STABILITY AND INSTABILITY IN QUANTUM MECHANICS
Jean BELLiSSABD* Centre de physique Théorique' CHRS, Luminy, D»e 907 13283-Morseilie Cedex 09,(Fronce)
©- INTRODUCTION. 8- SMALL DIVISORS. HE-STABILITY IN QUANTUM MECHANICS. B8B-THE PULSED ROTATOR:RESULTS. SSP-STABILITV OF THE PULSED ROTATOR: A PROOF. ^-INSTABILITIES IN THE LARGE:PROBLEMS. V2-ANAL06IES WITH RANDOM MEDIA. TO-CRITICAL INSTABILITIES: A RENORMALISATION GROUP APPROACH.
» Ufeonrtoirt propre du C.N.R.S. Peeember 1984
•Université de Provenoe ,Marseille(France)ond ZIF, Bielefeld (Germany)
CN^- CPT-64/P.1699 Introduction-1
0» INTRODUCTION:
This work is an attempt to describe a program which seems to be one of the topics to be developed in mathematical physics during the next decade. It concerns the problem of stability and instabilities of quantum systems . This subject was launched six years ago [ô], but very few is actually known {9l. The physics which is behind belongs to a class of problems involving very delicate experiments, both in molecular physics or in solid state physics . Evidences exist that a kind of transition to chaos indeed occurs, and few theorical models have been proposed to investigate such a transition . However, nothing really convincing, nor precise schemes are available yet. This is the reason why we prefer to use the term "instabilities" waiting for the time when it will be worth calling such an effect «quantum chaos ». Nevertheless, there are several similarities between the behavior of classical systems and of the quantum ones, from the point of view of instabilities, provided we use the right dictionnary between these fields . This is the spirit in which this report is organised. Firts of all, quantum instabilities will not occur if we perturb an hamiltonian which has a discrete spectrum, even when the corresponding classical system is extremely chaotic . For instance the Laplace operator on the stadium which has been the focus of attention of several theoricians (69,71] will be out of the frame of this study . The reason is that eventhough in the semi classical Mrodtretwn-2 region it is possible to recognise the occurence of classical instabilities by liking at the shape of the eigenfunctions or at the level spacing distributions, there is no sudden change in the nature of the spectrum after perturbation ; therefore the chaotic behavior is mostly a question of order of magnitude when comparing classical and quantum effects. On the contrary, if one starts from an operator which has a dense point spectrum, a perturbation may or may not modify in an essential way the nature of the spectrum . In particular, the time behavior of the observables will be qualitatively different when a continous spectrum appears. There are two fields where experimental evidences of quantum instabilities exits . From the dynamical point of view, the most striking phenomena is the experiment of Bayfield and Koch [57], which is described in Section V. It concerns the ionisation of a beam of hydrogen atoms by a microwave electromagnetic field . The originality of the effect comes from the fact that it is neither due to a Stark effect in the approximation where the frequency of the microwave field is considered as equal to zero, nor to a multiphotonic ionization as it occurs in laser experiments . The ionisation occurs with an high rate only at high amplitude of the external field, whereas it is almost négligeable at small amplitude. Therefore, the result is non linear and non perturbative. When analysing the theorical background, one may argue as it is done in the Section V, that the unperturbed hamiltonian has indeed a spectrum which may be assimilated to a dense point spectrum . Then a long range tunnelling effect 160] is a scheme which might explain a qualitative change is the time behavior of the observables. Among the relevant observables, either the position or the kinetic energy of the electrons in the atom become unbounded in time, instead of being quasi-periodic. As a consequence the electrons are extracted from the atoms with a high probability, and the ionization is observed. We emphasize however that an another explanation of this Introduction- 3
* phenomena may be relevant as pointed in (10-4}: it is possible [ . that the ionization observed in the Sayfield and Koch experiment ^ ' is a classical effect due to the fact that the time during which the beam is driven by the external field is short compare to the time after which the quantal effect are relevant. There is also a possibility that the coupling between the bound states and the continuum provided by the external field is essential in creating the instability. We are unable at the present stage of the knowledge to give an insight of what the answer is. However we believe that it is important to examine the possibility of a transition in the spectrum of th* evolution for high perturbation. This belief is motivated by the analogy with the incommensurate system in solid state physics as it is explained now. Indeed the second field where such instabilities occur, concerns the solid state physics of incommensurate crystals . As an example, the Peierls instability [14ô] of linear organic chains is responsible for a qualitative change in their electric properties 136). A metal insulator transition takes place when the amplitude of the Peierls distorsion increases beyond a critical value . From a theoritical point, of view, one model, the almost Mathieu equation, has been proposed by S.Aubry to describe such a transition . A theoritical argument by SAubry and GAndré [45] supplemented by a rigourous proof and some precautions in stating the results (65] shows that indeed such a transition occurs in this model. A bridge between the two problems has been proposed by the Maryland group S.Fishman, DR.Grempel and RJtPrange [96,1001 and it constitutes a very important remark leading to the use of analogous technics in dealing with these questions . The main idea consists in using the Fourier analysis . If one starts from a hamiltonian with a dense point spectrum, admitting a complete family of commuting operators, each of which having a discrete spectrum, one considers an orthonormal eigenbasis. This situation corresponds to quantizing a completely integrable system . The set of indices labeling the eigenvalues can be viewed as points in Introduction-'»
a lattice describing a crystal . One can interpret the spectral properties of the original problem in term of properties of this crystal . The matrix elements of a perturbating observable will therefore represent couplings between neighbouring sites, whereas the original hamiltonian, being diagonal in this basis, will represent a potential energy of the crystal. A qualitative change in the nature of the spectrum of the perturbed hamiltonian will lead to the question of the existence of localized, versus extended, I states in this spectral representation. ; In classical mechanics, one may investigate the asymptotic properties of the hamiltonian flow . One important object is its spectrum! 11]. If p is an invariant measure, the Liouville measure on the enery shell, for instance, the flow generates a one parameter group of unitary operators in the Hilbert space L2(d|i). The ergodic properties of the flow are classified via the spectrum of this group . Usually a point spectrum is associated to the existence of quasi-periodic orbits (invariant torii), whereas a Lebesgue spectrum is associated to an extreme ergodicity like K-systems I 111. In quantum mechanics, the hamiltonian also generates a one parameter group unitary operators. If one uses the orthodox description of the measurment, one can also interpret a point spectrum in term of stability, whereas a continuous spectrum (singular or Lebesgue) will lead to instabilities (See Section II). However, the spectrum of the hamiltonian flow is only a very rough information. We would like to know more about the nature of the classical orbits. Correspondingly, we need more information about the nature of the quantal wave functions. We want also to know if the energy shell is "filled" by an orbit or not, a question related to the ergodicity. Correspondingly, the question of simultaneous existence of different kind of spectras, will be raised in quantum systems. In the light of this analogy, we can describe the scenario leading Introduction-5
to the different kind of instabilities. First of all, the instabilities "in the small", which appears for nearly integrable systems, are related to resonances in classical mechanics . The perturbation theory is difficult because of the small divisors . A treament using the algorithm proposed by A.HJJolmogorov [22,131], V.I.Arnold [41,42,43], and J.Moser [1391, permits to get a large set of invariant torii, namely a stability result. A similar treatment can be done in quantum mechanics . The small divisor problem will be illustrated by a simple example fully described in the first Section of this report . The KAM. algorithm will be used also to prove a stability result for the "pulsed rotator problem", in the Section III and IV . Aside from this result, the instabilities "in the small" are also investigated in both models . Here, an essential difference between the classical and the quantum mechanics comes from the eastence of natural quantum frequencies which depends on the physical parameters of the theory: resonances may be exhibited between these frequencies leading to instabilities. The problem of instabilities "in the large" is closer to the question raised by the experiments . We discuss these questions in sections V & VI. The section V is devoted to the ].E.Bayfield & P.Mloch experiment [20,57,50] and to the theoritical explanations given by J.Gleopold & I.CPercival [133,134] and by RJensentl 10,119). We also raise the question of wether or not instabilities in the large take place in the kicked rotator problem. The Section VI is devoted to the description of the result known for the incommensurates systems as well as the analogies with the previous problems. Since reviews of these questions east already [34,59,66,120,161], we have restricted this discussion to the Almost Mathieu equation. At last, the behavior of these systems near a transition is discussed in the last Section . We give a short review of the renormalization group technics used in classical mechanics to htroduction-6
evaluate the behavior of a classical system near a transition point. One also described simplified models in quantum mechanics for which the renormalization group analysis can be performed exactly, as well as the behavior of a system close to the fixpoin t of the renormalization group.
We have tried to present here as many new results as it was possible to produce while writing this report. The results of Section I, eventhough almost known, are essentially published nowhere else . Some results in the Section III are new and we have given the proofs of them . We have just mentionned without proofs the results published elsewhere by others authors . The Section IV is completely new . It is a version of the K.A.M. theorem applied to quantum mechanics, which establiches a result of stability . As it will be seen, the formalism needed is rather heavy. Most of the sections V & VI is a report on already known results as well from the experimental size as from the theoretical or mathematical size. Several results of the Section VII are also new eventhough they have been partially solved by other authors. Since a complete proof of this new part will hopefully appear later on, only a sketch of it has been given here. At last we have also added a list of references covering more than the subject strictly treated here. A list of monographies on classical mechanics [1,2,3,5,15,24,25,26,27,28, 33,351 and a list of articles covering the topics presented here have been quoted for the convenience of the reader. Acknowledgments: to write this paper I have benefited from discussions with several physicists and mathematicians. I want especially to thank MSarnsley, G.Casati, Sfishman, D.Grempel, R.Jensen, jjrfather, PAloussa, Y.Pomeau, R.Prange, R.Rammal, B.Simon, K.Yajiraa. It is also a pleasure to thank Elieb and A.Wightman for inviting me at the Mathematical Department of the Princeton University, where this work was cooked. Let me thank the ZIF, where this report was finally written, and especially L.Streit, for financial support and scientific facilities. Chapter 1-7
8.SMALL DIVISORS:
E.S A simple esamp|e in classical mechanics:
In his book "Elements of Mechanics", G.Gallavotti proposed the following simple exercise [ l5,p.290,Pb. 1]: Let #(A,(p,t) be the hamiltonian given by (1) #=ocA + %t) (ctsE ) (A, (2) y(t) = mT+n0 «) %.»=Sûn,)vn/ Then we get: (4) A(t)=A+{to-5 /tytyJ+F ^•tp/qAJ-F^t)} Chopttr 1-8 where (5a) '!,,(?)«£, I, , e*» m + rrt) (5b) F MHY nimf /(mp+nq)e* * Therefore unless f. =0 W1 € 1, A(t) diverges linearly as t*<» k),-lp » for all initial data^ (A,y) such that Vty^ f(( i l,(nra+,,) (6) Aft^A-Z ,,2imf /(ma+n)e ^{e -l} Eventhough the right hand side may converges uniformly on each time interval [-T,+T], it does not mean that it is quasi periodic, for the denominator mcc+n can be fairly small and may produce a serious divergency as &•<». In order to avoid such an effect, we will select cc among the set of diophantine numbers 116,19,21,32): Definition.-ct is a diophantine number of order a (a>2) if there is $>0 such that |oc-p/q| > i$ -a is a Liouville number if it is not diophantine. The reader will find in [61] a proof of the following classical result: Theorem 1: (i) Let f be a C°* function on T2. Then if a is a diophantine number of order o, the solution of (2) is quasi periodic. (ii) There is a dense G-set n in (f°(T2), and for each f h in Ù, there is a dense G-set Z(f) in R, of zero Lebesgue measure, such that if oceHf), the solution (6) is unbounded in time. • Remark: the set Q can be chosen to be the set of functions with non vanishing Fourier components. On the other hand, the set 2(f) depends highly on how the Fourier components of f behave at infinity. Consequently if a is badly approximated by rational numbers, one gets a stable motion, because each orbit is quasi periodic and is supported by an invariant torus. On the other hand when a is a Liouville number, depending of how smooth f is, we get unstable orbits. Actually, their structure can be quite complicate involving strong recurrences. I=g A quantum analog: As we will see, some analogy occurs between the t- previous example and the analogous model in quantum Chapter 1-10 mechanics. Let % be the Hilbert space L20r,dx/2Ti). We will quantize (A, (Ô) H(t)=-aia,'âx + f(x,t) The study of this example goes back to a work of Sfishman, D. Grempel & Prange[99].It has been supplemented by several other papers[64,?2,97,101,160]. However, to our knowledge, these results do not concern the evolution of the wave function but rather a transformed operator acting on 12(2) which represents an incommensurate crystal. We have adapted these results to the present situation here. The solution of the Schrôdinger equation: (9) ity/Jt'-ctiiy/et + ftetty is given by: ( 10) y(x,t) ' expfijg'dsf(x-cxt+as,s)} y (x-at,0) In particular, since H(t)= H(t+2n), we get: (11) y (x,27in)= Ufyx) y(x)=y(x,0) where U is the Floquet operator, namely the evolution operator between t=0 and t=2n: (12) Uf9>(x)= expJiJ^dsKx-cct+cc^s)} y(x-2na) Clearly U is unitary. The long time evolution of the wave function is Chapter 1-11 described in term of the spectral properties of Uf. If X is an eigenvalue of V with eigenstate y, then yN(x,t) will be quasiperiodic in time, namely: n (13) yx(x,27in)=X yx(x,0) and 1X1=1 In particular the mean value of any bounded function g(A) in this state will be quasiperiodic in time. On the contrary if U has some continuous spectrum, and if y belongs to the corresponding eigenspace, the mean value of g(A) in the state y will converge to zero in Cesaro's mean as t-><» I30,p.340]. This can be interpreted as the fact that A will not stay in a bounded region of the phase space, an instability statement. In this section we want to prove some rigourous result needing several technical hypothesis. For convenience we will impose some restrictions on f. However we emphasise that, they could be easily weakened. 1=3 Some rigourous results: We will choose f (x,t) of the form: (15) f(x,t) = 7 •°vfe)Sa>2iin) with v(x) a holomorphic function in the strip B •{ xeC/2JiZ ; |Imx| z ï+0 J0+ ' (16) UfMy(x) = e*** y(x-2tict) We will assume that: Choptn-1-12 (17a) v&)»E *•*•*. v=fv(x)dx=0 (17b) limsup, . |v |,/W.*T>0 Let us dénote by .# the space of such functions, endowed with the norm of uniform convergence on each compact subset of B . On the other hand we will set[ 1: (10) L(cO= limsup l/lml Lnll/sinuctml The set of ot's for which L(cO=0, has a full Lebesgue measure, since it contains the diophantine numbers, whereas the set of oc's for which L(a)=°°, is a dense G.in E. Theorem2: Let cc=p/q be a rational number. (x) there is a dense open set 0, in M such that if ve 0 ( the spectrum of U is absolutely continuous, (ii) if ve 0. the spectrum contains also finitely many eigenvalues of infinite multiplicity. Oii) if llvll =sup lv(x)| is small enough the spectrum is made of q disjoint components containing espfi27il/q) (l=0,l...,q-l). Theorem^) Let oc be irrational, and let us define 2 Var Then if 2nVar(v) GO For almost all oc in [0,11 if v is twice continuously differentiate, Uf has no absolutely continuous spectrum. Theorems If vs ff and L(cc)<«, then Uc has a pure point spectrum of multiplicity one. JheoremS: if in addition to ( 17) w© assume (19a) liminf |v |1/|rrWir- ttrxa m (19b) 3*S_<2g then if L(ct)>0$_, U has a purely continuous spectrum. If in addition 2nVar(v) 8°4i Proof of the results: Proof of the theorems- Let oc be given by p/q where p and q are prime to each other. As in 1117llet us define: q 2,,knp q yk(x)-Pky(x). 1/q £ns0 "V ' Yfe-2inp/q> An inverse Fourier transform allows us to reconstruct y from the yk,s. We also set: r1 M (X) = 1/q V e«2tffr(ft-1 Vrkty/q+vCx+îpnp/q» Then we get easily: Qearly the matrix M(x)=«Mkk.(x))) is unitary and smooth as a function of x. Therefore all its eigenvalues are smooth with respect to x, and there is a piecewise smooth map x->S(s), where S(x) is a qxq unitary matrix, such that S(x)M(x)S(x)* is diagonal for all x € T. Therefore : where the X's are the eigenvalues of M. Hence Uf is unitarily equivalent to the direct sum of the operators of multiplication by. X Cm-o,l..,q-l) (Obviously IX (x)|-l). If none of the X "s is a constant, we get an absolutely continuous spectrum. Since X is an algebraic function of the matrix elements of M, it follows that X =const. defines a closed subset of H whose complement is dense in H. Thus the set O Chapter 1- 15 of v's €# for which X *const. Wm, is dense and open. Hence T m * fi J O, -Û is a dense G-set and if Fs# Ur has a purely p/q p/q l F absolutely continuous spectrum for all p/qsQ. If HO. there is an m such that X =const. Thus X is an eigenvalue; the m m corresponding eigenspace is also infinite dimensional since it is generated by the functions y of the form yk(x)=Smk*(x) n 1 UF>(x)=exp{i2îî2k=0 " v(x-2irctk)}Y(x-2îtan) Let us assume that y has a derivative in L2(T,dx/2n). Let p/q be a rational approximation of cc such that loc-p/qli 1/q2. Then by the Denjoy-Koksma inequality [cf.l7,p.73J we get (recalling that J^vfcOdxsO): <20> C^'v&^naUliVartv) On the other hand: 2,T 2 ,/2 (J0 dx/2jr ly(x.2Ji l limsup >eo tylU^y»! > l-2nVar(v) > 0 Using the spectral theorem, the absolutely continuous spectrum is empty. The other part of the theorem is proved in the same way by using the Carleson inequality [75]: for almost all a in [0,1], there 2 2,, are C>0 and (3>0 such that if veC we get (if J0 v(x)d2=û ) p E^0Vv 2 a ,/,n, L(a) limsup tv (e- w ".i)-'| .e-^ g belongs to H Therefore the eigenvalue equation Uy = \y Chapter 1-t7 is equivalent to: y(x-2nct) = Xy(x) ; yte) =ei9W.yfe) Thus we get \= e*2Wam for some meZ, and the corresponding lmx i(s0y+rnx) eigenfunctions are y fe)=e ; consequently ijim(x)=e is an eigenfunction of U Clearly {fm;meZ } is a complete 2 orthonormal basis in L (Tids/2n) and Up has only a point spectrum. - D Proof of the theorem S: The aim of the argument is to prove that if e «e21*"*. ft (21) litninf 1/N7 ""V |Ur e >P=0 ymeZ if v and a satisfies the hypothesis . For if this is true, the point is a spectrum is certainly empty I30,p.340], for ty ) 2 complete basis. If in addition Var(v) S («)= 2 ISn (oc)-Sn m(p/q)l < 2jr|a-p/ql(lmkn llJv/asll^) < 2n I oc-pA} I n21 |m| + liav/Jxiy Since v € H, lldv/èxll is certainly finite. The hard part of the Chopterl- 18 proof is the following lemma: (see Appendix) Lemma 1: if v satisfies ( 19) and if s>0, there is A >0 and q € W, such that if q > q [v +€,, 1/2 k |UcVq)e >I q • 1/N £ ""'IS 2 2 2lir +sl(| 4 1/N 7 '"IS (a)l £2LnN{A /Ne- - + |a-p/qlBN }. From the definition of L(ot), there is an infinite sequence pk/qk L(a)q such that : \a-1>k/\\ <- e" * Let us choose N =i~e EGr.+O+UcO] "K^. Then we. get some D >0 such that : 1 2 L 1/K.S «V IS (oc)l 1D q e<\/S<^.- *">*> k > kfe) k"n=0 n,rn g^k If z is chosen so small that L(ot) - cft_> it, the right hand side converges toward zero as k-» «>. D Appendix- 19 Appendix A: Proof of the Lemma 1 Let v be a function on JR. which is real and 2u -periodic . We assume that v has a Fourier stries of the form ta vfe)= V1 *n m=-eo m *• ms-M m such that: VI) v0«0 V2) limsup !v \Um = fT* m m V3) liminf |v |'*-••»- * mm V4) 0 < -Q i $_ < 2$ < +<» One wants to prove the following results: Proposition 1: Let v satisfy V1-4. Then. there is a constant C(v) depending only on v, and Q € K^ such that, if p/q € Q, q i Q and n € Z: ! j 2* dx/2ir expti £ V with C(v)< 1+1/2 •^Vm Imllmr I. In order to prove this result we shall need the following intermediate step. Let f be a C2 function on R such that : Fl) f is real valued and 2v periodic F2) f has two critical points only in I0,2JT) F3) f has two inflexion points only in (0,2ir) Then we set, for N > 0 : Appwidix - 20 att JSMnf {I nam SeK andlf (£)-f<£±)< n/Hj Where £ are the critical points of f. Let also g be any C1 function on R, 2n-periodic. Then we get : Proposition 2: Let f satisfy F1 -3, and g be as before. . Let N be a positive number such that a=a(f, NQ) >0. Then if n> N we nave: 2 n lj0 *exp(inf (0)gl.iM./2ii 11 llgllcl0r W Let us denote by £_,£+ the critical points of f and by lvlx its inflexion points. We will assume : (1) Oi^VW*-*211'V2* We will choose the sign of f in such a way that (2) f_=f(£J 2 0) H0 *exp £+ £ +2ff 2TTJ= Wi; =(Js. +J5+ - )exp(inf^))g(Ç)dÇ Since the treatment of I" is similar to those of I, let us restrict n n the study to I. By (1) f is monotone on (ZZJ and its derivative does not vanish . Therefore, it has an inverse function denoted by y, and defined on (f_,f+). The derivative of y diverges at f . Let us perform the following change of variable : (4) f «) = T) Appendix - 21 We get : Since n > N^ we have a(f,n) > 0, and therefore f " does not vanish on the set IK_XJ u të £ 1 where : (6) lt- y(f±±ir/n) In particular we get (7) i_+ji/n <ô) K^f-'^^h-^eiM) go(p/ro t( n )+llg /nI (2) (10) 2!!nl< !lgllJ(VV SA^ C° V n Using Fubini's theorem and replacing -j by (pfy+cnr/n), we get: (,) f+ ïï/n , 3 (11) ln =Jf_ ' d'9 J0 dciny('3*an/n)|/rof(i)+c'n/n)t where: (12) f(C+to)W+-(l-o)n/n and Appendix - 22 f(£,(o))= f_ + OJt/n From(7), the rits. of ( 11) is given by: m (13) in ^Min%(o)huv(l.M)-2/m0)} Using the same procedure, we get: (2) f+ n/n 2 (14) In =Jf. " d-r) \J do 1/ f'oy^+OT/n) for f " is increasing on KJU and decreasing on (Çtf?+l By the definition 12) of Ç (a) we get: (15) ov/a- f«.(o» - f«_) - «.(O-ï/^'dsf 1-B) f'«.to» where for f'(Ç_)=o . On the other hand we have: UUczhUohK. and therefore: l/2f"(S_(o.s))>a(f,n)>a Thus: (16) 11 (oh l_\ < fofl/na)1, z ; 0 < o < 1 We also remark that: (17) f( lia)h( l(o)- Ç.)J0'dc f(Ç.(a,î)) Using ( 15) we get easily: U2 1/2 (10) f( Uo)) >. {ïïa/n JQ^zS"(KJotz))} >. (2ana/n) Appendix-23 The same estimate holds for ZJa), f'(£+(cO), provided a is replaced by l-o. Patching together the estimates <10),{13),(l4),(lo),Uô),weget: ,/2 (19) llJidiW^i^îlIgll^ • (Ti/na) (?+- VWQ, Going back to the definition of I', the same estimate holds for it, n provided we replace Ç+- $_ by 2n-(£+- ?.). Dividing them by 2n and adding up, we get the result by remarking that 1+V2 (20a) 7* ^'vte^nlp/q) = nv (x) + q( V (x)-V (x-2*np/q)) Where (20c) VW-E'T /q(l-e-2Wmp/q)eim* here in £' we sum up on me Z\qZ . Appendix-24 Let-us set: (21) G (x)=esp(iq(V (x)-V (x-2irnp/q))) Now, if v =|v |exp(i (22) v (x) = |v |{ 2cos(qx+(p VR (qs+f )} = Iv |f(qx«p ). q q * Jq q 4 s 1 On the other hand we have: 2n ,, 1 Jn=Jo dx/2n esp(i2|=0 " v(x-2iilp/q)) 2n >Je dx/2n expUn |v If(qx+ Changing x into £=(qx+y ) leads to the expression of the proposition, provided n is replaced by n|v I and q (23) g«>- 1/q E|=0 "' Gwfc-2JTlp/q) We need now to estimate "a" and II g II, .From (21),(23) we get: (with x=x-2n lp/q) (24) g-W-l/q^1 ilV^)-^x^Trnp/q)]^). Therefore: «5) feyi fcV2BV'^ Appendix-25 Since msq2 in (20c) we get easily: qU_e-2itf mp/q| > 2qisinn/ql 14 and therefore: IIV B i £ Im.v 1/4 Consequently, for any qeW : (26) IfeB i l+l/27Jmlm.vml = C(v) In order to estimate "a", we remark that R ($) converges rapidly q to aero as q-»°°. For we have from (20b) (22): From(Vl-V4)/ if 0 < z < 2ç-tf_ < ç ,there are constants Cf*>0 and QjfetelN^ such that if rn> Q (s) : (27) C " ero(-(7j +s/2)lm|) < |v I < C + «tp(-(c+s/4)|m|) In particular if q > Q fe) we get (n:i + IR (Ç)/n! I < 2%.^™f/n! C€ /C€"exp(-j(ç-s/4)q+<ç+E/2)q) Appendix-26 If we use the inequality jn/n!e")'J (2$) Max^lR^J/nll < C(s)exp(-q(2e-??_-s)) = s(q) where Cfe) < +°° if q > Q2(s) > Q] te}. Therefore provided q is big enough f(£) looks like 2cos? (eq.22). Clearly in this case F2-F3 will be sastisfied; in particular £_ is close to zero, £+ is Close to n, 2^ and £ are close to TT/2 and 3IT/2. More precisely we get: o=rcn=R'(n-2smE ± q ± t Therefore: (29) l^_( < ÎT/4 s(q) ; l£+-n| < nlA j(q) On the other hand if UK)-f«t±)l£n/N0 we get: I cos* - C0SÇJ < l/2lf(S)-f(£±)l + s(q) * s(q) • */2NQ In particular: Appendix-27 IT<Ç)/2| >.Icos ? I - s(q)/2 > I cos?,±l - 3s(q)/2 - n/2»0 and from (29) we get 2 |T(£)/2I >1 - l/2(ns(q)/4) -Xq>/2 - n/2KQ Choosing N =2JT, there is Qfe) > Q,(s) such that if qjQJs), D 3 2 J lf(0/2l H/2 when |f(x)-f(£ )l < ÎÏ/N0. Therefore we can use the proposition 2 with q>Q3te), a=l/2, together with (25). This gives: IJI » IL2SdC/2TT expttnlv |f£)) g(x) < (2n/nlv I)'n C(v) provided nlv I x 2JT=N„. Since for nlv I £ 2u we have from (24): q O q ,/2 IJnl s llgllco £ C(v) < (2ir/nlv)|) C(v) we get the proposition for every value of n. D Proof of the lemma 1: It is clear that: Using the proposition 1 and the hypothesis V2 for each s>0 there is a constant C " and Q(s) > Q.fe) such that if q > Q(s) we have c 3 Appendix-28 IT I > C " exp(-(ç +s)q) • 4 I This gives imrnediatly the result of the lemma 1. Chopt«-2-29 88. THE STABILITY PROBLEM IN QUANTUM MECHANICS: 88=8 Stability in classical mechanics: The stability problem in classical mechanics could be settled as follows: given a completely integrable, hamiltonian H (A ,...,Ay) for a system with v degrees of freedom, in the action-angle variables, and given f (A. open set which is connected for v>J. As shown by Arnold [44] a slow diffusion takes place in this subset, which produce an instability [10,141]. Thus the answer should be no for v*3. SE=§ Stability in quantum mechanics: In quantum mechanics, we consider a quantized version of ( 1): for £=0 the operator (2) Hn = ff (-ii/is,,..., -iS/Jx ) 0 0 1 v has point spectrum; namely H (m ,...,m ) • E are the eigenvalues of IL and e : x€Zv->eiro'x are the corresponding eigenstates for m € Zv. We can ask the following question: is this point spectrum stable under a perturbation of H ? Obviously, if the E 's are discrete the answer will be ves provided the m' perturbation is bounded. However the answer is unknown in general when H has some dense point spectrum. We could also settle the stability problem in the same way as in classical mechanics; namely, if y is a normali2ed vector in L2(Tv,dvx/(2i7 )v ), does the mean value: (3)- Proof: letc (t) be defined by: (4) c (t)= 2 ® 2m62vlcm(t)l =l In the other hand, we assume that y belongs to the continuous spectral subspace. Then, by the Wiener critérium (30,pMû), we get : T 2 (6) limT1 1/Tf lc (t)| dt = 0 T->co J0 m If we set a (T)= 1/TLTI c (t)l 2dt, given s >0, and M € M there is T >0 such that if T > T „ . then T. „M a (T) *M2Z ,M» (T)->M2(I-S) Since s and M are arbitrary, we get Therefore, if we want to get a stability result, we need to answer the same question: does the hamlltonian H admit a pure point spectrum? We must emphasize that a positive answer is not sufficient to insure that I 2/TT2 V v v (9) H^-fy, b etE|t must belon to e Then yt"2k2 i ^i 8 * domain of A. at all Chapter 2- 32 time. It may happen that some of the eigenstates of H are not in the domain of A. leading to a divergency as t->«. The previous stability problem can also be studied on time dependent hamiltonians. We will reduce ourself to the case where the perturbation is periodic in time. Namely we consider: (10a) #(&&)=ffb(A) •sffAijrt) where (10b) f(A, (12) H(t)-HQ+V(t) where H is a self adjoint operator and V(t) is a bounded operator with V(t+T>V(t), sup l|V(t)!l<+ooandt-W(t)isnorm continuous, then the evolution of the wave packet is given in term of a unitary family {U^s^seR^IR} with the following Chapter 2-33 properties: (13) 0 Uv(s,t)-Uv(s,u).Uv(u,t) 3 ii) Uv(s,t)*- Uv(t,s) » Uv(s,t)-' y (s,t,u)€E iii) Uv(s+T>T) = Uv(s,t) Actually Uvfe,t) can be built from a Dyson expansion (14) s iH ( s) ...e-«V*- iMs,^o^i^VCs^e-^k-rV Vfek)e' o V . which is uniformly norm convergent. Thanks to (13) the evolution of a wave packet at large time is governed by U =Uv(T,0), the Floquet operator, for : n 1 (15) y(t+nT) = uv(t,o)uv uv(t,o)- y(t) Thus the spectral properties of U will determined the behavior of the wave packet as t-»w. Let us also remark that V=0 => For example, if we assume that y (0) is an eigenvector of U with eigenvalue e,u, we get : (16) yfoT^e*"* (p(o) which means that point spectrum for o < s small enough ? Amazingly we will see in the next chapter that in contrast with the classical mechanics, this question may have a positive answer in a certain number of cases The calculation of U is not usually simple, and Howlands 111*], Yajima [170] proposed to consider instead the self adjoint operator . (17a) Kv = -iâ/èUH(t) acting on (17b) L2(JR/TZ )<&#=£ Theorem 1: (Yajimall70]): There is a unitary operatorW on S. such that (10) esp(-iy) = W(SeUv)W"' Thus instead of investigating U it will be easier to investigate Kv which is selfadjoint. We remark that up to the change of variable x =2Ttt/T, K looks like the quantized version of -£(£A + ;yjp ) in the equation (11). A special case will be a kicked version of ( 12) namely: +00 (19 H„(t) = Hn4VV 6(t-nT) ||VII<*«> Even though this operator is not well defined in this way, we can define its Floquet operator by regularizing the Dirac measure and taking the limitas the régularisation converges to zero. Using the Dyson expansion (14) it is easy to check that: iH T WT (20) Uv = Uv(T+0,+0) = e" o e' and the same question will arise. 88=€ Spectral type and long time behavior: Let us finish this section with the physical interpretation of the diopter 2-35 different type of spectra which may occur here. If 9 is a unit vector in our Hilbert space, belonging to the point spectrum subspace of the evolution (or the Floquet) operator, then the mean value of an observable B:. "recurrent subspace " . By definition BJtr contains a dense subspace B of vectors for which ye U^ => < 9 | The subspace 7f is made of vectors for which «pl |2dt = 0 t->oo u ' ' Therefore if ESC- An example: foe pulsed rotator: In this section we will give an example of model for. which stability does occur in quantum mechanics. In I960, Taylor [166] and in 1971 Chirikov 110) proposed to xmsider a kicked rotator in classical mechanics as a standard model to be studied for the problem of transition to chaos. The classical hamiltonian is: (1) Mk,fX) = A2/2 • 2nk/T cosy ^.J™ Ut-nV The corresponding equations of motion can be easily integrated between two kicks, and only the change in the action variable at each kick gives rise to a non trivial evolution. If we set : (2) q = 1/2TT a 2 (5) Uk= exp(i2itaè /8x ). exp(-i2ukcosx) As we will see, the properties of U depend in an essential way Chapter 3-38 on the arithmetical nature of cc. For if fc*0, U lias only a point spectrum given by the set (6) I(ct)={exp(i2jiccm2) ; meZ } which is finite for o: a rational number, whereas it is dense in the unit circle if oc is irrationalfcf.il 1,14,231). The rational case has been studied rigourously for k*0 by Chirikov, Casati, Ford, Israelev [02] for ccsZ, and later on by Israelev, Shepeliyanski [117] for aeQ. As a result we get the following proposition: Proposition 1 Assume a=p/qeQ, and p,q are relatively primeJor any keR, the Floquet operator given by the equation (5) has an absolutely continuous spectrum. There is k depending on p/q , such that if lkl For example if aeZ, U is just the operator of multiplication by e-i2jjkeosx^ vrtiicb is obviously a non constant absolutely continuous function, giving rise to an absolutely continuous spectrum. If k< 1/2 this spectrum is not the full circle. For a irrational, the Maryland group [90,100] showed that in some sense this model is equivalent to a discrete Schrôdinger operator in one dimension describing a disordered luear chain, a solid state physics interpretation. They reduced the eigenvalue equation: i2,, (7) Uky=e "y to the following one (cf .section V): 2 (Ô) tanjr(cc-am ) y(m) +2m>s2.W(rn-m) y(rn) » 0 Chapter 3-39 where { W(m) ; meZ ) is the Fourier series of the function tan(irkcosx). This reduction is allowed without any further restriction only if k On the other hand, the equivalence between the two models is not so easy to use mathematically, for the quasi energy co in (7) is transformed into the random variable in (8). Since most of the known results on the disordered system ($) occur only for almost all w (with respect to the Lebesgue measure) it is perfectly possible that they are irrelevant in view of the spectral properties of U as shown by the proposition 2. Let us however give a rigourous result, which answer a question of [90]: Proposition^: If a«Q, and keR, the spectrum of U is the full circle. Proof: Let W be the unitary operator on L2(T,dK/2ii) given by: Chopta- 3-40 nx (9) Wmy(x)=e-* . {am2}. A famous result of H.Weyl[23l shows that given ^elCU) there is a subsequence [m ;leZ} of integers such that. Since exp(2ji?5/3x) is strongly continuous with respect to £ we have: (H) s-lini W UW *=e12,,1)U l->oo ml k ml Ic The spectrum does not increase under strong limits. Thus we get (15) e^oOycottycS, yge[0,l) ( S is the unit circle). Since 15 is arbitrary, we achieve the result D Every effort to prove rigourously a stability statement for this model have failed up to now. However, if we smooth out the diopter 3-41 kicks, it is possible to achieve this result. Let us consider instead of (1), the following model which we will call the p.ulsed rotator model: (16) *KA/p,t)=aA2 + nV( Theorem 1: Let E(a^i) be given by ( 16), whith V as before (cf .eq. (17)). Then giving s >0, and 0< roo< r, there is a set flf contained in [ 1,»), and u >0, such that: 1 C G) n is open with a Lebesgue measure smaller than s. (ii) u. ->0 as s->0 or r ->r. (iii) for cc€[l, nx+n trigonometric polynomials, e =ew O. The corresponding eigenvalues co(m,n) satisfy: Chapter 3-42 o)(m,n) = n+ocm2+ g(oc,(i;m) where {g(cc,n;m);meZ} is a bounded sequence of Lipshitz continuous functions of cc, and continuous functions of \i, converging uniformly to zero as \x->0. (iv) In addition, The proof of this result will be the containt of the nest section. It shows that a smooth perturbation of the free rotator depending periodically on the time does not give rise to an instability provided we are far from a resonance (namely a is chosen in a "good" set ). Chapter 4-43 W~ STABILITY OF THE POINT SPECTRUM: B^=S The strategy The aim of this section is to prove the theorem 1 of the section III. The strategy to prove it is given by the Koimogorov \rnold-Moser method. In our case, we start from a matrix K<\x) acting on 12(22) (identified with L2(T2,a2x/4ji2) via the Fourier transform) as follows: (1) K(ix)e = (n+ocm2)e +u£. , ,2 V(m-m',n-n')e , . We adress the question wether or not K(^i) admits a pure point spectrum. As explained in the previous section, the answer will depend on the arithmetical nature of «. For n=0, K(0) is diagonal. If the minimal distance between two eigenvalues were positive, the usual perturbation theory would be sufficient. However the spectrum of K(0) has a lot of accumulation points since it is the real line, and this argument does not work. The usual perturbation theory eventhough well defined in tern of formal power series, will diverge because of the small divisors. To get a perturbative result we will use an accelerated convergence method as in the Newton method for computing the roots of a polynomial. The first idea consists in finding a unitary operator R(y), such that _, (2) R(n)K(n)R(n) = K(0)+ng(n) where g(\x) is diagonal. A first order expansion gives: 2 (3) R(n>=fl+nW+0(n ) with: (4) W(m,n,^nVn')=V(m-m^n-n•){n-n'•^cc(m2-m'2)}~, provided (m,n)*(±m',n') ; up to the fiist order we get: Chapter 4-44 (5) g(m,n;±m,n)=V(m-±m,,û) Thus g is not diagonal but block diagonal in the sense that it can be decomposed into a direct sum of lxl or 2x2 matrices. Since diagonalizing 2x2 matrices is elementary, the result will be essentially the same. The main problem in (4) is th» small divisor. The only way to get rid of it consists in forcing the numerator to be even smaller. Using the fact that V comes from a holomorphic potential we get the following estimate: (6) |V(m-m>n'M < const.e_K,m_m'l+|r,",>'l) We must remark here that the denominator depends only on 1ml and Im'l. For this reason we will restrict ct to satisfy a diophantine condition of the following type: (7) In-nVato^m'^liçfln-n'MIml-lmlir (m.m-AnOeZ4 for some o>0 j?>0. We also introduce in the set of matrices on Z2 the following norm: (6) IIAII =sup V . jA(m,m;n,n-)lerC,n-n1+l|m|'|rn'll) px a From (7) and supx>0xV < to/epf= C(o)p' (p>0), we get: (9) HWII^Cto/eplVI^ , w Now we remark that the operator RJ(^)=e* is unitary and agree with R(p) up to the second order. Following NJIBogolioubov and NMJCrylov[73l we use this operator to diagonalize K(\J.) up to the second order: 1 (10) K, (y) = R, (n)K 2 Vf(n)= 0 (n ) Chop*«r4-45 The next idea due to A.N.Kolmogorov[l31], consists in replacing D=K(0) by D, =K(0)+g1 (^) on the one hand and \N by V1 (p) on the other hand, then to iterate the procedure. We will get in this way R Viith sequences Kjy)- kW> g^^H 1 £ 11) Kfc(H) = \t>x) KM (y) R^)" = UOH^x) +Vk(ta) where Vk(^)=0(n ). If \x is small V fy.)wil l converge to zero quite rapidly as k->» , whereas g (\i) will converge to g^ and Rk+1 (y)- 8 =0( H ) which permits the infinite product (12) R^O^.^^/H)..^,^) to converge as well. Finally the goal will be reached : ( 13) KCOVgJiu) • RM^) K(n) R^^)"' However there are several difficulties in the course of this algorithm. The first one comes from the estimate (9): at each step of the recursion the rate of decreasing r will be lowered by a small amount. We will get a sequence f'VV ">rk • w^11 ^ hopefully converge to some r >o. Therefore ^-r^,-^ and °"s produce a divergence in (9). As was observed by V.I Arnold and JMoser, this divergence is compensated by the extreme rapidity with which Vk converge to zero: if s =||V llrwe will get an estimate of the following type: (14) s^^const^-r^r.^ Choosing r -r^t - ^dLnsT' ), we get: (15) (a) * ,iconst.|LnsT s,2< const. £,3/2 k+1 k k k « (b) rk-rk+|= 0((W*) Therefore (14) is strong enough to take the divergence into Chopter 4-46 account. The second difficulty comes from the fact that the diagonal term changes at each step of the recursion. It forces to reconsider the diophantine estimate (7) at each value of k. For a given k, we will define a subset Q of values or a via an inequality similar to (7) on which it will be possible to continue the algorithm. We must check that CI is not empty. In order to do so we follow the idea of AH.Kolmogorov (cf. also (33,1591) namely we will evaluate the Lebesgue measure of the part of Ù which we eliminate to get 0 • we will get an estimate of the type: (16) lnk\ak+)l 1 choose again 5k=0(|Ln s^ ), than!; to (9) we will get 2 s, . < const w, ' ' (r, -r, i( )"'.s, k+1 wk k k+1 k instead of ( 14) but the same conclusion will hold. The last difficulty concerns the proof of the estimate (16). For we have to evaluate the Lebesgue measure of the set of oc's such that: 2 2 (17) ln-n+a(m -m' )+gk(a,TO)-gk(a;m')l > ^{In-nl+llml-lm'll}"" If we have no information on g. there is no way to control the small divisor. Thus we will prove step by step that it is a c' function of a with a small c' norm. For in this case every Urne the left hand side of (17) is close to zero, its derivative with respect to a will be m2-m'2 + O(^) which is big enough to control the size of the set of points not fulfilling (17). These heuristic ideas explain the technical choices made in the next paragraphs. chtjptf r 4- 47 2^=8 The algebra of matrices: On the set Z2 let us define th* pseudo metric (1) d(m,n;m',n>!n-n,|+l!ml-lm'll It satisfies for x,y in Z2: (2) (i) dfe,y)€W Oi) d(x,x) = 0 (iii) d(x,y) = d(y,x) (jv) d(x^> < dfey) + d(y,z) However it is not a distance since dfe,y)=0 does not imply x=y. As usual d divides Z2 into equivalence classes, namely fcn,n) is equivalent to (-m,n), corresponding to cluster of points with aero distance among them. We can therefore identify the class of (m,n) with (|mi,n)elMxZ. An equivalence class contains one point if the first component vanishes, otherwise it contains two points. Since d depends only on n-n" we will use the notation d(m,m",n-n') instead of (1). For fi a closed subset of [ l,<») and r>0, let *#Er,fl) denote the set of matrix valued functions cisfl-»A(a;m,m',n-n') depending only on n-n" and continuously differentiable with respect to ct, such thatllAll Q is finite where: (3) rd<:m m k) ...19 A(«;m,m",k)l+ia A(a;m>,-k)l}e < '' a a It is easy to see that y#r,Q) is a Banach space. In addition it becomes a Banach *-algebra if it is endowed with the product and the involution defined as follows: chapter 4- 48 (4) AB(a;m,m',k) = 2(v. .jeZ2A(a;m,m",k*)B(a;m".m,k-]c') (5) A*(a;m,m',k)=A(c{;m,,m,-k)* It is a simple calculation to check that : (6) llABIIr>0^ll0IIB|lriQ HA*llr^llA||)0 A special subalgebra is made of the set of block diagonal matrices £((l) namely the matrices the elements of which vanish unless d(m,m'Jc)=0. In order to simplify the calculations let us group the elements of Ae*-?(r,Ci) into blocks with equivalent indices namely: - (7) lAtejm.in*) A(a;m,-m',k) I A(a;tail,lm'|,k)= I I |A(a;-mJm'Jc) A(a;-m,-m',k) | is a 2x2 matrix if neither m nor m" vanish, whereas it is a s xs m m' matrix in general where s = 1, s =2(m*o). With this convention. As S{Q) if and only if A(a;lml,lm'U)»0 unless k=0,and Imklm'l. Chapter 4- 49 W"% The small divisor problem; We now denote, by n(m,m',k) the set: (1) Q(m,m',k)-{ ccsll,*>) ; I i+a(m2-m,2)\ < 1} This is an interval centered at oc=-k/(m2-m'2) of width UtaP-m*). However this interval may be empty. It is the case if either Iml-lm'l or if ImMm'land (2) Max^-kil^AnrW2)* 1 On fi(m,m',k) the function D , (oc)= k+oc(m2-in'2) is monotone rn,m',k and its derivative is equal to m2-m'2 ; its absolute value is greater than one. Let us consider the space c'{ll,oo)x Z2) of functions g(oc,m,m') on ll,«>)x Z2, such that : (3) (i) « -> g(am,m') is C1 for each (m,m')eZ2 , , • (ii) llgllcl=supasQCm llgllcls 1/4 and T) be a positive number with yi 1/2. For (nun*) e Z3, let I9 . M denote the set: (5)I9 ^.^Mael1."»> ;lk+a(m2-m,2)*g(cc. m. m')!^} Then Chapter 4- 50 (6) 7'II9 ...MliloNOLnN+lh where 2" means that we sum on the triples (m,myc) belonging to r(H) Wegetimmediatly: Corollary: Let g satisfy the hypothesis of the Lemma 1, and if 02M us define: (7) 0 * r«g {«e[ l.oo); Ikta^-m'^+gCot^^iJIiC/dkltllml-lm'll)0} where 5 < 1/2. Then, (Ô) 11l,o») \afv\ sC,to)ç. where 16 (lom+ccta^-m^ + gfanxmJh i-2ffgl!cl >t/2 r$ and a $« I's .(-n, 1 ) then, the derivative of D . '(oc) of (9) is bounded belo>wbyw by;: (11) 13 D ««(«) l > !m2-m,2l -1/2 > 1/2 a m.m'jk because m2-m'2 * 0 ThusIThus I*' .ty) is an interval included in n(m,m» with width bounded by (12) |1H'« ,n./D>U 2y(Im2-m,2| - 1/2) 1*9 m 9 Actually I a ty) is empty unless we get : (13) (i) m2*m2 Chapter 4- St (ii) -t/(m2-m'2) >0 and (|kM/2).'!ni2-ffi'2l>l From ( 13) it follows that, if (m,m-,k) € T(N) (14) (i)ImklnVI< (lkl+1/2)/llml-lra'l!< N/ llinl-iffill (ii) 1 < llml-lmll < H Thus setting p= llml-lm'll (15) Tfl^jfifl ^E.,""**» * l^Li'N+lKj The corollary is proved by remarking that [1/«)\Q is the union over nyn'.k of the I? , Ws where -g = ©/(Ikl+Hml-lmll)" . For (m,m',k) € IXN) we get ij-^/N0. Therefore using the lemma 1 we have:' (16) |[l.oo)\n UE^te/N'Jl^NanN+l) sl6 1 eE„=l°° onH+u/ir The right hand side converges for a > 2 and: C0 W=fl The recursion process: Let us consider Q a closed subset of [ 1,00), and the operator (1) K«K(0)tg + V where g € £{Q), Il g II < 1/4 and V € A (r,n) with II V II. = s small. In addition we will assume (2) g = g* V = V* The first result of this section is the following: Lemma l:Let s > 2,5 < 1/4. The following equation in the variables W,Sg Chopfrr 4- 52 (3) lyg,Wl=V-8g possesses at least one solution W€»3.(r-p,fl), Sge -0(0) lor any p>0, where: (i) 0* is a subset of G such that (4) IfftOftiC^ff (ii) for some C5(a) >0 (Ui) IISgll0illV|lr0 Gv) W = -w* and Sg = Sg* Proof: to write (3) let us use the block matrix form: (7) [k+a(m2-m*)]W(ccMtaï|,k> + ••• , ..€(a,lml)W(a,|mllm'U:)-W(o:,|ml)|m|,k)g(«Jm'l) -V(o:,lm|,|mU)-8g(a.lml)Sw>^0 It follows that we can choose. (ô) Sg(a,lml) = V(a,|mlJml,û) now if k * 0 or Imlslm'l, (7) can be written as: 2 ,2 (9) {k+a(m -m )+LWJm.1(a)} w(aMta»"U> = VtaMlm'UO where L. ,. ,(a) is ;. linear operator on the set of s xs Kni,Hrr| * mm* matrices depending linearly on g .This operator is self adjoint if we endow the space of matrices with the Hilbert-Schmidt norm. Consequently, since g€.ff(û), L_. ...(a) is C1 with respect to a, and so do its eigenvalues r.(a;lm|,|m'l). Moreover, Chapter 4- S3 (10) llr.UmUmDILi < IIL, ., JIi i C lml,|m| C and clearly: (1I) ilL o,l ,s2su {| ta tolI)l +l,8 (0C lml N>i c Pc^ te ' lp og ' \ where II . II is the operator norm in the space of matrices. Comparing with the norm II. Ilr it is easy to see that the right hand side of (11) is dominated by llgll (eqj §IV-2). We set: (12) Q^ 2 ,2 a {cc€0;lk+ato -m )+r.(a;lml,|m1)|>g(|kkllml-|mlir ,ym^i'Jk4} Since i runs from 1 to s .s <4, the Corollary in the section IV-3 m m' leads to : (13) lOVyn^foto provided o>2,g (15) iWfotWJmUJIk* (16) C2II.IUII.|^«C3II.|| We also need to compute J W(«;|ml,lm'l,k) which has two terms. The first one contains J V(cc;lml,lm'Uc) and can be estimated as before. The second one is more singular for it diopter ...{k+ctûn^m-^L., .(cOr'VtaMlmU) Ki>l,|m | Writting (10) m2-ms=ik.*a(m2-mayL. ,, .(a)] /a -lk+1. .. ,(a)]/a and using : (19) (i) lkl< Ifcl + limi-trnfl (ii) II L .. „(cOII< 2llglL we get, for some C >0, the following estimate: (20) ||8oV(a;|m|Jm'U)ll< 2o+, C4 ft [ |kl+ Umi-m'll ] {IIV(a;lml,|m-|,k)IWI5aV(a;lniLlm'l,k)lll Using again the inequality xpe"p* Lemma2: Under the assumptions of the lemma 1, let us assume that (w,5g) is a solution of (3), then: w w (22) K#=e Ke" =K(0)+g$+v$ where: (23) g,-g*SgeJ(0,) MW^Q V e.f4(r-p,ù )and: o+! 2 2 2 +! 2C5to)/5y .||V|lr0 esp(2C5to)/e p ' .llVllr<)) At last g* and V* are both self adjoint. Proof: It is enough to expand the left hand side of (22) in power expansion in W: (25) Kt-KO)ig*V*lWJC(0)igJ»_. ... [W,[...[W,{[W,K(0)+g]/(n+ l)!+V/n!}]...]+... Using (3) v/e get: V • [W^Oj+g] = Sg The remainder in (25) is called V* and since *%x-p,Qj is an algebra, we get Using the lemma I,eq.5, we get the result Finally, since W is antiselfadjoint by construction, ev is unitary. On the other hand Sg is also self adjoint as well as g . D BT°5 Convergence: As explained in the section IV-1, we define recursively Chapter 4-56 (1) gQ=0 V0=V Vr n0=[l,O and v (2) %>,*%?* w"«w, ^i*V. Correspondingly we choose a sequence x >0, pk>0 such that: (3) © «T "B^O T! °°p=r-r «» Let us choose: 2 1 (4) Qk^/2™ WJ' * Since we need $ From the §IV-4 lemma2 we also deduce: (6) (a) Ite^lfe^ To use the lemma 1 we will need: (7) g < T "s i 1/4 Let us assume that A>0 has been chosen such that for 0 L L 2 2o+1 (L 1) (9) 2" LnEL£2' Ln(2Cs/çLH pt.) )i-2- ' LnsL.t By recursion, from 1 /2+...+1 /2Li 1, we get (10) 2"lLn£, < A*(4a*3)Ln4*Ln(2CL/ç 2p 20+,.O L 3 00 '00 U Chapter 4-57 In order to get the convergence of i to zero we need: do «V^'v1 Returning to(6 ) we must have: (12) Vlnfjto 2p 2B+VA/CJ<5 2p 2e+,/2g1V... 0 k>0 v» ' co 6 voo ' to 5 ...(2"ft+,)(2<'+3)/2k)) This implies that for some C9: (14) 8„ < en 2p 2"+1 0 9vDO r©0 Finally if 0 =fl, J*Q, ,we remark that: ' oo k»0 k (15) lli.»)\flj i c.fo)!^ - ç^to) Therefore, ç is just a measure of the Lebesgue measure of the forbidden set If s denotes the minimum values of the right hand sides of ( 13) c & (14), the process converges if: The unitary operator R(cqi) in the theorem 1§3 is given by: R(a,H)=lim,, evk(a) e Wa) ...evoCa> ~ k->eo Chapter 4-58 which converges in Âfx ,n ). In particular it must satisfy (iv) in the theorem 1§3. Actually we have: RfayHKCOtyVjRta-F)"1- K(0>g(a^) where g(a^)gj&(Q ).Thus g(a^) is a block diagonal matrix. A further diagonalisation is needed. But it does not affect the estimate (iv) since the diagonalisation involves only the subspace generated bye ande . 0 Choptero-59 V= QH THE INSTABILITY IN THE LARGE: PROBLEMS: V=8 The multiphotonic ionization experiment: In 1974 J.E.Bayfield and PM.Koch £7lfeee also [20,561) performed a suggestive experiment on the microwave ionisation of hydrogen atoms. A beam of Hydrogen atoms in a highly excited state (typically 63 external periodic force. They concluded that the classical approach rewodvce fatrl^ w?l< the 9w*w* behaviour '*>t the system. However it does not predict the many resonances occuring in the quantum problem. The first successful! and precise theoretical work was due to Leopold & Percival [133,134). They considered the classical motion of a three-dimensional particle described by the hamiltonian: 2 2%q) = P /2-l/lql + F(t)q3 where F(t) reproduces the effective field seen by a sample atom of the beam. Namely: F(t)= A(t)F coscot max where A(t) is equal to 1 during the time te [t,tf], and is exponentially increasing to 1 before t., and decreasing to zero after t. This transient term was added to simulate the ingoing and the outgoing motion in order to take into account the effects of a sudden rise and fall of the field. The initial conditions at t=0 are chosen randomly via a Monte-Carlo calculation using a microcanonical distribution at the energy given the classical equivalent of an equal population of all states of principal quantum number n. The corresponding trajectories were computed numerically, and the authors performed the statistics of the trajectories ending with a positive energy. Their result is shown in the figure 1 below and compared with the experimental results of Bayfield and Koch. In this picture $ represents the "Keldysh" parameter, namely the ratio (co/co A)/(F /F J where oc» is the Bohr ' at max at at pulsation of the electron on the circular orbit of principal quantum number n, and F the Coulomb field that it sees in this state. As we observe on the fig. 1 the agreement is extremely good. In Chapter 5-61 addition Leopold and Percival gave a description of the individual orbits; they classified them into four classes: (i) the trajectories leaving on an invariant torus, which never ionise (ji) the trajectories ionizing very rapidly (iii) the trajectories passing through one or several extremely highly states (EHE) i.e. with quantum number greater than 5n, and a sharp transition between them before ionization (iv) the trajectories passing through a sequence of EHE without ionization during the passing time. This intermediate resonances have been actually observed in a FMJ.I- lonisotion rote os o fonction of the "Keldysh" porometer. Closed circles: measured probability for Ionization of H (with 63inl69) by o 9.91GHz microwove field Iref.57].ûpen circles: results of the Monte-Carlo calculotiofts thot employed the classical theory to model the experiment of ref.il 34](taken from ref. [20) ).- Chopter5-62 later experiment by Bayfield, Koch & Gardner(5cU A semi classical picture is therefore rather convincing. ^=i A theoretical approach: In 19Ô2 R.Jensen [116,119] proposed to take seriously the semi classical approach and tried to treat as exactly as possible the transition to a chaotic behavior in the classical system. During the last decade many improvements have been performed in the study of hamiltonian systems with two degrees of freedom. However the situation is far less clear for more than two degrees of freedom as in the previous example. For this reason R.Jensen proposed to consider a one dimensional hydrogen atom. Experimentally it can be realized through a surface state electron (SSE) near a bath of liquid Helium. The polarizability of the liquid, even though very weak is sufficient to produce an electric image of the electron of charge Ze through the surface of separation. The Coulomb binding force gives rise to a quantum hamiltonian of the form: 2 2 (1) HQ = p /2m- Ze /q q>0 where q denotes the distance of the electron to the surface of the helium, and the wave function y is submitted to the Dirichlet boundary condition y(0)=0. The specfrum of H is given by: (2) E=-Z2R/n2 n= 1,2,3,.. n R=13.6...eV For the liquid Helium Z is fairly small Z « 7.id3. Consequently the binding energies and the characteristic frequencies are four orders of magnitude smaller than those for the hydrogen atom. The spectroscopy of the SSE have been studied by C.C.Grimes et al. [108] and by DJÎ.Lambert & P.L.Richard [132]. The experimental results agree with the predictions given by the hamiltonian of a one dimensional hydrogen atom. R.Jensen proposed to study the effect of an external periodic Chapter 5-63 electric field perpendicular to the surface, on the quantum behavior of the SSE. Because of the small sise of Z it should be possible to observe an extraction effect even for the state with quantum number n= 1. The classical system has one degree of freedom in a time varying external field. As explained in the section II, it is equivalent to a system with two degrees of freedom. The transition to chaos can be studied by using the methods developed during the seventies. Among them, the contributions of B.V.ChirifcovIlOl, I.CPercival[149l, I.Greene 1107], F.Doveil & DBcande 113,92,93,041, LPIadanoff & J.Shenker[122l, from the theoretical side may be considered as the most valuable. The «gourous study leaded to the beautifull work of SAubryHM?], and J.Mather[i3U3?,13cU giving a necessary and sufficient condition for existence of invariant circles. The time dependant K-hamiltonian in this case is: (3) Ef=-ifiô/ôt+H0+eqFcosC2t 2 1 2 acting on L(E/27iD"2,dt)©LÇ0,<*,dq). For F=0, E0 split into two parts K =K*K Ode where K has a point spectrum whereas Ec has only a continuous spectrum. The eigenfunctions of Kd are given by: (4) e (x,t)=w (x)ein0* meW neZ 2 where v is the eigenfunction of Hn of energy -^B/m , and we get: (5) Ke - (uhQ-Z2R/m2)e » hfi(n-oc/m2)e with (6) o = Z2R/nG Chapter 5- 64 A typical value of a in toe context.oi the u»e rsaynelc. & Kocn experiment is oc«4.105. Therefore eventhough K has a discrete spectrum, cc is so big, or the level spacing is so small that it mimicScs a dense point spectrum. However the fieid couple this system to the continuum and this will be certainly a source of difficulties in the mathematical study of this model If we ignore the positive energy contribution we are lead to consider the operator K , the matrix of which in the basis {e } being: (7) 2 2 (6) JfU/fX) • -Z /?! + F £k=_J* Vk(l).cos(k^-Qt) where I»m/(2R)"2 and k»m-m«m. The estimate provided by Jensen [ 1191 is (9) V, (I ) = .411I2/(Zfc5/s) as k->» k The classical approach of Jensen consists in computing the critical value F of F which corresponds to the classical transition to chaos. Namely for F«F , the K.A.M. theorem asserts that (Ô) admits invariant torii which disconnect the phase space into cells out of which the orbits do not escape. The theorem by Aubry & Mather asserts that they survive until F=F and disappear for F>F . In any case however, there exist for each rotation number C special orbits called Cantorii (1491 Mather sets! 124,125), or broken torii[3U which are homeomorphic to the Denjoy set[17,l$l Chapter S- 65 of a diffeomorphism of the circle. They are labelled by a rotation number, and they are presumably unstable. When F>F only these Cantorii survive and they no longer prevent the other orbits to wander in the phase space and to eventually get to infinity. Applying this scheme to (6), I-»» means reaching the positive energy region, i.e. the ionization of the atom.The evaluation of Fc gives an insight of the threshold observed by Bayfield and Koch. The initial data of the semi classical system are actually random .The real motion of the system is closer to a diffusion than to a deterministic motion, thanks to the randomness of the quantum motion. An evaluation of the diffusion coefficient and of the diffusion time in the phase space is also possible! 1191 Actually the practical scheme used to compute F is called resonance overlap critérium [13] . It has been proposed by Chirikov and refined by Doveil and Escande via a renormalization group analysis. The idea is that if F is small, the equation of motion from (8) is given by: 2 3 (10) dy/dt - 2 /I = fJo(I) dl/dt*0 Thus, 372 (11) O=kfi0(D <=> M «(zVn) keN Thus if we keep only the résonnant term in (6) we get: 2 2 + (12) X(ljf.Q * -Z /I F Vk(l).cos(k?-ût) This approximate hamiltonian is completely integrable and if we expand it in a Taylor expansion up to the second order around Ifc, it looks like the hamiltonian of a pendulum. It gives rise to the existence of a trapping region where the motion is nearly harmonic separated from the remainder by a separatrix. Varying k, we get a decomposition of the phase space into Chapter 5 -66 various resonant re?!ort (13) ttouD-^kp *t/f*9 will be extended in this representation, namely y(m,n) will not converges to zero as m-><». V=S A possible mechanism: a long range tunelling effect: As explained in the previous section, the instability in the large, which is tied up v/hith the non linear classical behaviour, can also be interpreted quantum mechanically via a long range tunnelling effect in the lattice of indices indexing the eigenstates of the unperturbed system. An illustration of this fact may tx> given on the kicked rotator model via the Maryland transformation^, 100,102,1601. We recall that the hamiltonian is given by: (i) H(t)=-«a2/a^*v(x)T "set-am) The corresponding Floquet operator is 2 (2) Uv=esp(i2naa /â^)exp(i2JiV) Let us consider a wave function y(s) as the sequence (y (m)) given by its Fourier transform. The eigenvalue equation Choptw5-6? arito (3) Uvy = e y is given by i2,,v , 2 W SmsZe where ^"(m) is the as* Fourier coefficient of e0l,v. If V is smooth, namely if it is 2 i2,,v 2 , 2, v {l-exp(i2n(«-ctô ))}{l*e }y={l*esp(i2n{W-aà ))J{l-e' ' hj; If we set (5) y.{I*COT)Y we get (6) tamKoxxn2)? (m) + J^ wta-m>ûn') = o where (7) tan*V(x)-7 Wfci)e1mx The Maryland group proposed to see (6) as a discrete Schrôdinger equation on Z with a pseudo random potential given 2 by vm =tanir(w-am ). It is known indeed that for cc irrationnal, the sequence {can2} is uniformly distributed on the tora[23L and it is used to compute random numbers. An especially nice choice for Vis (8) tanuV(x) = a+ 2bcosx which corresponds to the discrete Laplacian: (9) W(m) = aS „+b(S , •& ,) Chapter 5 -68 Then (10) e«OTûi)= (i-i(a+2bcoss))/(l+i(a+2bcosK)) This was actually the model studied in [90]. However, since W(m) is short range for each value of a and b, in this latter case, we do not expect it to exhibit any kind of global instability, in the x-space this is due to the singularities of tann(«-ccffi2) giving rise to high energy barriers. Therefore apart from the usual instabilities "in the small" when a is a Liouville number, we expect the wave function y to be localized in the m-space. Thanks to (5), y should be localized too and U should have a point spectrum. However if V(x)=kcosx, and k>i/2 the transformation becomes singular: 9 belongs to the domain of tarmV, but the sequence W(m) is no longer rapidly decreasing. On the contrary, W(m)«m_> 1/lml, and a long range interaction appears in the Schrodinger equation (6). Therefore for k large the qualitative behavior of the wave function should be different Let us illustrate this claim on a solvable case which is however certainly not generic. More precisely we consider the potential (11) V„(x) = x/2n if -7i« 2 ix (12) UHr=exp(i2Jid )e Performing the Maryland transformation gives: (13) tanJiV(x) = tan(x/2) Its Fourier expansion does not decrease to zero at infinity. We actually get the following result: Chopter 5 -69 proposition l:The Floquet operator U has a Lebesgue spectrum. Proof: we set e (s) = eimx m Then U,. e = exp(i2Jict(m+l)2)e ^, In m r m+1 Recursively we get for N>0 : U N = 2 + 2 Kr, *m *M&™Uto* l) +..+to K) ]}ein+N In much the same way for N>0 • N 2 2 U,. " e = exp{i2ncc[m +...+(m-N+ l) ]}e „ lui m r m-N Therefore 2,r iNo (14) Eventhough this trwhari'Rrn etm\A produce a transition in the nature of the spectrum of the Floquet operator (2) when V(x)» k.cosx at high k, from a point to a continuous spectrum, there is another possibility, as pointed out by the Maryland group in 1104]. Namely the classical behaviour dominates at times t< tc| where (15) t.iCMBtfi» and n is the Planck constant. This estimate is obtained in [1041 from a renormalisation group analysis. After this time the quantal behaviour dominates. It may perfectly happen that the quantal motion does not exhibit any instability in the "long run" as it is claimed in [104]. In this latter case, assuming that this model is qualitatively relevant for the photoionisation by microwaves, there would be however no contradiction with the Bayfield & Koch experiment since toe beam of atoms stay only for a short while in the microwave chamber. All these considerations lead to ask the following questions: - is there any kind of transition in the nature of the spectrum of the Floquet operator of the standard kicked rotator model for k big? - if yes, what kind of continuous spectrum occurs? singular continuous ? absoluetely continuous ? transient or recurrent? - if no, in what sense the semi classical behavior approximate the quantal one at small t? We may summarise these question by asking: -does the quantum chaos easts? Chapter 6-71 tyll" ANALOGY WITH INCOMMENSURATE CRYSTALS: V2=8 The almost Mathieu equation: As we have pointed out in the sectionV-3 there is an analogy between a time dependant quantum problem and a Schrôdinger operator on a lattice, via the Fourier transformation. In this case the lattice points represent the quantum phase space coordinates using the analogy described in the iV-2 ( eq. 7 & 6 ) In this representation the potential at the point neZ was given by a n-periodic function of nan2 where a is some irrationnai number. This suggests an analogy with another model called the almost Mathieu operator [34]. It acts on Z and is given by the hamiltonian: (1) Hy(m)=y(m+l)+y(m-l)+2^cos27ife-am)y(rn) yel2(2) In this operator, ^ represents a coupling constant, a is a frequency modulation giving rise to a quasi-periodic force, s is a phase which represents a generic translation of H. Actually, if T is the operator of translation by 1: (2) Ty(m).y(m-l) Then: (3) • Tfifca^.r1 = H(x+a,a^t) These equations appear in several problems of the condensed matter physics. First of all it describes the Hamiltonian for the quantum Hall effect in a perfect crystal in the tight binding approximation [109,135.140.153,154]. For let us consider a 2-dimensional cubic lattice the lattice sites of which being labeled by (m,n) € Z2. The free motion of a single electron is described via the discrete laplacian: Chopter6-72 (4) Ay(m,n)=y(m+ l,n)+y(m- l,n)+^{^(mji- l)+y(m,n+1)} We have added the anisotropy constant \x here. If a magnetic field B is switched on, this equation is modified via a IKD lattice gauge field, representing the discrete version of a vector potential. Choosing a Landau gauge, this gives: (5) à y(m,n)= 2Wn 2Wn y(m- l,n)+Y -Y (m+l)+y (m-l)*2ncos2fl(x-matyx(m)- (H(x,a^)fx)(m) Therefore A is just the direct integral over xeK/Z of the family {H(x,cc,n);X€R/Z}. The isotopic crystal corresponds to \x= 1. The formal derivation of (1) in the Hall effect, goes back to Harper [1091 ( see alsoRauh 1153,11,4]) eventhough the equation has been pointed out first by Peierls in 1933 I Hô! and investigated by Luttinger in 1951 11351 in tne limit where ct is very small, (very small magnetic field) In the late fiftie's, the russian school realised that the spectrum of (1) has a structure related to the continuous fraction expansion of ct. (Zil-bermann 1171,172,1731 >. .,3,541). Azbell in 1964 conjectured that it was actually a Cantor Set. It was not until 1976 that D.Hofstadter[112] computed Chapter 6-73 numerically this spectrum and found a very beautyfu! fractal set, looking like a butterfly,if we draw it as a function of a. (Fig. 2) F«9.2-The Hofttodtoi- spectrum: the block points represent the votes of the energy belonging to the spectrum of (1) or o function of More recently, the equation (7) has been used to describe the critical line in the phase space (T,B) for a superconductors Chopter6-74 network [3ô,40,Ô?,144,1451. Namely, let y(m,n) represent the order parameter at the vertex (m,n) of the superlattice. Via the Landau theory, it has been shown [30,67] to be the ground state of toe eigenvalue equation A y = EQy where E is related to the critical temperature Tç of the transition from the conducting to the superconducting phase, and (f is the magnetic flux through a unit cell of the lattice. The numerical results agrees fairly well with the most accurate experimental datas (Pannetier et al.[ 144,1451). ISO 100 / \ / ' 50 v y v> * . 1 ,1 ] 1 4, H (Oil Fig.3- Left: Critical Mr* (H,T) of o regular honeycomb network of superconducting Indium. The arrows indicate the value of the magnetic field corresponding to one quantum flux ip =hc/2e. Right : (a mognificotion of the left viev) plot of the measured resistance (sconst.T ) as function of the c applied magnetic field in the range 0«p< It is not yet known rigorously if this spectrum is a Cantor set for each irrational oc's. However the following partial result has been proved [67]: Theorem 1: There is a generic set r in K x T/Z, such that if the Omt*r6-75 pair QJI.CO belongs to r, the operator H(x,ot,H> has a Cantor spectrum for every x. ¥j}=8 The Metal Insulator transition: In 1976, SAubry and G Andre [45] using a trick of Derrida and Sarma [ôôl proposed a scheme for exhibiting a metal insulator transition in the Almost Mathieu Model. For p « 1, a formal perturbation expansion allows us to write every eigensolutions of ( 1) in the form: (9) *(m> - eaWcm V. f e2"***"™"0 with (10) 'Ù = E(k) = 2cos2îik + %i) Let us replace y(m) by the previous expression in (1) ; we get: (11) f +J+f _1+(2/n)cos2Jt(k-pa)fi)= (EOO/n).^ M we assume that (9) is actually convergent, then y(m) describes a Bloch wave, namely an extended state. Therefore, the corresponding electron is free to travel across the chain, and we get as metalic behavior. Whereas, in ( 11 ), the coupling constant is 1/H » 1, and since (9) is convergent, the wave function {f ;peZ} is now localized. The system becomes an insulator. The question is to check if such an argument is correct. One defines the Lyapounov exponent as the exponential rate of increasing of a generic eigensolution of ( 1), namely, if Hy = Ey. (12) limm >w (l/2|ml)Ln [|y(m)P+ty'(m+l)P] = sfE^oc.) 5(E^,a,)=|0'ds5(E^,a,) One has actually the following result[45,49,174]: ii.op;w o - 7û proposition l: ç(E^,a) > 0 and çŒ^ocU Lnl[jtl for ail E, \i, The first proof of this inequality was given by Aubry and André [451 using the Herbert-Jones-Thouless (111,1671 formula which leads actually to. (13) g(E,w,«) = Ln^.l + îj(l/Vi,E./n,«) pi A rigourous proof of the H.J.T. formula was provided by Avron & Simon [491. A very elegant proof of the proposition 1 was given by MJternian 1174] using a subliarmonicity argument. Corollary 1: If a is irrationnal and \x> 1, H has no absolutely continuous spectrum This corollary results from the Pastur-Ishii [115,116,147) theorem. However $ > Lnl^tl >0 does not mean that a point spectrum occurs, with exponentially localized states. Actually an argument of Gordon gives:I52,106] Proposition 2: Assume that loc_ / ! s 1/n< (14) Pn % ^ Then, for any JJ , Hy - Ey has no solution vanishing at infinity. Corollary 21521: if a satifies ( 14) and \t> 1, the operator H(X,OC^JI) has only a singular continuous spectrum for almost alls. This result shows that Aubry & Andre argument does not lead to exponentially localized states in general. However, we suspect diopter 6-77 that the origin of the previous results may fee found in a small divisor problem similar to those of the section 1. Therefore we expect that if cc is reasonably irrational (namely diophantine) the Aubry André duality leads indeed to a metal insulator transition. Actually, this was proved by J£ellissard, LLima & D.Testard 165]. Theorem 2: Let a be a diophantine number. 1) Tnere easts \x >0 such that if \\>\ below by 4(l-o(D) as^-*0. 2) If in addition oc admits a fraction erç>ansion (a„,a,...,a ...) with limsupla |=w, then there is y >1 0 I n ' n n 2 such that for almost all x in K/Z, and for \\x\> \x2, H(x,oc,n) as infinitely many eigenvalues ; the corresponding eigenstates are exponentially localized. The closure of the eigenvalues has a Lebesgue measure bounded from below by 4^1(1-0(1)5 as n*» Let us mention that the proof of this theorem requires all the Gwpter6-78 machinery of the K.A.M. theorem. It has been used prior to 1651 for Schrôdinger operator with quasi-periodic potential by Dinabufg and Sinai [691, Belokolos [6ôl Rûssmann[15Ôl to get informations on it spectrum. The previous set of results provides actually a list of technical tools to investigate the properties of systems with small divisors. Some of them can be used directly for getting results on similar systems. For instance, let us consider the following model on I: (16) Hy (m) = f(rn+1 ) +y (m-1 ) + 2^i.cos2 jr(cm2+y ).f Cm) Then, using Herman's proof of the proposition ! together with the Gordon theorem (proposition 2) we get: Proposition ^:Let H be given by ( 16) lj The Lyapounov exponent for H is bounded below by Ln |[.*l. Thus for \>bl H has no absolutely continuous spectrum. 2) If a satisfies (14) then Hy-Ey has no solution vanishing at infinity. Thus for |*|>1, H has only a singular continuous spectrum. The same argument holds if cos2n£ is replaced by any trigonometric polynomial. These methods actually provided a way of getting results described in sections I, II & III. The technics used in the theorem 2 are more involved. The strategy is simpler for the almost Mathieu equation. However there is a great similarity between it and the result investigated in the section IV to prove the stability of the pulsed rotator. The question we adress now is the following: is the metal insulator transition in the almost Mathieu model a good guide towards a" transition to chaos in quantum mechanics? Chopt*r7-7S HU- CRITICAL INSTABILITIES: A aSMuRMALlSAllUK OROlfr APPROACH. VSS-8 Transition pointe in classical mechanics Let us go back to the standard map (cf.§I!l-eq.3 * ( 1 ) F(q,p) = (q+p+ksin2uq, p+ksin2nq) for which the situation is quite well understood. The K.A.M. theorem asserts that if k is small enough, there are F-invariant circles. The circles having a diophantjne rotation number are Cw. By a computer assisted proof, M.Herman showed that this situation subsists for k < 1/40 [1751. For k>4/3 J.Mather proved [137] by using a result of GK..Birkhoff [lô! that there is no F-invariant circle. These rigourous results are however far from the results obtained with a computer analysis. The properties of this model have been investigated with great details by J.Greene [107). Using accurate numerical methods he computed that there is a critical value (2) k..97l6... such that if ksk there are invariant circles, whereas if k>k c « none of them survive. He also gave a set of rules inspired by the numerical observations, concerning the way the successive circles break down one after another in term of the arithmetic nature of their rotation number. The last surviving circles being those the rotation number of which has a continuous fraction expansion of the form JaJ#..,a^l,...l,...l, i.e. ending only by a sequence of Is. These numbers are equivalent under the action of SL(2,Z) to the golden ratio: (3) a=(/5-D/2= 1/(1+1/(1+.)) The next step towards a theoretical computation of k in a more general situation was accomplished by F.Doveil&D.Escande, who Ctraptfr7-80 proposed a renormalisation group analysis (R.GJ of the phase space. Guided by an experiment on the heating of a plasma by two electromagnetic plane waves they investigated the following model [13,92,93,941: 2 (4) iS[M(q,p.t} = p /2 + L.cosZîiq + M.cos2n(q-£3t) in term of the parameters L and M. As in the section V-2, tlie phase space can be decomposed into resonances, by keeping in tt, tot each p, the dominant contribution to the motion. Near p=0 the M=0 term dominates and H is approximated by the hamiltonian of a pendulum. Near p=Q, the term L=0 dominates and we get another pendulum. However H has a lot of periodic orbits. In the vicinity of one of them the dominant contribution is different Looking at the shape of the orbits in the phase space it is not difficult to recognize that they organise themselves at a smaller scale as they do at the initial stage: we get séparatrices like for the pendulae corresponding to L=0 or M=0. To exploit this situation Doveil&Escande performed a canonical transformation in order to select the dominant contribution in the vicinity of the chosen orbit. The hamiltonian is transformed into ff of the form: * (5) H (q',p",t) «p'2/2 + L .«^nqVM .cos2iï(q'-fl t) +... They claimed that the other contributions are negligible which means that, by another canonical transformation, it should be possible to reduce H to the form H at least on a large subset of the phase space. In this way we get a renormalisation group transformation J?(L,MML , M ). Actuary the sequence Cl-*Ci is defined by the rotation number of the invariant torus we focus on. It is not very hard to see that (0,0) is attracting for St and the evaluation of its domain of attraction gives rise to a diopter 7-81 numerical estima» oi Hi* viiue o« cu* parauwUtf» 101 waicn ciw given torus breaks down. Indeed if (L,M) belongs to this basin, there is an new big enough for which WiLJA) = (LVMl is small enough to apply the KAM. theorem and to prove that the given torus is staMe. VSS=8 Scaling properties of the almost Mathieu operator: Such an analysis may be applied to the almost Mathieu equation. Looking at the Hofstadter spectrum (§VI fig.2) we realise that it has the structure of a régulai- fractal with self reproducing properties. As pointed out by YaAzbell [531 s»d also by DHofstadter [1121 this self similarity is related to the fraction expansion of a namely the action of the modular group SL(2,2) on [0,11 generated by the map G(c0= I/a-tl/ccI. Several attempts bave been made to use a P..G. analysis. Let us mention the numerical work of E.N.Economou8e CM.Soufcoulis (9H a« well as the contribution of J£.Sokoloff (162,1651. A more detailed analysis using a semi classical approximation have been given by M.Wilkinson [1691 and lMSuslov|l651 . We must also mention the works of LPiCadanoff et al.t 123,1301 and S.0stlund et al. 1142, M31 in a different context These works lead to the conclusion that |i«0 and \x= We want to preseti* Vj'-r^ ». posait»'*1 sfbe'ri^ Vivwrtf 9 r'amrnno study of a R.G. transformation. As we will see in the next section this.scheme is powerfull in the study of other models but it seems to us that it should be the way of investigating the properties of the almost Mathieu equation. We consider a Jacooi matrix acting on 12(2) by: (6) Jyta)=t y(m+ lMmy(m- D+vjKm) where we assume (7) v e K t >0 V me 2 We assume that A-{m;keZ} is an infinite subset of Z and is labelled in such a way that mk< mk+J. Let DA be the operator defined as: (85 DAw (k) = w(mk) This is a partial isometry: (9) D.I>*=& D * D« X. ('projection on 12(A)) n A ft n n We claim that partial isometries are the quantal equivalent to the canonical transformations of the classical mechanics. For, whereas they preserve locally (i.e. in a subspace) the equation of motion, they forget the part of the information which is outside the region we are interested in. Going back to our problem we get : Lemma 1: Given J as in (6) with t «0, v eZ and D. as in (5). There is a Jacobi matrix valued holomorphic function z->j (z) defined by (10) D^z-jr'D^fe-J^z))'1 such that zea(J) <=> zea(JA(z)) Proof: First of all, given J any matrix, the equation (10) defines a new matrix J (z) called the Schur complement! 176] which is Chapter 7-83 given by (11) J^'DJD^DJQ^-^JQ/QJDA* where QA*8-XA* provided z«cKQAJQA) Because t *0 W m, the space D *J? is cyclic for ], namely if yej? (* 12(Z)) satisfies (12) Now. we remark that QAJQA is a Jacob; matrix given from (6) by replacing tm by (l-XAXm)tffi(«-XA>ta- » and vm by (&-XA)(m)vm. In other words Q JQA decouples the intervals I^Cnyn^) from each others. We get: (i3) OA-^W^.11» From each ksZ, J is just a finite dimensional matriK which is the restriction of J to the L with zero boundary conditions. For this reason the second term in the right hand side of ( 11 ) coupies only nearest neighbouring blocks: DAJQA couples each site k with either L or L ; (z-Q JQ )_1 leaves either of these blocks invariant, and Q p * connects L with k or k*l and 1^, with k A A •* or k-1. On the other hand DA JDA * is diagonal unless there is k such that m, =m, + l, in which case we get nevertheless only k+1 k nearest neighbours interactions. Thus JA fe)i s a Jacobi matrix. D A more precise computation of the coefficients of J (z) can be Chapter 7-64 done using (10). In the case where m >m +1 \f li€'£ we get: (14) t (k;2) = t(m, • l)t(m1, ,+2)...t(mL )/T> , (z) A k-1 k-1 k k-1 (15) VA(k;z) = v(m ) |t(m+l)|2Q,(s)/P (z) lt(m-1)|\ ,(zVP. ,(z) k + k k k + k k-1 k-l where (16) Q (z)/P (z) = (10) Pk(z) = det(z-Jk) Clearly, Pk is a rnonic polynomial of degree d^ll^l whereas Qk and R are monic polynomials of degree d-1. They can be computed recursively from the matrix elements of J via a two terms recursion relation. It is possible to apply this algorithm to toe almost Mathieu equation; for p 1 let us define (H as in( I)): s (19) J=(l/(.i)H: V ^ vm=2cos2ir(x-ma) For ki big, the potential v creates strong barriers at the points m for which x-ma is close to zero. More precisely let A(x) be the set of m's in 1 for which -a/2 Lemrna2: if A(x) is labelled as {m ; keZ} with ny mfc+] we have mk+]-m e{a a+l}wherea)=[l/a]. Thanks to ( 14) we get: d +1 (20) t (k;z)=l/u k .l/Pl,l(z) A * K-1 Chapter 7-85 which indicates that the effective coupling for Jftfe) is l/^iV instead of 1/y; therefore the iteration of the map J->JA(z) seems to give a sequence of Jacobi matrices converging quite rapidly to a diagonal one when \x > 1 : a situation leading to a point spectrum. A duality argument of Aubry-André's type would allow to conclude to the existence of Bloch waves and presumably to an absolutely continuous spectrum when \i in the meromorphic function J f .(z). For each k, Pk has only finitely many zeroes, which actually move with k (or equivalents with x) in such a way that their union is dense in a closed set of perhaps a non zero Lebesgue measure. Even though it should be possible to show that this set does not meet the spectrum of ] it is necessary to have a better knowledge of its location in order to get an estimate on 1 /P and to show that the effective coupling is really of order 1/(A+1. More difficult again is the case jjt=l. It would be convenient to find a "fispoint" of the equation (10), namely JA(z; F^cyOKoc'X) + F2(z;ct;i), where a'=l/a-[l/cf] and x'=f(cc,3rt for some f. In the next section we intend to describe models for which such a fixpoint exists. • ?S8=S A solvable model: In [62] J.Bellissard, DSessis & PMoussa described a solvable model for which the previous scheme works. They considered a Jacobi matrix H on H instead of Z defined as in the sectionVII-2 eq.( 10) by: Chapter 7-66 (1) Dy(m)=y(2rn) men, y€l2(IN) (2) D(z-H)"'B*=2.(22A-Hr1 From the previous analysis we get: (3) V =0 t =t *=R(m),/2 m mm provided \>2, where the R's satisfy the following recursion relation: (4) R(0)=0 R<2m) + R(2m+1)-1 R(2m-l).R(2m)=R(m) As shown in [62] (see also [140]), the operator H has the following properti-;; i)-the sp^ti.tm of H . for \>2 is the Julia set 2 [12,74,95,121] Jf of the map F: zee •+ z -X e C. In this case JF is defined as the set of zef such that n (5) supnIF (z)l < « As shown by G.Julia [1211 and P.Fatou [951 (<*• also H.Brolin [74]) Jf is a Cantor set of zero Lebesgue measure since the critical point 2=0 of F does not satisfy (5). ii)- J can be constructed as the set of limit points of F~1. Namely F admits two inverses maps (6) E(o)=lirn y o oo ' oo ' homeomorphism of the set {-1,+1}" onto Jr such that (7) FoE=EoS Chapter 7-87 provided S is the one-sided shift. iii)- It follows that the spectral measure relative to the state |0> defined by: (Ô) jdu(E} fCE) = Hy (E) = E y(E) EeJf are built recursively as follows: y (E)=P (E).{R(l)...R(m)}-,/2 where P (E) is the monic polynomial of degree m defined by : (11) P ,(E) = EJ> (E)-R(m)J> ,(E) In particular the P's are the orthogonal polynomials associated with du. From (2) it is possible to show that they satisfy (12) P„.(E) = P (E^XJ'P (FŒ)) If now BeJ 3 oe{-1,+1}" such that E=E(a), and we get: Chapter 7-86 (13) y, Thus if E is chosen randomly according to d[)., the sequence k> k (E); fceW} is random. This does not prove that u» (E) itself is '2 m r 'm random. For example the sequence cos27ictm is almost periodic wnereas the subsequence cos(2Troc2km) is random [1291. However for \>2, one can show that u' (E) cannot be of Bloch type m i.e.: m ym(E)*e* Xfa.-E) (some keE) with Xto-E) limit periodic with respect to m, with the same frequency module as R. This model exhibits a lot of interesting properties which are, we believe, analogous to properties of the time dependent quantum systems at the critical point if any. In particular the wave packets evolve in time in a very recurrent way namely. (a) the total time spent in ?ny finite region of K by y(t)= e",tHy is infinite C4 2 J0 dt|(m|y(t)>| =0o ymeZ This is because H has a singular spectrum. (b) The average time it spends in any finite region is however equal to 2ero: T 2 ^T-x» I/Tjo dt| This is because H has a continuous spectrum. Roughly speaking it means that the wave packet goes back and force infinitely many times at each site before eventually get to infinity. VB8=4 On the ftepoint of the R.G. equation: The equation (2) in the previous section can be generalized as follows, let D be a partial isometry such that: (1) DDM D*D=H-Qsfi We also suppose that there is a D-invariant vector y. Let H be a bounded self adjoint operator which is cyclic with respect to the subspace image of D* . The R.G. equation we intend to investigate here is (2) Dfe-H)*D, * = Gfe3) Here G(z,x) is a rational function of 2 and x behaving like 1/z as z tends to infinity. Let us give few examples which appeared in the littérature. (3) Dfe-H)"1D* = 1/NP'(2).(P(Z)-H)"' where P is a monic polynomial of degree N. This is an obvious generalization of the model described in the section VII-3. It has been investigated in some cases by Bainsley, Geronimo, Harrington 155.561. B£ (4) D(z-H)"V = 3(3z-H)/{(3z-H)*-4} It gives rise to an operator the spectrum of which being the usual tryadic Cantor set on t-1,+ 1]. Moreover it can be represented in term of a Parisi matrix [146] (cf .theorem 1 below). 1x3 (5) D(z-H)"1D*= (z- l)(z-2)/(z+2> {z(z-3)-H}"1 This is the R.G. equation for the lattice Laplacian on the triangular Serpinsky gasket in two dimensions. This equation has been investigated almost in this form by Rammal [1521, and by Alexander 139]- Let us note that this model is an idealization of a percolation cluster of superconductors. An analogous equation exists for a d-dimensional Serpinsky gasket. « Our first result concerns the example 2. Chopt«-7-90 Theorem I: i) Every solution of (4) haï a spectrum given by ii) The spectral measure of any solution of (4) is equivalent to the Cantor measure on C„. iii) The equation (4) admits a unique solution as infinite matrix acting on f(2) wl:en D is the dilation operator : Dy(rn)=y(2m) ye 12(Z) This solution is given as a Parisi matrix (cf.eq.(6) below). The corresponding Parisi matrix is given by: , or equivalently by (6b) 0 2/3 2/9 0 2/27 0 0 0 2/3 0 0 2/9 0 2/27 0 0 2/9 0 0 0 0 0 2/27 0 0 2/9 0 0 0 0 2/27 . 2/27 0 0 0 0 2/3 2/9 0 0 2/27 0 0 2/3 0 0 2/9 . 0 0 2/27 0 2/9 0 0 2/3 . 0 0 0 2/27 0 2/9 2/3 0 VTe will not give the proof of this theorem for it is just a matter of calculation. Cho(rt«r7-91 The main result of this section which completes the work of I5&] is the following: Theorem2: Let P be a monic polynomial of degree N with real coefficients. We suppose that its Julia set J is contained in R and does not contain any critical point of P. we also suppose that there is a D-invariant vector which is cyclic for H. Then : (i) The equatio:i(3) admits one solution at least. Any solution of (3) admits J as spectrum. (ii) The spectral measure of any solution of (3) is equivalent to the balanced measure on Jp. (iii) There is a unique solution of (3) on 12(Z) when D is the dilation by H: Dy(m)=y(Nm) yei2(£) This solution is a limit periodic Jacobi matrix the frequency module of which being the set N-adic rational numbers (mod. 1). D Let us recall that the Julia set of P is the set of points of the Riemann sphere Cu« having no neighbourhood on which the family {P" ; n*0} is normal. The reader wilt find a monography in tj on the subject. The main result of G.Julia & P.Fatou is that J is completely disconnected with a zero Lebesgue measure if it contains no critical point of P. Moreover, it is invariant both by P and P"1. When it is contained in E we can choose the determinations of the inverse of P. w,(x),...,w„te) to be 1 « holomorphic in a neighbourhood of the convex hull IE_£+I of J Then any point in Jp is uniquely given in term of a coding namely: (7) E(o)=lim w »w 0...0W (x) n->» 90 ?1 on Chapter 7-92 where a •= [1.H1 (.?•! i ) and the limit i* uniform in snd indépendant of x. The P-balanced measure u is given by: (6) fdu (E)fiE)=lim WT . ,f(E(o)) Let us recall that a matrix A indexed by N is limit periodic if there is a sequence of periodic matrices converging in norm to A. For a Jacobi mstrixas in §VII eq. (6) this means (9) Jim .sup . . |v(m+q, r)-v(m)l=0 k rm>O,r>0 *k for some sequence q of integers. The proof of the limit periodicity of the jacobi matrix associated to the Julia set of a polynomial P was given in (621 for polynomials of degree 2, in [56] for polynomials of the form P(z)=l/aTH(az) (a>l) where T is the Tchebyshev polynomial of degree ti and a is big enough. We will give here a sketch of a global proof usin:, the properties of the R.G. equation (1). A sketch of the proof -if the theorem2: l)Eastence of a solution: H Let us consider the set n=[l,Nl of sequences a=(cO. 0 where o.e[ 1JNJ yi. a is a coding. S will denote the one sided shift (10) S(o).-a. , Let do be the corresponding Bernoulli measure on Q. We denote by B? the Hilbert space L2(fi,dc). If fej& we set: (11) Hf(a)=E(o)f(a) We will also introduce the operator D as follows: (12) D*f(o)=f(So) It is easy to check that D,H satisfy the equation (2)&(3). The Chapter 7-93 swtnifn f,( H t* «ins* tuiv&n h'r thfl ItnoCA ^f <~> linger * »•'a^'"lt,' * and its spectral measure with respect to the cyclic function f=const. is exactly \x 2)Spectrum of a solution: Since there is yeTSf which is D-invariant and cyclic for H, z belongs to the résolvant set of H if and only the Green function G(2)= (15a) v(nN)=-p(/H (15b) if H is the N-lxN-1 matrix defined by knN+O t(nf<+2) ... 0 I H(nN+2) v(nN+2) ... 0 | \= '. > 10 ' 0 ... «nN+N-01 I 0 0 ... ((nN+N-1) v(nN+N-1)| Chopt«r7-94 we get detfe-Hn)= 1 /P'fe) (150 {t(nH+l) 2 2 (15d) t(nN+lM(nN)=p( l/>;i-l/N)-2p, /N (15e) t(nN+l)t(nN+2)...t It is a tedious but elementaty calculation to check that this set of relations defines entirely the sequences v and t by recursion. We also get as a consequence (16) RD = DP-(H) a relation already pointed out in [62]. 4) Limit periodicity: To prove that the previous Jacobi matrix is limit periodic we proceed as follows. Let us consider the equation (17) Dfe-Hr'D» sl/RPmfe-K)'' where K=K* is bounded and H is an unknown operator. V:e assume that the spectrum of K is contained in the interval E+ « Min {P(z) ; P'(z)=0, P"(z)<0} E_ - Mas {P(z) ; P"(z)-0, P"(z)>0} If E+- sup Jp and E_» inf J the hypothesis on P imply E_ sum equal to 11 commuting with K. Since wn(E_JE+) c (E_,E+), H is self adjoint with a spectrum included in (E_,E ). In particular if we define R as the map such that H> JUK) P we also get by replacing P fcyit s a* iterate Pw n JlD (K)=V.w,o...oW (E)n . m P *-tn\ ol an ol.-fln where again the n's constitute an orthonormal family of If projections with sum equal to I. Since sup, „ |w o...oV7 .(x)-E(cn,o,,..,o ,,...)! rE-4xiE+ tx, on-t 0 1 n-1' converges to zero as n->» the sequence ÏL"(K)«K converges in norm to an operator K which is obviously a f ixpoint of ( 17). In addition if K is a Jacobi matrix it is possible to check (again a non trivial fact) that the previous solution can be chosen as a Jacobi matrix by using the Schur complement method. Eventhough (17) has generally many solutions there is a unique solution as a Jacobi matrix. From this uniqueness if K is periodic of period L, the solution H of (17) is periodic of period NL. Therefore starting from K=E 8 with E_ «Êtessta MSfflspsgiSê®: tl J R.ABRAHAM, J.E.MARSDEH, Foundotior. of Meehonics, Benj Reoding, Moss. 2nd Ed. (1976) 12] V.I.ARNOID, A.AVEZ, Problem» srgodiques en Mécanique Classique, Paris (1967). 13] V.I.ARNOLD, Chapitres Supplémentaires de la théorie des equations différentieOes ordinaires. Ed. Mir, Moscou (1980). [4] P.BILLINGSLEY, Ergodic theory and information, Chop.1 ,§4, J.Viley & Sons, New York, London, Sydney (1965) |5) G.D.BIRKHÛFF, Dynamical Systems, AMS Coll. Publiootions, Vcl.9, (t 927) 16] BDBIRKHOFF, CePected Mathematical Vorks, Vol.2, Dover, New Vork (1968). 17] E.B0REL, Leçons 'sur les Fonction Monegènes uniformes d'une Variable Complexe, Gauthier-Villors, Poris (1917). [8] G.CASATI, J.FDRD Eds., Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Sprinaer, Berlin, Heidelberg, New Vork, Lectures Notes in Physics, 93, (1979). [9i G.CASATI Ed., Chaotic Behavior iri quantum Systems, Plenum Press, New Vork, (1964). |10) B.V.CHIRIKOV, A Universol Instability of Manu Dimensional Oscillator Systems, PhysJ5ep.,52,263,(1979). 111] I.P.C0RNFELD,S.VFORMIN,Yo.û.SlNAI, Ergodio Theory, Grundlerhen Bd245. (1982), Springer Verlag, Berlin, Heidelberg, New Vork. M 2] A.D0UADV, Systèmes Dynamiques Holomorphes, Séminoire Bourboki, 599. (1982), 39-63. [13] F.D0VEIL, Groupe de Renormalisation pour les Systèmes Homiltoniens non Integrates: Application au Mouvement d'une Particule Chargée dans un Paquet d'Ondes Longitudinales, Thèse d'Étot, Univ. Poris VI, (1982), Paris. 114] H.FÛRSTENBERG, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, N.J., (1981). |15] O.GALLAVOTTI, Elements of Classical Mechanics, Springer Texts and Monographs inPhysfc3,(1983). References-97 [tel GJHUARCsV, E.M.WRIGHT, An Introduction to the Theory of Numbers, 5th Ed., C'foid, Cloivû'ijù rr»;, (i37S). [17Î M.R.HERMAN, Sur In Coniuga*so.i Differentiate des Oifféomorphismes du Cerd* à des Rotations, Pub. de l'I.H.E.S., 49, (1979). [t8] M.R.HERMAN, Sur les Courbes Invariantes par les Diffé (IQl A.Va.KHINTCHINE, Continuous Fractions, Chicago Univ.Press, (1964), Chicago. [20] P.MJCOCH, Interactions of Intense Microwaves with Rydberg Atoms, J.Phus. (Paris) Colloques, C2 43 , (1982), 187. (21 ] J.F.KOKSMA, Diophantische Approximation*», Chelsea. (22] A.N.KOLMOOORQV, Théorie Oénérale des Systèmes Dynamiques en Mécanique Classique, Proc. Intern. Congress of Math., Amsterdam (1957), English Translation in App.D of ref. (1 ]. [23] IXUIPERS, H.NIEC>EREITERS, Uniform Distribution of Sequences, Wiley & Sons, NewYork,(1974). [24] A.J.LICKTEHBERG, M.AJ.IESERMANH, Regular and Stochastic Motion, Springer Verloq, Berlin, Heidelberg, New York, (1985). [25] .A.M.LYAPOUNQV, Problème Général de la Stabilité des Mouvements, Ann. Foc. Soi. Toulouse (2), 202-474, (1907), reprinted in Ann. Moth. Studies, (1972), Princeton University Press, Princeton, N.J. (26] J.MOSER, Stable and Rondom Motion in Dynamical Systems, Ann. Math. Studies, (1973), Princeton Univ. Press, Princeton N.J.. [27] H.POINCARÉ, Les Nouvelles Méthodes de la Méconique Céleste, Tomes 1,11,111, Cauthiers-Villors (1S92-1894-1899), Reprinted by Dover, NevYork (1957). [28] H.POINCARÊ, Oeuvres, tome VII, Gauthier-VOIars, Paris, (1952). [29] Proceedings of the!981 Les Houches Summer School, "Chaotic Behavior in Deterministic Systems", G.boss, R.H.G.HeHemori, RStoro Eds., North Hollond, Amsterdam (19S4). [30] MJ5EED, B.SIMON, Methods of Modern Mathematical Physics, Vol.111 : Scattering Theory, Academic Press (197S) London, New York. [31] VP.REINHARDT, Classical Chaos, The Geometry of Phase Space and Semi-Classicol Quantization, to appear in "The Mothemoticol Analysis of Physical Systems", R.Miekens Ed., Van Nostrand Reinhold, Mew York, In Press. References - 98 132] V.SCHMIDT, Diophontine AppnBd-notions, Lectures Notes ir Math, nS 7Ë5. (1980), Springer Verlag, Beriin, Heidelberg, New Vork. [33] CSIEGEL, J.MOSER, Lectures on Celestial Mechanics, Orundlehrsn Ed 187, (1971) Springer Verlag, Berlin, Heidelberg, New Vork. [34] B.SIM0N, Almost Periodic. Schrodinger Operators, Adv. Appl. Moth, 3, (1932), 463-490. 135] S.SMALE, The Mathematics of Time, Springer Verlag, Berlin, Heidelberg, NewV'irk,(1930). [36] Cf. for instance: (0 Quasi One Dimensional Conductors, S£arisic et al. Eds., Lect. Notes inPhys. n?95&96.(1979),Springer Verlag,5erlin,Heicelberg, New Vork; (ii) Physics in One Dimension, J.Bernosccni £ T.Schneider Eds , Springer Verlog, Berlin ,Heide!berg, New York, (19SI). [37] P.V.ANDERSON, Absenoe of Diffusion in Certain Random Lattices,Phy5.Rev., 109 .(1958). 1492-1505. [38] S.ALEXANDER, Superconductivity of Networks. A Percolation Approach to the effect of Disorder, Phys. Rev., B27, (1983), 1541-1557. [39] S.ALEXANDER, Some Properties of the Spectrum of the Serpinski Gosket in a Magnetic Field, Preprint 1984. [40] S.ALEXANDER,E.HALEVI, Superconductivity of Networks II: the London Approach, J.Physique, 44, (1963), 805. [41 ] V.I.ARNOLD, Small Divisors I. On the Mappings of the Cirt!< v itself, fev. Akad. Nouk., SSSR Hot., 25.(1), (19*3), 21-26. [42] V.I.ARNOLD, Small Divisors II. Proof of o ANKolmogorov Theorem on conservation of conditionnais Periodic Motion under Small Perturbations of the Homiltonian Function, Usp.Mat. Nauk., 18 (5), (1963), 13-40. [43] V.I.ARNOLD, Small Divisors Problem in Classical and Celestial Mechanics, Usp. Mot. Nouk., 18 (6), (1963), 91-192. [44] V.I.ARNOLD, Instability of Dynamical Systems with many Degrees of Freedom, (Russian), Dokl. Akad. Nauk SSSR, l£è, (1964), 9. [45] S.AUBRV, G.ANDRÉ, Analyticity Breaking and Anderson Localization in Incommensurate Lottices, Ann.of Isroel Phys. Soo., 3 ,(1980), 133. [46] S.AUBRV, P.V. Le DAERON, The Discrete Frenkel-Kontorovo Model ond «s References - 99 Applications, I, Exact Results for the Ground State, Physicc 8D, (19S3),33t-422. [47] S.AUERV, P.Vie DAEROH, S.ANDRÉ, Classical Ground State of a or* dimensions Model for Incommensurate structures. Preprint Saolay, France, (1982). [4SI J.AVRON, B.SIMON, Almost Periodic ScrBdinger aerators I : Limit Periodic Potentials, Comm. Math. Phys., §2, (19S2), 101-120. 1*9] J.AVRON, B.SIM0N, Almost Periodic Sehrbdinger Operators II : «he Density of States, Duke Math. J., 5fJ, (19S3), 369-391. ISO) J.AVRON, B.SIM0N, Transient and Recurrent Spectrum, J. Funct. Anol., 43, (1980,1-31. t51] J.AVRON, B.SIMON, Almost Hill's Equations and the Rings of Saturn, Phys.Rev. Lett., 46.(1931), 1166-1168. 152] J.AVRON, B.SIMON, Singular Continuous Spectrum for a Class of Almost Periodic JacoMMatrices,Bull. AMS,6.,(I932),S1-S6. [53] M.Va.AZEELL, Energy Spectrum of Conductivity Electrons in a Magnetic Field, Sov. Phys. JETP, 19, (1964), 634. [54] M.Va.AZBELL, Quantum Particle in One-Dimensional Potentials with Incommensurate Periods, Phys. Rev. Lett., 43, (1979), 1954. [55] MJ.E.-ARNSLEV, J.S.GERÛNIMO, AN.HARRINGTOH, Infinite Dimensional Jocobi Matrices associated with Julio Sets, Proc. AMS, SS, (1933), 625-630. 156] MJ.BARNSLEY, J.S.0ER0NIMO, A.N.HARRINGTON, Almost Periodic Jocobi Matrices Associated with Julia Sets for Polynomials, To appear in Comm. Moth. Phys.,(l9S5) [57] J.E.BAYFIELD, P.M.KOCH, Multiphotonic Ionization of Highly Excited Hydrogen Atoms, Phys. Rev. Lett, 33, (1974), 25S. [58] JIBAVFIELD, L.D.GARDNER, P.M.KOCH, Observation of Resonances in the Microwave-Simulated Multiphotonic Excitation and Ionization of Highly Excited Hydrogen Atoms, Phys.Rev. Lett. 39,(1977), 76. [59] J.EELLISSARD, Almost Periodic Schrb'dinger Operators: an Overview, Lecture Notes in Physics, 153, (1982), 356, Springer Verlag, Berlin, Heidelberg, NewYork. [60] J.BELLISSARD, Small Divisors in Quantum I lechanics, in ref. [9] [61 ] J£ELLISSARD, Stability and Instability in Quantum Mechanics, CIME conf of Como, Seot. 1984. References- 100 [62] J.BELLISSARD, D.BESS1S, P.M0USSA, Chaotic States of Almost Periodic Sctrosfnger Operators, 49, ( 1982), 701 -704 163] J.BELLISSARD, E.SC0PPOLA, The Density of States for Almost Periodic Schradinger Operators arid the Frequency Module: a Counter Example, Cecum. Math. Phys., 85, (1932), 301-303. [64] J.BELUSSARD, R.LIMA, E.SCOPPOLA, localizator» in v-dimensiona! hcommeraurote Structures, Coram. Math. Phys., g£ ,(1983), 465. [651 J.BELLISSARD, R.LIMA, D.TESTARD, A Metol Insulotor Transition for the Almost Mathieu Model, Comm. Math. Phys.JS, (1933) 207. see also J.BELLISSARD, AfGRMOSO, R.LIMA, D.TESTARD, Quasi Periodic Interaction with o Metal Insulotor Transition, Phys. Rev. 626,(1982), 3024-3030. [661 J.6ELLISSARD, R1IMA, D.TESTARD, Almost Periodic Schrodinger Operators, Preprint, ZIF, Bielefeld, Nov. 84. [67] J.BELLISSARD, B.SIMON, Cantor Spectrum for the Almost Mathieu Equation, JJunct. Anal., 48,(1982), 408-419. [66] E.D.BELOKOLOS, Quantum Particle in One Dimensional Distorted Lattices. Estimate of the Sfee of Lacuna? in the Spectrum, Their. Moth. Phys., 25, (1975,1,1176. [69] M£ERRY,inref. [8], [70] M.BERRY,inref.[9], [71] M.BERRV,inref.[29]. [72] M.BERRV, Incommensurability in on Exactly Soluble Quantal and Classical Model ftor a Kicked Rotator, Physico pJO, (1984), 369. [73] N.N.BOG0LIÛU6OV, N.M.KRVL0V, Sur quelques Formules de Développement en Séries dons la Mécanique non Linéaire, Ukranin. Akad. Nauk. Kiev, i, (1934), 56, see also Ann.of Math. Studies, II, (1947), 106. [74] M.BROLIN, Invariant Sets under Iteration of Rational Funotions, Arkiv for Math., 6, (1966), 103-144. [75] L.CARLESON, A Remark on Denjoy 's Inequality and Herman's Theorem, Pub. IHES. nM9, (1979), 235-242. [76J O.CASATI, I.GUARNERI, Chaos and Special Features of Quantum Systems under External Perturbations, Phys Rev. Lett., 50, (1983), 640. [77J G.CASATI, I.GUARNERI, Non Recurrent Behavior in Quantum Dynamics, Comm. Moth. Phys., 95, (1984), 121-127. References- 101 [781 > ttefc (• .CASAT! ° !.SUAF™EP! ':" "A ttzar!. [79] G.CASATI, B-VjCHKIrBV, F.M.ISRAELEV, JfORB, in ref. [8l. [80] G.CASATI, I.6UARNERI, F.VALZ-&RIS, Degree of Rondomness of the Sequence o' Eigenvalues, Phys. Rev., A3C1. (1984), 15S5. (81 ] G CASATI, Quantum Dynamics cf Classical Stochastic Suit*:.*, This iss-e. [82] B.V.CHIRiKOV, F.M.ISRAELEV. D.L.SHEKELS'ANSKY, in Soviet Scientific Review, Ç2,(1981). [63] F.H.CLARO, G.H.VANNIER, Maanetic Subband Structure 1n Hexagonal Lattices, Phys. Rev., Big, (1979), 6063. (84) V.CRAIG, Dense Pure Point Spectrum for the Almost Periodic Hill's Equation, Comm.Math. Fhys., 8£, (1983), 113. [831 Y.CRAIG, B.SIMON, Subharmonicity of the Lgapounov Index, Duke Math. J., JO, (19S3),551-560. [S6] R.J.DAMBURG, V.V.KOLOSOV, A Hydrogen Atom in a Uniform Electric Field III, J. Ph'js.B12, (1979), 2637-2643. ' [871 P-O.De SENNES, CRAS, B292, (1980,9; CRAS, B292, (1980, 279. [EC] B.DERRIDA, SARMA, unpublished, (197S). I thank the Physicists from the DPhT (Socloy) for pointing out to me this contribution quoted in (451. [89] E.I.DINABURG, Vo. SINAI, On the One Dimensional Schrb'dinger Operator with 3 Quasi Periodic Potential, Funct. Anal. Appt.,9, (1973), 279. [90] B.DDREZI, B.GRAMMATICOS, Y.POMEALI, The Periodically Kicked Rotator: Recurrence and/or Energy Gtwth, J. of Stot. Phys., 37, (19S4), §3-100. [91 ] E.NXCONOMOU, C.H. SOUKOUUS, Localization in One-Dimensional Lattices in the Presence of Sicommeiisurote Potentials, Phys. Rev. Lett., 48, (1982), 1043 (92) D.F.ESCANDE, in Intrinsic Stochosticity in Plosmos, GLaval & D.GrésSIan Ed., Editions de Physique, Courtoboeuf, Fronce, (1979). [031 D.F.ESCAHDE, F.DOVEIL, Renormaliiotion Method for the Onset of Stochostioity in a HomHtonian System, Phys. Lett., 83A. (1981), 307 (94) DJ.ESCANDE, FJMWEIL, Renormalcotion Method for Computing the Threshold of the Large Scale Stochastic Instability in Tvo Degrees of Freedom Hamiltonion Systems, J.Stat. Phys., 26, (1981), 257-284. References - 102 (95) P.FATÛU, Mémoire sur les Equations Fonctionnelles, B.S.M.F., 47, (1919).. 181-271 ;22,'.'.°2£.H-y. cr.i2C8-ÎM. [96] M.FE1N60LD, SJISHMAN, D.GREMPEL, R.E.PRANGE, Statistics or Quasi Energy Separation in Chaotic Systems, Preprint, Univ of Maryland, (1984). (971 A.L.FIG0TIN, L.A.PASTUR, An Exactly Solvable Model of a Multidimensional Incommensurate Structure, Comm. Motli. Phys., 95, (1984), 401-425 (98) S.FISHMAN, D.R.GREMPEL, R.E.PRAN0E, Chaos, Quantum Recurrences and Anderson Localization, Phys. Rev. Lett., 49, (1982), 509-512. [99! S.FEHMAIi, D.R.GREMPEL, R.E.PRAM&E, Locoicotion in an Incommensurate Potentlol: an Exactly Solvable Model, Phys Rev. Lett., 49, (1982), 833. [100) S.FISHMAN, D.R.6REMPEL, R.E.PRANC-E, Quantum Chaos and Anderson Localization, in ref. (9). [101) S.F1SHMAN, D.R.GREMPEL, R.E.PRAK&E, Vov* Functions at a Mobility Edge: an Example of Singular Continuous Spectrum, Phys. Rev., §28, (1983), 7370. [1021 S.FISHMAN, O.R.GEEMPEL, R.E.PRANOE, Quantum Dynamics of a nor. Intejrable System, Phus. Rev., A29, (19S4), 1639-1647. [103) S.FBHMAN, D.R.GREMPEL, R.E.PRANGE, Long Range Resonance in Anderson Insulators: Finite Frequency Conductivity of Random and Incommensurate Systems, Preprint Univ. of Maryland, (1984). [1041 S.FISHMAH, D.R.GEEMPEL, R.E.PRANGE, Finite Plonck's Constant Scaling at Stochastic Transition of Dynamical System, Preprint Univ. of Maryland (1984). (103) S.GELTMAN, Ionisation of a Model Atom t>y a Pulse of Coherent Radiation, J.Phys., HO, (1177), 831-840. [1061 A.Va.GORDON, The Point Spectrum of the One Dimensionol Schrb'dinger Operator, (russian), Usp. Mot. Nauk., 3J., (1976), 257. [1071 J.M.GREENE, A Method for Determining o Stochastic Transition, J. Moth. Fhys., 2£, (1979), 1183-1201. [10SJC.C.6RIMES, THROWN, M.L£URNS, C.L2IPFEL, Spectroscopy of Electrons in Image-Potential-Induced Surfaces State Outside Liquid Helium, Phys. Rev., BI3, (1976), 140. [109) P.G.HARPER, Single Band Motion of Conduction Electrons in a Uniform Magnetic Field,Proe. Phys. Soc. London, A68, (1955), 874. [110) Y.KHAYMAN, Meromorphic Functions, Clarendon Press, Oxford, (1994), Chap.6-7,p.172. References- 103 [ill] D.;ilR££R(, ft.JCir.wi. Locolsa ïtotei .1 Infor*:»* Systems, J.Fhys., Ç4, (1670,1145. (112] D.HÛFSTADTER, Energy Levels and Vove Functions of Eloch Electrons in a Rational on* Irrational Magnetic Field, Phys. Rev., Elf, 0976), 2239. [113] J.SJWVLAN!), Stotionnory Scattering Theory for the Time Dependent HomiKanioft, Moth. Ann., 207» (1074), 315-325. [114] J.S.HOWLAND, Scotterino Theoru for Komiltonions Periodic in Time, Indiono Uni/. Moth. J., 26, (1979), 471-494." [113] K.ISHII, Locolfzatfon of Eigenstotes and Transport Phenomena in the One Dimensions Disordered System, Suppl to Progress Theor. Phys., 53, (1973), 77. [116] K.ISHII, HMATSUDA, Localization of Normal Modes ond Energy Transport in Disordered Harmonic Chains, Progress Theor. Phgs. Supp., 45, (1970), 45. [117] F.M.IZRAELEV, D.L.SHEPELVANSKI, Quantum Resonances for 0 Rototor in 0 Non Lineor Periodic Field, Theor. Mot. Fiz., 43, (19S0), 553-560, (english translation). [118] R.JENSEN, Stochastic Ionization of Surface Stote Electrons, Phgs. Rev. Lett.,49, (19S2)„ 1365. [119] R.JENSEN, Stccnastic Ionization of Surface Slate Electrons: Classical Theorg, Phgs. Rev., A30, (1984), 3S6. (1201 R.A.JOHNSON, A Review of Recent Yorks on Almost Periodic Differential and Difference Operators, Acta Appl. Math., I, (19S3), 241-161. [121] G JULIA, Itération de; Applications Fonctionnelles, J.Math. Pures et Appl., (1918), 47-243. (1221L.P.KADAN0FF, J.SHENKER, Critical Behavior of 0 KAM Surface I. Empirical Results, J. Stat. Phys., 22, (19S2), 631-656. 1123] LPXADANOFF,M.KOHMOTO, C.TANG, Localisation Problem in One Dimension: Mapping and Escape, Phgs. Rev .Lett., 50, (1983), 1870. [124] AKATOK, Some Remarks on Birkhoff and Mother Twist Map Theorem, Erg. Theo. ondDgn. Syst.,2, (1982), IS5-194. [125] A.KATOK, More about Eirkhoff Periodic Orbits and Mather Sets for Twist Maps, Preprint l,IHES, (1982), Preprint II, Univ. Maryland, (I9S2). (126] L.V.KELDVSH, fonfsation in Ihe Field of a Strong Electromagnetic Vove, Sov. Phys. JETP, 20, (1963), 1307-1314. Reterences- 104 (127) J.KITADA, K.YAJIMA, A Scattering Theory for Time Dependent Long Ronge Potentiels, Duke Moth. J., 4a, 0932;, «41-^76 [12SÎ P.M.KOCH, D.R.MAR1ANI, Precise Measurements of the Static Electric Field Ionization Rote for Resolved hydrogen Stark Substances, Phys. Rev Lett., 46. (1980,1275. [12B] I thonk M.KOHMOTÙ for this Remark. [130] MKOHMOTO, V.OONO, Cantor Spectrum fcr on Almost Periodic Schrodinger Equation ond a Dynamical Map, preprint Univ. of Minois Urbane, (1983). !131] A.N.KOLM0GOR0V, Ori the Conservation of Conditionnai^ Periodic Motions under Small Perturbations of the Hamiltonion Function, Dokl. Akad. Nouk. SSSR, 68, (1954), 527-530. [132] D.K.LAMBERT, P.L.RICHARD, Far Infrared and CapotItance Measurments of Electrons on Liquid Helium, Phys. Rev., §23, (1980,32S2. [133] J.O.LEOPÛLD, I.C.PERCIVAL, Microv/ove lonfaatior, ond Excitation of Rydberq Atoms ,Phys. Rev. Lett., 41., (I97S), 944. [134] J.G.LEOPOLD, I.C.PERCIVAL, Ionization of Highly Excited Atoms by Electrfc Fields III: Microwave lonfeotion ond Excitation, J. Phys., ÈV2, (1979), 709-721. [135] J.M.LUTTINOER, The Effect of a Magnetic Field on Electrons in o Periodic Potential, Phys.Rev.,§4,(195D,S14. [136] J.N.MATHER, Existence of Quasi Periodic Orbits for Twist Homeomorphisms of the Annulus, Topology, 21_, (19S2), 457-467. [137] J.H.MATHER, Hon Existence of Invariant Circles, Erg. Theor. Dyn Syst., 4, (1934), 301-303. [13S] J.NJ1ATHER, A Criterion for the Non Existence of Invariant Circles, Preprint Univ. Princeton (16S2). [139] J.MOSER, On Invariant Curves of a Area Preserving Mapping of on Annuiius, Naohr. Akad. Viss. Ob'ttingen Math. Phys. Kl, Ma n*l,(1962), 1-20. [140] P.MOUSSA, Un Opérateur de Schrodinger Presque périodique à Spectre Singulier, Associé aux Itérations d'un Polynôme', Proc. RCP 25, CNRS n£34, (1QS4), 43, Strasbourg. [141]W.NNEKH0R0SHEV, The behavior of Hamiltonian Systems that ore Close to JMegroM» Ones, Funct. Anal. Appl., 5, (1970,33S-339. [142] S.OSTLUNDT, R.PANDIT, Renormalizotion Group Anoiysis of a Discrete Quasi References- 105 Periodic Schrodinger Equation, Preprint (1984). [143) E.OSTLUHDT, R .PANDIT, D.RANDT, H.SCHELINHUBER, E.D.SIGGIA, One Dimensional Schrb'dinger Equation with an Almost Periodic Potential, Phys. Rev .Lett., 50, (1983), 1873. 11441 B.PANNETIER, J.CHAUSSV, RJiAMMAL, Experimental Determination of the (H,T) Phase Diagram of a Superconducting Network, J. Phusique Lettres, 44, (I9S3), L653-L858. [145J B.PAHNHIER, J.CHAUSSV, R.RAMMAL, J.C.V1LLEOIER, Experimental Fine Tuning of the Frustration: 2D Superconducting Network in a Magnetic Field, Sub. to Phus Pw.Lett.,(19S4). [146] GPARKI, Infinite Number of Order Parameters for Spin Glosses, Phys. Rev. Lett., 43, (1979), 1754-1756. [147] L.A.PASTUR, Spectral Properties of Disordered System in One Body Approximation, Comm. Math. Phys., 75, (1980), 179. [14SJ R.S.PEIERLS, Quantum Theory of Solids, Oxford Clarendon Press, (1955) Chop. 5-3; see olso Z.Phys., 80, (1933), 763. [1491I.C.PERCIVAL, Variational Principles for Invariant Torii and Cantorii, in Symposium on "Non Linear Dynamics and Beam-Beam lnteraction",Amer. Inst, of Physics, Conf. Proc. nS57, M.Month & J.CHerrero Eds.,(1980), 302-310, see olso J. Phys., M2, (1979), L57. [1501 A.MKREMOLÛV, V.S.PÛPOV, M.V.TEREHTEV, Ionization of Atoms in an Alternote Electric Field, Sov.Phys. JETP, 23, (1966), 924-934. [151 ) J.POSCHEL, Examples of Discrete Schro'dinger Operators with Pure Point Spectrum, Comm. Math. Phys., SS, (19S3), 447. [152] R.RAMMAL, Spectrum of Harmonic Excitations on Fractals, J. Physique, 45, (1984), 191-206. 1153] A.RAUH, Degenaracy of Landau Levels in Crystals, Phys. Status Sol., §65, (1974),K131. [154] A.RAUH, On the Broadening of Landou Levels in Crystals, Phys. Status Sol., 569. (1975), K9. [135] HJÎÙS3MANN, On Optima! Estimotes for the Solutions of Linear Partial Differential Equations of First Order with Constant Coefficients on The Torus, Lecture Notes in Phys. 38, (1975), 598-624, Springer Verlag, Berlin Heidelberg, New York. [156] H.RÛSSMANN, Notes on Sums Containing Small Divisors, Comm. Pure and Appl. Moth., 29, (1976), 7S5-75S. References- 106 [157] H.RÛSSMANN, On Optimo! Estimates for Solutions of Linear Difference Equations: or. the Circle, Ceiestia! Mechanics, U, (1976),33-37. [15S] H.RÙSSMAI*!, Or. the On* Dirnensionol Schrodinger Equotion with a Quasi Periodic Potential, Arm.N.V. Acad, of Sa., 357, (1980), «0. [159] CJ..SIEGEL, Iteration of Analytic Fundi:-*, Anr.. Math.,43, (1942), 607-612. [160] B.SIMON, Almost Periodic Schrijdipger Operators IV: the Maryland Model, to Appear in Arm. Phys. (1984). 1161 ] B.SIHON, B.SÛU1LLARD, Franco-American Meeting on the Mathematics of Random and AhMst Periodic Potentials, J. Stat. Phys., J6, (1984), 273-288. [162] JÊ.SOKOLOFF, Electron Localization in Cry «tot vith Quasi Periodic Lottice Potentials, Phys. Rev., |22, (1SSÛ), 5S23. [163] J.E.SOKCLOFF, Bond Structure ond Locolfeotion in Incornroensuroie Lattice Potentials, Phys. Rev., |2S, (1980), 6422. [164] J.6.S0K0L0FF, Unusual Band Structure tfove Functions ond Electrical Conductance inCrystols with Incommensurate Periodic Potentials, Preprint Northeastern Univ. Boston Moss., 0984). [165] I.M.SUSLClV, Localisation in One Dimensional Incommensurate Systems, Sov. Phys. JETP, 56, (1982), 612-C! 7. [166] J.B.TAVLOR, unpublished (196S), see ref.[10r ] [167] D.THOULESS, A Relation between the Density of States and Range of Localization for One Dimensional Random Systems, J. Phys., Ç5, (1972), 77-81. [168] R.B.WALKER, R.K.PRESTON, Quantum versus Classical Dynamics in the Treatment of Multiple Photon Excitation of on Anhor-nomc Oscillator, J.Chem. Phys., 67, (1977), 2017-2028. [169] M.WILKINSON, Critical Properties of Electron Eigenstates in Incommensurate Systems, to appear in Proc. Roy, Soc. London, Preprint (1983). [170] K.VAJIMA, Scattering Theory for SchrBdinger Equations vith Potential periodic in Time, J.Math. Soc. Japan, 29, (1977), 729-743. [171]G.E2IL'BERMAN, Behavior of an Electron in a Periodic Electric ond a Uniform Mognetic Field, Sov. Phyi. JETP, 32©., (1957), 208. [172] G.E.2IL'EERMAN, Electron in a Periodic Electric and Homogeneous Magnetic Field II, SOY. Phys. JETP, 33(6), (1958), 299. "f References-107 [173] GliU-'BERMATl, Energu Spectrum in Open Period» Trojectories, Sov. Phijs ŒTP, SiQ)., 095B), 515-516. |I74] M.RJIERMAN, Une méthode pour minorer !e.: exposonts de Lu,opounw et quelques exemples montrant le eoroctère local d'un théorème d'Arnold et Moser sur le tore * dimension deux. Preprint Ecole Polytechnique Poloisê-M, Fronce, (19S2). [17S] I thank J.Mother for this information. 117611 thank Méarnsleu, fer this information. 1 =V ,v m a (T)