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Quantum systems with fractal spectra
Chaos, Solitons and Fractals 14 (2002) 799–807 www.elsevier.com/locate/chaos
Quantum systems with fractal spectra I. Antoniou a
a,b,*
, Z. Suchanecki
a,b,c
International Solvay Institute for Physics and Chemistry, CP 231, ULB Campus Plaine, Bd. du Triomphe, 1050 Brussels, Belgium b Theoretische Natuurkunde, Free University of Brussels, Brussels, Belgium c Institute of Mathematics, University of Opole, Opole, Poland Accepted 15 January 2002
Abstract We study Hamiltonians with singular spectra of Cantor type with a constant ratio of dissection and show strict connections between the decay properties of the states in the singular subspace and the algebraic number theory. More specifically, we study the decay properties of free n-particle systems and the computability of decaying and non-decaying states in the singular continuous subspace. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The decay of a quantum system depends on the nature of its spectrum. It is well known that if a Hamiltonian system has a purely absolutely continuous spectrum then each state decays. If the Hamiltonian has point spectrum then each state is non-decaying. If a Hamiltonian system consists of both point and absolutely continuous spectrum then the underlying Hilbert space can be decomposed as a direct sum of two Hilbert spaces. Each of these spaces reduces the Hamiltonian and one of them consists only of decaying states while the other only of non-decaying. However, the spectrum of an arbitrary Hamiltonian does not necessarily consist of these two parts only. In addition, singular continuous spectrum may be present. It is therefore natural to ask whether the Hilbert space of an arbitrary Hamiltonian H can also be decomposed into two parts, which reduce H , in such a way that one Hilbert space consists of decaying states and the other of non-decaying. It is astonishing that although a clear definition of quantum decay and decaying states is known since 1920s [1,2] the above question has been raised and resolved affirmatively only recently in our publications [3–6]. We have shown that the division line between decaying and non-decaying states goes through the singular part of the spectrum, which may contain both decaying and non-decaying states. For a long time singular spectra have been regarded as ‘‘unphysical’’ (see for example [7, p. 23]) and neglected. However it is easy to find genuine ‘‘physical’’ systems for which singular spectra not only appear but they are even generic [8–14]. The simplest of them are Cantor-like (or fractal) sets which we study in this paper. Quantum systems with singular spectra have many interesting and sometimes even counter intuitive properties. Singular spectra may behave like point spectra but may also behave as absolutely continuous spectra. What is however really surprising is the comparison of two formulations of quantum mechanics – Hamiltonian and Liouvillian in the presence of singular spectrum. If the singular spectrum is absent then these two formulations are equivalent. However, as we have shown [6,15–17] this is no longer true for system with singular spectra. It is always possible to construct Hamiltonians with singular spectra for which no scattering states exist, because of the absence of absolutely continuous spectrum, while their corresponding Liouvillians may have scattering states because of the emergence of the absolutely continuous spectrum, contrary to the general belief that scattering theory is equivalent in both Hilbert and Liouville spaces [18].
*
Corresponding author. Tel.: +32-2-650-50-48; fax: +32-2-650-50-28. E-mail address: [email protected] (I. Antoniou).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 2 4 - 3
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This paper is devoted to study one of the simplest class of singular spectrum, namely a Cantor type set with a constant ratio of dissection. Such spectra in spite of their simplicity exhibit all the above-mentioned complexities of the behavior of quantum systems. They provide constructive examples of fractal spectra and show strict connections with algebraic number theory. In Section 2 we discuss the problem of decay and our recent result concerning the characterization of decaying states. Section 3 is an introduction to Hamiltonian systems with fractal spectra. In Section 4 we study the connections between decay and algebraic number theory. We show the one-to-one correspondence between decaying states from systems with Cantor type spectra and the algebraic S-numbers. We also analyze the question of separability of decaying and non-decaying states. Section 5 is devoted to the analysis of transitions from non-decaying to decaying states on an example of n-free particle system. We also analyze here the decay rates in systems with fractal spectra. Among other results we give a constructive example of a Hamiltonian with singular spectrum which gives rise to the Liouvillian with non-empty absolutely continuous spectrum.
2. Decay of pure states in quantum mechanics The pure states of a quantum mechanical system are wave functions regarded as elements of a separable Hilbert space H in the von Neumann formulation [19] of quantum mechanics. The time evolution of a wave function w 2 H is governed by the unitary group Ut ¼ eitH ;
t 2 R;
on H, which is the solution of the Schroedinger equation ot w ¼ iH w;
h ¼ 1:
The Hamiltonian H is a selfadjoint operator on H. Denoting by r the spectrum of H and by fEk g its spectral family we can write Z k dEk ; H¼ r
Ut ¼
Z
eikt dEk : r
A pure state w 2 H is called a decaying state if its survival amplitude decays asymptotically, t ! 1: Z hw; Ut wi ¼ eikt dhw; Ek wi ! 0: r
The survival probability, i.e. the probability that at time t the state w has not yet decayed is df
pðtÞ¼jhw; Ut wij2 : The survival amplitude and probability are correctly defined for each vector w 2 H. Therefore in the space H we can distinguish the class HD of all decaying elements. Our purpose is to give the spectral characterization of the class HD . It turns out that HD is a closed linear subspace of H reducing H . This allows us to introduce and characterize the decaying and non-decaying spectra of the Hamiltonian. Moreover we can also relate the decaying spectrum to the point, singular continuous and absolutely continuous spectra of the Hamiltonian. Before doing so it is however necessary to recall some basic facts concerning the spectral analysis of selfadjoint operators. Denote by Hp the closed linear hull of all eigenvectors of H . The continuous subspace of H is Hc ¼ H Hp . Recall that the singular continuous subspace Hsc of Hc consists of all w 2 Hc for which there exists a Borel set B0 R of Lebesgue measure zero such that B0 dEk w ¼ w. By Hac ¼ Hc Hsc we shall denote the absolutely continuous subspace of Hc . Recall also that Hp , Hc , Hsc and Hac are closed linear subspaces of H which reduce the operator H . The spectra of the corresponding reductions of H will be called, respectively, point, continuous, singular continuous and absolutely continuous spectrum of H , and will be denoted by rp , rc , rsc and rac correspondingly [20]. For any given w 2 H we denote by l ¼ lw the spectral measure on r determined by the non-decreasing function Fw ðkÞ ¼ hw; Ek wi for k 2 R:
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Let w ¼ wp þ wsc þ wac be the decomposition of w corresponding to the direct sum Hp Hsc Hac . Putting lp ¼ lwp , lsc ¼ lwsc and lac ¼ lwac we obtain the Jordan decomposition of l: l ¼ lp þ lsc þ lac
ð1Þ
onto the point, singular continuous and absolutely continuous component. Conversely, given any three finite Borel measures lp , lsc and lac , where lp is concentrated on a countable set of points, and the other two measures are, respectively, singular and absolutely continuous, one can always construct a Hilbert space H and a selfadjoint operator H such that these measures are spectral measures associated with some w 2 H. The above-presented decomposition of the spectral measures and the Hilbert space H can be further refined in such a way that the decay can be incorporated in a natural way. Proposition 1 [5,6]. The space HD is a closed linear subspace of H. Let us now introduce the space HND , of non-decaying states, as the orthogonal complement of HD . By Proposition 1 we have the direct sum decomposition H ¼ HD HND : We have the following. Proposition 2 [5,6]. The spaces HD and HND reduce the Hamiltonian operator H . An important consequence of above propositions is that the reductions of the Hamiltonian to HD and HND have their own disjoint spectra rD and rND such that r ¼ rD [ rND : The obtained decomposition of the spectrum of the Hamiltonian concerns in fact its singular continuous component since, obviously, each absolutely continuous component decays, i.e. rac rD and discrete does not, i.e. rp rND . In the same way as above we can also obtain a refinement of the singular continuous part in the spectral decomD position of the Hamiltonian. Namely, let us denote HD sc to be the set of all decaying states in Hsc . The space Hsc consists of all vectors w 2 Hsc such that the corresponding measure l ¼ lw is singular with respect to the Lebesgue measure and its Fourier transform is 0 in infinity. HD sc is also a closed linear subspace Hsc . This allows to define, as the ND D orthogonal complement of HD sc in Hsc , the space Hsc consisting of all non-decaying singular states. Both spaces Hsc D ND and HND also reduce the Hamiltonian H . Therefore we have the direct sum decomposition H ¼ H H . This sc sc sc sc leads to the following direct sum decomposition of the whole space H: ND H ¼ Hp HD sc Hsc Hac ;
HD ¼ Hac HD sc ;
HND ¼ Hp HND sc ;
H ¼ HD HND :
As it is well known [20] the other two spaces Hp and Hac also reduce H . Therefore denoting the corresponding spectra ND of reduced operators by rp , rD sc , rsc and rac , respectively, we obtain a refinement of the Jordan decomposition (1) of the spectrum r of a selfadjoint operator which takes into account decay: ND r ¼ rp [ rD sc [ rsc [ rac :
3. Fractal spectra Singular spectra may already appear in Hamiltonian systems of the form ðd2 =dx2 Þ þ V , where the potential V is an almost periodic function or even when V is a uniform limit of periodic functions. Each period in the potential creates a gap in the spectrum. In result the spectrum of such an operator is in general a Cantor type set. For example, one can show that there is an absolutely summable sequence fan g such that the spectrum of the Hamiltonian H ¼
1 d2 X þ an cosðx2n Þ 2 dx n¼0
is the Cantor set (see, for example, [9] and references therein).
ð2Þ
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A vast literature has been devoted to the study of the classes of potentials that lead to singular spectra. We shall not review these important results here. Instead, given a Hamiltonian with a specified singular spectrum, we shall study its properties. Let us, therefore, show first how to correspond a Hamiltonian to a given spectrum. Suppose that the required spectrum r is a Borel subset of R and a measure l, which we would like to regard as a spectral measure, is a Borel measure on r. In the spectral representation, the Hamiltonian with spectral measure l is the multiplication operator Hf ðkÞ ¼ kf ðkÞ
ð3Þ
on the Hilbert space L2 ðr; lÞ. The spectral resolution Ek of H is Z H¼ k dEk ; r
where Ek f ðk0 Þ ¼
f ðkÞ; 0
0 6 k0 < k; otherwise:
The spectral measure lðdkÞ ¼ dF ðkÞ corresponds to the distribution function F associated with the cyclic vector w ¼ 1: ð4Þ
F ðkÞ ¼ hw; Ek wi:
We introduce now an important class of Cantor type sets which will serve as the supports of singular measures. In order to simplify the notation we restrict our considerations to the interval ½a; b. Let frn g1 n¼1 be a sequence of real numbers, 0 < rn < 12. In the first step divide the interval ½a; b into three parts of the lengths proportional to r1 , 1 2r1 and r1 , respectively. Then remove the middle open interval. In the second step divide each of the two remaining intervals into three parts of lengths proportional to r2 , 1 2r2 and r2 , respectively. Then remove the middle open intervals, and so on. In this way we obtain T in the kth step a closed set rk consisting of 2k disjoint intervals, each one of the length ðb aÞr1 r2 rk . Denote r ¼ k rk and observe that r is a closed set with points of the form x ¼ a þ ðb aÞ½e1 ð1 r1 Þ þ e1 r1 ð1 r2 Þ þ þ ek r1 rk1 ð1 rk Þ þ ; where ek ¼ 0 or 1. An important particular case is a Cantor type set with constant ratio of dissection when r1 ¼ r2 ¼ ¼ r, where 0 < r < 12. In this case the set r consists of the points x: x ¼ a þ ðb aÞð1 rÞ
1 X
ek rk1 :
ð5Þ
k¼1
P 1 k Cantor’s ternary set on the interval ½0; 1 with points x ¼ 2 1 k¼1 ek =3 , is obtained by putting a ¼ 0, b ¼ 1 and r ¼ 3. Let us focus our attention on Cantor type sets on the interval ½0; 2p with a constant ratio of dissection. Therefore each point x 2 r has the form x ¼ 2pð1 rÞ
1 X
ek rk1 :
ð6Þ
k¼1
On the set r define the distribution function F putting for the points x of the form (6) F ðxÞ ¼
1 X ek : k 2 k¼1
We extend F on ½0; 2p putting F ðxÞ ¼ sup F ðyÞ: y2r y6x
It is easy to see (cf. [21]) that F is non-decreasing and continuous with F 0 ðxÞ ¼ 0 for almost all x 2 ½0; 2p. Therefore the df corresponding normalized Borel measure l, where lð½a; bÞ ¼ F ðbÞ F ðaÞ, is singular with respect to the Lebesgue measure. According to the above prescription we can define a Hamiltonian system on the Hilbert space df H ¼ L2 ð½0; 2p; lÞ ¼ L2 ðr; lÞ putting as H the multiplication operator. It is easy to see that the function (state) w 1 is a cyclic vector for H , i.e. the set of all finite linear combinations of H n w, n ¼ 0; 1; 2; . . . ; is dense in H. Therefore, the operator H , considered as a Hamiltonian on H, has purely singular continuous spectrum.
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4. Decay and S-numbers We would like to know whether the constructed in the previous section cyclic state w ¼ wðrÞ is decaying. It turns out that the decay of w depends on the ratio of dissection. In [16] we have given some examples of decaying and nondecaying states. For example, if the ratio of dissection r ¼ 13, which leads to Cantor’s ternary set, the corresponding state wð13Þ is non-decaying. On the other hand, taking r ¼ 29 we can prove that the state wð29Þ is decaying. One may ask therefore what properties of the ratio of dissection r decide about the decay properties of the corresponding state w. Another natural question is whether decaying and non-decaying states can be in some sense separated. It is surprising that the answer to the second question is negative. As we shall see below for any dissection rate r determining a non-decaying state wðrÞ and any e > 0 one can find a dissection rate r0 with jr r0 j < e, such that the state wðr0 Þ is non-decaying. This will follow directly from an algebraic characterization of decaying states associated with the Cantor type sets with a constant ratio of dissection. First, however, recall some basic facts from algebraic number theory (see, for example, [22]). An algebraic integer is a root of an equation of the form an xn þ an1 xn1 þ þ a0 ¼ 0;
ð7Þ
where ak are integer numbers and an ¼ 1. If a is a root of the polynomial (7) which is irreducible, i.e. there is no polynomial of degree m < n with the integer coefficients and the leading coefficient having a as a root, then the other roots of (7) are called the conjugates of a. An algebraic integer a > 1 such that each conjugate a0 , a0 6¼ a, that satisfies ja0 j < 1 is called an S-number. We have: Theorem 1. Let wðrÞ be the cyclic state of the Hamiltonian H with the spectrum of Cantor type with a constant ratio of dissection r. The state wðrÞ is non-decaying if and only if 1=r is an S-number. Correspondingly, wðrÞ is decaying if and only if 1=r is not an S-number. In order to prove this theorem it is enough to show [16,21] that the Fourier transform of the spectral measure lwðrÞ , which is of the form Z 2p 1 Y 1 1 eitx dF ðxÞ ¼ epit cos ptrk1 ð1 rÞ; ð8Þ lwðrÞ ðtÞ ¼ 2p 0 2p k¼1 converges to 0 or not when 1=r is not or is, accordingly, an S-number. However, the behavior of (8) as t ! 1 is equivalent to the behavior of u 7!
1 Y
cos purk
ð9Þ
k¼1
as u ! 1 [21]. An elegant proof of the fact that (9) converges if and only if 1=r is not an S-number can be found in [23] (see also [21]). It p is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obvious that the S-numbers include all integers n > 1. It is also easy to verify that any number of the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðp þ p2 þ 4qÞ, p where p; q 2 N; q 6 p is an S-number (its only conjugate is 12ðp p2 þ 4qÞ < 1). For example the 2 ffiffi ffi golden number 12ð 5 þ 1Þ is an S-number. On the other hand, none of the irreducible rationals p=q with p; q 2 N n f1g is an S-number. In fact such p=q is not even an algebraic integer [22]. Therefore if the ratio of dissection is any irreducible rational number k=n < 1=2, where k and n are integers different from 1, then the corresponding cyclic state is decaying. Moreover, since the irreducible rationals are dense in R, they are also arbitrary close to S-numbers. It follows from the above considerations that it is impossible to isolate non-decaying states associated with Cantor type sets. On the other hand the decaying states can be separated from non-decaying. In fact we have: Proposition 3. For each ratio of dissection r, which determines a decaying state wðrÞ, there is d > 0 such that the nearest ratio of dissection r0 , which determines a non-decaying state wðr0 Þ is at the distant larger than d. In order to show the above proposition we use the fact that the set of S-numbers is closed [23]. Let r be a ratio of dissection which determines a decaying state. Since 1=r does not belong to the set of S-numbers, there is a neighborhood, i.e. some numbers a; b such that a < 1=r < b, and such that the interval ða; bÞ has empty intersection with the class S. This implies that the rate of dissection r is separated from the nearest ratio of dissection of a non-decaying state by at least d ¼ minfr 1=b; 1=a rg.
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5. Decay rate of states associated with fractal spectra In the previous section we have shown that in the singular continuous space with a spectrum of Cantor type there is no clear distinction between decaying and non-decaying states. The singular continuous space contains, however, also states that are very close to states associated with the absolutely continuous spectrum. Since the decay rate is in practice the only observed quantity a physically important question is: Can we judge from the decay rate what is the nature of the spectrum? In other words: How fast the decay in the singular and absolutely continuous spectra can be? One can find in literature [24] wrong statements that in the singular continuous case the decay is very slow compared with the absolutely continuous case. This may be true in some examples, however, we shall also demonstrate the opposite in another example constructing one state in the absolutely continuous subspace and another state in the singular continuous subspace that decay with almost the same rate. It is surprising that those decaying states from the singular continuous space that have decay rates similar to the rate of some states from absolutely continuous space can be obtained by taking consecutive iterations of non-decaying states associated with Cantor’s ternary spectrum and a very simple continuous monotonic transformation. In the sequel we shall consider only Hamiltonians with bounded spectra and, for simplicity, confine ourself to the study of the Fourier–Stieltjes coefficients which reflect the behavior of the whole correlation function. The construction is partly based on a method of Wiener and Wintner [25] and its elaboration by Schaeffer [26]. Let us consider a free n-particle system. Suppose that the one particle system is described by the Hamiltonian H on the Hilbert space H. The states of the compound system are elements of the tensor product space H¼H H: |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} n-times
The corresponding n-particle Hamiltonian is H ¼ H I I þ I H I I þ þ I I H:
ð10Þ
If the spectrum of H is r, then the spectrum of H is rH ¼ fx ¼ x1 þ þ x2 : x1 ; . . . ; xn 2 rg: Suppose w is a cyclic state for H and F the corresponding distribution (4). Then the spectral distribution F associated with the state W ¼ w w |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} n-times
is the convolution F n . Similarly the corresponding spectral measure is l n . Assume, as before, that rH ½0; 2p, which is the case if r ½0; 2p=n. Define the following transformation of the interval ½0; 2p onto itself: KðkÞ ¼
1 2 ðk þ pkÞ: 3p
ð11Þ
We have the following. Theorem 2. Suppose that the spectrum of the Hamiltonian H is the Cantor ternary set on the interval ½0; 2p=n and w its e ¼ KðHÞ be the function of the n-particle Hamiltonian H and f e e . Then the cyclic state. Let H E k gk2R the spectral family of H spectral measure m ¼ mW determined by the distribution k 7! hW; e E k Wi has singular continuous spectrum and for each e > 0 there is n ¼ nðeÞ such that the rate of convergence of the Fourier coefficients of m is of the order nð1=2Þþe . Before proving the theorem, denote by fEk g the spectral family of H, i.e. Z 1 k dEk ; H¼ 1
and by FðkÞ the distribution function associated with W FðkÞ ¼ hW; Ek Wi:
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Since FðkÞ ¼ F n ðkÞ, the Fourier transform of F satisfies ^ðtÞ ¼ F^n ðtÞ: F Since F is the distribution function of Cantor’s ternary set, as we know from the previous section, F is singular continuous and its Fourier transform F^ðtÞ does not converge to 0 as t ! 1. This implies that FðkÞ is also singular (see [25] ^ for details) and, of R 1course, FðtÞ also does not converge to 0 as t ! 1. e are Since KðHÞ ¼ 1 KðkÞ dEk , the spectral projectors e E k of H e E k ¼ E ðK1 ðkÞÞ: Therefore hW; e E k Wi ¼ hW; E ðK1 ðkÞÞWi ¼ FðK1 ðkÞÞ; which implies that FðK1 ðkÞÞ is the distribution function corresponding to the spectral measure m. The rest of the proof follows exactly [25]. For the sake of completeness, we remind the reader the crucial steps. Firstly, since the map K is one-to-one and its derivative has positive bounds, FðK1 ðkÞÞ is also singular continuous. Secondly, denote by ck the Fourier–Stieltjes coefficients of FðkÞ: Z 2p eikk dFðkÞ: ck ¼ 0
One can prove (it is the most difficult part of the proof [25]) that for each e > 0 there is N ¼ N ðeÞ such that N X
jck j ¼ OðN e Þ as N ! 1:
ð12Þ
k¼N
It remains to prove that (12) implies Z 2p eink dFðK1 ðkÞÞ ¼ Oðnð1=2Þþe Þ as n ! 1: j^ mðnÞj ¼
ð13Þ
0
Remark 1. It follows immediately from Theorem 2 that the rate of convergence of the order nð1=2Þþe can be as close as we please to the rate of convergence of states from an absolutely continuous subspace. Indeed it is enough to consider the sequence an ¼ nð1=2Þd ; n ¼ 1; 2; . . . ; d > 0. Because fan g is square summable then, by the Riesz–Fischer theorem [21], there is a function f 2 L2½0;2p such that the Fourier coefficients of f coincide with fan g. Then define the measure m1 ðdxÞ ¼ f ðxÞ dx, which is, of course, absolutely continuous. As we already know, the measure m1 can be regarded as the spectral measure associated with a cyclic state of a Hamiltonian with absolutely continuous spectrum. Remark 2. The decay rate of states associated with Cantor type spectra can actually be of the order rn n1=2 , where frn g is an arbitrary slowly increasing to infinity sequence of positive numbers. To construct a spectral measure with this property one can apply Salem’s method [27]. This method amounts to the construction of a Cantor type set on ½0; 2p with varying both ratio of dissection and the number of intervals removed on each step. On the set constructed in this way one can define a Cantor distribution function GðkÞ such that their Fourier–Stieltjes coefficients satisfy (12) with OðN e Þ replaced by rN . The final distribution function F ðkÞ having Fourier–Stieltjes coefficients cn satisfying cn ¼ Oðrn n1=2 Þ, as n ! 1, is obtained through transformation (13), i.e. F ðkÞ ¼ GðK1 ðkÞÞ. Then the corresponding Hamiltonian and decaying state can be constructed as in Section 3. Remark 3. Theorem 2 provides a constructive example of a Hamiltonian with purely singular spectrum which gives rise to the Liouvillian with non-empty absolutely continuous spectrum (see also [15–17]). Indeed, let H be the Hamiltonian e from Theorem 2 but with the spectrum shifted to the interval ½p; p. Let w be a state with the spectral measure H l ¼ lw satisfying the thesis of Theorem 2. Consider the Liouvillian L¼H I I H acting on H H and the state df
W ¼ w w:
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Since in this case the measure l is symmetric with respect to 0, the spectral measure m ¼ mW associated with W is the convolution square of l: m ¼ l l: The Fourier–Stieltjes transform of m is therefore mðtÞ ¼ jlðtÞj2 : In particular the Fourier–Stieltjes coefficients of m are squares of cn thus of the order n1þd where d is an arbitrary small positive number. Taking d < 12 we see that the Fourier–Stieltjes coefficients of m are square summable which implies that m is absolutely continuous (see Remark 1).
6. Concluding remarks (1) Proposition 3 shows how inappropriate the computational modeling of decaying and non-decaying states can be. In the case of Hamiltonians with fractal spectra the construction of decaying states amounts to the construction of a Cantor type set with a given ratio of dissection. According to Proposition 2 it is possible to construct such decaying states wðrÞ for which the distance d of r from the nearest inverse of an S-number is within the computing accuracy. However any construction of a non-decaying state wðrÞ for which r has an infinite dyadic expansion is completely unreliable. The reason is that we can not perform computations on numbers with infinite dyadic expansion. Therefore any truncation of the dissection rate give us, in general, a decaying state instead. Physically speaking any finite approximation of such nondecaying state is a decaying state. Only in the infinite limit we obtain non-decay. Moreover the possibility of construction of decaying states is also rather theoretical because very little is known about the localization of S-numbers. (2) It is interesting to observe how by increasing the number of particles in a free n-particle system we can change the decay properties of the compound state w w. The state w with the Cantor spectral measure lw is not decaying. Moreover it follows from the estimation of the Fourier coefficients fck g of lw [15,21] that the time evolution df
t 7! wt ¼ eitH w;
t ¼ 1; 2; . . . ;
P is not even weakly mixing [28]. We only know that ð1=NÞ Nk¼N jcn j is bounded as N ! 1 (weak mixing means that the latter averaged sum converges). However, for a given 0 < e < 1 there is n such that the Fourier coefficients f~ ck g of the spectral measure lww of the compound state satisfy (12). In particular, the time evolution of the compound state is mixing. By further increasing the number of particles, we obtain stronger Cesaro convergence although the compound state is still purely singular and non-decaying. The latter follows from the fact that its spectral measure is in fact an infinite convolution of purely discontinuous Bernoulli distributions and the dichotomy theorem of Jessen and Wintner [29]. Nevertheless (12) means that the events when c~k jumps away from 0 become scarce as the number of particles increases.
Acknowledgements We thank Professor I. Prigogine for his interest and support, and Professor H. von Weizs€ acker, and Dr. D. Lenz for their interest and useful references. This work enjoyed the financial support of the Belgian Government through the Interuniversity Attraction Poles.
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Reed M, Simo B. Methods of modern mathematical physics, vol. I: Functional analysis. New York: Academic Press; 1980. Pearson DB. Singular continuous measures in scattering theory. Commun Math Phys 1978;60:13–36. Bellisard J. Schr€ odinger operators with almost periodic potential: an overview. Lect Notes Phys 1982;153:356–63. Bellisard J, Iochum B, Scoppola E, Testard D. Spectral properties of one-dimensional quasicrystals. Commun Math Phys 1989;125:527–43. Damanik D, Lenz D. Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues. Commun Math Phys 1999;207:687–96. Damanik D, Lenz D. Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent. Lett Math Phys 1999;50:245–57. Damanik D, Killip R, Lenz D. Uniform spectral properties of one-dimensional quasicrystals, III. a-Continuity. Commun Math Phys 2000;212:191–204. Lenz D. Aperiodische Ordnung und gleichm€aßige spektrale Eigenschaften von Quasikristallen. Berlin: Logos; 2000. Antoniou I, Gadella M, Suchanecki Z. General properties of the Liouville operator. Int J Theor Phys 1998;37:1641–54. Antoniou I, Shkarin S, Suchanecki Z. The spectrum of the Liouville–von Neumann operator in the Hilbert–Schmidt space. J Math Phys 1999;40:4106–18. Antoniou I, Suchanecki Z. Quantum decay in Liouville space. In: Doebner H-D, Hennig J-D, L€ ucke W, Dobrev VK, editors. Quantum theory and symmetries. Singapore: World Scientific; 2000. p. 179–90. Prugovecki EE. Quantum mechanics in Hilbert space. New York: Academic Press; 1981. von Neumann J. Mathematical foundations of quantum mechanics. Princeton, NJ: Princeton University Press; 1955. Weidmann H. Linear operators in Hilbert space. Berlin: Springer; 1980. Zygmund A. In: Trigonometric series, vol. I, II. Cambridge: Cambridge University Press; 1968. Stewart IN, Tall DO. Algebraic number theory. London: Chapman & Hall; 1979. Salem R. Algebraic numbers and Fourier analysis. Belmont, CA: The Wadsworth Mathematics Series; 1983 (Reprinted from: Orig. publ. 1963 by Heath, Boston). Milek B, Seba P. Singular continuous quasienergy spectrum in the kicked rotator with separable perturbation: possibility of the onset of quantum chaos. Phys Rev A 1990;42:3213–20. Wiener N, Wintner A. Fourier–Stieltjes transforms and singular infinite convolutions. Amer J Math 1938;60:513–22. Schaeffer AC. The Fourier–Stieltjes coefficients of a function of bounded variation. Amer J Math 1939;61:934–40. Salem R. On some singular monotonic functions which are strictly increasing. Trans Amer Math Soc 1943;53:427–39. Antoniou I, Suchanecki Z. Spectral characterization of mixing evolutions in classical statistical physics. In: Lumer G, Weis L, editors. Evolution equations and their applications in physical and life sciences. New York: Marcel Dekker; 2000. p. 301–10. Jessen B, Wintner A. Distribution functions and the Riemann zeta function. Trans Amer Math Soc 1935;38:48–88.
Quantum systems with fractal spectra I. Antoniou a
a,b,*
, Z. Suchanecki
a,b,c
International Solvay Institute for Physics and Chemistry, CP 231, ULB Campus Plaine, Bd. du Triomphe, 1050 Brussels, Belgium b Theoretische Natuurkunde, Free University of Brussels, Brussels, Belgium c Institute of Mathematics, University of Opole, Opole, Poland Accepted 15 January 2002
Abstract We study Hamiltonians with singular spectra of Cantor type with a constant ratio of dissection and show strict connections between the decay properties of the states in the singular subspace and the algebraic number theory. More specifically, we study the decay properties of free n-particle systems and the computability of decaying and non-decaying states in the singular continuous subspace. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The decay of a quantum system depends on the nature of its spectrum. It is well known that if a Hamiltonian system has a purely absolutely continuous spectrum then each state decays. If the Hamiltonian has point spectrum then each state is non-decaying. If a Hamiltonian system consists of both point and absolutely continuous spectrum then the underlying Hilbert space can be decomposed as a direct sum of two Hilbert spaces. Each of these spaces reduces the Hamiltonian and one of them consists only of decaying states while the other only of non-decaying. However, the spectrum of an arbitrary Hamiltonian does not necessarily consist of these two parts only. In addition, singular continuous spectrum may be present. It is therefore natural to ask whether the Hilbert space of an arbitrary Hamiltonian H can also be decomposed into two parts, which reduce H , in such a way that one Hilbert space consists of decaying states and the other of non-decaying. It is astonishing that although a clear definition of quantum decay and decaying states is known since 1920s [1,2] the above question has been raised and resolved affirmatively only recently in our publications [3–6]. We have shown that the division line between decaying and non-decaying states goes through the singular part of the spectrum, which may contain both decaying and non-decaying states. For a long time singular spectra have been regarded as ‘‘unphysical’’ (see for example [7, p. 23]) and neglected. However it is easy to find genuine ‘‘physical’’ systems for which singular spectra not only appear but they are even generic [8–14]. The simplest of them are Cantor-like (or fractal) sets which we study in this paper. Quantum systems with singular spectra have many interesting and sometimes even counter intuitive properties. Singular spectra may behave like point spectra but may also behave as absolutely continuous spectra. What is however really surprising is the comparison of two formulations of quantum mechanics – Hamiltonian and Liouvillian in the presence of singular spectrum. If the singular spectrum is absent then these two formulations are equivalent. However, as we have shown [6,15–17] this is no longer true for system with singular spectra. It is always possible to construct Hamiltonians with singular spectra for which no scattering states exist, because of the absence of absolutely continuous spectrum, while their corresponding Liouvillians may have scattering states because of the emergence of the absolutely continuous spectrum, contrary to the general belief that scattering theory is equivalent in both Hilbert and Liouville spaces [18].
*
Corresponding author. Tel.: +32-2-650-50-48; fax: +32-2-650-50-28. E-mail address: [email protected] (I. Antoniou).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 2 4 - 3
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This paper is devoted to study one of the simplest class of singular spectrum, namely a Cantor type set with a constant ratio of dissection. Such spectra in spite of their simplicity exhibit all the above-mentioned complexities of the behavior of quantum systems. They provide constructive examples of fractal spectra and show strict connections with algebraic number theory. In Section 2 we discuss the problem of decay and our recent result concerning the characterization of decaying states. Section 3 is an introduction to Hamiltonian systems with fractal spectra. In Section 4 we study the connections between decay and algebraic number theory. We show the one-to-one correspondence between decaying states from systems with Cantor type spectra and the algebraic S-numbers. We also analyze the question of separability of decaying and non-decaying states. Section 5 is devoted to the analysis of transitions from non-decaying to decaying states on an example of n-free particle system. We also analyze here the decay rates in systems with fractal spectra. Among other results we give a constructive example of a Hamiltonian with singular spectrum which gives rise to the Liouvillian with non-empty absolutely continuous spectrum.
2. Decay of pure states in quantum mechanics The pure states of a quantum mechanical system are wave functions regarded as elements of a separable Hilbert space H in the von Neumann formulation [19] of quantum mechanics. The time evolution of a wave function w 2 H is governed by the unitary group Ut ¼ eitH ;
t 2 R;
on H, which is the solution of the Schroedinger equation ot w ¼ iH w;
h ¼ 1:
The Hamiltonian H is a selfadjoint operator on H. Denoting by r the spectrum of H and by fEk g its spectral family we can write Z k dEk ; H¼ r
Ut ¼
Z
eikt dEk : r
A pure state w 2 H is called a decaying state if its survival amplitude decays asymptotically, t ! 1: Z hw; Ut wi ¼ eikt dhw; Ek wi ! 0: r
The survival probability, i.e. the probability that at time t the state w has not yet decayed is df
pðtÞ¼jhw; Ut wij2 : The survival amplitude and probability are correctly defined for each vector w 2 H. Therefore in the space H we can distinguish the class HD of all decaying elements. Our purpose is to give the spectral characterization of the class HD . It turns out that HD is a closed linear subspace of H reducing H . This allows us to introduce and characterize the decaying and non-decaying spectra of the Hamiltonian. Moreover we can also relate the decaying spectrum to the point, singular continuous and absolutely continuous spectra of the Hamiltonian. Before doing so it is however necessary to recall some basic facts concerning the spectral analysis of selfadjoint operators. Denote by Hp the closed linear hull of all eigenvectors of H . The continuous subspace of H is Hc ¼ H Hp . Recall that the singular continuous subspace Hsc of Hc consists of all w 2 Hc for which there exists a Borel set B0 R of Lebesgue measure zero such that B0 dEk w ¼ w. By Hac ¼ Hc Hsc we shall denote the absolutely continuous subspace of Hc . Recall also that Hp , Hc , Hsc and Hac are closed linear subspaces of H which reduce the operator H . The spectra of the corresponding reductions of H will be called, respectively, point, continuous, singular continuous and absolutely continuous spectrum of H , and will be denoted by rp , rc , rsc and rac correspondingly [20]. For any given w 2 H we denote by l ¼ lw the spectral measure on r determined by the non-decreasing function Fw ðkÞ ¼ hw; Ek wi for k 2 R:
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Let w ¼ wp þ wsc þ wac be the decomposition of w corresponding to the direct sum Hp Hsc Hac . Putting lp ¼ lwp , lsc ¼ lwsc and lac ¼ lwac we obtain the Jordan decomposition of l: l ¼ lp þ lsc þ lac
ð1Þ
onto the point, singular continuous and absolutely continuous component. Conversely, given any three finite Borel measures lp , lsc and lac , where lp is concentrated on a countable set of points, and the other two measures are, respectively, singular and absolutely continuous, one can always construct a Hilbert space H and a selfadjoint operator H such that these measures are spectral measures associated with some w 2 H. The above-presented decomposition of the spectral measures and the Hilbert space H can be further refined in such a way that the decay can be incorporated in a natural way. Proposition 1 [5,6]. The space HD is a closed linear subspace of H. Let us now introduce the space HND , of non-decaying states, as the orthogonal complement of HD . By Proposition 1 we have the direct sum decomposition H ¼ HD HND : We have the following. Proposition 2 [5,6]. The spaces HD and HND reduce the Hamiltonian operator H . An important consequence of above propositions is that the reductions of the Hamiltonian to HD and HND have their own disjoint spectra rD and rND such that r ¼ rD [ rND : The obtained decomposition of the spectrum of the Hamiltonian concerns in fact its singular continuous component since, obviously, each absolutely continuous component decays, i.e. rac rD and discrete does not, i.e. rp rND . In the same way as above we can also obtain a refinement of the singular continuous part in the spectral decomD position of the Hamiltonian. Namely, let us denote HD sc to be the set of all decaying states in Hsc . The space Hsc consists of all vectors w 2 Hsc such that the corresponding measure l ¼ lw is singular with respect to the Lebesgue measure and its Fourier transform is 0 in infinity. HD sc is also a closed linear subspace Hsc . This allows to define, as the ND D orthogonal complement of HD sc in Hsc , the space Hsc consisting of all non-decaying singular states. Both spaces Hsc D ND and HND also reduce the Hamiltonian H . Therefore we have the direct sum decomposition H ¼ H H . This sc sc sc sc leads to the following direct sum decomposition of the whole space H: ND H ¼ Hp HD sc Hsc Hac ;
HD ¼ Hac HD sc ;
HND ¼ Hp HND sc ;
H ¼ HD HND :
As it is well known [20] the other two spaces Hp and Hac also reduce H . Therefore denoting the corresponding spectra ND of reduced operators by rp , rD sc , rsc and rac , respectively, we obtain a refinement of the Jordan decomposition (1) of the spectrum r of a selfadjoint operator which takes into account decay: ND r ¼ rp [ rD sc [ rsc [ rac :
3. Fractal spectra Singular spectra may already appear in Hamiltonian systems of the form ðd2 =dx2 Þ þ V , where the potential V is an almost periodic function or even when V is a uniform limit of periodic functions. Each period in the potential creates a gap in the spectrum. In result the spectrum of such an operator is in general a Cantor type set. For example, one can show that there is an absolutely summable sequence fan g such that the spectrum of the Hamiltonian H ¼
1 d2 X þ an cosðx2n Þ 2 dx n¼0
is the Cantor set (see, for example, [9] and references therein).
ð2Þ
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A vast literature has been devoted to the study of the classes of potentials that lead to singular spectra. We shall not review these important results here. Instead, given a Hamiltonian with a specified singular spectrum, we shall study its properties. Let us, therefore, show first how to correspond a Hamiltonian to a given spectrum. Suppose that the required spectrum r is a Borel subset of R and a measure l, which we would like to regard as a spectral measure, is a Borel measure on r. In the spectral representation, the Hamiltonian with spectral measure l is the multiplication operator Hf ðkÞ ¼ kf ðkÞ
ð3Þ
on the Hilbert space L2 ðr; lÞ. The spectral resolution Ek of H is Z H¼ k dEk ; r
where Ek f ðk0 Þ ¼
f ðkÞ; 0
0 6 k0 < k; otherwise:
The spectral measure lðdkÞ ¼ dF ðkÞ corresponds to the distribution function F associated with the cyclic vector w ¼ 1: ð4Þ
F ðkÞ ¼ hw; Ek wi:
We introduce now an important class of Cantor type sets which will serve as the supports of singular measures. In order to simplify the notation we restrict our considerations to the interval ½a; b. Let frn g1 n¼1 be a sequence of real numbers, 0 < rn < 12. In the first step divide the interval ½a; b into three parts of the lengths proportional to r1 , 1 2r1 and r1 , respectively. Then remove the middle open interval. In the second step divide each of the two remaining intervals into three parts of lengths proportional to r2 , 1 2r2 and r2 , respectively. Then remove the middle open intervals, and so on. In this way we obtain T in the kth step a closed set rk consisting of 2k disjoint intervals, each one of the length ðb aÞr1 r2 rk . Denote r ¼ k rk and observe that r is a closed set with points of the form x ¼ a þ ðb aÞ½e1 ð1 r1 Þ þ e1 r1 ð1 r2 Þ þ þ ek r1 rk1 ð1 rk Þ þ ; where ek ¼ 0 or 1. An important particular case is a Cantor type set with constant ratio of dissection when r1 ¼ r2 ¼ ¼ r, where 0 < r < 12. In this case the set r consists of the points x: x ¼ a þ ðb aÞð1 rÞ
1 X
ek rk1 :
ð5Þ
k¼1
P 1 k Cantor’s ternary set on the interval ½0; 1 with points x ¼ 2 1 k¼1 ek =3 , is obtained by putting a ¼ 0, b ¼ 1 and r ¼ 3. Let us focus our attention on Cantor type sets on the interval ½0; 2p with a constant ratio of dissection. Therefore each point x 2 r has the form x ¼ 2pð1 rÞ
1 X
ek rk1 :
ð6Þ
k¼1
On the set r define the distribution function F putting for the points x of the form (6) F ðxÞ ¼
1 X ek : k 2 k¼1
We extend F on ½0; 2p putting F ðxÞ ¼ sup F ðyÞ: y2r y6x
It is easy to see (cf. [21]) that F is non-decreasing and continuous with F 0 ðxÞ ¼ 0 for almost all x 2 ½0; 2p. Therefore the df corresponding normalized Borel measure l, where lð½a; bÞ ¼ F ðbÞ F ðaÞ, is singular with respect to the Lebesgue measure. According to the above prescription we can define a Hamiltonian system on the Hilbert space df H ¼ L2 ð½0; 2p; lÞ ¼ L2 ðr; lÞ putting as H the multiplication operator. It is easy to see that the function (state) w 1 is a cyclic vector for H , i.e. the set of all finite linear combinations of H n w, n ¼ 0; 1; 2; . . . ; is dense in H. Therefore, the operator H , considered as a Hamiltonian on H, has purely singular continuous spectrum.
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4. Decay and S-numbers We would like to know whether the constructed in the previous section cyclic state w ¼ wðrÞ is decaying. It turns out that the decay of w depends on the ratio of dissection. In [16] we have given some examples of decaying and nondecaying states. For example, if the ratio of dissection r ¼ 13, which leads to Cantor’s ternary set, the corresponding state wð13Þ is non-decaying. On the other hand, taking r ¼ 29 we can prove that the state wð29Þ is decaying. One may ask therefore what properties of the ratio of dissection r decide about the decay properties of the corresponding state w. Another natural question is whether decaying and non-decaying states can be in some sense separated. It is surprising that the answer to the second question is negative. As we shall see below for any dissection rate r determining a non-decaying state wðrÞ and any e > 0 one can find a dissection rate r0 with jr r0 j < e, such that the state wðr0 Þ is non-decaying. This will follow directly from an algebraic characterization of decaying states associated with the Cantor type sets with a constant ratio of dissection. First, however, recall some basic facts from algebraic number theory (see, for example, [22]). An algebraic integer is a root of an equation of the form an xn þ an1 xn1 þ þ a0 ¼ 0;
ð7Þ
where ak are integer numbers and an ¼ 1. If a is a root of the polynomial (7) which is irreducible, i.e. there is no polynomial of degree m < n with the integer coefficients and the leading coefficient having a as a root, then the other roots of (7) are called the conjugates of a. An algebraic integer a > 1 such that each conjugate a0 , a0 6¼ a, that satisfies ja0 j < 1 is called an S-number. We have: Theorem 1. Let wðrÞ be the cyclic state of the Hamiltonian H with the spectrum of Cantor type with a constant ratio of dissection r. The state wðrÞ is non-decaying if and only if 1=r is an S-number. Correspondingly, wðrÞ is decaying if and only if 1=r is not an S-number. In order to prove this theorem it is enough to show [16,21] that the Fourier transform of the spectral measure lwðrÞ , which is of the form Z 2p 1 Y 1 1 eitx dF ðxÞ ¼ epit cos ptrk1 ð1 rÞ; ð8Þ lwðrÞ ðtÞ ¼ 2p 0 2p k¼1 converges to 0 or not when 1=r is not or is, accordingly, an S-number. However, the behavior of (8) as t ! 1 is equivalent to the behavior of u 7!
1 Y
cos purk
ð9Þ
k¼1
as u ! 1 [21]. An elegant proof of the fact that (9) converges if and only if 1=r is not an S-number can be found in [23] (see also [21]). It p is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obvious that the S-numbers include all integers n > 1. It is also easy to verify that any number of the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðp þ p2 þ 4qÞ, p where p; q 2 N; q 6 p is an S-number (its only conjugate is 12ðp p2 þ 4qÞ < 1). For example the 2 ffiffi ffi golden number 12ð 5 þ 1Þ is an S-number. On the other hand, none of the irreducible rationals p=q with p; q 2 N n f1g is an S-number. In fact such p=q is not even an algebraic integer [22]. Therefore if the ratio of dissection is any irreducible rational number k=n < 1=2, where k and n are integers different from 1, then the corresponding cyclic state is decaying. Moreover, since the irreducible rationals are dense in R, they are also arbitrary close to S-numbers. It follows from the above considerations that it is impossible to isolate non-decaying states associated with Cantor type sets. On the other hand the decaying states can be separated from non-decaying. In fact we have: Proposition 3. For each ratio of dissection r, which determines a decaying state wðrÞ, there is d > 0 such that the nearest ratio of dissection r0 , which determines a non-decaying state wðr0 Þ is at the distant larger than d. In order to show the above proposition we use the fact that the set of S-numbers is closed [23]. Let r be a ratio of dissection which determines a decaying state. Since 1=r does not belong to the set of S-numbers, there is a neighborhood, i.e. some numbers a; b such that a < 1=r < b, and such that the interval ða; bÞ has empty intersection with the class S. This implies that the rate of dissection r is separated from the nearest ratio of dissection of a non-decaying state by at least d ¼ minfr 1=b; 1=a rg.
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5. Decay rate of states associated with fractal spectra In the previous section we have shown that in the singular continuous space with a spectrum of Cantor type there is no clear distinction between decaying and non-decaying states. The singular continuous space contains, however, also states that are very close to states associated with the absolutely continuous spectrum. Since the decay rate is in practice the only observed quantity a physically important question is: Can we judge from the decay rate what is the nature of the spectrum? In other words: How fast the decay in the singular and absolutely continuous spectra can be? One can find in literature [24] wrong statements that in the singular continuous case the decay is very slow compared with the absolutely continuous case. This may be true in some examples, however, we shall also demonstrate the opposite in another example constructing one state in the absolutely continuous subspace and another state in the singular continuous subspace that decay with almost the same rate. It is surprising that those decaying states from the singular continuous space that have decay rates similar to the rate of some states from absolutely continuous space can be obtained by taking consecutive iterations of non-decaying states associated with Cantor’s ternary spectrum and a very simple continuous monotonic transformation. In the sequel we shall consider only Hamiltonians with bounded spectra and, for simplicity, confine ourself to the study of the Fourier–Stieltjes coefficients which reflect the behavior of the whole correlation function. The construction is partly based on a method of Wiener and Wintner [25] and its elaboration by Schaeffer [26]. Let us consider a free n-particle system. Suppose that the one particle system is described by the Hamiltonian H on the Hilbert space H. The states of the compound system are elements of the tensor product space H¼H H: |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} n-times
The corresponding n-particle Hamiltonian is H ¼ H I I þ I H I I þ þ I I H:
ð10Þ
If the spectrum of H is r, then the spectrum of H is rH ¼ fx ¼ x1 þ þ x2 : x1 ; . . . ; xn 2 rg: Suppose w is a cyclic state for H and F the corresponding distribution (4). Then the spectral distribution F associated with the state W ¼ w w |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} n-times
is the convolution F n . Similarly the corresponding spectral measure is l n . Assume, as before, that rH ½0; 2p, which is the case if r ½0; 2p=n. Define the following transformation of the interval ½0; 2p onto itself: KðkÞ ¼
1 2 ðk þ pkÞ: 3p
ð11Þ
We have the following. Theorem 2. Suppose that the spectrum of the Hamiltonian H is the Cantor ternary set on the interval ½0; 2p=n and w its e ¼ KðHÞ be the function of the n-particle Hamiltonian H and f e e . Then the cyclic state. Let H E k gk2R the spectral family of H spectral measure m ¼ mW determined by the distribution k 7! hW; e E k Wi has singular continuous spectrum and for each e > 0 there is n ¼ nðeÞ such that the rate of convergence of the Fourier coefficients of m is of the order nð1=2Þþe . Before proving the theorem, denote by fEk g the spectral family of H, i.e. Z 1 k dEk ; H¼ 1
and by FðkÞ the distribution function associated with W FðkÞ ¼ hW; Ek Wi:
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Since FðkÞ ¼ F n ðkÞ, the Fourier transform of F satisfies ^ðtÞ ¼ F^n ðtÞ: F Since F is the distribution function of Cantor’s ternary set, as we know from the previous section, F is singular continuous and its Fourier transform F^ðtÞ does not converge to 0 as t ! 1. This implies that FðkÞ is also singular (see [25] ^ for details) and, of R 1course, FðtÞ also does not converge to 0 as t ! 1. e are Since KðHÞ ¼ 1 KðkÞ dEk , the spectral projectors e E k of H e E k ¼ E ðK1 ðkÞÞ: Therefore hW; e E k Wi ¼ hW; E ðK1 ðkÞÞWi ¼ FðK1 ðkÞÞ; which implies that FðK1 ðkÞÞ is the distribution function corresponding to the spectral measure m. The rest of the proof follows exactly [25]. For the sake of completeness, we remind the reader the crucial steps. Firstly, since the map K is one-to-one and its derivative has positive bounds, FðK1 ðkÞÞ is also singular continuous. Secondly, denote by ck the Fourier–Stieltjes coefficients of FðkÞ: Z 2p eikk dFðkÞ: ck ¼ 0
One can prove (it is the most difficult part of the proof [25]) that for each e > 0 there is N ¼ N ðeÞ such that N X
jck j ¼ OðN e Þ as N ! 1:
ð12Þ
k¼N
It remains to prove that (12) implies Z 2p eink dFðK1 ðkÞÞ ¼ Oðnð1=2Þþe Þ as n ! 1: j^ mðnÞj ¼
ð13Þ
0
Remark 1. It follows immediately from Theorem 2 that the rate of convergence of the order nð1=2Þþe can be as close as we please to the rate of convergence of states from an absolutely continuous subspace. Indeed it is enough to consider the sequence an ¼ nð1=2Þd ; n ¼ 1; 2; . . . ; d > 0. Because fan g is square summable then, by the Riesz–Fischer theorem [21], there is a function f 2 L2½0;2p such that the Fourier coefficients of f coincide with fan g. Then define the measure m1 ðdxÞ ¼ f ðxÞ dx, which is, of course, absolutely continuous. As we already know, the measure m1 can be regarded as the spectral measure associated with a cyclic state of a Hamiltonian with absolutely continuous spectrum. Remark 2. The decay rate of states associated with Cantor type spectra can actually be of the order rn n1=2 , where frn g is an arbitrary slowly increasing to infinity sequence of positive numbers. To construct a spectral measure with this property one can apply Salem’s method [27]. This method amounts to the construction of a Cantor type set on ½0; 2p with varying both ratio of dissection and the number of intervals removed on each step. On the set constructed in this way one can define a Cantor distribution function GðkÞ such that their Fourier–Stieltjes coefficients satisfy (12) with OðN e Þ replaced by rN . The final distribution function F ðkÞ having Fourier–Stieltjes coefficients cn satisfying cn ¼ Oðrn n1=2 Þ, as n ! 1, is obtained through transformation (13), i.e. F ðkÞ ¼ GðK1 ðkÞÞ. Then the corresponding Hamiltonian and decaying state can be constructed as in Section 3. Remark 3. Theorem 2 provides a constructive example of a Hamiltonian with purely singular spectrum which gives rise to the Liouvillian with non-empty absolutely continuous spectrum (see also [15–17]). Indeed, let H be the Hamiltonian e from Theorem 2 but with the spectrum shifted to the interval ½p; p. Let w be a state with the spectral measure H l ¼ lw satisfying the thesis of Theorem 2. Consider the Liouvillian L¼H I I H acting on H H and the state df
W ¼ w w:
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Since in this case the measure l is symmetric with respect to 0, the spectral measure m ¼ mW associated with W is the convolution square of l: m ¼ l l: The Fourier–Stieltjes transform of m is therefore mðtÞ ¼ jlðtÞj2 : In particular the Fourier–Stieltjes coefficients of m are squares of cn thus of the order n1þd where d is an arbitrary small positive number. Taking d < 12 we see that the Fourier–Stieltjes coefficients of m are square summable which implies that m is absolutely continuous (see Remark 1).
6. Concluding remarks (1) Proposition 3 shows how inappropriate the computational modeling of decaying and non-decaying states can be. In the case of Hamiltonians with fractal spectra the construction of decaying states amounts to the construction of a Cantor type set with a given ratio of dissection. According to Proposition 2 it is possible to construct such decaying states wðrÞ for which the distance d of r from the nearest inverse of an S-number is within the computing accuracy. However any construction of a non-decaying state wðrÞ for which r has an infinite dyadic expansion is completely unreliable. The reason is that we can not perform computations on numbers with infinite dyadic expansion. Therefore any truncation of the dissection rate give us, in general, a decaying state instead. Physically speaking any finite approximation of such nondecaying state is a decaying state. Only in the infinite limit we obtain non-decay. Moreover the possibility of construction of decaying states is also rather theoretical because very little is known about the localization of S-numbers. (2) It is interesting to observe how by increasing the number of particles in a free n-particle system we can change the decay properties of the compound state w w. The state w with the Cantor spectral measure lw is not decaying. Moreover it follows from the estimation of the Fourier coefficients fck g of lw [15,21] that the time evolution df
t 7! wt ¼ eitH w;
t ¼ 1; 2; . . . ;
P is not even weakly mixing [28]. We only know that ð1=NÞ Nk¼N jcn j is bounded as N ! 1 (weak mixing means that the latter averaged sum converges). However, for a given 0 < e < 1 there is n such that the Fourier coefficients f~ ck g of the spectral measure lww of the compound state satisfy (12). In particular, the time evolution of the compound state is mixing. By further increasing the number of particles, we obtain stronger Cesaro convergence although the compound state is still purely singular and non-decaying. The latter follows from the fact that its spectral measure is in fact an infinite convolution of purely discontinuous Bernoulli distributions and the dichotomy theorem of Jessen and Wintner [29]. Nevertheless (12) means that the events when c~k jumps away from 0 become scarce as the number of particles increases.
Acknowledgements We thank Professor I. Prigogine for his interest and support, and Professor H. von Weizs€ acker, and Dr. D. Lenz for their interest and useful references. This work enjoyed the financial support of the Belgian Government through the Interuniversity Attraction Poles.
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