Physics Cantor Spectrum and Singular Continuity for a ... - Project Euclid

Communications in Commun. Math. Phys. 122, 643-679 (1989)

Mathematical

Physics

© Springer-Verlag 1989

Cantor Spectrum and Singular Continuity for a Hierarchical Hamiltonian* Herve Kunz1, Roberto Livi1'**, and Andras Sϋto2 1

Institut de Physique Theorique, Ecole Polytechnique Federate de Lausanne, CH-1015 Ecublens, Switzerland 2 Institut de Physique Theorique, Universite de Lausanne, BSP, CH-1015 Lausanne, Switzerland

Abstract. We study the spectrum of the Hamiltonian H on \2(E) given by (Hψ)(ri) = ψ(n + l) + ψ(n — 1)+ V(ri)ψ(ri) with the hierarchical (ultrametric) potential V(2m(2l+\)) = λ(\-Rm)/(\-R\ corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 02. In fact, we find this last behaviour for a periodic subset of sites x but we believe this characterizes the envelope of the nonperiodic solutions. All these results follow from a remarkable property of renormalization of the equation Hψ = Eψ, due to the form (1.1) of the potential. This, together with the choice (1.2) leads to an autonomous nonlinear recurrence equation for the traces τ n , similar to the one existing for the Fibonacci Hamiltonian [16-19] and exploited already in [10, 11]. The spectral variable E enters the recurrence via the initial condition and determines the asymptotic behaviour of τn as n goes to infinity. We show that for any EeΊR. either {τn} is bounded or τn-» + oo and τJτ^^-^R (if jR0 the spectrum is a Cantor set: (i) There are no isolated points in the spectrum. For R < 1 this comes from the limit-periodicity of V(ri) while for R ^ 1 the spectrum is purely continuous. (ii) The eigenvalues of the Dirichlet problems Hψ = Eφ, ψ(0) = ψ(2n) = 0 for n^. 1 are dense in σ(H) and they are at the border of spectral gaps of H. Therefore the gaps are dense everywhere. Thus σ(H) is a closed nowhere dense nonempty set without isolated points: it is a Cantor set. The absence of eigenvalues for .R ^ 1 and also for R < 1 in a large part of the spectrum follows from a detailed analysis of the renormalization map. The rest of the paper is organized as follows. In the next section we discuss the potential (1.1). We show that it is limit periodic for R< 1. It can be written as a ^-dimensional Coulomb potential in the valuation metric of 2-adic numbers with d varying with R. The elements of the hull of the potential are labelled by the 2-adic integers. Section 3 presents the renormalized Schrόdinger equation and the recurrence equation for the traces τn. The asymptotic behaviour, as n goes to infinity, of the sequence τn is analyzed in Sect. 4. Sections 5 and 6 contain the main results of the paper: the determination of the spectrum via the asymptotics of {τπ} and the results on the spectral measures in Sect. 5, the Cantor property, IDS and gap labelling in Sect. 6. Sections 7 and 8 deal with the asymptotic behaviour of the wave functions: they contain respectively the results on the Lyapunov exponent and on the gap-edge states. In Sect. 9 we give an explicit formula for the Green's function. 2. The Potential: Ultrametric Properties and Limit Periodicity A bi-infϊnite sequence V={V(n)}™ao is called limit periodic if there exist periodic sequences VN = {VN(n)}™:=_00 tending uniformly to V: n

Consider now the potential (1.1). Since ord(/ 2« +fc)= ord(fc)

if fc = l,2, ...,2"-l,

we have for any leZ and n ^ l integer. Choose VN to be periodic with period 2N and VN(n)=V(n),

n = l,2,...,2". N

Then VN differs from V only at the sites n = l 2 ,leZ. Therefore \\V-VN\\00=sup\V(l-2N)-V(2N)\ ίeZ

= |A|sup|/(ord(/-2N))-/(N)|

(2.2)

Spectrum of Hierarchical Hamiltonian

647

where we used Eqs. (2.1) and (2.2), ord(0) = oo and V(n)= V(-n\ With the choice (1.2)for/andforRl, one can introduce a map by the definition \n\s =S-°rά(n\

(2.4)

where 2ord(M) is the highest power of 2 dividing n.\ \s is called a valuation, \ | 2 is the standard valuation. The valuation has properties similar to the absolute value: 1. |n|s^0 with equality iff π = 0,

2. |— n|s = |n|s, 3. |n + m| s ^max{|ft| s , |m|s}, 4. \nm\s = \n\s\m\s. Property 3 implies \n -f m\s ^ |n|s H- |m|s. Valuations with Property 3 are called nonarchimedian or ultrametric. | |s induces a metric by d(n,m) = \n — m\s. The metric space (Z, | |s) is precompact. By completing Z with respect to | \s with any s > 1 one obtains the ring of 2-adic integers, J2, satisfying Properties 1 .-4. There is a one-one correspondence between elements ω of I2 and symbol sequences S(ω)= ...ω2colω0, where ω k e{0,1}. S(ω) with a finite number of 1 is the diadic fe representation of the nonnegative integer ω = £ ωk2 . A sequence with a finite 1 number of 0 represents a negative number: S" (...lll)=—1 and n—1

1

J

oo

fc

fe

S~ (...lllω M _ 1 ...ω 0 )= £ ωfc2 — 2". One may use the formal series £ ω k 2 asa

notation for ω. The order of ωe/ 2 is obtained by extension from Z as ord(ω) = min{fc|ωk = l}. Arithmetics on I2 is extended from Z. If S'(ω) + S(ω'), S(ω)S(ω') are defined according to the addition and multiplication of integers in diadic representation then S(ω + ω') = S(ω) + S(ω')9 S(ωω') = S(ω)S(ωf). Let us return to the potential. It naturally extends to x e I2 via l(i-Rord(x)W-R),

1 x

1 V( )—

1 ord(Λ;)-l

if Λ Φ 1 ,

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H. Kunz, R. Livi, and A. Sΰtδ

Therefore, by the extension to / 2 of Eq. (2.4), (1-|*lι/*)/(l-K), -Iog 2 |x| 2 ,

if R1,

(2.5a) xφO.

Thus, one may recognize in V a 1-, 2-, and 3-dimensional Coulomb potential for R!, respectively. Alternately we can write /(1 2V)

if v φ 0 , - ' if v=o,'

(

(Z 25b) ™>

where v = \og2R. Notice the continuity of V in the valuation metric: if \x— y\s < min {|x|s, \y\s}, then F(x) — V(y) = 0. From Eq. (2.5a) and by using Property 4 one easily verifies

This symmetry of dilation is at the origin of the renormalizability of the Schrδdinger equation. Let now fix .R < 1. Then V is limit periodic and the set of its translates V(k\ where V(k\n)=V(n + k), is precompact in /^ . The closure of { V(k)] with respect to the l^ norm is the hull of V (see [12]). Proposition 1. The hull of V is given as

cl({ where cl stands for closure and

Notice that for any n, V(ω\n) can be calculated by a finite algorithm, thanks to Property 3. Proof. Fix any ω e I2 and choose a sequence of integers kt converging to ω in / 2 :|ω — ^|s->0 as /->oo. By the triangle inequality, for any Thus, with s=l/R we find

as so that F(ω) is in the hull. Suppose now that W is in the hull, then there is a sequence of integers k{ such that W(h)= limF i->oo


649

where the limit exists and is uniform in n. The sequence {fcj has at least one accumulation point ω in 72, s° that W(n)=V(n + ω), or equivalently

all neZ

(ω)

W= F .

More is true: the above equality holds with a unique ω e I2. Indeed, assume there exist ω,ω'e/ 2 such that '\s9

all

neTL.

Then by Property 3, \ω — ω'\s = \n + ω — (n + ω')|s ^ \n -f ω\s , all

n e TL .

We can choose a sequence of integers HJ such that n^—ω in /2, i.e. l^. 12 with the operation + forms a compact topological group. It carries a unique + invariant probability measure μ, called the Haar measure. (72,μ, +) is a dynamical system with the "dynamics" ωi—>ω-fl. Lemma 1. 77ze Haar measure on the topological group of 2-adίc integers is dμ(ω)= X dP(ωk)

(2.6)

fc = 0

with P(0) = P(1) = \. (/2>μ> +) is an ergodic dynamical system. Proof. Equation (2.6) defines a probability measure on I2. Consider the cylinder sets ΩΛ(ωJI) = {x6/ 2 |x J I = ωJI}, ω w e{0,l}

and for ωe/ 2 , 5(ω,n)-Ω0(ω0)n ... nί^K^).

(2.7)

Clearly ,n))="πP(ω i ) = l/2-.

(2.8)

For AcI2 let A + l = {ω + l \co o = τm follows from the comparison of (3.12) with Eq. (3.18). It happens that with the choice (1.2) for /, τm satisfies an autonomous difference equation (cf. [10]). Proposition 2. Let f(m)=

m-l

X Rk. Then .

(3.19)

Proof. Take the trace of Eq. (3.15) and use then Γ/0 L(θ

(3.20)


653

On the other hand, from (3.15) (MJ 2 1 =(Mi_J 2 1 =τ M _ 1 (M m _0 2 ι = ^T(τi-ι-2-τJ,

(3.21)

where in the last equality we applied (3.20) with m— 1 replacing m. Equation (3.20) together with (3.21) gives (3.19). Π Remarks. 1. If we suppose only Eq. (1.1), we obtain a formula similar to (3.19) but R replaced by (f(m+1)—/(m))/(/(m)—/(m — 1)) which generally depends on m. This makes Eq. (3.19) non-autonomous and introduces unnecessary complications in the further discussion. Therefore we fix Af(m)/Af(m — 1) = const = R and solve this equation for /. The general solution is /(m) = α -f bRm which gives (1.2) with a = — b = 1/(1-Λ). 2. Another useful form is obtained by breaking the iteration (3.19) into two steps. Introducing , (3.22) Eqs. (3.20) and (3.21) read as τ

=τ

2

2 λ T

λ =Rλ

ι

(323)

This two-dimensional recurrence was studied in [11]. 3. The initial conditions for (3.19) are τ0 = tτA1=E,

τ1=trA2A1=E(E-λ)-2 = τ%-2-λτ0.

(3.24)

Comparison with (3.23) shows that λ0 = λ and thus λm = λ R w τ M _ 1 . . . τ 0 .

(3.25)

1

4. Let t/?° and ψ be the solutions of Hψ = Eψ with initial conditions t/Λθ) = 0,

™°(1) = 1,

(3.26)

Then ιp°(2m)

ιpl(2m)

(3.27)

so that (MJ21 = φ°(2m). From (3.22) and (3.25) we get λm = λRmψ°(2m)

(3.28)

ψ°(2m) = τ m _ 1 . . . τ 0 .

(3.29)

and

The last equation can be obtained directly from Eq. (3.18). Let us return to Eq. (3.6). It can be solved easily under the assumption (1.2). From αm,o = τ m , α m,ι=τ w -/l m , and α m ? 2 = τ m -/

654

H. Kunz, R. Livi, and A. Sύtδ

we make the guess (3.30) and verify it by replacing (3.30) into Eq. (3.6b). As a consequence, αm(n) = τ m -^F(n).

(3.31) 00

This formula is valid also for n = 0 provided that 7(0) = λ £ Rk. Later on, we will m=0

need a general formula for αm(0), showing its dependence on F(0). Such a formula is easily obtained from Eq. (3.6 a). Set n = 0, iterate the equation until descending at α0(0) = £- F(0) and use Eqs. (3.8) and (3.29). This gives αm(0) =ΨQ(2m) \E - V(ϋ) - £ 2/t/;0(2fc)l .

L

J

fc=ι

(3.32)

We have a particular interest in αm(0). The potential is even, V(n)= V( — n\ so that any solution of the Schrόdinger equation can be written in the form αφ° + bψe. Here ψ° is the odd solution introduced in Eq. (3.26) and ψe is the even solution chosen such that ψe(0)=\. Writing Eq. (3.11) for ψe and setting n = 0 we find ψe(2">) = ±vm(Q).

(3.33)

Notice that for -R^l, Hψ = Eψ has no even solution because F(0) = oo. A comparison of Eq. (3.31) with Eqs. (3.11) and (3.4) shows that τm and λm play the role of the renormalized energy and potential amplitude, respectively. Intuitively one expects that both should be bounded in the continuous spectrum. Our results confirm this for τ m with the exception of the set S for jR < 1 . On the other hand, Eq. (3.25) will imply that λm vanishes for large m in (some) gap-edge states. 4. Asymptotic Behaviour of the Sequence of Traces

In this section we classify the sequences {τn} which solve Eq. (3.19) according to their asymptotic behaviour. The classification is similar to the one valid for the Fibonacci Hamiltonian [17] with some additional difficulties due essentially to the fact that bounded sequences are not uniformly bounded or at least we cannot prove it. Our main concern is to describe the unbounded sequences. As to the bounded sequences, we will make use only of the following simple result. Lemma 2. Let τn = 0. Then τn+i= —2 and τn+k = 2 for k>\. Proof. This comes immediately from Eq. (3.19).

Π

We say that a sequence {an} tends monotonically to A if it is monotone for n sufficiently large, and an-+A. Definition 1. A solution {τn} of Eq. (3.19) is l^-type if there exists some N^ 1 such that — τN>2. (4.1)


655

Lemma 3. // {τn} is U^-type then τn tends monotonically to -f oo. Proof. Suppose that (4.1) holds true. Then Eq. (3.19) shows that

for any n^N.

Π

Lemma 4. {τn} is U^-type in the following cases: (i) τ N < —2 for some AΓ^l. (ii) |T N _!|^2 and τN>2 for some J V ^ l . Proof. In both cases Eq. (3.19) implies τN+l>τ2N-2>2.

D

The above list is not exhaustive but we do not need more to prove Proposition 3. Let {τn} be unbounded. Then τn tends monotonically to -hoo. Proof. Lemma 3 proves it for L^-type sequences. Suppose now that {τπ} is unbounded but not 171. Then

for all n ^ 1, as one sees from the definition of U\ and from Lemma 4. One can improve the lower bound for τn: Let us notice that Eq. (3.19) can be written in the form J?

/

9_Lτ

\

(4.2)

Since the last term in the parenthesis is positive, the inequality

L

R .,

holds equally true for all n ^ 1 . Now {τn} is unbounded, whence there is some N ^ such that τN>2 + 2/R. Then

2R therefore the estimation can be repeated for τN+k, all fe^l, which gives the result. Π Definition 2. {τn} is (72-type if there exists some N ^ 1 such that τ j V >.Rτ^_ 1

and

(4.4)

τN>ί/R.

Lemma 5. // {τn} is U2-type then τn tends monotonically to +00.

Π

656

H. Kunz, R. Livi, and A. Suto

Proof. Suppose (4.4) holds true and define bn by τn = bn/R. Write Eq. (3.19) in the form - + -?-*-

(4.5)

Equations (4.4) and (4.5) imply that

Since (4.4) holds for N + 1 replacing N9 the estimation can be repeated. We find for all n>N. This shows monotonicity and divergence.

Π

We denote by 171? C72 the families of sequences of Ux - or t/2-type, respectively. From the proof of Lemmas 3, 5 it is clear that U1CU2

for £ 0, dg/dx > 1 for x = mml,R and oo. The unstable fixed point may, however, become attractive from below (case #< 1) or disappear (case R^ 1):

657


Fig. 2. The function g(x) and the fixed points for R < 1

Proposition 4. For {τn} unbounded the iteration (4.9) is convergent and xn tends either to R or to 1. For R^ίthe only limit point is R and {τj is U^-type. For R 0 increases and εn decreases monotonically for n^N. We distinguish between two cases: (i) x w ^min {!,#}. Then due to the facts that max{l, R} is an attractive fixed point of Eq. (4.10), and εn|0, xn tends to max{l,#}. (ii) xN 0 for all n ^ N and g'(x) > 0 for x > 0. Suppose that xn+1 1. If xn-^R, then xn> 1 for large n. This and τπ-κx) give (4.1) which makes {τn} ί/^type. Below we show that for R = l9xn cannot converge to 1 from below, and for R>\9 xn cannot converge to 1 at all. This proves that each unbounded {τn} is 17rtype for R ^ 1 and, in particular, is (72-tyρe for # = 1. In fact, from Eqs. (3.23) and (3.25), 9

T

9

;j?w

τ0x1x2...xn

(4.11)

658


This immediately shows that for R > 1, τn-> oo and xπ-> 1 are incompatible, and so are τΛ->oo and x Λ fl for R = l. Π By the above proposition we finished the classification of the solutions of Eq. (3.19) for the purposes of the present paper. For #

for any mΞΐO, φ(n)ΆO as n-χχ> and Lemma 7 applies: E ε σ(H)\σ (ii) R^ί. Let φ(0)φO. From Eq. (3.18), ψ(2m+1)-τmψ(2m)=-ψ(0),

(H).

PP

(5.5)

all m ^ 0. Let now φ(0) = 0, then ψ(l ) φ 0. Take at first R = 1 , then Eq. (3.17) reads for all mÔ as

(5.6)

Take J?>l.If m

m+1

λR ψ(2

(5.7)

)^-ψ(ί)

then ψ(n)-^0 asn->co, as it is seen immediately from Eq. (3.1 7). Suppose therefore

as Then

R-\

-fo(l)

as

m-κx).

(5.8)

660

H. Kunz, R. Livi, and A. Sϋtό

All the Eqs. (5.5)-(5.8) imply that ψ(ri)^*Q as w-»oo. Therefore Lemma 7 applies, Eeσ(H)\σpp(H). Now from the argument above and from Lemma 2 it follows that the zeros of the polynomials τn(E) are in the spectrum. By Eq. (3.29), these are also the zeros of the polynomials ψ°(2n)(E). For R < 1, when the spectrum is bounded, a theorem on orthogonal polynomials [21, Theorem 6.1.1] assures that these zeros are dense in the spectrum. We anticipate and prove in the next section that this holds also for #^1. D Proposition 6. Let R 0, and sup σ(H) < 2 if λ < 0. For a self-adjoint operator A let R\σμ).

(5.20)

Lemma 9. For R < 1

Q(H)= U f Π Q(Hn)\nt = lim ( f| Q(Hn)\nt> where Λ

mt

(5.21)

means the interior of Λ .

Proof. For #'(m 2',n\|2

^ k =Σ2

n+ί

which goes to zero as n goes to infinity, because ιpf e D(H\ the domain of H. (iii) Using (ii) and Theorem VIΠ.24a of [23] one obtains UfΠ

N \n^N

r

But ρ(H n) = ρ(Hn) as we learned from (i), whence (5.27) follows.

Π

Theorem 2. Let jR^l. Then ρ(H)=U, i.e. (H) = B = {E e R | {τn} is bounded} ,

σ

(5.28)

and the spectrum is purely singular continuous. Proof. lϊR^l then R = £u 17. Since ρ(#) C I/ [see Eq. (5.24)], it remains to prove that UCρ(H).

Spectrum of Hierarchical Hamίltonian

665

For R ^ 1 we obtain from Eq. (4.6) and Definition 1 that ί/ = {£eR|{τ M }is I7rtype} = (J QN= lim QN, N-+OO

where βN = {£eR|τ j V >max{τJ_ 1 -2,2}}. The sets QN are open which shows that U is open. Let E e 17, then EeQN for TV sufficiently large. Since QN C Q(Hn) for all n ^ N, we find

Equation (5.27) then implies QN C ρ(H) for all N, from which U C ρ(H) is obtained as N goes to infinity. B is a closed set and, according to Proposition 5, it supports a purely continuous spectral measure. A recent theorem by Simon and Spencer [26] asserts that H0 + W has no absolutely continuous spectrum if both { W(ή)}^ x and [W( — n)}™= i are unbounded. This theorem applies to our case, thus the spectrum is purely singular continuous. Π We may summarize what is common in Theorems 1 and 2 as follows. Corollary of Theorems 1, 2. For any R>0, σ(H) = R\l7,

(5-29)

and there is no isolated point in the spectrum.

6. Cantor Spectrum, IDS and Gap Labelling

The entire discussion of this section is based on the fact that we know explicitly a dense set in the spectrum: the zeros of the polynomials τn(E). We stress that such a knowledge is quite unusual in direct (not inverse) spectral problems exhibiting Cantor spectra. Let us consider the Dirichlet problem on the internal [1,2n — 1] assigned to H. More precisely, let H% be given by the semi-infinite matrix

(H

^y={θ,y'

otherwise.'

The solution of H»=Eip in the interval [l,2n —1] corresponds to the solution of H\p = E\p with boundary conditions ψ>(0) = ψ(2n) = 0. Due to the translational symmetry (2.1) of the potential and to the "decoupling" at the sites / - 2", H% is the infinite direct sum of the copies of a finite rank operator with 2Π — 1 eigenvalues of multiplicity 1. The spectrum of H% consists of these 2" —1 points, each being a simple zero of the polynomial ψ°(2n)(E): the condition ψ(0) = 0 singles out ιp° given by Eq. (3.26). Let Hr denote the restriction of H to {ψ(n)} ™=x with boundary condition φ(0) = 0. By our convention, Hr = H for R^l.

666

H. Kunz, R. Livi, and A. Sϋtό

Lemma 11. For all R>0, the spectrum of Hr is purely continuous and Proof. Any eigenvalue of Hr would belong to U or S, and ψ° of Eq. (3.26) would be the corresponding eigenvector. However, diverges if E e U or E e S. Therefore σ(Hr) is continuous. The potential V is even, hence σess(H) = σess(Hr). The result follows from the absence of isolated points in both σ(H) and σ(Hr). Π Lemma 12.

(i)

σ(HΪ)= V {EeR|τ f e (EHO}.

(6.1)

fc = 0

(ii) For any R > 0, H% tends to Hr in the strong resolvent sense. (iiϊ) For any R > 0, Off] Q(HΪ)}ίnίcρ(H).

(6.2)

Proof. Equation (6.1) is an immediate consequence of φ°(2II) = τ I I _ 1 ... τ 0 [see Eq. (3.29)]. The proof of (ii) is analogous to that of (ii) of Lemma 10. One uses

(H (w

D

or y = / 2»,

,

and the estimate ||(H'-ff>Ίl = Σ

1=1

-»0 as rc->oo, r

1

where ψ' = (z — H )~ ψ, ImzφO and φe/2(N). Equation (6.2) follows from (ii) via Theorem VIIL24a of [23] and from Lemma 11. Π Proposition 9. For any R>0 and any N ^ 0, the zeros of the polynomials {τn(E)}n ^N are dense in the spectrum of H. Proof. According to Eq. (6.1) the claim is cl(0 σ(HΪ)\=σ(H). \n*N

(6.3)

)

Now σ(H%) C σ(H) for all n, see Proposition 5. The spectrum is closed, thus C holds in Eq. (6.3), and 3 follows by complementing Eq. (6.2). Π Let En(k\ /c = l,2, ...,2" be the zeros of τn(E) in increasing order. Such an ordering is possible, because of Lemma 13. The zeros of τn are real and simple, £„(!)< E Π (2)0 we find that it is an upper gapedge. Π Theorem 3. For all R > 0 the spectrum of H is a Cantor set. Proof. Combining Propositions 9 and 10 one obtains that the gaps oϊH are dense in the spectrum: For any Eeσ(ίf), inf |E-E'| = 0.

£'e 17

Therefore, the gaps are dense everywhere. Π We keep exploiting Proposition 10 to describe the gap structure in detail. n

Proposition 11. The zeros of ψ°(2 ) separate the zeros of τn, En(k) < en(k) < En(k +1),

k = 1,.., 2" -1.

(6.6) +1

The two sets of zeros together form the set {en+^(k) \ k = 1,..., 2" -1} of the zeros of ιp°(2n+1). This leads to the following

668

H. Kunz, R. Livi, and A. Sϋtδ

Corollary. For rc^l, m

En(k) = en+m(2 -\2k-l)),

all m ^ l .

(6.7) n+ l

Proof. By noticing that in Eq. (6.6) we order, in fact, the zeros of ψ°(2

), we find

and

Replacing in the latter equation n by n+ 1 and k by 2k — 1 we obtain

as claimed.

Π

Proof of Proposition 11. We use the fact that Pk(E): = ιp°(k + !)(£) form a set of orthogonal polynomials relative to some measure (the spectral measure for Hr generated by dj. By Theorem 3.3.3 of [21], in the open intervals (-00, £„(!)), (en(\\ en(2)\ . . ., (en(2n - 1), + oo) there is at least one zero of ψ°(2n +1). Since the zeros of this latter are en(l),...9en(2n-ί) and £„(!), ...,En(2n\ one obtains en(2n-l) 0. It follows from Proposition 1 1 that sgnφ°(2-)(£ll(0) = (-l)1,

/=!,.. .,2".

(6.10)

Replacing λn in Eq. (6.5) by its expression (3.28) gives (x)l

||«Foll=ι

[cf. Eq. (3.14)], the Lyapunov exponent yields the maximal rate of increase of wave vectors as x->oo. In our case, x-> — oo gives the same result. Let

Proposition 14. γ = lim ym exists and m->oo

lim sup α(x)/x = y . χ-» oo

(7.4)

Proof. Taking the norm of Eq. (3.15) and dividing by 2 m + 1 we obtain

ym+^ym+Amβm+i with

We define a sequence xm by setting 7m = x m + m

fc =

Replacing this into (7.5) we obtain •^rn+1 =xm '

For any #>0, ^ Δk/2k+i is convergent. By Eq. (7.3) y w ^0, therefore

(7.5)


673

and limx m exists. Thus, 2k+^0

(7.6)

fc = 0

exists as well. Let now x ^ 0 integer. In diadic representation, x= £ xk2k,

x f c e{0,l}.

Due to the symmetry (2.1) of the potential, (7.7) k =0

/

\k>n

for any nÔ. With a repeated application of this factorization we get, by setting T(x)=T(x02°)T(x121)....

(7.8)

After taking the norm, passing to the logarithm and dividing by x we find X

(7.9)

X k^

X kÔ

Let ε > 0 be arbitrary, then there exists an integer n such that yk ^ y + ε if k> n. We can continue the estimate (7.9) as °^ί- Σ xka(2")+- Σ xk2kyk^ + ε+ί- Σ a(2k}. X

X

k

X k=0

X k>n

=Q

Take now the limits x-ôo and then ε->0 to find limsup - rgy. x-ôo

X

However, 7 = limα(2")/2", so that (7.4) is verified.

Π

Theorem 6. The Lyapunov exponent exists and is zero: in σ(H) if -R^l, and in σ(H)\Sif R 0) or upper (if λ < 0) gap boundary. From the Floquet theory it is known that among the gap-edge states of any one dimensional periodic Schrodinger equation there is a unique [up to factor] periodic or antiperiodic solution, and the others increase linearly. We may compare this with Proposition 15. Let τm(E) = 0 and ψ be a solution of Hψ = Eψ. (i) // φ(0) = 0 then φ(/c + 2 m + 1 ) = -ψ(k) for all keZ. (ii) // tp(0) φ 0 then ψ(2l 2m) = ( - 1 )zφ(0) and

Hog 2 /,

ϊ-»oo

Λ = 2,

(8.1)

o2

Proof. According to the assumption, and

0 = τ m = α m , 0 = α m (2/+l), all ιp(2l 2m)= -ψ((2l-2)2m)= ... =(-l)V(0),

(8.2)

m+l

see Eqs. (3.5), (3.8), and (3.1 1). Set at first ψ(Q) = 0, then ιp(l -2 ) = 0, all I e Z, and ψ does not depend on the value of the potential at the sites / 2 m+1. Thus, ip solves the 2 m+ x -periodic Schrodinger equation Hm+,ιp = Eψ.

(8.3)

By Lemma 2, τ m + 1 = — 2, so that (8.3) has an, up to factor, unique antiperiodic solution [24]. From Eqs. (2.1) and (2.2) VH(l 2n-k)=Vn(l'

2» + k)

(8.4)

for 1,1'eZ and fc = 1 , . . ., 2" — 1. Writing Eq. (8.4) for n = m + 1 one sees easily that ψ is in fact antiperiodic. Let now φ(0) φ 0. Then Eq. (8.2) shows that ψ is bounded oscillating at the sites 2/ 2m. On the other hand, from Eqs. (3.11) and (8.2), 2") = αm(2/)φ(2/ - 2m)-ιp((2l- 1)2™)


675

and after repeated application one obtains k=l

Replacing am(2k) by its expression (331) and using τm = 0 this equation becomes φ((2/H-l)2m)-ιp(2m)-tp(θμm(-l)z Σ /(ord(2fc)).

(8.5)

fe=l

The density of integers of order m among the even numbers is l/2m [cf. Eq. (2.8)], therefore [log 2 2I]

I

Σ /(ord(2fc))=

k=l

I

Σ /(w) Σ W*>.»

m=l

k=l

[log 2 2J]

z

« / Σ /N/2m.

"*°°

(8.6)

m=l

m-1

Setting/(m)= Σ ^fc? Eq. (8.1) follows after simple calculation.

Π

That something happens at 11 = 1, can be noticed only by looking at the corrections to the asymptotic formula (8.1). These can still be obtained from Eq. (8.6) and are respectively of the order of I//, Iog2ί and Zlog2* for R< 1, R = 1 and 9. An Explicit Formula for the Green's Function

We know that the diagonal part of the Green's function is limit periodic when R < 1, see the remark subsequent to Lemma 9. This implies that it should be possible to write GJ(x,x) = Σ Gΐ\x,x), n =0

(9.1)

where G(?\x, x) is a function of period 2" of the variable x. We are now going to derive such a formula which will give an "explicit" solution for the Green's function. At this stage, however, we will make no claim of rigor, and our expression will be a formal one. Let us define the function F(x, θ) by 2π Λ/3

Gz(x,x + r)= J — f(χ,0y». 0 2M

(9.2)

It should satisfy the linear equation -z)F(x,θ) = l.

(9.3)

Note that if the potential is almost periodic then so is F, with the same frequency module. Our potential has the form oo

7(x)= Σ vn(x),

(9.4)

676


where vn(x) is of period 2", V0(X)

=

λ 1-R'

(9.5)

We look for a solution of (9.3) of the form Σ /»(x,0)

F(x,θ)=

(9.6)

with fn(x) of period 2". Using the explicit form (9.5) of the potential one can see that V(x)F(x)=V(x)F(0)+

Σ [/B(x)-/»(0)] Σ u/*)

n=0

(9.7)

Equation (9.3) then gives after some computation

Φn(*) = Σ [#»(#) - ^] " '(^ V) {(z - 2 cos θ)fn(0) - vn(y)F(0)} . y =0

(9.9)

Here HΛ(Θ) is the Hamiltonian defined by P(x-l)+Wn(x)φ(x)

(9.10)

Π

in [0,2 —1] with periodic boundary condition, and n

Σ v{x). J

(9.11)

Σ (Hn(θ)-zΓ\x,y)

(9.12)

(H^-zJ-Hx^Kίy),

(9.13)

Wn(x)=

j=o

Defining

Kn(x)= and

**(*)= V we see that for n^l,

Inserting this expression and (9.8) into the definition (9.6) we get the following expression for F(0), oo

T

C\\

'o-z.

(9.15)

Spectrum of Hierarchical Hamiltonίan

677

This "solves," in principle, the problem we posed. We now express F(0) in a more explicit form. For this, let us note that φn(x) is the solution of the inhomogeneous equation, = (z-2 cos 0)/ΛO) - υJίx)F(0)

(9.16)

with φw(0) = 0. Since

when n^2, we can make a renormalization transformation of this equation by eliminating odd sites. For n ^ 2, we get e-i2θφn(2x + 2) + e2ίβφn(2x - 2) + [Wn_ ,(χ) - z^φn(2x) = (z2-4cos2θ)/n(0)-ι;;_1(x)F(0).

(9.17)

Here z' = z2 — 2 — λz and W£_ ^ and v'n _t are the same as W£ _t and t;n _t except that λ has been replaced by λ' = zRλ. The variables z and i undergo the same transformation as τm and λm earlier, see Eq. (3.23). The condition φn(0) = 0 gives now fjft,θ;λ,z)

1

1,^(0. 20; A'. zQ ' ' '

l

;

Let us define gn(θ; λ9 z) =

' '

(z — 2cosθ).

(9.19)

Then from Eq. (9.14),

and Eq. (9.18) transforms into the recursion formula gn(θ; λ, z) =

g«-1(20; A', z').

(9.21)

Using U)=~>

(9-22)

after n steps of iteration we obtain :.

(9.23)

678

H. Kunz, R. Livi, and A. Sύtό

Since {z(m\ λ(m}} satisfy the recurrence equations (3.23), we may keep the earlier notations, _ z— τ0 , 7

τ

(m) _r

7

z

— τm ,

(m) _

A

— Λm .

With this identification we arrive at

For the Green's function this gives 2π

dθ Γ

/I

1

QO

ii-l

Ί-l

,r)= J — 2COS0-Z+—- + -— Σ λn Π (τm + 2cos2^)-1 0 ^π L

1— K

1— K n=ι

m = o

x e "*, (9.24)

a formula in terms of the renormalized energies, τ π and coupling constants, λn. To give a rigorous meaning to this expression one should analyze the RG transformation on the complex plane, a task which seems not to be easy.

References 1. Frόhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151-184 (1983) 2. Huberman, B.A., Kerszberg, M.: Ultradifϊusion: the relaxation of hierarchical systems. J. Phys. A18, L331-L336 (1985) 3. Teitel, S., Domany, E.: Dynamical phase transitions in hierarchical structures. Phys. Rev. Lett. 55, 2176-2179 (1986) 4. Maritan, A., Stella, A.L.: Exact renormalization group approach to ultradiffusion in a hierarchical structure. J. Phys. A19, L269-L273 (1986) and Exact renormalization group for dynamical phase transition in hierarchical structures. Phys. Rev. Lett. 56, 1754 (1986) 5. Teitel, S.: Transition to anomalous relaxation; localization in a hierarchical potential. University of Rochester preprint (1988) 6. Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Quantum particle in a hierarchical potential with tunneling over arbitrarily large scales. J. Phys. A17, L635-L638 (1984); Multiple tunnelings in ^-dimensions: a quantum particle in a hierarchical potential. Ann. Inst. Henri Poincare 42, 73-108 (1985) 7. Roman, H.E.: Electronic properties of a one-dimensional hierarchical system. Phys. Rev. B 36, 7173-7176 (1987) 8. Ceccato, H.A., Keirstead, W.P., Huberman, B.A.: Quantum states of hierarchical systems. Phys. Rev. A 36, 5509-5512 (1987) 9. Ceccato, H.A., Keirstead, W.P.: Critical eigenfunctions in a quantum hierarchical system. J. Phys. A 21, L75-L83 (1988) 10. Schneider, T., Wurtz, D., Politi, A., Zannetti, M.: Schrόdinger problem for hierarchical heterostructures. Phys. Rev. B36, 1789-1792 (1987) 11. Livi, R., Maritan, A., Ruffo, S.: The spectrum of a 1 — D hierarchical model. J. Stat. Phys. 52, 595-608 (1988) 12. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrόdinger operators. Berlin, Heidelberg, New York: Springer 1987, Chaps. 9 and 10 13. Bellissard, J.: Almost periodicity in solid state physics and C* algebras. Talk given at the Harald Bohr Centenary Conference on Almost Periodic Functions, April 25, 1987 14. Pearson, D.B.: Singular continuous measures in scatterings theory. Commun. Math. Phys. 60, 13-36 (1978)


679

15. Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. AMS 6, 81-86 (1982) 16. Casdagli, M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrόdinger equation. Commun. Math. Phys. 107, 295-318 (1986) 17. Sύtό, A.: The spectrum of a quasiperiodic Schrόdinger operator. Commun. Math. Phys. Ill, 409-415 (1987) 18. Kohmoto, M., Kadanoff, L.P., Tang, C: Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50, 1870-1872 (1983) 19. Ostlund, S., Pandit, R., Rand, D., Schellnhuber, H.J., Siggia, E.D.: One-dimensional Schrόdinger equation with an almost periodic potential. Phys. Rev. Lett. 50,1873-1876 (1983) 20. Mahler, K.: p-adic numbers and their functions. Cambridge: Cambridge University Press 1981 21. Szego, G.: Orthogonal polynomials. Providence, RI: American Mathematical Society 1939 22. Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys. 75, 179-196 (1980); see also ref. 12 23. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I. New York: Academic Press 1980 24. Eastham, M.S.P.: The spectral theory of periodic differential equations. Scottish Academic Press 1973 25. Bellissard, J., Scoppola, E.: The density of states for almost periodic Schrόdinger operators and the frequency module: a counterexample. Commun. Math. Phys. 85, 301-308 (1982) 26. Private communication by B. Simon

Communicated by J. Frόhlich

Received August 30, 1988