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STABILITY and INSTABILITY in QUANTUM MECHANICS Jean 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.
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