtain a constant C(ψ, p, h) > 0 such that for all T > 0, ( ) 2+2β |hφ, ψi| hh|X | p iiψ,T ≥ C(ψ, p, h)L ψ (T ), with L ψ (T ) = sup β , φ∈Hψ kφk2 Uφ,ψ (T ) where Hψ is the cyclic subspace spanned by ψ and H , Z Uφ,ψ (T ) = dµψ (x) dµφ (y)R T (x − y) , R2

and R is some bounded fast-decaying function defined by (3.5) in Section 3. (One 2 should think of R(w) as of the Gaussian e−w /4 .) Thus, as in [6] and [18], one can choose, for each T , a T -dependent vector φ in the decomposition ψ = φ + χ, which contains enough spectral information to approximate the supremum in L ψ (T ). The second improvement then consists of the way one chooses this particular vector φ. We show (see Theorem 4.1) that a judicious choice enables one to connect, up to a logarithmic factor, the quantity L ψ (T ) to the integral Z h i−( p/d)/(1+ p/d) 1 Iµψ , T −1 = dµψ (x) µψ x − T −1 , x + T −1 1 + p/d R which defines the generalized fractal exponents Dµ±ψ 1/(1 + p/d) . Our method also applies to the previous approaches in [15], [16], [8], [27], and [6], where the crossed term in (1.6) was not treated well. It yields, respectively, as stated in Appendix B, the fractal dimensions Dµ±ψ (1 + β)/(1 + 2β) and Dµ±ψ (1 +


165

2β)/(1 + 3β) . Since the functions Dµ±ψ (q) are nonincreasing functions of q, our Theorem 2.1 gives a better lower bound. This means that a better estimate of the crossed term in (1.6) does provide an improvement (see Theorem B.3). Finally, using extra assumptions on the decay of the generalized eigenfunctions u(n, x) of H (in the spirit of [23], [25]), it is possible to improve the above bounds. In particular, suppose that there exists a constant C such that for µψ a.e. x, P 2 d ± ± |n| 0, Iµ takes the value +∞. Remark 2.2 (i) For our purpose it is sufficient to discuss the case q ∈ (−∞, 1) (see, e.g., [10] for the general case). (ii) There actually exists a wide number of generalized fractal dimensions. For example, they can be defined with the help of the so-called “singularity spectrum function” f µ of the measure µ (see [20], [30]), or as a solution of an implicit equation (see [20, Formula 2.8], [30]). The resulting dimensions coincide with each other in certain very specific cases, like, for example, cookie-cutter measures in R (see [30]).


167

In order to state our results, we also define the following integrals that could be considered as approximations of the quantities Iµ (q, ε). The function R is a bounded even function with fast decay properties at ±∞ and is precisely defined in (3.5): K µ (q, ε) =

Z

Z supp µ

dµ(y)R (x − y)/ε

dµ(x)

q−1

.

R

In Lemma 4.3 we prove that for any measure µ verifying the condition (H) of our theorem, taking K µ (q, ε) in (2.2) instead of Iµ (q, ε) leads to the same values for the generalized fractal dimensions. We review below some properties of the fractal dimension numbers Dµ± (q) that are of interest for us. 2.1 Let µ be a Borel probability measure. (i) Dµ− (q) and Dµ+ (q) are nonincreasing functions of q ∈ (−∞, 1). (ii) For all q ∈ (−∞, 1), Dµ− (q) ≥ dim H (µ). (iii) For all q ∈ (−∞, 1), Dµ+ (q) ≥ dim P (µ). (iv) If µ has a bounded support, then for all q ∈ (0, 1), 0 ≤ Dµ− (q) ≤ Dµ+ (q) ≤ 1. PROPOSITION

Proof Statement (i) is already known (see, e.g., [10]). This is a straightforward consequence of concave and convex Jensen inequalities (for the proof of (ii) and (iii), see Appendix A). Statement (iv) follows from Corollary C.1. Note that (iv) does not necessarily hold any more if one lets q vary in (−∞, 1) (see Appendix D). From now on, each time we refer to the exponent β it should be understood that β = p if H is a general separable Hilbert space with basis {en }n∈N , and β = p/d in the specific case H = `2 (Zd ) equipped with its canonical basis {δn }n∈Zd . Our main result is the following one. THEOREM 2.1 Let H be a self-adjoint operator on a separable Hilbert space H , and let ψ be a vector in H , kψkH = 1. Assume that the spectral measure µψ associated to ψ is such that (H) Dµ±ψ (s) < +∞ for any s ∈ (0, 1).

Then for hh|X | p iiψ,T defined by (1.1) and with β as described just above, the following holds: loghh|X | p iiψ,T 1 − lim inf ≥ β Dµψ , T →∞ log T 1+β

168

and


loghh|X | p iiψ,T lim sup ≥ β Dµ+ψ log T T →∞

1 1+β

.

Remark 2.3 (i) As a consequence of Proposition 2.1(i)–(iii), this result does improve previous known bounds of [27], [1], and [18]. (ii) As we show in Appendix D, Theorem D.1, for any β = p/d > 0 and any δ > 0 there exist a bounded self-adjoint operator H and a vector ψ such that limT →∞ loghh|X | p iiψ,T / log T = 0, but at the same time Dµ±ψ ((1/(1 + β)) − δ) > 0. Therefore one cannot hope to obtain a general (i.e., without additional assumptions on µψ or on generalized eigenfunctions) lower bound of the form β Dµ±ψ q(β) with some q(β) < 1/(1 + β) like, for example, Dµ±ψ (1 − β). Proof of Theorem 2.1 The proof of Theorem 2.1 is the combination of Theorems 3.1 and 4.2, which are proved, respectively, in Sections 3 and 4. We now discuss a model of Jacobi matrices, the Julia matrices. In this case, the upper bound derived in [5] together with Theorem 2.1 above enable one to prove for small p that the increasing exponents of the moments of order p are entirely controlled by the generalized fractal dimensions. Julia matrices There exists a class of models for which nontrivial (i.e., nonballistic) upper bounds for the moments of order p are derived in terms of generalized fractal dimensions: the Julia matrices. They are constructed by considering polynomials, disjoint iterated function systems (IFS), giving rise to a real hyperbolic Julia set J (see, e.g., [5], [19], and references therein for details). Given such an IFS, one considers the balanced measure µ of maximal entropy on J and then constructs a Hamiltonian H (H corresponds to the Jacobi matrix associated to µ) on `2 (N) as follows. Let Pn , n ≥ 0, denote the orthogonal and normalized polynomials associated to µ. The family (Pn )n∈N forms a Hilbert basis in L2 (R, µ) and satisfies a three terms recurrence relation E Pn (E) = tn+1 Pn+1 (E) + vn Pn (E) + tn Pn−1 (E), n ≥ 0, where vn ∈ R and tn ≥ 0 are bounded sequences, and P−1 = 0. Therefore the isomorphism of L2 (R, µ) onto `2 (N) associated with the basis (Pn )n∈N carries the operator of multiplication by E in L2 (R, µ) into the self-adjoint finite difference operator H defined


169

on `2 (N) by H ψ(n) =

tn+1 ψ(n + 1) + tn ψ(n − 1) + vn ψ(n), t1 ψ(1) + v0 ψ(0),

n ≥ 1, n = 0.

(2.3)

Then µ is the spectral measure of H associated to the state δ0 located at the origin. For this model the upper and lower fractal dimensions Dµ± (q) are equal (:=Dµ (q)), and continuous for q ∈ (0, 1); furthermore, we have Dµ 1/(1 + p) = Dµ (1 − p) + O ( p 2 ) (see [5] and references therein). It is established in [5] that there exists a critical value pc ≥ 2 such that for all p ∈ (0, pc ), α + (δ0 , p, 1) ≤ Dµ (1 − p). Therefore, putting together Theorem 2.1 and [5, Theorem 1], we get, for the exponents of any moments of order p ∈ (0, pc ) and for the initial state δ0 , that 1 Dµ ≤ α ± (δ0 , p, 1) ≤ Dµ (1 − p), 1+ p and thus for small p > 0, Dµ (1 − p) + O ( p 2 ) ≤ α ± (δ0 , p, 1) ≤ Dµ (1 − p). This is the first model of a Schrödinger-like operator treated rigorously for which such bounds are derived. If Dµ (q) were known to be strictly decreasing in some interval (0, δ), one would get intermittency for hh|X | p iiψ (T ), with small p. However, to our best knowledge, this fact is only emphasized by numerics on the generalized fractal dimension (see, e.g., [28], [29]), and no rigorous results are provided. 3. A general lower bound This section can be regarded as the first part of the proof of our main result, Theorem 2.1. Let H be a self-adjoint operator in Hilbert space H , and let {en } be an orthonormal basis in H labelled by n ∈ N or by n ∈ Zd . Let ψ be some vector in H such that kψk = 1. We are interested in lower bounds for the moments of the abstract position operator associated to the basis {en }, defined as X p |X |ψ (t) = |n| p |he−it H ψ, en i|2 . (3.1) n

`2 (Zd ),

In particular, if H = one can take the canonical basis en (k) = δnk , n, k ∈ Zd , to obtain the moments of the usual position operator. Our results can also be extended to the case H = L2 (Rd ) (see [3]) and Z p |X |ψ (t) = |x| p |ψ(t, x)|2 dx, ψ(t, x) = (e−it H ψ)(x). Rd

170


It can be done using Theorem 3.2 in the same manner as in [6] (see [6, Corollary 2.4 and Theorem 2.5]; see also [27]). However, for the sake of simplicity, we consider in this paper only |X | p given by (3.1). We derive a lower bound for the time-averaged moments of position operator in terms of an abstract quantity in the spirit of [6], namely, |hφ, ψi|2+2β , kφk2 Uφ,ψ (T )β (3.2) where Hψ is the cyclic subspace spanned by ψ and H . The exponent β = p if n ∈ N, and β = p/d if n ∈ Zd . Notice that if φ = χ (H )ψ, where is a Borel set, then L ψ (φ, T ) reads as kφk2+4β L ψ (φ, T ) = . (3.3) Uφ,ψ (T )β L ψ (T ) = sup L ψ (φ, T ), φ ∈ Hψ , hψ, φi 6 = 0 ,

L ψ (φ, T ) =

The quantity Uφ,ψ (T ) is defined as follows: Z Z Uφ,ψ (T ) = dµφ (x) dµψ (y)R (x − y)T ,

(3.4)

R R

where R(w) is a bounded and fast-decaying function defined in (3.5). Throughout the paper we use the notation (ε playing the role of 1/T ) Z x−y (R) b (x, ε) := dµψ (y)R , ε R R so that Uφ,ψ (T ) = R dµφ (x)b(R) (x, 1/T ). The quantity Uφ,ψ (T ) is crucial since one may consider that it codes the determinating part of the spectral information that is involved in the dynamical behaviour of the considered quantum system. The quantity in (3.4) should be compared to other quantities such as Z Z Z 2 2 dµψ (x) dµψ (y) or dµψ (x) dµψ (y)e−(x−y) T /4 , |x−y|≤1/T

R R

which already appear, in the limit ε = T −1 → 0, as key quantities in order to discuss the nature of the spectrum (see [26], [27], [34], [6]). The main result of this section is the following. THEOREM 3.1 Let H be a self-adjoint operator on H , and let ψ be a vector in H , kψk = 1. Let {en } be some orthonormal basis in H and Z 1 T X p −it H p hh|X | iiψ (T ) = |n| |he ψ, en i|2 dt. T 0 n


171

R1 Let h ∈ C0∞ ([0, 1]) be any positive function such that 0 h(z) dz = 1, and define ( 1 if |w| ≤ 1, R(w) = (3.5) 2 ˆ |h(w)| if |w| > 1, where hˆ stands for the Fourier transform of h. Then, for all p > 0 and with L ψ (T ) as defined in (3.2), there exists a constant C(ψ, p, h) such that for all T > 0, hh|X | p iiψ (T ) ≥ C(ψ, p, h)L ψ (T ). Remark 3.1 We point out that h seems to be the necessary trick to take into account averaging on time between [0, T ] only, instead of [−T, T ]. This trick actually allows one to deal with the crossed term in (1.6) and to recover a function R with fast decay at ±∞. 2 Replacing h by the usual Gaussian e−z /4 in (3.15) and thus in (3.6) leads to the same p result, but with |X |ψ (t) averaged over [−T, T ]. As is now well known (see [15], [16] [8], [27]), a key point in the proof of lower bounds for hh|X | p iiψ (T ) is a good control on the behaviour of the wave packet inside a ball of radius N , namely, Z 1 ∞ X Bψ (T, N ) = |he−it H ψ, en i|2 h(t/T ) dt. (3.6) T 0 |n|≤N

To that end, we need Theorem 3.2, which is a generalization of [6, Theorem 2.1]. In order to state it, we first recall well-known facts involving the spectral theorem for the self-adjoint operator H and the chosen vector ψ (see, e.g., [33] and references therein). Namely, there exists a unitary map Wψ from the cyclic subspace Hψ into the space L2 (R, dµψ ) such that Wψ (ψ) = 1 and Wψ (e−it H ψ) = e−it x . We denote by Pψ the orthogonal projection on Hψ . The map Wψ has a kernel u(n, x) defined by u(n, ·) = Wψ (Pψ en )(·), so that Z −i H t (3.7) he ψ, en i = e−i xt u(n, x) dµψ (x), and, more generally, for any vector ξ ∈ H , one has Z hPψ ξ, en i = Wψ Pψ ξ (x)u(n, x) dµψ (x).

(3.8)

In the case H = `2 (Zd ) and en (k) = δnk , for each fixed x ∈ R the vector u(n, x)n∈Zd may be seen as a generalized eigenfunction of H (i.e., in a distributional sense). This observation is of interest for some applications (see [26] and Theorem 4.3).

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The expansions (3.7) and (3.8) are of course also possible with any vector φ ∈ H and corresponding kernel v(n, y) = Wφ (Pφ en )(y). (Actually, if φ = f (H )ψ, with f ∈ L2 (R, dµψ ), then one checks that v(n, y) = f (y)u(n, y).) We are now ready to formulate Theorem 3.2. 3.2 Let h(z) be some function in L1 (R), let H be a self-adjoint operator acting on H , and let A be a Hilbert-Schmidt operator in H . For any couple of vectors ψ, φ from H , define the quantity Z 1 +∞ (h) hAe−it H φ, e−it H ψih(t/T ) dt. Dφ,ψ (T ) = T −∞ THEOREM

Let (h) Uφ,ψ (T )

Z Z

ˆ dµφ (x) dµψ (y)|h((x − y)T )|2 .

= R R

The following estimate holds: 1/2 (h) (h) |Dφ,ψ (T )| ≤ kAk2 Uφ,ψ (T ) , where kAk2 is the Hilbert-Schmidt norm of A. P In the special case A = |n|≤N h·, en ien , one has kAk2 ≤ C N d/2 , and therefore Z +∞ X 1 (h) |Dφ,ψ (T, N )| = he−it H φ, en ihen , e−it H ψih(t/T ) dt T −∞ |n|≤N

1/2 (h) ≤ C N d/2 Uφ,ψ (T ) . Here d = 1 if one considers the abstract position operator associated with the base {en } labelled by n ∈ N, and d ≥ 1 in the case H = `2 (Zd ) equipped with the canonical basis en = δn . Proof Since A is Hilbert-Schmidt, there exist two orthonormal bases { f n }n∈N and {gn }n∈N of P 2 H and a monotonely decreasing sequence {E n }n∈N , E n ≥ 0, such that ∞ n=1 E n = P ∞ 2 kAk2 < +∞ and A = n=1 E n h·, f n ign . Therefore (h)

Dφ,ψ (T ) =

Z

1 T

∞ +∞ X

E n he−it H φ, f n ihgn , e−it H ψih(t/T ) dt.

(3.9)

−∞ n=1

Then (3.7) reads as −it H

he

φ, f n i =

Z

dµφ (x)e−it x u(n, x), R

(3.10)


173

where u(n) = Wφ (Pφ f n ) ∈ L2 (R, dµφ ). The similar formula holds for ψ with v(n) = Wψ (Pψ gn ) ∈ L2 (R, dµψ ). One obtains from (3.9) and (3.10) that Z Z (h) dµφ (x) dµψ (y)hˆ (x − y)T S(x, y), (3.11) Dφ,ψ (T ) = R R

where S(x, y) =

∞ X

E n u(n, x)v(n, y).

n=1

The sum converges in L2 (R2 , µφ × µψ ). Applying the Cauchy-Schwarz inequality to (3.11), one gets (h)

(h)

|Dφ,ψ (T )|2 ≤ Uφ,ψ (T )kSk2L2 (R2 ,dµ

φ ×dµψ )

.

(3.12)

One can easily see that kSk2L2 (R2 ,dµ

φ ×dµψ )

∞ X

=

E n E k ank bnk ,

n,k=1

where Z ank = R

dµφ (x)u(k, x)u(n, x) = Wφ (Pφ f k ), Wφ (Pφ f n ) L2 (R,dµ

φ)

= hPφ f k , f n iH ,

(3.13)

where we used an analog of (3.8), and in the same manner bnk = hPψ gk , Pψ gn iH = hgk , Pψ gn iH . We have also used the fact that both Pφ and Pψ are orthogonal projections. By Parseval equality, ∞ X

∞ X

|ank |2 = kPφ f k k2 ,

n=1

|bnk |2 = kPψ gn k2 .

k=1

Therefore, as k f k k = kgn k = 1 for all k, n, kSk4L2 (R2 ,dµ ×dµ ) φ ψ

≤

∞ X

E k2 kPφ f k k2

k=1

∞ X

E n2 kPψ gn k2

n=1

≤

X ∞

E n2

2

= kAk42 .

n=1

(3.14) The first statement of the theorem follows from (3.12) and (3.14). The proof of the second part is essentially the same. The only difference is that in the case H = `2 (Zd ) the sums are taken over n ∈ Zd : |n| ≤ N . In particular, estimate (3.14) reads X X kSk4L2 (R2 ,dµ ×dµ ) ≤ kPφ ek k2 kPψ en k2 ≤ C N 2d . φ

ψ

|k|≤N

This ends the proof.

|n|≤N

174


One should stress that the proof we presented here is simpler than the proof of [6, Theorem 2.1] because we do not use the product space H ⊗ H . In the case ψ = φ and h(z) = exp(−z 2 /4), the result of Theorem 3.2 is equivalent to that of [6, Theorem 2.1]. Proof of Theorem 3.1 R1 Pick a positive function h(z) ∈ C0∞ ([0, 1]) such that 0 h(z) dz = 1. The role of 2 . Note that one trivially has, for any ˆ h is to supply a fast-decaying function |h(w)| z ∈ [0, 1], h(z) ≤ khk∞ χ[0,1] (z). Then one verifies that Z ∞X 1 dt p hhX iiψ (T ) ≥ |n| p |he−it H ψ, en i|2 h(t/T ) khk∞ 0 T n ≥

Np kψk2 − Bψ (T, N ) , khk∞

(3.15)

with Bψ (T, N ) defined in line (3.6). As usual, one needs a control of this quantity Bψ (T, N ) which represents the behaviour of the wave packet in a ball of radius N . Decompose the vector ψ as φ + χ, with hφ, χi = 0 and φ 6= 0. (One should think of φ = χ (H )ψ.) Thus Bψ (T, N ) = Bφ (T, N ) + Bχ (T, N ) Z ∞ X 2 + Re he−it H φ, en ihe−it H χ, en ih(t/T ) dt T 0 |n|≤N

= −Bφ (T, N ) + Bχ (T, N ) Z ∞ X 2 Re he−it H φ, en ihe−it H ψ, en ih(t/T ) dt. + T 0 |n|≤N

R∞ Then, taking into account that 1/T 0 h(t/T ) dt = 1 and h(z) ≥ 0, we have P Bχ (T, N ) ≤ kχ k2 = kψk2 − kφk2 . Let A = |n|≤N h·, en ien . Then (h)

Bψ (T, N ) ≤ kψk2 − kφk2 + 2 Re Dφ,ψ (T, N ), (h)

where Dφ,ψ (T, N ) was defined in Theorem 3.2. The second statement of this theorem immediately gives 1/2 (h) Bψ (T, N ) ≤ kψk2 − kφk2 + C N d/2 Uφ,ψ (T ) , (3.16) where, as in Theorem 3.2, (h)

Uφ,ψ (T ) =

Z Z R R

ˆ dµφ (x) dµψ (y)|h((x − y)T )|2 .


175

2 ˆ ˆ As |h(w)| ≤ 1 for all w, and by definition (3.5) of R, we clearly have R(w) ≥ |h(w)| (h) for all w. Therefore Uφ,ψ (T ) ≤ Uφ,ψ (T ), and estimate (3.16) is valid with Uφ,ψ (T ) ˆ 2. defined by (3.4), that is, with the function R instead of |h|

We are now in position to finish the proof of Theorem 3.1. The basic strategy is standard: let N be the largest integer such that C N d/2 Uφ,ψ (T )1/2 ≤ kφk2 /2; it yields kφk2 . 2 Inequalities (3.17) and (3.15) yield, with some positive constant C(ψ, p, h), Bψ (T, N ) ≤ kψk2 −

hh|X | p iiψ,T ≥ C(ψ, p, h)

kφk2+4β = C(ψ, p, h)L ψ (φ, T ), Uφ,ψ (T )β

β=

p . d

(3.17)

(3.18)

One recovers L ψ (φ, T ) as given in line (3.3). It is this latter lower bound that is used in the proof of the lower bound in Theorem 4.1. However, in the more general case where φ is any function of H with hψ, φi 6= 0, one gets the bound with L ψ (φ, T ) :=

|hφ, ψi|2+2β . kφk2 Uφ,ψ (T )β

(3.19)

Indeed, take such a φ, and then define φ˜ = (hφ, ψikφk−2 )φ as in [6]; one thus ˜ then hφ, ˜ χi = 0 and one is able to apply the result checks that if χ = ψ − φ, ˜ Taking into account that kφk ˜ = |hφ, ψi|kφk−1 and that line (3.3) to ψ and φ. 2 −4 Uφ,ψ ˜ (T ) = |hφ, ψi| kφk Uφ,ψ (T ), one finds the announced expression (3.19). To optimize the lower bound, we should take the supremum of L ψ (φ, T ) for a given T over all possible φ. One can show in the same manner as in [6, Lemma 3.1] that it is sufficient to take φ only from the cyclic subspace Hψ . This gives us L ψ (T ) defined in line (3.2). 4. Toward the fractal dimensions This section deals with the connection between the dynamic quantity L ψ (T ) introduced in the previous section and the fractal dimensions defined in Section 2, called the generalized fractal dimensions. We prove the following. THEOREM 4.1 Let H be a self-adjoint operator on H , and let ψ be a vector in H , kψkH = 1. Assume (H), namely, the spectral measure µψ associated to ψ is such that

Dµ±ψ (s) < +∞

for any s ∈ (0, 1).

176


Then, for all β > 0, there exists a constant C1 > 0 such that, for all ε > 0: 1+β 1+β C1 1 1 −1 K , ε ≤ L (ε ) ≤ K , ε , µψ ψ µψ 1+β 1+β | log ε|1+β

(4.1)

where K µψ

Z Z −β/(1+β) 1 ,ε = dµψ (x) dµψ (y)R((x − y)/ε) . 1+β supp µψ R

This actually implies the following. 4.2 Under the same hypothesis (H) as previously stated, one has log L ψ (T ) log L ψ (T ) 1 1 − + = β Dµψ , lim sup = β Dµψ . lim inf T →∞ log T 1+β log T 1+β T →∞ THEOREM

Remark 4.1 (i) We point out that the first inequality in (4.1), that is, the lower bound on L ψ (ε−1 ), is sufficient to prove our main result, Theorem 2.1. However, the right side of (4.1), that is, the upper bound, is of interest too. In particular, it says that once one derives the lower bound L ψ (T ), one cannot hope for a better result than the one we stated. We also take advantage of the right part of (4.1) in Appendix B, Theorem B.3. (ii) Note that the proof of the right-hand side inequality in (4.1), that is, the upper bound, is true for any measures µψ . (iii) In Appendix C we show that (H) holds for all measures verifying the condition Z |x|λ dµψ (x) < +∞ for any λ > 0. (4.2) R

This is true in particular if µψ has a compact support. Moreover, if (4.2) holds, Dµ±ψ (s) ∈ [0, 1] for any s ∈ (0, 1). We start by providing a proof for Theorem 4.1. Throughout this section, integrations must be systematically understood on the support of the measure µψ or µφ we consider. Proof of the upper bound in Theorem 4.1 The upper bound of (4.1) in Theorem 4.1 is proved if, taking any function f ∈ L2 (R, dµψ ), one shows 1+β Z 1+β 1 L ψ ( f (H )ψ, ε−1 ) ≤ K µψ ,ε = dµψ (x)b(R) (x, ε)−β/(1+β) , 1+β (4.3)


177

R where b(R) (x, ε) = dµψ (y)R((x − y)/ε). This result follows from CauchySchwarz and Hölder inequalities. Pick φ = f (H )ψ, with f ∈ L2 (R, dµψ ). Therefore Z dµφ (x) = | f (x)|2 dµψ (x) and hφ, ψi = dµψ (x) f (x). R Remember also the equality Uφ,ψ (T ) = dµφ (x)b(R) (x, ε). Then, rewriting −1 L ψ (( f (H )ψ, ε ) as given in (3.2), one gets R dµψ (x) f (x) 2+2β −1 L ψ ( f (H )ψ, ε ) = R R β . (4.4) dµψ (x)| f (x)|2 dµψ (x)| f (x)|2 b(R) (x, ε) One starts with a Cauchy-Schwarz inequality applied to the numerator and to the functions b(R) (x, ε)−β/(2(1+β)) and f (x)b(R) (x, ε)β/(2(1+β)) . It yields Z 2+2β Z 1+β (R) −β/(1+β) dµψ (x) f (x) ≤ dµ (x)b (x, ε) ψ Z ×

dµψ (x)b

(R)

β/(1+β)

(x, ε)

| f (x)|

2

1+β

= K µψ (1/(1 + β), ε)1+β Z 1+β (R) β/(1+β) 2 × dµψ (x)b (x, ε) | f (x)| .

(4.5)

A Hölder inequality applied to the last term, and with the coefficients p = 1 + β and p 0 = (1 + β)/β, leads to Z 1+β (R) β/(1+β) 2 dµψ (x)b (x, ε) | f (x)| Z

dµψ (x)(| f (x)| )

Z

dµψ (x)| f (x)|2

= ≤

2 1/(1+β)

Z

β/(1+β) 1+β 2 (R) | f (x)| b (x, ε)

β dµψ (x)| f (x)|2 b(R) (x, ε) .

(4.6)

One thus recovers exactly the denominator term, and (4.3) holds. Remark 4.2 We stress the very strong link that comes out between L ψ (ε−1 ) and the integral K µψ (q, ε) with the particular value q = 1/(1+β). The same appears to hold with the other lower bounds L 1 (T ) and L 2 (T ) defined in Appendix B and coming from the former approaches (see [15], [16], [8], [27], [6]). This shows how relevant the fractal dimensions Dµ± (q) are with regard to the time behaviour of the quantities that have been studied for many years in quantum dynamics.

178


We now turn to the second and main part of Theorem 4.1, that is, the lower bound. Our basic strategy to get the lower bound in (4.1) is to estimate the quantity L ψ (φ, T ) in (3.19), with a vector φ = χ(r ) (H )ψ. And (r ) is a “thin" set of the form (r ) = {x ∈ supp µψ | εr +A/N < b(R) (x, ε) ≤ εr }, but which supports, roughly speaking, a significant part of the mass of the integral K µψ (1/(1 + β), ε). The constant A(q) is fixed by Lemma 4.1. The integer N stands for the integer part of log T . Before going on with the proof of Theorem 4.1, we need the following two lemmas. 4.1 ± Let q ∈ (0, 1), and suppose that (H) holds for µψ ; that is, Dµψ (s) < +∞ for any R −1 s ∈ (0, 1). Define b(x, ε) = dµψ (x)g (x − y)ε , with g(w) = χ[−1,1] (w) or R(w). Then there exist A = A(q) and ε0 (q) > 0 such that for all ε ∈ (0, ε0 ), Z b(x, ε)q−1 dµψ (x) ≤ ε. (4.7) LEMMA

{x∈supp µ | b(x,ε)≤ε A }

LEMMA 4.2 Let µψ , q, and A = A(q) be as in the previous lemma. Let N > 0 be an integer. Then, for b(x, ε) defined as in Lemma 4.1, there exist an r0 and a set (r0 ) = {x ∈ supp µψ | εr0 +A/N < b(x, ε) ≤ εr0 } such that, for all ε small enough, Z Z 1 q−1 b(x, ε)q−1 dµψ (x). b(x, ε) dµψ (x) ≥ 2N (r0 )

Lemma 4.2 is the key lemma to get Theorem 4.1 since it is this lemma that supplies the set (r ), and so the vector φ = χ(r ) (H )ψ, that is needed to prove Theorem 4.1. Lemma 4.1 does not enter explicitly in the proof of Theorem 4.1 but is an important ingredient that we use twice: once while proving Lemma 4.2 and then in Lemma 4.3. Proof of the lower bound in Theorem 4.1 Let N be the integer part of − log ε. For the sake of simplicity we use N = − log ε (rather than the integer part). Take A as in Lemmas 4.1 and 4.2. Here q = 1/(1 + β), and thus q − 1 = −β/(1 + β). For fixed ε we choose φ = χ(r0 ) (H )ψ with r0 given by Lemma 4.2. Thus from the definition of (r0 ) we obtain Z −1 Uφ,ψ (ε ) = b(R) (x, ε) dµψ (x) ≤ εr0 µψ (r0 ) . (r0 )


179

Therefore, using expression (4.4) of L ψ f (H )ψ, ε−1 with f = χ(r0 ) , one has L ψ (ε

−1

1+2β Z 1+β µψ (r0 ) (Aβ)/N (R) −β/(1+β) )≥ b (x, ε) dµψ (x) β ≥ ε (r0 ) εr0 β µψ (r0 ) Z 1+β e−Aβ (R) −β/(1+β) b (x, ε) dµψ (x) ≥ (−2 log ε)1+β 1+β C1 1 K , ε , (4.8) = µψ 1+β | log ε|1+β

where in the last inequality we used Lemma 4.2 and N = − log ε. This holds for ε small enough. Remark that the fact that r0 disappears in relation (4.8) is crucial. It is of course related to the particular value q = 1/(1 + β) of the fractal dimension that enters into account. This is the place where the deep link between L ψ (T ) and Dµ±ψ (1/(1 + β)) shows up. We are left with the proofs of Lemmas 4.1 and 4.2. Proof of Lemma 4.1 For any A > 0, ε > 0, define B(A, ε) = {x ∈ supp µ | b(x, ε) ≤ ε A }. Let 0 < s < q < 1. As b(x, ε) ≥ µ([x − ε, x + ε]) whatever b(x, ε) is, we can estimate Z Z q−1 b(x, ε) dµ(x) = b(x, ε)q−s b(x, ε)s−1 dµ(x) B(A,ε) B(A,ε) Z A(q−s) ≤ε b(x, ε)s−1 dµ(x) B(A,ε) Z A(q−s) ≤ε µ([x − ε, x + ε])s−1 dµ(x) B(A,ε)

≤ε

A(q−s)

Iµ (s, ε).

(4.9)

Let us take, for example, s = q/2. As Dµ+ (s) < +∞ for any s > 0, for ε small enough one has (Dµ+ (s)+1)(1−s) 1 Iµ (s, ε) ≤ . ε Taking A = (q − s)−1 (Dµ+ (s) + 1)(1 − s) + 1 in (4.9), we obtain the result of the lemma.

180


Proof of Lemma 4.2 To simplify the notation, denote B A = {x ∈ supp µψ | b(x, ε) ≤ ε A } and also B A = {x ∈ supp µψ | b(x, ε) > ε A }. Then, using the bound of Lemma 4.1, one has for ε small enough Z Z Z b(x, ε)q−1 dµψ (x) = b(x, ε)q−1 dµψ (x) + b(x, ε)q−1 dµψ (x) BA BA Z ≤ b(x, ε)q−1 dµψ (x) + ε. BA

Remark that one always has b(x, ε) ≤ 1 (since R(w) ≤ 1). Thus one can divide the remaining set B A into N parts: (k A/N ) = {x ∈ supp µψ | ε(k+1)A/N < b(x, ε) ≤ εk A/N }, with k = 0, 1, . . . , N − 1. At least one of these N sets (k A/N ) gives rise to an integral bigger than 1/N times the integral over the whole set B A . And so k0 , r0 = Ak0 /N are picked, and thus the set (r0 ) of the lemma. R To end the proof, remark that b(x, ε)q−1 dµψ (x) ≥ 1 since q < 1. Therefore, for ε ≤ 1/2 one gets Z Z 1 q−1 q−1 b(x, ε) dµψ (x) − ε b(x, ε) dµψ (x) ≥ N (r0 ) Z 1 b(x, ε)q−1 dµψ (x). ≥ 2N We now turn to the proof of Theorem 4.2, which is actually a consequence of Theorem 4.1 and of the following Lemma 4.3 that relates the integrals K µψ (q, ε) to the integrals Iµψ (q, ε) that enter into account in the definition of the generalized fractal dimension Dµ±ψ (q). More precisely, Lemma 4.3 says that under assumption (H) on µψ , both K µψ (q, ε) and Iµψ (q, ε) have the same growth exponents Dµ±ψ (q). This is of course because of the fast decay properties of the function R we have chosen (via the choice of h ∈ C0∞ ([0, 1])). 4.3 Let q ∈ (0, 1). Suppose that (H) holds for µψ . Then for all ν ∈ (0, 1), LEMMA

1 Iµ (q, ε1−ν ) ≤ K µψ (q, ε) ≤ Iµψ (q, ε), (4.10) 22−q ψ where the left inequality holds for ε small enough (ε ≤ ε(ν)). As a consequence, log K µψ (q, ε) 1 lim inf = Dµ−ψ (q) 1 − q ε→0 − log ε and log K µψ (q, ε) 1 lim sup = Dµ+ψ (q). 1 − q ε→0 − log ε

(4.11)


181

Remark 4.3 We strongly believe that (4.11) holds in full generality for any µψ . Proof of Lemma 4.3 Throughout this proof we denote B(x, ε) = [x − ε, x + ε]. One has Z q−1 Iµψ (q, ε) = µψ B(x, ε) dµψ (x) Z =

dµψ (x)

Z

χ[−1,1]

x−y ε

q−1 dµψ (x) .

Since, by (3.5), χ[−1,1] (w) ≤ R(w) and q − 1 < 0, one has K µψ (q, ε) ≤ Iµψ (q, ε). So we need to get the lower bound of (4.10). First, notice that Z x−y (R) b (x, ε) = dµψ (y)R ε Z Z x−y x−y = dµψ (y)R + dµψ (y)R ε ε |(x−y)/ε| 0 as small as one wants. Proof of Theorem 4.2 Theorem 4.2 is a direct consequence of Theorem 4.1 and of Lemma 4.3. As already mentioned at the end of Section 2, our main theorem, namely, Theorem 2.1, is a direct consequence of Theorem 3.1 (see Section 3) and of Theorem 4.2. We end this section with a few words concerning the case where, as in [23] and [25], one assumes some further properties on the spatial behaviour of the kernel u(n, x). Assume that for some constant C independent of the energy x, and for γ ≤ d, one knows that for µψ a.e. x, X |u(n, x)|2 ≤ C N γ , (Hypothesis (HS)); |n|≤N

then one can not only get a better result taking β = p/γ instead of β = p/d, but it turns out that one can push our approach one step further to reach the dimension numbers Dµ±ψ (1 − β). The gain takes place in inequality (3.12), where a direct estimate is made possible thanks to Hypothesis (HS). This thereby enables us to avoid the Cauchy-Schwarz inequality we made in order to split the integral into a spatial part and a spectral part. This therefore leads to the lower bound hh|X | p iiψ,T ≥ C L 0ψ (T, φ) := C((kφk2+2β )/(Uφ,ψ (T )β )). Then the same technique as the one used to prove Theorem 4.1 supplies the following result. THEOREM 4.3 Suppose in addition to the hypotheses of Theorem 2.1 that the spatial hypothesis (HS) above holds for some γ and constant C. Then if β = p/γ < 1, one has

lim inf

log L 0ψ (T ) loghh|X | p iiψ,T ≥ lim inf = β Dµ−ψ (1 − β), T →∞ log T log T

lim sup

log L 0ψ (T ) loghh|X | p iiψ,T ≥ lim sup = β Dµ+ψ (1 − β). log T log T T →∞

T →∞

and T →∞


183

Appendices A. Complement of Section 2 For the reader’s convenience, we provide the proof of statements (ii) and (iii) of Proposition 2.1. Proof of Proposition 2.1(ii) We first note that from the convex Jensen inequality, and since q − 1 < 0, we have, for any set A ⊂ R, Iµ (q, ε/2)

1/(q−1)

Z

µ(x − ε/2, x + ε/2)

q−1

=

1/(q−1) dµ(x)

R

Z

µ(x − ε, x + ε)

q−1

≤

1/(q−1) dµ(x)

A

Z

µ(x − ε, x + ε) dµ(x).

≤

(A.1)

A

Consider now, for all ν ∈ (0, 1), the set dim H (µ)−ν ) A(H }. ν (ε) ≡ {x ∈ R | µ(x − ε, x + ε) < ε

We have − ) ) A(H := lim inf A(H ν (ε) ⊃ {x ∈ R | γµ (x) > dim H (µ) − ν}. ν ε→0

By the definition of dim H (µ) given in line (2.1) we get µ {x ∈ R | γµ− (x) > (H ) (H ) dim H (µ) − ν} > 0. Furthermore, lim infε→0 µ(Aν )(ε) ≥ µ(Aν ) ≥ µ {x ∈ (H ) R | γµ− (x) > dim H (µ) − ν} > 0. Thus, letting A = Aν (ε) in (A.1), we obtain Dµ− (q) = lim inf ε→0

≥ lim inf

log Iµ (q, ε/2)1/(q−1) log(ε/2) (H ) log µ Aν (ε) εdim H (µ)−ν

ε→0

log(ε/2)

= dim H (µ) − ν.

Since the above inequality is valid for all ν ∈ (0, 1), Proposition 2.1(ii) is proven. Proof of Proposition 2.1(iii) We define, for εk = e−k , dim P (µ)−ν

A(P) ν (εk ) := {x ∈ R | µ(x − εk , x + εk ) < εk

}.

184


Since limk→∞ log εk / log εk+1 = 1, we have lim supk→∞ log µ(x − εk , x + εk )/ log εk = lim supε→0 log µ(x − ε, x + ε)/ log ε. Thus we get (P) A(P) ν := lim sup Aν (εk ) k→∞

⊃ {x ∈ R | lim sup = {x ∈ R |

k→∞ γµ+ (x)

µ(x − εk , x + εk ) > dim P (µ) − ν} log εk

> dim P (µ) − ν}.

Therefore, from the definition of dim P (µ) given in line (2.1) and the above inclusions, (P) we get µ(Aν ) > 0. Using the Borel-Cantelli lemma (as done in [18]) implies that P (P) k µ Aν (εk) = ∞, and thus there exists a subsequence εk(n) & 0 of εk such that (P) (P) µ Aν (εk(n) ) ≥ k(n)−2 = (log εk(n) )−2 . Thus, letting A = Aν (εk(n) ) in (A.1), we obtain dim P (µ)−ν (P) 1/(q−1) log µ A (ε ) εk(n) ν k(n) log Iµ (q, ε) Dµ+ (q) = lim sup ≥ lim n→∞ log ε log εk(n) ε→0 = dim P (µ) − ν. Again, since the result is true for all ν ∈ (0, 1), Proposition 2.1(iii) is proven. B. Relation to other lower bounds Getting a growth exponent in terms of fractal dimensions Dµ±ψ (q) is not specific to our lower bound L ψ (T ) (3.2). It is also possible to get such relations from the lower bounds formerly derived, either directly (Barbaroux and Tcheremchantsev’s lower bound L 2 (T )) or by improving them (Guarneri, Combes, and Last’s lower bound, improved by optimizing in φ for each single T ; see L 1 (T )). We explain this point briefly in this section and thereby propose to the reader a link between the present work and the former methods. In particular, this appendix illustrates how actually deeply connected are the generalized fractal dimensions Dµ±ψ (q) to the lower bounds of hh|X | p iiψ,T studied for the past ten years, that is, since Guarneri [15], [16]. Since we focus on the relation between those lower bounds and the fractal dimensions Dµ±ψ (q), and for the sake of simplicity, we only consider vectors φ of the form χ (H )ψ instead of general φ ∈ Hψ as in (3.2). We first consider the lower bound that appears in [6]. For some constant C(ψ, p) > 0, it is shown that for all T > 0, hh|X | p iiψ,T ≥ C(ψ, p)L 2 (T ), with kφk2+8β L 2 (T ) = sup L 2 (φ, T ) := , φ = χ (H )ψ , Sφ (T )β


where Sφ (T ) =

Z R2

185

dµφ (x) dµφ (y)R T (x − y) .

Here R(w) can be chosen as the usual Gaussian e−w /4 . Then one can mimic the proof of Theorem 3.1 and relate the quantity L 2 (φ, T ) = kφk2+8β /Sφ (T )β to the integral K µψ ((1+2β)/(1+3β), T −1 ). More precisely, the same kind of Hölder inequalities as in (4.5) and (4.6) supply an upper bound for L 2 (T ), and using a set (r ) in the same spirit as in the proof of Theorem 4.1 yields the lower bound, again up to a logarithmic factor. The following theorem then holds. 2

B.1 Under the same hypotheses as in Theorem 2.1, one has log L 2 (T ) 1 + 2β − = β Dµψ lim inf T →∞ log T 1 + 3β THEOREM

and lim sup T →∞

log L 2 (T ) = β Dµ+ψ log T

1 + 2β 1 + 3β

.

The second lower bound we want to discuss in this section is the improved version, using [6], of the first kind of quantity that has been considered in order to bound from below hh|X | p iiψ,T , and that comes from Guarneri [15], [16], Combes [8], and Last [27]. Roughly, the basic idea is to take into account the function G φ (T ) := R µφ − ess sup dµφ (y)R T (x − y) where R(w) = exp(−w2 /4). (Notice that G φ (T ) is smaller than C T −α if µφ is uniformly α-Hölder continuous; see (3.10) in [27].) RT One then uses G φ (T ) in order to bound Bφ (T, N ) defined as in (3.6) but with 1/T 0 R +∞ rather than 1/T 0 h(t/T ). Using, for instance, 2|v(n, x)v(n, y)| ≤ |v(n, x)|2 + |v(n, y)|2 , Z Z X Bφ (T, N ) ≤ C dµφ (x) dµφ (y)R (x − y)T |v(n, x)v(n, y)| R R

≤ C G φ (T )

|n|≤N

Z

dµφ (x) R

X

|v(n, x)|2 ≤ C 0 G φ (T )N d .

|n|≤N

Following [27] and [6], this leads to the lower bound hh|X | p iiψ,T ≥ C(ψ, p)L 1 (T ), with kφk2+4β L 1 (T ) = sup L 1 (φ, T ) := , φ = χ (H )ψ . G φ (T )β One should compare the obtained lower bound L 1 (T ) to expression (6.10) in [27].

186


Then one can again mimic the proof of Theorem 4.1 and relate the quantity L 1 (T ) to the integral K µψ ((1 + β)/(1 + 2β), T −1 ). This leads to the following theorem. THEOREM B.2 Under the same hypotheses as in Theorem 2.1, one has log L 1 (T ) 1+β − lim inf = β Dµψ T →∞ log T 1 + 2β

and lim sup T →∞

log L 1 (T ) = β Dµ+ψ log T

1+β 1 + 2β

.

One should also compare the expression of L 1 (T ) and L 2 (T ) to the expression of L ψ (T ) given in line (3.3), that is, where as above the supremum is taken over the set of vectors φ = χ (H )ψ, namely, the supremum of kφk2+4β /Uφ,ψ (T )β . One may have the right to wonder whether there are some links between these quantities and also whether L ψ (T ) is effectively a better lower bound than L 1 (T ) and L 2 (T ). Theorem B.3 answers these questions. This theorem is actually a consequence of Theorems 4.1, B.2, and B.1, and of the nonincreasing property of the functions Dµ±ψ (q) (see Proposition 2.1(i)). THEOREM B.3 Under the same hypotheses as in Theorem 2.1, one has

lim inf

log L ψ (T ) log L 1 (T ) log L 2 (T ) ≥ lim inf ≥ lim inf , T →∞ T →∞ log T log T log T

lim sup

log L ψ (T ) log L 1 (T ) log L 2 (T ) ≥ lim sup ≥ lim sup . log T log T log T T →∞ T →∞

T →∞

and T →∞

Remark B.1 (i) It is worthwhile to point out that such a comparison is made possible thanks to the upper bounds obtained for L ψ (T ), L 1 (T ), and L 2 (T ); upper bounds that are for the first time derived for such quantities. (ii) Theorem B.3 tells us that it is worthwhile to deal with the crossed term in (1.6) while one develops Bψ (T, N ) with ψ = φ + χ . Dealing with the crossed term does lead to an improvement, with regard to the former approaches (see [15], [16], [8], [27], [6]) where this term was not treated well. (iii) One can show that L ψ (φ, T ) ≥ c1 L 1 (φ, T ) with some constant c1 > 0 uniform in T, φ. The latter is not true as one compares L ψ (φ, T ) and L 2 (φ, T ) for some φ. We confess that the inequalities involving L 2 (T ) in Theorem B.3 are quite a surprise


187

for us since we were expecting the exponents of L 2 (T ) to be bigger than those of L 1 (T ). (iv) We believe a stronger version of Theorem B.3 to be true, namely, L ψ (T ) ≥ c1 L 1 (T ) ≥ c2 L 2 (T ). C. A sufficient condition for Hypothesis (H) The following statement gives a sufficient condition for (H) to hold, which can be useful for applications. PROPOSITION C.1 Let 0 < q < 1. Assume that

1−q , one has (H1) for some λ > q

Z

|x|λ dµ(x) < +∞. R

Then 0 ≤ Dµ± (q) ≤ 1. Proof of Proposition C.1 Let b(x, ε) = µ([x − ε, x + ε]). Define (ε) = {x ∈ supp µ | b(x, ε) ≤ ε}, and I j = [( j − 1/2)ε, ( j + 1/2)ε), j ∈ Z. For any x ∈ I j , µ([x − ε, x + ε]) ≥ µ(I j ) ≥ µ I j ∩ (ε) := b j .

(C.1)

We first remark that for any j, b j ≤ ε. Indeed, if b j = 0, it is trivially true. And if b j > 0, there exists x0 ∈ I j ∩ (ε). Inequality (C.1) and the definition of (ε) imply b j ≤ µ [x0 − ε, x0 + ε] ≤ ε. Next, inequality (C.1) yields Z X Z b(x, ε)q−1 dµ(x) = (ε)

b(x, ε)q−1 dµ(x)

j:b j >0 I j ∩(ε)

≤

X

q−1 X q µ I j ∩ (ε) µ I j ∩ (ε) = bj .

j:b j >0

(C.2)

j∈Z

One can rewrite this summation as follows: X j∈Z

q

bj =

+∞ X k=0

Sk ,

Sk :=

X j∈Jk

q

bj ,

(C.3)

188


where Jk = { j ∈ Z | I j ⊂ [ek , ek+1 ) ∪ [−ek+1 , −ek )} for k > 0 and J0 = { j | I j ⊂ [−e, e)}. To be rigorous, equality (C.3) holds if one replaces in the definition of the sets Jk the quantity ek by [ek /ε]ε + ε/2, where [·] stands for the integer part, but for the sake of simplicity we use the definition of Jk given above. Let γ ∈ (0, 1). For k ≥ 0, Hölder inequality and the fact that Card Jk ≤ 2(ek+1 − k e )/ε ≤ 4ek /ε and b j ≤ ε imply Sk =

X

q bj

=

j∈Jk

X

γ q (1−γ )q bj bj

≤

X

j∈Jk

≤ ε(1−γ )q

4e ε

j∈Jk

γ q

k 1−γ q X

bj

bj

γ q X

((1−γ )q)/(1−γ q) bj

1−γ q

j∈Jk

= 4εq−1 ek(1−γ q)

X

j∈Jk

γ q bj

.

(C.4)

j∈Jk

From Assumption (H1) it is straightforward that there exists C(λ) < ∞ such that for all k, X X bj ≤ µ(I j ) ≤ C(λ)e−kλ , (C.5) j∈Jk

j∈Jk

with λ from the statement of Proposition C.1. Then one can find γ ∈ (0, 1) such that λγ q − (1 − γ q) > 0. Then (C.3)–(C.5) imply X

b j ≤ 4C γ q (λ)εq−1 q

j

+∞ X

e−k(λγ q−(1−γ q)) ≤ D(q, λ)εq−1 .

(C.6)

k=0

The proof is completed as follows. One can write Iµ (q, ε) as Z Z Iµ (q, ε) = + b(x, ε)q−1 dµ(x) =: I1 + I2 . (ε)

R\(ε)

The bounds (C.2) and (C.6) give I1 ≤ Dεq−1 . Furthermore, due to the definition of (ε) and q ∈ (0, 1), we have I2 ≤ εq−1 µ R \ (ε) ≤ εq−1 . We so obtain Iµ (q, ε) ≤ (D + 1)εq−1 and thus Dµ± (q) ≤ 1. COROLLARY C.1 Let µ be such that

Z

|x|λ dµ(x) < +∞ for any λ > 0. R

Then Dµ± (s) ∈ [0, 1] for any s ∈ (0, 1) and thus (H) holds. In particular, this is true if µ has a compact support.


189

D. An example of finite pure point measure with nontrivial generalized fractal dimensions For λ > 1 and α > 0, let an = a/n λ , xn = 1/n α , where a > 0 is a normalization P constant. We define the finite pure point probability measure µ = ∞ n=1 an δxn on R. As the measure µ has a bounded support, 0 ≤ Dµ± (q) ≤ 1 for any q ∈ (0, 1) (see Proposition 2.1(iv)). We denote by B(x, ε) the closed ball of center x and radius ε. For any given ε > 0, take N to be the integer part of ε−1/(1+α) . Thus there exists a constant c > 0 uniform in N (and ε) such that for all n ≤ cN , µ B(xn , ε) = µ({xn }) = an . Let q < 1/λ. Then we have Iµ (q, ε) =

X

cN q−1 X q−1 an µ B(xn , ε) ≥ µ B(xn , ε) an

n

=

cN X

n=1 q

an ∼ C ε−((1−qλ)/(1+α)) .

(D.1)

n=1

Therefore

1 − qλ . (1 − q)(1 + α) This implies that for 0 < q < 1/λ, one obtains strictly positive generalized fractal dimensions Dµ± (q) for the pure point measure µ. Moreover, by taking q and α small, one can render these dimensions as close to 1 as one wants. Estimate (D.1) is actually also valid for q < 0 (and one can show that the dimensions Dµ± (q) are finite for any q < 0). However, the behaviour of the fractal dimensions is rather strange: they can be greater than 1, and the bigger λ is, the bigger Dµ± (q) are as q → −∞: Dµ± (q) ≥

lim inf Dµ± (q) ≥

q→−∞

λ >1 1+α

if α is small enough. Furthermore, note that if µ is now defined with an = ae−λn , then Dµ± (q) = +∞ whenever q < 0. This shows that one should be very cautious when considering possible physical applications of Dµ± (q) with q < 0. To conclude, we give the example of a self-adjoint operator H and a state ψ, as mentioned in Remark 2.3(ii). We prove the following. THEOREM D.1 Let β > 0 and δ > 0. There exist a bounded self-adjoint operator H on H = `2 (N) and a state ψ ∈ H such that, with p = β, loghh|X | p iiψ,T 1 = 0, but Dµ±ψ − δ > 0. lim T →∞ log T 1+β

190


Proof of Theorem D.1 Let β > 0 and δ > 0 be as in the theorem, with ν > 0 to be chosen later on. Let (en )n≥1 be the canonical basis of `2 (N∗ ): en (k) = δnk , k, n ≥ 1. Define a self-adjoint operator H in H as follows: H en = xn en ,

xn = n −α ,

n ≥ 1.

It is easy to see that the spectral measure µψ associated to the vector ψ(k) = √ −λ/2 ak , λ = 1+β+ν, is exactly the measure µ defined above. (So supp µψ ⊂ [0, 1] and Theorem 2.1 applies.) Notice that ψ(t, k) = exp(−it xk )ψ(k). Then, as d = 1 one has β = p, and one checks that X hψt , |X | p ψt i = ak β−λ = C(ν) < +∞. k≥1

Therefore limT →∞ loghh|X | p iiψ,T / log T = 0. On the other hand, as we have seen above, Dµ±ψ (q) > 0 for any q < 1/λ = 1/(1 + β + ν). Taking ν small enough, for example, such that 1/(1 + β + ν) = (1/(1 + β)) − δ/2, one gets the required example. Remark D.1 One can easily construct such an example on H = `2 (Zd ) too by numbering the canonical basis (en )n∈Zd in a “spiral" way and taking β = p/d as usual. Indeed, one then gets en (k) = 1 for some k ∼ n 1/d . Note added in proof. Since this article was accepted, there have been new developments that lead to concrete applications. In [37] our lower bound is combined with results from [4] on equivalence of generalized fractal dimensions to prove intermittency for the model with sparse potential studied in [9]. Note that the belief stated in Remark 4.3 is proved in [4]. We also refer the reader to [36], where lower bounds in the spirit of ours but involving the behaviour of the generalized eigenfunctions are obtained. Acknowledgments. F. Germinet is grateful to S. De Bièvre, S. Jitomirskaya, and H. Schulz-Baldes for enjoyable and stimulating discussions on this subject matter. F. Germinet acknowledges the hospitality of the University of California, Irvine, where part of this work was done. The authors are also grateful to the referee for remarks and suggestions.


191

References [1]

[2]

[3]

[4] [5]

[6]

[7] [8]

[9]

[10]

[11] [12]

[13] [14] [15] [16]

J.-M. BARBAROUX, J.-M. COMBES, and R. MONTCHO, Remarks on the relation between

quantum dynamics and fractal spectra, J. Math. Anal. Appl. 213 (1997), 698–722. MR 98g:81028 162, 168 J.-M. BARBAROUX, F. GERMINET, and S. TCHEREMCHANTSEV, Nonlinear variation of diffusion exponents in quantum dynamics, C. R. Acad. Sci. Paris. Sér. I Math. 330 (2000), 409–414. MR 2000m:81225 165 , “Quantum diffusion and generalized Rényi dimensions of spectral measures” in Journées “Équations aux dérivées partielles” (La Chapelle sur Erdre, France, 2000), exp. no. I, Univ. Nantes, Nantes, 2000. MR 2001f:81042 165, 169 , Generalized fractal dimensions: Basic properties and equivalences, to appear in J. Math. Pures Appl. (9). 190 J.-M. BARBAROUX and H. SCHULZ-BALDES, Anomalous quantum transport in presence of self-similar spectra, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 539–559. MR 2001a:82068 163, 168, 169 J.-M. BARBAROUX and S. TCHEREMCHANTSEV, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), 327–354. MR 2000h:81060 161, 164, 169, 170, 171, 174, 175, 177, 184, 185, 186 J. BELLISSARD and H. SCHULZ-BALDES, Anomalous transport: A mathematical framework, Rev. Math. Phys. 10 (1998), 1–46. MR 99b:81046 162 J.-M. COMBES, “Connections between quantum dynamics and spectral properties of time-evolution operators” in Differential Equations with Applications to Mathematical Physics, ed. W. F. Ames, E. M. Harrel, and J. V. Herod, Math. Sci. Engrg. 192, Academic Press, Boston, 1993, 59–68. MR 94g:81205 161, 162, 163, 164, 171, 177, 185, 186 J.-M. COMBES and C. MANTICA, Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, preprint, 2000, http://ma.utexas.edu/mp_arc-bin/mpa?yn=00-365 190 C. D. CUTLER, Some results on the behaviour and estimation of the fractal dimensions of distributions on attractors, J. Statist. Phys. 62 (1991), 651–708. MR 92b:58137 166, 167 S. DE BIÈVRE and G. FORNI, Transport properties of kicked and quasi-periodic Hamiltonians, J. Statist. Phys. 90 (1998), 1201–1223. MR 99g:82046 162 S. DE BIÈVRE and F. GERMINET, Dynamical Localization for the random dimer Schrödinger operator, J. Statist. Phys. 98 (2000), 1135–1148. MR 2000m:82053 162 D. H. DUNLAP, H.-L. WU, and P. W. PHILLIPS, Absence of localization in a random dimer model, Phys. Rev. Lett. 65 (1990), 88–91, http://prola.aps.org/PRL/v65/i1 162 K. J. FALCONER, Techniques in Fractal Geometry, Wiley, Chichester, England, 1997. MR 99f:28013 166 I. GUARNERI, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95–100. 161, 162, 163, 164, 171, 177, 184, 185, 186 , On an estimate concerning quantum diffusion in the presence of a fractal

192

[17]


spectrum, Europhys. Lett. 21 (1993), 729–733. 161, 162, 163, 164, 171, 177, 184, 185, 186 I. GUARNERI and H. SCHULZ-BALDES, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317–324. MR 2001c:81044 165


[18]

[19]

[20]

[21]

[22] [23]

[24]

[25]

[26] [27]

[28]

[29]

[30] [31]

[32] [33]

193

, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Electron. J. 5 (1999), no. 1, http://ma.utexas.edu/mpej/MPEJ.html#v5. MR 2000d:81033 161, 162, 164, 165, 166, 168, 184 , Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra, Rev. Math. Phys. 11 (1999), 1249–1268. MR 2000k:81082 163, 168 T. C. HALSEY, M. JENSEN, L. KADANOV, I. PROCACCIA, and B. SCHRAIMAN, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A (3) 33 (1986), 1141–1151. MR 87h:58125 166 H. G. E. HENTSCHEL and I. PROCACCIA, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), 435–444. MR 85a:58064 166 S. JITOMIRSKAYA and Y. LAST, Power-law subordinacy and singular spectra, I: Half-line operators, Acta Math. 183 (1999), 171–189. MR 2001a:47033 163 R. KETZMERICK, K. KRUSE, S. KRAUT, and T. GEISEL, What determines the spreading of a wave packet?, Phys. Rev. Lett. 79 (1997), 1959–1963. MR 98g:81012 162, 165, 182 R. KETZMERICK, G. PETSCHEL, and T. GEISEL, Slow decay of temporal correlations in quantum systems with Cantor spectra, Phys. Rev. Lett. 69 (1992), 695–698, http://prola.aps.org/toc/PRL/v69/i5 161 A. KISELEV and Y. LAST, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125–150. MR 1 741 780 165, 182 H. KUNZ and B. SOUILLARD, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201–246. MR 83f:39003 170, 171 Y. LAST, Quantum dynamics and decomposition of singular continuous spectra, J. Funct. Anal 142 (1996), 406–445. MR 97k:81044 161, 162, 163, 164, 168, 169, 170, 171, 177, 185, 186 G. MANTICA, “Quantum intermittency in almost periodic systems derived from their spectral properties” in Lattice Dynamics (Paris, 1995), Phys. D 103, North-Holland, Amsterdam, 1997, 576–589. 162, 169 , “Wave propagation in almost-periodic structures” in Physics and Dynamics Between Chaos, Order, and Noise (Berlin, 1996), Phys. D 109, North-Holland, Amsterdam, 1997, 113–127. 162, 169 L. OLSEN, A multifractal formalism, Adv. Math. 116 (1995), 82–196. MR 97a:28006 166 Y. B. PESIN, Dimension Theory in Dynamicals Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, 1996. MR 99b:58003 166 F. PIÉCHON, Anomalous diffusion properties of wave packet on quasiperiodic chains, Phys. Rev. Lett. 76 (1996), 4372–4375, http://prola.aps.org/toc/PRL/v76/i23 162 M. REED and B. SIMON, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. MR 58:12429a 171

194

[34]

[35] [36] [37] [38]


R. DEL RIO, S. JITOMIRSKAYA, Y. LAST, and B. SIMON, Operators with singular

continuous spectrum, IV: Hausdorff dimensions, rank one pertubations, and localization, J. Anal. Math. 69 (1996), 153–200. MR 97m:47002 162, 170 S. J. TAYLOR and C. TRICOT, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699. MR 87a:28002 166 S. TCHEREMCHANTSEV, Mixed lower bounds in quantum dynamics, preprint, 2001, http://ma.utexas.edu/mp_arc-bin/mpa?yn=01-227 190 , in preparation. 190 M. WILKINSON and E. J. AUSTIN, Spectral dimension and dynamics for Harper’s equation, Phys. Rev. B 50 (1994), 1420–1429, http://prola.aps.org/search 162

Barbaroux Unité Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Unité de Formation et de Recherche de Mathématiques, Université de Nantes, 2 rue de la Houssinière, F-44072 Nantes Cédex 03, France; [email protected] Germinet Unité Mixte de Recherche 8524 du Centre National de la Recherche Scientifique, Unité de Formation et de Recherche de Mathématiques, Université de Lille 1, F-59655 Villeneuve d’Ascq Cédex, France; [email protected] Tcheremchantsev Unité Mixte de Recherche 6628 - Mathématiques, Applications et Physique Mathématique d’Orléans, Université d’Orléans, B.P. 6759, F-45067 Orléans Cédex, France; [email protected]