Mathematical Physics Electronic Journal 5&6 (v. 5 & 6

MATHEMATICAL PHYSICS ELECTRONIC JOURNAL Print Version Volume 5,1999

Volume 6,2000

Chief Editors

Chief Editors

J.-P. Eckmann

P. Collet

G. Gallavotti

H. Koch

H. Koch

C. E. Wayne

MATHEMATICAL PHYSICS ELECTRONIC JOURNAL Print Version

This page is intentionally left blank

MATHEMATICAL PHYSICS ELECTRONIC JOURNAL Print Version

Volume 5,1999

Volume 6,2000

Chief Editors

Chief Editors

j.-P. Eckmann G. Gallavotti H. Koch

P. Collet H. Koch C. E. Wayne

Technical Editors

R. de la Llave H. Koch C. Radin Department of Mathematics University of Texas at Austin USA

V

fe World Scientific •

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Preface

The main goal in the creation of MPEJ was to have a journal that was distributed as widely as possible in the community and that had as its only criterion for acceptance the quality of the papers. In particular, we did not want that considerations of the number of pages to be published every year entered into decisions of acceptance. These goals are well served by the free electronic distribution of the papers and the electronic supplements available from: • http://www.ma.utexas.edu/mpej • http: //mpe j . unige. ch/mpe j • http://www.maia.ub.es/mpej and the about 20 mirrors organized by the European Mathematical Information Service (EMIS) whose main server is: • http://www.emis.de

We think that adding a print version of the papers to the electronic distribution will increase the availability and, given that there is no commitment of pages nor any financial incentive for editors or authors, it will allow MPEJ to keep quality as the sole criterion. We thank World Scientific and, specially Dr. Sen Hu for the organization of this print version.

v

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m^Ej MATHEMATICAL PHYSICS ELECTRONIC JOURNAL*

ISSN 1086-6655

Volume 5,1999 Chief Editors J.-P. Eckmann Universite de Geneve Geneva

G. Gallavotti Universita di Roma La Sapienza Rome

H. Koch The University of Texas Austin

Editorial Board J.E. Avron Israel Institute of Technology Haifa

G. Benettin Universita di Padova Padua

P. Collet Ecole Polytechnique Palaiseau

P. Constantin The University of Chicago Chicago

J. Feldman University of British Columbia Vancouver

R. Kotecky Charles University Prague

A. Kupiainen Helsinki University Helsinki

R. de la Llave The University of Texas Austin

H. Spohn Universitat Munchen Munich

H. Tasaki Gakushin University Tokyo

C.E. Wayne Boston University Boston

For an electronic version of this volume, and for information on subscriptions and other matters, see http://www.ma.utexas .edu/mpej/ or h t t p : / / m p e j .unige. ch/mpej/ or send an empty e-mail message to mpejQmath.utexas.edu for instructions.

Vll

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Contents Volume 5 (1999)

I. Guarneri H. Schulz-Baldes

Lower Bounds on Wave Packet Propagation by Packing Dimensions of Spectral Measures

1

G.D. Raikov

Eigenvalue Asymptotics for the Dirac Operator in Strong Constant Magnetic Fields

17

S. De Bievre J.V. Pule

Propagating Edge States for a Magnetic Hamiltonian

39

F. Bonetto G. Gentile

On a Conjecture for the Critical Behaviour of KAM Tori

56

Y. Lacroix

Local Perturbations of Energy and Kac's Return Time Theorem

65

Stability of the Brown-Ravenhall Operator

76

G. Hoever H. Siedentop

M A T H E M A T I C A L PHYSICS E L E C T R O N I C J O U R N A L

ISSN 1086-6655 Volume 5, 1999 Paper 1 Received: June 19, 1998, Accepted: Jan 6, 1999 Editor: J. Avron

:

"••-.TT.Taran

Lower bounds on wave packet propagation by packing dimensions of spectral measures I. Guarneri 1,2 , H. Schulz-Baldes 3 1

International Centre for the Study of Dynamical Systems, Universita di Milano, sede di Como, via Lucini 3, 22100 Como, Italy 2

Istituto Nazionale di Fisica della Materia, Unita di Milano, Via Celoria 16, 20133 Milano, Italy

3

Technische Universitat Berlin, Fachbereich Mathematik, Strafie des 17. Juni 136, 10623 Berlin, Germany

Abstract We prove that, for any quantum evolution in £2(ZD), there exist arbitrarily long time scales on which the gth moment of the position operator increases at least as fast as a power of time given by q/D times the packing dimension of the spectral measure. Packing dimensions of measures and their connections to scaling exponents and box-counting dimensions are also discussed.

1

2

1

Introduction.

Let H be a selfadjoint operator in a separable Hilbert space %. Let \4>) be any vector in H, /j, its spectral measure relative to H and B = {|n)} n eN a Hilbert basis in %. As time t increases, the wave packet e~%tH\^)) spreads out over the basis B. In particular, it is known [Gl, G2, La, Co] that, for q > 0, the gth moment of the probability distribution associated with the Fourier expansion of e~'tH\tp) over the basis B increases, in time average, at least as fast as a power of time given by q times the Hausdorff dimension of fx. This qualitatively means that propagation is faster the more continuous the spectral measure is, the degree of continuity being measured by the lower pointwise dimension of the spectral measure. The present work is summarized by the qualitative remark that, at any given time t, the wave packet can only probe continuity of the spectral measure on the level of spectral resolution achieved at time t, hence on a spectral scale of the order of 1/i. This remark is relevant to the case of non-exactly scaling spectral measures. For such measures, the upper and lower pointwise dimensions do not coincide, meaning that there are arbitrarily small spectral scales at which the measure is somewhere in the spectrum scaling faster than expressed by the Hausdorff dimension. We accordingly obtain that on arbitrarily long time scales a larger lower bound is valid, given by the packing dimension of the spectral measure. There is at least one abstract example of a zero-Hausdorff dimensional spectral measure with packing dimension equal to 1, which leads to ballistic transport [G3]. Spectral measures of a similar nature are likely to occur also for concrete Schrodinger operators. Such may be the case with the quasi-ballistic dynamics exhibited by the Harper model with Liouville incommensuration [La]. The Hausdorff dimension of a measure is related to its lower pointwise dimensions by the theory of Rogers and Taylor [Ro], whose relevance in the present context was advocated by Last [La]. Our present result calls upper pointwise dimensions into play. Apart from [G3], these have not yet found their way in the theory of quantum transport. In the appendix, we therefore elaborate on results by Cutler [Cu] and develop a treatment, in a sense dual to Rogers' and Taylor's, which connects such dimensions to the packing dimension of a measure. Furthermore, we show that these packing dimensions can also be calculated by a box-counting procedure. In the next two sections we establish preliminaries and notations. In Section 4 we state and prove our main result Theorem 1. The main element of its proof is Proposition ? which is basically a restatement of existing results. However, we give here an alternative a b initio derivation which makes no use of Strichartz theorem, but is based on Proposition 1 combined with Last's argument [La]. A modified version of Proposition 1 allowed to prove also upper bounds for a special class of Hamiltonians [GS]. Finally, in Section 5, we transpose the main result to Hamiltonians on a D-dimensional lattice. A useful discussion with Y. Last is acknowledged.

3

2

Growth exponents.

For given time T and e G (0,1), we define the minimal carrier n(e, T) of the wave packet on the basis B as follows: n(e, T) = min

SeN

£p»C0oo g log(T)

(3)

We have /J+ < /?,f whenever g < 0 and /?" > fif whenever q > 0. Lower bounds on /J* convey stronger information than lower bounds on /?*, 9 > 0. For instance, if i f has a pure point spectrum then ($ = 0, because minimal carriers remain bounded in time; still, moments may display nonvanishing growth exponents. The lower growth exponents (3~ characterize the minimal spreading of the wave packet: for all T > 0, Mq(T) > C{5)Ti^~5S> holds for all J > 0 with appropriate constants C(S). The upper exponents /?+ give the fastest possible spreading on sequences of times: for all 5 > 0, Mq(T) < C(5)T 9( ^ ++ *' holds at all times T, but there exists a diverging sequence (7*)*>i such that Mq(Tk) > C (6)Tf0* ~s) for all k > 1.

3

Spectral dimensions.

Given a (Borel) probability measure p on R, we define its lower and upper pointwise dimensions at B e supp(/i) as follows:

4

US) = l £i m i n f ^ - f M ->°

log(e)

+ t

») ,

3MV^ ) =

l i m s u p log(M[^-^^])) e->o

log(e)

(4) while for E g supp(//), d^E)

= rfM(£J) = oo.

The (upper) Hausdorff dimension dimg(yu) and the (upper) packing dimension dimj(/x) of the measure /x can then be defined as follows: d i m ^ ^ ) = /z-esssup dJE) , dimp(/j) = ju-esssup dJE) , (5) EeR sea where, for a real function / , ^ - e s s s u p £ ; e R / ( £ J ) denotes the /i-essential supremum of / , i.e. the infimum over all sets A of full /^-measure of the quantity s u p B e A f{E). Although the above definitions are optimally suited to our present purposes, dimensions of Borel measures are more properly defined and discussed in the appendix. In the rest of this article, we shall use dyadic partitions of the real axis in intervals If = (C? — l)2 _ A r , J2~N], j € Z. Of course, any other hierarchic partition could be used. For E 6 R, we shall denote /jY™ the dyadic interval of the N t h generation to which E belongs. We shall in particular make use of the fact that

MV

iV^oo

log(|/j^,|)

^cc

Similar equalities hold for the upper pointwise dimension d^E), lim sup.

4

K

N

with liminf replaced by

Lower b o u n d s on growth exponents.

The following proposition expresses in a quantitative way the fact that the time evolution up to time T does not resolve details of the spectrum on scales smaller than 1/X (h = 1). Up to time T it is thus possible to work with an approximate Hamiltonian with discrete spectrum the eigenvalues of which have a spacing of order 1/T. We use the notation \xf)

= xfiH)^),

where xf

IS

the characteristic function of

if.

Proposition 1 Given e > 0, we associate to any time T a generation index N so that: 2

Putting this into (9) and recalling (7), we directly get •

P r o p o s i t i o n 2 For integer N and a G (0,1), let Ajv,a be the union of all the dyadic intervals of the Nth dyadic generation which have measure ^(Ij1) < 2~Na. Set bpr = H(AN,O)- If bit > 0, then the following holds true for T satisfying b^2N~l < 9T < b^2N: n(^-,T\ v 2

> C(a)b%-aTa

,

(10)

where C(a) > 0 is only dependent on a. Proof. Let us define \ipN) = XAN^.{H)\II>), SO || IV'JV)!!2 = bN > 0. We wish to apply Proposition 1, with the following specifications: replace \ip) by \ipri) and choose T as the integers smaller than a given m, finally set e = (bN/9)2. Condition (7), which makes Proposition 1 applicable, then becomes precisely bu2N~x < 9T < bN2N. The spectral measure of \ipN) is dfiN(E) = XAN,a(E)d^(E). Hence the sum on the right-hand side of (8) is restricted to the set JNiU of indices j € Z for which pilf) < 2~Na. The sum can be estimated as follows:

6

E

E IWxf>l2 < E E

llxfll2llxf(tf)|n>ll2

= E E

M^llx^tfOWII




and denote Pm = £ „ < m |n)(n|. Substituting all the above in Proposition 1, we obtain

9 J llrt

'

m

where Pm(t) = e Pme~' . Now we continue as in [La, Theorem 6.1]. Let \ip'N) W — IV'JV}- Then \ip'N) is orthogonal to IVW), and

2

T

2

[ ^ iipm(i)iv)ii < [ f \\pmm*)\\ +2ii i^)ii / § \\pm{t)\4>»)\\ + II I^>II2 o

< (^) 2 +in^>ir+^Whence, recalling || |^AT)|| 2 + || \ip'N)\\2 = *> w e S e t :

^Tf||(l-Pm(t))|^||2 > M r f - ( ^ )

2

9

-

2 '

2

because || |V>JV)|| = bpf < 1. It therefore follows from the definition of a minimal carrier that 7j(6;v/2, T) > m. The proof is concluded on recalling the definition of m and e as well as the connection between N and T. • A link to spectral dimensions is provided by (we also use lim sup and lim inf in their set-theoretic meaning; further we suppress the subscripts N —> oo):

limsup^

=

Umsup{£€R|/*(-$£))
a}

2 N

~ "}

-logoffs)) w

1 >

a

\ (12)

7 In a completely analogous way, we get liminf AN,a

= {E € R \dfi{E) > a)

(13)

At this point, we derive lower bounds. First of all, if a < dining)) then (5) and (13) show that liminf fi(ANa) > /^(liminf^4jv,a) > 0, so b^ is larger than some 60 for all N. Inserting such a 60 in Proposition 2, we immediately get n(b0/2, T) > const Ta for all T. This proves the already known lower bound, Pa ^ dim H (/i). In addition, we can now prove the existence of a sequence of times on which the large packing dimension of the " thin" part of the spectral measure forces the wave packet to travel possibly farther than imposed by the Hausdorff dimension. Theorem 1 For all positive q, the upper growth exponent of the q-th moment of the position operator satisfies /?+ > dimp(fx). Proof. It is obviously sufficient to consider the case dimp"(/z) > 0. Then, if a < dim P (/i), it follows from the definition (5) of the packing dimension and from (12) that p,(lim sup ANza) > 0. Therefore, from the Borel-Cantelli lemma we get that £jv fi(ANta) = co. This in turn implies t h a t there is a sequence of integers Nk —> oo such that 6 ^ =

n{ANk,a) > Nk2.

In this situation, Proposition 2 says that

•{%*)

>

In this inequality, k G N is arbitrary, C(a) sequence of times Tk satisfies bNk2Nk~l < all that it follows that Tk —> oo as k —> oo f{x) = 2 I /(18x 2 ). Hence we get that, for b

Nk

rr,\

2'v

^6-2*

J

* "

is a numerical factor only depending on a, the 9Tk < bNk2Nk for all k, and bNk > Nk2. From and that Nk < f~l(Tk) where / is the function all k £ N ,

^

1

C a

()

(z- ^))6-2"

k

•

We denote the right-hand side by Ck. Since a > 0, Ck will be eventually larger than 1. From the definition (1) of a minimal carrier, we obtain that, at all large enough k, the total probability supported by basis states |n) with n > Ck — 1 is larger than 6jvt/2. For such fc's, the Chebyshev inequality yields: b

Mq{Tk) >

-f{ck- \y .

2

Whence, replacing Ck and using bNk > Nk , M(T) M

"^

k>

>

>

X

a

we get (

C{a)T

?

2(/-i(Tfc)) UZ-MW" 2 *

1

Since lim r _ > 0 0 log(/ _ 1 (T))/log(T) = 0, Theorem 1 follows from the definition of the exponent /?+. •

Remark 1 The present proof does not allow to bound /3Q" below by means of dimp(/it). In order to do that one needs to know that limsup^t(Aiv,a) > 0 . We do not know whether that follows from dim~p((i) > a. Nevertheless we can prove the weaker result given in Proposition 3 below. For our next result, let us define TV \ r lQg 2 (.A/'fa^,6)) X>(/x) = sup hmsup —

(14)

where, for given N and e, we define Af{n, N, e) as the minimal number of dyadic intervals of generation N which support more than 1 — e of the measure \i. It is shown in the appendix that V (fi) is bigger than or equal to the box-counting information dimension of fi and smaller than or equal to its packing dimension. Proposition 3 /#" > D(/x). Proof. It is enough to show that, if a is less than the expression on the rhs of (14), then limsup/i(j4jv,a) > 0. Let us choose e and a sequence Nk so that a < lim

1

°g2(^JVfc'e)) .

(15)

If lim/i(>ljv,a) = 0, then the sequence of characteristic functions XAN „ converges to 0 in fimeasure. So there is a sequence N^ such that XAN ,„ converges to 0 /i-almost everywhere. We can then find a compact K, with n(K) > 1 — e, so that XAN

a

converges uniformly to

0 in K. Hence, K is eventually a subset of all the ACN . Thus, for all sufficiently large j , K has a covering by dyadic intervals of the iV^th generation, everyone of which is not smaller than 2~a k' in ^-measure. There cannot be more than (1 — e)2aNk' intervals in those coverings. Therefore, Af(/j,, Nkj,e) < (1 - t)2aN"i, which contradicts (15). • Remark 2 An explicit albeit abstract illustration of the lower bounds proven above is given in [G3]. There H is multiplication by E in L2((0,2-K),II), \n) = exp(2mnF(E)), where F(E) — fi((Q,E)), and the measure n has dimj(/i) = 0 = dim7(/i), dimp(/i) = 1 = dim]!" (^4). That measure is constructed following [RJLS]. The motion in this model is ballistic, in the sense that /?+ = 1 for all q > 0.

5

Lower bound for covariant lattice Hamiltonians.

Here we transpose the results of the last section to quantum diffusion of dynamics governed by covariant Hamiltonians on the lattice. Because disordered media and quasicrystals can be described by these models, this situation is of particular physical interest. We do not

9 furnish the technical proofs for the various statements in this section because they can be completed along the lines of [SB]. Let the space of disorder or quasicrystaline configurations fi be a compact and metrizable space on which is given an action T of the group ZD. We suppose that to each configuration u 6 Q there is a bounded operator Hw : (?(ZiD) —> (?{2,D) and that this operator family is strongly continuous in w and covariant with respect to a projective representation U of the translation group ZD on P[ZD), t h a t is U(a)HuU{a)*

= HT^

a e ZD .

,

Finally we fix an invariant and ergodic probability measure P on CI. Let now I/JU € £2(ZD) be a cyclic vector for Hu and /i u its spectral measure. It can then be shown by the same techniques as in [SB] that the packing dimension dimp(/Ltw, A) in the Borel set A C R (see the appendix for the definition) is P-almost surely constant and thus defines the packing dimension of the local density of states in A. It is smaller than or equal to the packing dimension of the density of states M which is defined to be the disorder average of the spectral measure pu,|o) of the state |0) localized at the origin, namely we have dim P (/z u ,A) < dimP(A/", A) ,

P-a.s. .

Next we define the time averaged moments of the position operator X on £2(ZD) by

MqAA?)

= j f %-> d^(E) and E H-> d^(E) are Borel functions in L°°(R, /J,) taking values in [0,1] and only depending on the measure class of /x. Let us further introduce as in [Ro] the upper and lower 7-derivative of fj, at E by

W^E)

= l i m s u p ^ - ^ ^

,

= liminf^-£»

DUE)

g + e])

.

Links between lower pointwise dimensions and upper 7-derivatives are given by 4(£) A}) < 5_ ,

MZ({E

e A | £ 7 ( £ ) > A})
Ae7 V e < J. Therefore, if (B(Ek, r*))fceN is a 5-packing of .RJ(