Sparse Schrödinger Operators

We study spectral properties of a family , indexed by a non-negative integer p, of one-dimensional discrete operators associated to an ergodic dynamical system (T,X,ℬ,μ) and defined for u in l2(ℤ) and n in ℤ by , where Vx(n)=f(Tnx) and f is a real-valued measurable bounded map on X. In some particular cases, we prove that the nature of the spectrum does not change with p. Applications include some classes of random and quasi-periodic substitutional potentials.


Introduction
The one-dimensional discrete Schrôdinger operator 11, sometimes called the Jacobi matrix, is defined for u belonging to Pç27 $ne Hilbert space of square summable sequences), and for any integer n by (Hu)(n): u(n -1) + u(n + 1) + l'(n)u(n) (1.1) where (V(n)).ez is a bounded rcaI potenti,a/. There has been a lot of interest for Schrôdinger operators to be associated with a dynamical s-"-stern f = (T.X.B,p) as follows. For all r in X, let 11" be defined by (H,u)(n) :u(n -1)+u(n+1)+I".(n)u(n) Yn eZ and Vu Q (2(Z), (1.2) where t'',(n): f Q"r) and / is a real-valued bounded measurable function on X.
Under the ergodicity of T , the invariance of the spectral properties of such operators is true for p-almost every operalor H,, which means that Ér.-almost all operators have the same spectrum and spectral components (see [1,2,4,5,6,9,13. 14, 18] for more details). In this paper. we introduce more general operators defined on P(27. associated with the dynamical system T and indexed by a non-negative integer p. More precisely, we put for all r in X, (He"u)(n) : u(n -p) + u(n+p) + v,(n)u(n), v n €Z and vu € t2(v') (7-3) 316 C. CUILLE,BIEL with, th,e dynarn'ical systetn T and utitlt. the Ttotent'ial I'.. Our purpose is to study spectral properties of such operators. according to tiie vaiues of p.
In Sec. 2, we first set up notations and terminologies of dvlamical systems. spectral theorv and random operators. \&'e can also see that the notion of sparse Schrôdinger operators is inclucled in the more generai theory of random ergodic operators described in details bv A. Figotin and L. Pastur iu [g] . According to [9] and uncler the ergodicity of 7. immediate spectral properties like inr.ariance /r-almost ever)'where of the spectrunt and the spectral compoDents. or absence palnost surely of the cliscrete component. can be deduced for sparse Schrôdinger operators.
The specific form of psparse Schrôdinger operators allows a more accurare understanding of their spectra and spectral components. In Part 3. the natural decomposition of {2(Z) in a direct sum of orthogonal subspaces, which are stable under each 1{, permits the study of p classical discrete one-dimensional Schrôdinger operators associated fo H!. instea.d of studving f{ itself. \û'e thus obtain p families of associated operators, each of them being defined for anr. r. in -Y. In this case. we prove that each familf is associated with the dynamical s1'stem Tp : (Tp . X. B, tt).
Let us note that Tp is not necessarily ergodic. When the dynarnical system 7 is ergodic and rninimal, we can cut f into nz disjoint Zp-invariant closed subspaces, denoted by X0,...,I-_1, where nl is a non-negative integer depending on p and less than or ecltal to p. This result is due to W. H. Gottschalk and G. A. Hedlund (see 110]), ancl T. Karnae ([tt]) for substitutional dynamical systems. \À,'e also refer the reader to [7]. Section 4 of this article states this theoretn and discusses the spectral properties of each familv of associated operators, according to the r'aiue of zn. We prot'e in particular it suffices to restrict our attention to elements of Xn.
The forthcoming three sections concern special cases of potentials. In Part 5, wedealwiththeperiodiccase: f :V,lNV,, where,Àtrisanon-negativeinteger. For any r belonging to X, the sequences (i;(n)),E2 are l[-periodic. We prove that the spectrum Ir of any operator fff is purell'absolutely continuous and composed of ,Ày' not necessarily disjoint bands (closed iutervals of R.). This is exactlv the same result as in the classical case of discrete unidimensional Àr-periodic Schrôdinger operators, which can be found in [t0. Chap.4]. Moreover, we sltou'that Ip can be explicitly described and depends onlv ou p modulo ,\. More precisely, !p : s'pmod(N) ; !,\-(pnrod(N)), for all p € N*, and there exist exactll, tT] + t possible spectla, which are It, !t, ..., !t+l and !N. Section 6 treats the randorn case: we suppose that the (1"(n)).ez are independent identically distributed random variables. We prove that the general results stated for discrete one-dimensional Schrôdinger operators with such potentials, can be extended to sparse Schrôdinger operators. More preciselv. u'e prove arr analogue of the Kotani and Simon theorem: the absolutely component of p-almost all operators is empty (see [13,17], or [1]. for rnore details). X4oreover, when the density function is continuous on lR and compactly supported, the spectrum lp of !-almost all operators 1{f; is pure point and equal to [o -2,0 +21, if [o,rg] is the support of the density function. This is an analogue of the Kunz and Souillard theorem. In the same way we state an analogue of the Carmona. Klein and Martinelli theorem (random variables admitting a Bernoulli distribution). In these two extremal cases. we show that the nature of the spectrum does not change with p.
We analyze in Sec. 7 the case of substitutional potentials: the dynamical system Tisgeneratedbl'aprimitivesubstitution{onafinitealphabet,4:{0,...,r-1}. It is strictly ergodic, and we can apply the main theorem of Sec. 4. In the cases m:1. or m: p, we state that the spectrum Ip of any operator Iff is the union of the spectra of any associated operator on X6 and there is no absolutely continuous part almost everywhere. N{oreover, when { has a constant length (, and m.: 1 or p, the associated operators are classical Schrôdinger operators with substitutional potentials. Finally, if p : (,n for n ) 1, then this new substitution is again { and the spectra of almost all associated operators have same nature which is the same for all Il".

2.L. Dgnarnical systems and randorn operators
Let T : (7, X,B,p.) be a dynamical system: f is an non-empty. compact metrizable space, 6 denotes the o-algebra of Borel sets of X. trr is a probability measure on X and I : X * X is an automorphism (invertible transformation) of X. preserving the measure p (that is to say for any A € B, p(f-tA): p(A)).
The dynamical system 7 is said lo be ergodic if each Borel subset A of I such lhat T-rA: -,{ has a p-measure equal to 0 or 1. It is called uniquely ergodi,c if there exists a unique ?-invariant probability measure on X which turns out to be ergodic. If X has no closed T-invariant subspace other than 0 and X itself, then T is a m'inimal dynamical system. When 7 is uniquely ergodic and minimal it is called strictly ergod'ic. In this article, we will always suppose 7 ergodic.
Let us denote by (.,.) the inner product of Pç27 and by ll ll2 its associated norm. Moreover, S denotes the shift operator on 121V,1.
We recall that a random uariable on the probability space (X, B,p) is a realvalued B-measurable function on X taking infinite values on a subset of X of p-measure 1. A random operator A on the probabi,li,ty space (X,B,p) oï domai,n 12(Z) is a map defined on X into the set of linear operators on [2(Z) by A : r *4", (2 1) where A" is for p-almost every r in X a bounded linear operator on (2 1V,). and such thatforalluand u\n(2127, themap (Az,u) : r€X -(A,u,u) e Risarandom variable. If, in addition, p-almost all operators ,4, are self-adjoint, we say that A is symmetric. Moreover, if 7 is ergodic, and if there exists a homomorphism from the group {T' ; n € Z\ inio a group {U-,n eV,} of unitary operators or PIZ) such that, for p-almost all r in X.

Spectral theory
We recall that the s'pectrum, o(H) of a self-adjoiut continuous linear operator fl is defined as the complement in C of the set of values À for which (rl -À1d)-1 exists and is a boundecl linear operator on P(Z). By.-self-adjointness ancl contiuuity of .É/. the set o(11) is a non-empty compact subset of R. A real number À for which there exist u e (2@), u l0. r'erifying Hu = Àu is callecl an eigenuahre: o[ H. The set of all eigenvalues of f1 is called the point spectrurn or(H). The pure'po,LïL(. specu'urn1 denoted by ooo(H), is defined by orr(H l: optHl . *h"r" on,(11) denotes the ciosure of the set or(H) in R. The set o(f1) \ ao(fl) is the cont'inuous spectrurn It can be crrt into two parts, according to the Lebesgue decomposition of the spectral measure: the absolutely continuous spectru,m oo.(H), and the singular cont,inuous spectrum o",(H).We thus have o(H) : ooo(H)U"",(û [J o""(FI) , (2.3) and these sets are not necessarily disjoint. For more details, we refer the reader to Berthier ([3]) and to Dunford and Schwarz ([8]).
Proof. We have already seen that I1p is a svmmetric random operator. By Relation (2.5) and by induction, we deduce Eq. (2.2) with Lr, = ,5' for all n. The ergodicity of the dynamical system 7 implies the ntetric transitivity of Hp. n It is now possible to have information about the spectra and their compouents of operators IIf. Let us denote by o(H!) (respectively oop(H|).o",(H!) and o",(Hl)), the spectrum (resp. the pure point, absolutely continuous. and singular continuous parts of the spectrum), of each operator f/f. According to the previous part, we can remark that each o(Hl) is anon-empty compact subset of iR, included in the interval l-2sup.e I ll;(")1.2 + sup,., ll.(")l].
Nloreover, b1'Figotin and Pastur (see [9]). \r/e obtain directly some properties of the spectrum of symmetric ergodic random operators. Thanks to random operator theory, we have obtained interesting results about nonrandomness of the spectrum and its components. But we do not know exactly neither their form nor the nature of the spectrum. It is the object of the next part of this article.
Remark 2.2. The rândom rea.l-valued map V(n) defined on X by l/(rz)(.r) : lt"(n),for all r € -\ where n is fixed. can be viewed as random variable orr the probability space (X,6,p).
Thus (V(n))nev. is a sequence of random variables. Consequently ;trrsparse Schrôdinger operators are special cases of randorn finite difference operators introduced by H. Kunz  with KiLKI for i + j .
Let us now consider the behavior of the operators H! on each subspace Kt.  Instead of stud;'ing IIf on (2(V,), we will do the studv on each subspace K;. Before this, we have to look a little bit more at these subspaces.

3.L. Study of subspaces Ki
First of all notice that every subspace K, is isometrically isomorphic to P(Z). Indeed let us consider for each 0 < ? < p, the map ôl: o":K,-  I4oreover. @, and r/1, for 0 < i < p, are linear continuous maps of norm equal to 1. Thus each [; is isometrically isomorphicro P(V'). This a]so implies that every subspace /J, is isometrically isomorphic tc) any K, (0 S i, j < p) and in particular the following lemma can be deduced. This signifies that the following diagram commutes: Proof. The proof is left to the reader. n

Associated operators
\\'e have already noticed that we had to study each operator Ho,l",, for 0 < i < p.
Because each subspace K, is isometrically isomorphic to PlV.), we will lift this studv from ,L; to P(72) by putting for all i € {0,...,p -1} and ail r € X: It is an interesting fact that the ilf,' spectra are "globally" invariauts. that is to say their union is invariant pr-almost evervwltere. But we do not know anything about the behavior of each one of these operators. Remark 3.3. If moreover the dynamical system Z is minimal, then !p is equal to the union of the spectra of H!" for any:r in -{.

Imrnediate pntper-ties of associated opervtors
In this section, we see the strong links between the operators I1g''. We begin by a ner'"' characterization of them. Proof. The result follows inductivelv from the previous proposition. n Remark 3.4. In particular, for all r € X and ali i e {0. ...,p-1}, we deduce that Ë/f'' : Hi:. Remark 3.5. Corollary 3.10 implies for example fhai H+': anrl I1"''*1 are unitarill' ecluir.alent. They also have salre spectra and same spectral components. Proof. Since 0 ( i < p, Corollary 3.11 can be applied with n, = 1 and nt :'i. Thus the proof is conrplete. n To end this part, we prrt gi : f oI', for 0 < i < p. Then.gl is a bounded measurable function from X to lR. and V!,"(") : gi((Tq)'r) , yn eZ, Vr 6 ; (3.23) This permits us to conclude with a theorem.
Proof. According to Propositions 3.8 and 3.11 with Relation (3.23), we complete the proof. n Remark 3.6. \^e can also say that Ho'i is a symmerric random operator on the probability space (-f.B.fr) of domain (2(Z). But we do not know if it is ergodic or not: in the genelal case, the dynarnical system (fp,-\.tl,pt) is not supposed to be ergodic! This is the object of the following section.

The Decoruposition of a Dynamical System and fts Applications
Let T be an ergodic dynamical system. If p is a non-negative integer, we denote by fn the neu,dynamical system (7o,f. B,p,). In this section, we are concerning into the ergodicity of the dynamical system Tr'. In general case. we are uot able to give any answer; but when 7 is ergodic and minimal, the following theorem is a useful tool which is given in [7] (see also 111, 10]).  for any Borel subset ,4 of X. The partition is also said to be mtn'imal. Moreover, p(Xr) : llm.
Remark 4.2. The non-negative integer m defined in Theorem 4.1 depends on p and is less than or equal to it. From no\4r on wiil we denote ni : ô(p) and ô(.) is called the decompos'it'ion funct,ion of powers of 7. It is linked to p bv 6(p) | p.
In all this section, we will suppose 7 ergodic and minimal. We also state some corollaries, which can be found in [7].  We suppose that the dynamical system T is stri,ctlE ergod'ic. If. moreouer, 6(p) : p, then each, dynam,ical system (4L,Xo, B r^,,1,u) is strictly e.rgodic.
The following theorem yields information about spectral behavior of the associated operators. In our study, we have to consider each associated operator H\,i ort the dynamical system 7p. Theorem 4.1 induces us to restrict our attention to Éf't on eaclr "sub"-dynamical system T{ : (T(au,X6.61ru,pr). Now Proposition 4.4 permits us to stud1, the associated operators only on the dynamical system T/' : (II,,'ro, Blxu,Fo).
Proof. Let i be given in {0,...,p -1} and À be in {0,...,6(p) -t}. By mininraiity and ergodicifi of T, we know that !p is the spectrum of any operator I1f , and (a.1) is deduced.    We now will describe explicit cases.

The Periodic Case
Let us consider X = ZINZ, where À" € N*. We denote by 6 the o-algebra of Borel subsets of I and by g, the counting measure on X (defined for all 0 < j < À' byp({"r}) : +) Thetransformation ?: rr*l isaninvertiblemeasurepreserving translbrmation of X. The dynamical system T : (7,X.13, p) is strictly ergodic. If / is a rneasurable bounded map from X to R, we define the potential forallr€Xb1' I'"(n) :f(T"r), YneZ. (5.11) In addition. we suppose / such that the sequence (V,(n))" is exactll',4[-periodic. Let p be given in N*. In this section, we stud5r lFsparse Schrôdinger opel'ators associated with the dynamical system T and with the potential (5.1). For general results on discrete one-dimensional periodic Schrôdinger operators. we refer the reader to Toda [19].

First properties of the associated oper"ators
We can now introduce the associated operators Hg', definecl by Eqs. Proof. It is easy to see that ng'i is a periodic Schrôdinger operator. For the calculatiort of its period, which clearly does not depend on zl, we consider two cases: P : ctl{ or not. Remark that gcd(À,p) : N. Suppose tow p f aÀ. If I { p l-l{. the period ,\ is such that À; x p is the lou,est multiple of ,^J. which is of course a rlultiple of p. Therefot'e 1''' = lcm(À-'P) . 'p If p > N, let us consider the Euclidia' di'ision of 2 by À: p -qÀ +r. where 0 < r'( ÀI (pl aÀ'). It appears that f3,,1n; = f (rl_nr*z) sorhat ^; = lcm(À-,r)/r. Nloreover for all a and b norr-negative integers. ab : Icm(a, b) x gccl(a. b). Thus Q gcd(a, ô) Finally, notice that lbr all a ) 1. gr:d(o.b) : gcd((a + ob).ô), ancl gcd(À.r) : gcd(À,p). This concludes the proof. n Recall that the spectrum of a discrete one-dimensional P-perioclic Schrôdinger operator is ra'ell knorvn. Namelf it is purelv absolutely'continuous and coprposecl of P bands, which are closed intervals of IR.. These bands are not necessarily disjoint. For a treatment of this case, we refer the reader to [19, chap. 4]. we can now formulate a similar result for periodic sparse Schrôdinger operators. There are at most pfr, bands ,in the spectrum.
Proof. By rninimality and ergodicity of 7. the spectrum is the same for all z (corollary 2.3). From the previous proposition, the associated operators are ,\'operiodic. Thus, the spectrum of each associated operator A3'i is purehabsolutelv contiuuous and composed of Âo bands. Theorem 3.7 completes the proof. n
The nature of the spectrum of a family of psparse l{-periodic Schrôdinger operators does not change witli p. It is always purely absolutely continuous and always composed of Àr bands. À4oreover, the spectrum itself can change according to the values of p. Notice that Ip is exactly the spectrum of the classical ,Atr-periodic Schrôdinger operator wheu p = l mod (-N). Thus, there exist exactly two disjoint bands in the spectrum when p is odd, but in the case where p is even, one or two bands can appear.

Daearnples
In particular when a: -0, with d ) 0. we always have exactly two bands in the spectrum. In this case. notice that the spectrum is symmetric with respect to the origin. Thus 11 is always composed of 3 disjoint bands whereas 13 can have 1 or 3 bands according 1o c'r is strictly greater than 4 or not. Anyway, the spectrum is again symmetric 'n,itli respect to the origin.
In all this part, we will suppose the l/(n) are independent identically distributed random variables of product distribution p and of same law r(.). Remark that the dynamical s)'sterrl T : (7.X.6.trr), where ? is the sliift operator on -{. is ergodic. The discrete one-dimensiona,l Schrôdinger operator 11. associated with the sequence of potentials (I',(n))nez, is referred to as the Anderson model (see [5,Chap. 9], or f9]). According to this, the p-sparse Schrôdinger operator l/f, is called a1Èsparse Anderson ntodeL Proof.
Each V'f't is a realization of a sequence of the random variables (V(np+i)),,e2. This sequence is also an independent identically distributecl random variables sequence. whose common product distribution is again p, and law is r(.). This proposition directly leads to a more precise result as Theorem 3.7. we refer the reader to [13] and [17]). Corollary 6.2 concludes the proof. ! According to some peculiar properties of the common density function r(.). several results can be deduced. Firstly we can state an analogue of the I{unz and Souillard theorem (see for instance [14,9] or [5. 1,6]). Theorem 6.4. Let (HI),rx be a p-sparse Anderson model. We suppose that, the common density funct'ion of (V@)). is a non-negat'iue funct'ion such th,at there eristsareal 0<À<Iwith On the other an analogue of tl instance). p-almost all r, the spectrum of. Hf i is pure point and equal to l-2,2) + Supp(r). Applying Corollary 6.2 completes the proof. !
In the particular case where the common density function r(.) is continuous with compact support, we get Corollary 6.5. Corollary 6.5. Let (Hl),ex be a p-sparse Anderson model. We suppose that. the common density function oT (V(n)). uerifies the t'ollowing conditions: On the other hand. if the random variables l,-(n) are Bernoulli distributed, an analogue of the Carmona. Klein and Martinelli theorem is verified (see [9] for instance). Theorern 6.6. Let (â3)"e x be a gt-sparse Anderson m,od,el. We supl4ose that th.e. sequence (I'-(n)),,e2 acLmits u, Bernoulli distribution, that is to sa11: utith probabi,li,ty p ruith probab'ility 1 -p, anrl, o real.
Proof. IJsing the same arguments as in the proof of Theorem 6.4 and according to the Carmona, Iilein and Martinelli theorenr given in [9], we obtain the proof. n Remark 6.3. Let us mention that in the special cases corresponding to Theorems 6.4 and 6.6, and to C)orollary 6.5, neither the spectrum nor its nature chauges with ir.

Sparse Schrôdinger Operators with Substitutional Potentials
The class of ahnost periodic potentials lies between periodic and random cases. \&'e study here the subclass of substitutional potentials. Iu the first part. n'e recall some elementary results in substitutional sequences. For more details, we refer the reader to [16].

7.I. Substitutional dynamical sgsterrts
Let us consider a finite set .4: {0,..., r -1} called an alphabet. We denote by AL tl'te set of all biinfinite sequerces of letters from -4. A utord is a finite sequence of letters. We consider a substi,tu,tion { which associates to ànv letter n in ",{. a word {(a). Moreover { will be supposed pr'i,mitiue. which means that there exist a non-negative integer È such that for all pairs ofletters a and li in .4, the word {k(a) contains the letter b.
Under the primitivity condition, { admits fixed-points, that is to say there exist bilateral sequences u) : ... 'ur-1 tug ?.rr ... in -42 such that {(,u) : u (see [7,16]). Suclr a fixed-point u' is called a substitutional sequence.. it is an almost peri,odi:,c sequence, which means that every word of u occurs in u' with bounded gaps (the bound dependiug on the word). We denote by 7 the shift operator on,4l. A topological dynamical system can be assigned in a natural walr to the substitutional sequence tu. Precisely . LTrrder primitivity' of (, there exists a unique I-invariant probabilitv measure p, on I({), which turns out to be ergodic (for more details see [16]). Thus 7: (7.1f({). B.p.) ts a strictil'ergodic dynamical system. 7 is called lhe dynamical system gerlerated by substitution {. Bv strict ergodicitv of T, we can apply Theorem 4.1 for a given non-negative integer p. we find a partition of X({) into ô(p) parts. \Â,'e alnays choose -\6 to be the member of the partition containing ur. When ô(p) : p, we deduce irnurediately fi'om Corollary 4.3. the strict ergodicity of T{ = (!},,,f0, Byxo,Fo). Proposition 7 .L. If p is a non-negatiue 'integer such that 6(p) : 7, then TP : (To, X({), B, p) i,s a nrinimal ergodic dynam'ical systern.
Proof. According to [7], Te is minimal if and only if it is ergodic. n Now. if B : bo...bj-t is a word of letters from "4. then j is called the length of B and is denoted bl' lBl. When for any letter a of "4, the length of {(a) is equal to l, where I is a non-negative integer, the substitution { is said to have constant length or uniform length. Otherwise, it has non constant lengtlt. In the case where { is a substitution with constant length, we cân sav more about the dynamical system TP (we refer the reader to [7]).

Prcperties of the sparse Schrôdinger operators
We only suppose for instance that { is a primitive substitution. Let us consider the potential (V,(n))"çv given by V"(n):f(T"r),Yn€Z where / is a real-valued bounded measurable application on X({). Then fIf, is called a lrsparse Schrôdi,nger operator ut'ith substitutional potenti,al. In the theory of Schrôdinger ope lr,here o is a finite r over r.t is chosen so is aperiodic.
According to P nertts are the uni< spectral compone other words.
where E € {pp,ac pq-alurost every r equals 1. where rr is a finite real-r'alued map frorn "4 and r0 is the first component of r' \'{oreover o is chosel so that the resllting sequence of potential values J'; : (r,(u,,))"ç7 is aperiodic. According to Proposition 4.6, we knou,' that if ô(p) = p, then Ip alld its cornponents are the unions of p cornpact sets, which are respectivelv the spectra and the spectral components of the associated operators or the dvnamical sl'sten T,l'. In other u'ords, 11 the case where 6(p) equals p, we have to study the p families of the associated operators (Êl'i),exo, with 0 1i,< p, whose potentials are given for any r in -Xs andfor0<i<p.b5r i'3't(n) =v,(np+ i) = T'(r,p1,) ' vrt €v' ' ( 7 7) \['e are now able to state, in the extremal cases ô(p) = 1 and ô(p) : p, a theorem coucerning the absolutely continuous component of Ip.