INS-Rep.-909 INSTITUTE FOR NUCLEAR STUDY Jan. 1992 UNIVERSITY OF TOKYO Tanashi, Tokyo 188 JNS-PO9_ Japan JP9202239
Nuclear Collective Dynamics and Chaos
Fumihiko SAKATA and Toshio MARUMORit)
Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan
t) Institute of Physics, University of Tsukuba, Ibaraki 305, Japan
To appear in Direction in Chaos, Vol.4 (Series Editor: Bai-Lin Hao), World Scientific Publication Inc. To appear in Direction in Chaos, Vol.4 (Series Editor: Bai-Lin Hao), World Scientific Publication Inc.
Nuclear Collective Dynamics and Chaos
Fumihiko SAKATA and Toshio MARUMORlt)
Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan
t) Institute of Physics, University of Tsukuba, Ibaraki 305, Japan
ABSTRACT The present status and future problems in both the classical-level theory and full quantum theory of nuclear collective dynamics are discussed by putting special emphasis on their relation to the classical and quantum order-to-chaos transition dynamics, respectively. The non- linear dynamics between the collective and single-particle excitation modes of motion specific for the finite, self-sustained and self-organizing system as the nucleus is discussed within the time-dependent Hartree- Fock (TDHF) theory, the basic equation of which is shown to be formally equivalent to the Hamilton's canonical equations of motion in the classical nonlinear dynamical system. An importance to relate the structure of the TDHF symplectic manifold with an inexhaustible rich structure of the classical phase space in the nonlinear system is stressed. A full quantum theory of nuclear collective dynamics is proposed under a dictation of what has been developed in the classical-level TDHF theory. It is shown that the proposed quantum theory enables us to explore exceeding complexity of the Hilbert space. It is discussed that a resonant denomi- nator known as a source of the extraordinary rich structure of the phase space trajectories, also plays a decisive role in generating a rich structure of the quantum Hilbert space.
I CONTENTS
1. INTRODUCTION 1.1. Ordered and Chaotic Motions in Nuclei 1.2. Single-Particle Dynamics 1.3. Order-to-Chaos Transition Mechanism in Nuclei in TDHF Manifold 1.4. Quantum Order-to-Chaos Transition Mechanism in Nuclei
2. SINGLE-PARTICLE DYNAMICS OF NUCLEAR SYSTEM 2.1. Single-particle Algebra 2.2. Hartree-Fock Theory 2.3. Symplectic Structure of the TDHF-Manifold 2.4. Constrained Hartree-Fock Theory 2.5. Adiabatic Time-Dependent Hartree-Fock Theory
3. CLASSICAL-LEVEL THEORY OF LARGE-AMPLITUDE COLLECTIVE MOTION 3.1. Self-Consistent Collective Coordinate Method 3.2. Self-Consistent Constrained Hartree-Fock Method 3.3. Stability and Separability Conditions of Collective Submanifold 3.4. Resonant Interaction Between Relevant and Irrelevant Degrees of Freedom
4. QUANTUM THEORY OF NUCLEAR COLLECTIVE DYNAMICS 4.1. Concepts of Quantum "Integrability" and Quantum "Chaos" 4.2. Basic Idea of Quantum Theory of Nuclear Collective Dynamics 4.3. Basic Equations for Determining the Dynamical Representation 4.4. Dynamical Representation and Extraction of Collective Subspace 4.5. How to Evaluate Stochasticity of Individual Eigenfunctions 4.5. Resonant Quantum Number
5. APPLICATION OF QUANTUM THEORY 5.1. Simple Model Hamiltonian 5.2. Fractional Parentage Plot 5.3. Stochasticity of Ensemble of Eigenstates 5.4. Structure of Eigenfunctions 5.5. Inexhaustible Rich Structure of the Hilbert Space 1.Introduction
1.1. Ordered and Chaotic Motions in Nuclei The objective of this article is to review the recent development towa- rd the microscopic theory of nuclear collective dynamics, which intends to describe the large-amplitude collective excited states situated far from the "ground state"("vacuum") characterized by a certain local stable mean- field, and to investigate dynamical transition mechanism from one local vacuum to another local vacuum in a finite, quantum many-body system. As is generally recognized, it is one of the main subjects in various fields of contemporary physics to explore states of matter located at large- amplitude (large-fluctuation) regions, e.g. far from a vacuum, a ground state specified by a stable mean-field or a thermo-dynamical state of equi- libration, etc. The individual states of matter in small-amplitude regions have been so far extensively studied and are well described by means of the random phase approximation (RPA) or the linear response theory, etc. On the other hand, various experimental investigations on the states of matter in the large-amplitude region are being achieved by using the recent state-of-art experimental facilities in both accelerators and in complex detector systems. Since the states of matter in the large-ampli- tude region usually show a quite different character from those in the small-amplitude region, it is very promising subject to develop a quantum theory of many-body problem which gives dynamical explanations how some characteristic feature of the states of matter in the small-amplitude region persists, transfigures, dissipates or terminates as they evolve toward the large-amplitude region, and how a new characteristic feature specific for the large-amplitude region comes out. In this sense, we are now in a new era of the physical science, which enable us to explore evo- lution of matter [1,2], in addition to the conventional physics of individual states of matter, on the basis of the recent diverse experimental observa- tions on various states of matter. In exploring the above stated new field of quantum mechanics, it is very instructive to learn enormous progress which has been developed during the last few decades in the general the- ory of dynamical systems such as the theory of nonlinear oscillations for Hamiltonian systems of classical mechanics [3,41. A history of nuclear physics may be summarized by stating that there have been incessant struggles for finding a proper position of contradicto- ry concepts; regularity and complexity. Since the nucleus satisfies satura- tion property, each nucleon inside the nucleus is supposed to interact only with its neighboring nucleons through the short-range strong nu- clear force. Consequently, it was hard to expect any ordered regular motion of the nucleons inside the nucleus. Based on this idea of N. Bohr, there introduced the compound nucleus model [5] as well as the liquid drop model [6], which implicitly imply chaotic complex motions of indi- vidual nucleons inside the nucleus so that only collective behaviors of the nucleus as a whole can be treated as regular motions like the surface vibrations, the giant resonances, the nuclear fission process, etc. On the other hand, Mayer and Jensen (7,8] proposed the independent-particle model (the shell model) which is based on the idea that the nucleons inside the nucleus are moving regularly and independently within an average mean-field. The striking success of the shell model proposed two fundamental problems for the study of nuclear structure at that time. The first problem is to understand why the ordered, regular single-p>article motions are justified for such a many-body system of nucleons with strong interaction. The second problem is how to find a proper place for com- plementary modes of motion referring to the single-particle and collec- tive motions. The Bruckner theory [9] struggled against the first funda- mental problem and succeeded in deriving a self-consistent average mean-field characterizing the regular single-particle mode, by properly taking into account both the strong two-body scattering correlations and the Pauli principle among nucleons inside the nucleus. As for the second fundamental problem, the collective model of A. Bohr and Mottelson [10] was proposed by allowing the average mean-field to be time dependent and deformable. The basic idea underlying the collective model is the following [2J. In the shell model, the average mean-field is assumed to be of a spherical "equilibrium" shape from the outset. In the finite quantum system as the nucleus, however, one may expect large fluctuations of the mean-field around the equilibrium (stable) shape, because of the strong self-consis- tency between the single-particle motions and the mean-field. Such fluc- tuations of the mean-field are simply the collective modes of motion, which has been treated by the liquid drop model as the ordered collective motions. One of the outstanding features of the collective model of Bohr and Mottelson is in its explanation of the rotational spectra. When the fluctuations become large, there may occur an equilibrium (stable) defor-
- i - mation of the mean-field due to the strong nonlinear self-consistency between the single-particle motions and the mean-field. This stable deformation introduces a spatial orientation of the system and breaks the rotational invariance, so that the ordered collective rotation manifests itself in order to restore the broken symmetry. A further progress to accommodate the two contradictory concepts. regularity and complexity, was made by Strutinski [11] just after the dis- covery of the fission isomer. He considered the nuclear binding energy to be divided into two parts; one given by an "average" level density and the other due to the shell-structure correction around the former, and pro- posed to evaluate the former by means of the liquid drop model and the latter by means of the shell model. It was surprising that the inclusion of shell structure into the liquid drop model (shell correction method) brought about a successful result in reproducing the second minimum (another stable mean-field located far from the first local minimum) of the collective potential. This method turned out to give a reasonable explanation of the fission isomers, asymmetric fission, etc. Namely, the regular nucleonic motion plays a decisive role even in the large-amplitude region where the liquid drop model has been regarded to give a success- ful description.
1.2. Single-Particle Dynamics According to the recent experimental studies, it is becoming more and more clear that some of the collective excited states in the large- amplitude region are well understood by introducing a suitable phenome- nological stable mean-field. It is really astonishing that there exists a set of characteristic collective excited states, which are far from the ground state of the spherical shell model and are well distinguished from thou- sands of neighboring excited states, and they are simply understood by an appropriate stable mean-field on which a new shell structure is con- structed. In fact, one of the most remarkable progress in the nuclear structure physics in the last decades is an establishment of the rotating shell model [12-15], which explains various characteristic features of the collective rotational states in rapidly rotating nuclei and gives an informa- tion on the superdeformed nuclei [16]. The existence of various stable mean-fields in the nucleus tells us that the states in small-amplitude region are well characterized by a certain stable mean-field, whereas those in large-amplitude region may be speci- fied by another stable mean-field with different symmetries. In other words, the nucleus has various phases specified by a variety of stable mean-fields. In this sense, the nucleus provides us with a unique oppor- tunity of developing a microscopic theory of collective dynamics of finite, many-body quantum systems by means of the single-particle dynamics on the basis of the self-consistent mean-field theory. What should be clarifi- ed within the theory are as follows. How does a stable mean field with a fixed symmetry become unstable and how does a new mean-field with another symmetry become stable in accordance with the evolution of coll- ective motion? What kinds of single-particle dynamics are responsible for the large-amplitude collective motion which is under the influence of two stable mean-fields with different symmetries? What kinds of "optimum" collective variables are there in describing the variation of the self-orga- nizing mean-field? These investigations are main themes of the microscopic theory of nuclear collective dynamics which is discussed in the present article. Thus, in this article, we consider the large-amplitude collective motion associated with the single-particle dynamics, where the two-body collisi- ons are strongly hindered by the Pauli blocking. This means that we treat a system with the Hamiltonian where the Bruckner's prescription for the two-body scattering correlations among nucleons has been already applied and the residual interactions are effective (smooth) ones. In this sense, we may safely employ the Hartree-Fock-Bogoliubov (HFB) theory to obtain the stable mean-field. To simplify the presentation of the problems, throughout this article, we further assume a simple Hamiltonian without the pairing interaction. In this case one may start with the Hartree-Fock (HF) theory rather than the HFB theory. Although the pairing correlation is known to be one of the essential correlations in actual nuclei, the space of this article does not allow us to discuss this problem in detail. In §2, we will recapitulate some important theoretical background of the nucle- ar structure theory related to the single-particle dynamics. According to the HF theory, the stable mean-field is expressed by means of a product wave function called the single Slater-determinant which is composed of a finite number of nucleonic wave functions. On the other hand, the single-particle wave functions describing the nucleonic motion are in turn defined by the stable mean-field, self-consistently. This non-linear dependence between individual nucleons and an average mean-field originates from the self-consistency required in an isolated, self-sustained and self-organizing system, and is the most important rela- tion one should consider in developing the large-amplitude collective dynamics in connection with the single-particle dynamics. The HF theory gives the stable mean-field associated with the lowest energy configuration. Namely it gives local information in the vicinity of the local minimum. In order to get global information on the large-ampli- tude collective motion, as will be discussed in §2.4, one often uses the constrained Hartree-Fock (CHF) theory [17,18] which defines the stable mean-field specified by a set of given expectation values of some constra- ining collective operators. The wave functions of single-particle states in the CHF stable mean-field undergo a change as the magnitude of the expectation values of the constraining collective operators becomes large. Regarding the expectation values as collective variables to describe the collective motion under discussion, we can obtain an information how characteristic features of the single-particle states are varied depending on the amplitude of collective motion. Even though the constraining coll- ective operator has to be phenomenologically put in by hand, the CHF theory gives us a basic idea of the nonlinear dependence of the single- particle states on the collective amplitudes. Since the collective amplitu- de dependence of the single-particle energies are different from state to state, which is well known from the Nilsson model [19], there occur many level-crossings or quasi-crossings in the single-particle states. Concerning to the level-crossing dynamics, recently it has been pointed out by Swiatecki [201 that the order-to-chaos transition of "classical" nucleonic motion is strongly related with the characteristic property of the collective motion. On the basis of extensive numerical simulations with static shape deformations of the mean potential, he has stated: i) If the nucleonic motion inside the nuclear potential is inte- grable, one expects to see strong shell effects in nuclear structure, so that the response of the nucleus to the (collective) shape change should rese- mble that of an elastic solid, ii) If nucleonic motion inside the nucleus is chaotic, one expect smooth, statistical liquid-drop model approach to be of good approximation, so that the response to the (collective) shape change should be that of a viscous fluid, iii) In the intermediate region, an elastoplastic behavior is expected. Swiatecki has also stated as follows: In order to have a chance of making a theory to properly estimate trie nucle- ar system's response to shape change, whether elastic, elastoplastic or chaotic, one need to pay particular attention to the breaking of symmetri- es and to the inclusion of the residual interactions. In conventional appr- oaches, the symmetry of the nuclear potential (specified by shape param- eters) is imposed artificially from the outset and the residual interactions among the single-particle states are put in by hand. Although these appr- oaches may be adequate for static problem, they can not be expected to be a good guide to the dynamical effects. If one intends to explore the order-to-chaos transition dynamics of nucleonic motion, one has to prop- erly take account of the self-consistent dynamical relation between the deformable mean potential and the residual interactions responsible for the single-particle dynamics, by going beyond the adiabatic or diabatic approximations. In order to discuss dynamical aspects of collective motion, one has to use the time-dependent (TD) HF theory. If the characteristic period of the collective motion associated with time variations of the mean-field is much longer than that of the single-particle motion inside the mean-field, one may use the adiabatic (A)TDHF theory [21,22]. The most, simple and physically transparent ATDHF theory is formulated on the basis of the CHF theory and will be discussed in §2.5. The collective potential and collec- tive inertia parameter characterizing the collective motion of the mean- field will be derived by applying the CHF+ATDHF theory. Here, it should be mentioned that the CHF theory defines the most energetically favor- able single Slater-determinant with the given constraint for the collective amplitudes. Consequently, there occur sudden changes in the configura- tions of the occupied single-particle states of the CHF single Slater- determinant at the level crossing points. As will be discussed in §2.5, the inertia parameter of the collective motion obtained by the CHF+ATDHF theory is strongly affected by the level-crossing of the single-particle states. Namely, the properties of collective motion is decisively affected by the single-particle dynamics near the level-crossing region, which can never be treated under the adiabatic assumption, by definition. Thus, we inevitably encounter with exploring the non-linear dynamics between the single-particle and collective modes of motion near the level-crossing region. The dynamical effects of the level-crossing mechanism on the large- amplitude collective motion has been the main issue in the nuclear physics from its early stage. Actually, there have been many works [20,23 -28] on this problem by using the adiabatic or diabatic single-particle potential models, i.e. by means of the single-particle states in the mean- field which parametrically depends on a set of shape variables (with fixed symmetry) put in by hand. In spite of many important works, an essential role of the level-crossing dynamics on the nuclear collective dynamics has been still left unresolved because of a difficulty of going beyond the adia- batic approximation. Moreover, there are no dynamical reason why the collective motion should always be constrained on the most energetically favorable single Slater-determinant states defined by the CHF theory, which is the basic assumption of the ATDHF theory.
1.3. Order-to-Chaos Transition Mechanism in Nuclei in TDHF Manifold The important task in investigating the nonlinear collective dynamics (discussed in the previous subsection) in connection with the classical order-to-chaos transition of nucleonic motions is to go beyond the CHF+ ATDHF theory, and to reformulate the TDHF theory itself in a suitable form so as to be compared with the general theory of classical dynamical theory. The first step toward this purpose is to recognize the following important point: When a set of (time dependent) parameters specifying the TDHF single Slater-determinant is suitably chosen, the basic equation of the TDHF theory can be expressed as a set of canonical equations of motion which is formally equivalent to that of the classical theory of Hamiltonian system [29-32]. As will be discussed in §2.3, we call an expression using this parameter set a canonical-variable representation (of the TDHF theory) and call the parameter space a TDHF [symplectic) manifold. Although the TDHF manifold has no direct correspondence to the classical phase space, this formal relation enables us to develop a classical-level theory of large-amplitude collective motion within the TDHF theory, by exploiting recent fruitful results obtained in the classical dynamical theory [3,4). Namely, when the canonical-variable representa- tion of the TDHF theory is applied to the finite, quantum many-body syst- em, it is possible to explore its nonlinear dynamics in connection with the classical order-to-chaos transition mechanism. This gives us a sound basis for discussing the "classical-level" order-to-chaos transition dynam- ics in nuclear physics by using various experimental evidences on individ- ual quantized single-particle states. The second step to develop the nonlinear collective dynamics (in con- nection with the classical order-to-chaos transitions of nucleonic motions) is strongly related to the essential problem of how to properly specify the collective and intrinsic variables in the TDHF manifold, and the residual interaction between them in the most optimum way. In the conventional nonlinear dynamical theory, one starts with the Hamiltonian which is divided into two parts from the outset, an unperturbed integral part and the other non-integrable interaction part. In the self-organizing and self-sustained system as the nucleus, however, one does not know in advance the integrable part of the Hamiltonian. Since there exist many stable mean-fields in the nucleus and there are many collective states excited from various mean-fields, it is naturally expected that a division of the Hamiltonian into an unperturbed integrable part and interaction part is not unique, and depends on a specific collective motion under consid- eration. Therefore, we are inevitably forced to face a difficult problem of defining the optimum collective variables as well as the optimum unper- turbed collective Hamiltonian in the TDHF manifold. This problem forms a cornerstone in developing the nuclear collective dynamics and will be discussed in §3. We first discuss a general method called the self-consis- tent collective coordinate (SCC) method [33], which intends to extract an optimum collective submanifold for a given trajectory representing the collective motion. The collective submanifold is nothing but an approxi- mate integral surface of the collective trajectory, and the boundary condi- tion specifying the collective submanifold is derived from the trajectory under discussion. The optimum collective variables are defined so as to characterize the collective submanifold, and the optimum collective Hamiltonian is given on the approximate integral surface. Once the opti- mum collective variables are determined, we can precisely define intrin- sic variables in a way compatible with the SCC method. Thus, the whole nuclear dynamics is optimally described, without depending on the adia- batic assumption from the outset, in terms of a set of canonical variables consisting of the collective and intrinsic variables. This set is called a dynamical canonical-coordinate (DCC) system [34] in the TDHF manifold. The Hamiltonian in the DCC system provides us with a starting point to explore the nonlinear dynamics between the collective and single-particle modes of motion. It will be shown in §3.2 how the SCC method overcomes an essential defects of the CHF+ATDHF theory. §§3.3 and 3.4 will be devoted for the discussion how to analyze an inexhaustible structure of the TDHF manifold in terms of the SCC method, by making reference to recent enormous progress of the order-to-chaos transitions in the classical dynamical the- ory.
1.4. Quantum Order-to-Chaos Transition Mechanism in Nuclei As is well known from the nonlinear classical dynamical theory [3,4], the phase space trajectories exhibit an inexhaustible structure even in a simple system with only two degrees of freedom. A vital point in explor- ing the classical chaos has been to make clear why the phase space exhib- its an exceeding complexity of topology. A dynamical explanation for both an infinite series of island structure of the regular motion and an appear- ance of instability zone called a stochastic sea o( the chaotic motion has been one of the major problems in the theory of nonlinear oscillations. It is known [35-38] that the island structure is related to an existence of regular motion described by the action variables alone, and the resonant denominators play a decisive role in generating an inexhaustible structure of the phase space. From the previous discussion in §1.3, it is quite natural tc expect that the same extraordinary rich structure has to exist in the TDHF manifold describing the TDHF trajectories, and the resonant denominators play an essential role in exploring the single-particle dynamics near the level- crossing region associated with the large-amplitude collective motion. Since the TDHF manifold is defined for the finite quantum many-body sys- tem, we can further expect that there may be important concepts in the Hilbert space of the quantum system, which may precisely correspond to the concepts of integrability and chaoticity in the TDHF manifold. Thus, a fascinating investigation on quantum analogue of the inexhaustible struc- ture of the classical-level TDHF manifold is becoming a necessary and inevitable subject in understanding a variety of excited states in the Hilbert space of the finite, self-organizing quantum system as the nucleus. This subject will be discussed in §§4 and 5. An important clue toward this aim is found in the fact that the canonical-variable representation of the TDHF theory (defining the TDHF manifold) precisely corresponds to a c-number version of the (full quantum) boson expansion theory [39-43] for even number many-fermion systems. This clear-cut correspondence provides us with a unique way to construct a full quantum theory of many- body problem [44], which is capable of directly exploring the extraordi- nary rich structure of the Hilbert space in a parallel way to what has been
11 - developed in the classical-level theory in the TDHF manifold. The concepts of quantum "integrability" and quantum "chaoticity" may be then clarified as the quantum correspondents of those concepts discussed in the classical-level theory.
- 12 - 2. Single-Particle Dynamics of Nuclear System
2.1. Single-Particle Algebra According to the steady development of the nuclear structure theory, an importance of the single-particle dynamics is being widely recognized in exploring nonlinear dynamics of the self-sustained and self-organizing finite many-body quantum system like the nucleus. Consequently, in this section, we shortly review some important theoretical background of the nuclear structure theory related to the single-particle dynamics.
Let ( lg lg (J)a(x) = IDap(g)-9p(g;x) . where Dap(g) = (e )ap , D^(g) = (e- )ap , satisfying ag)D+(g) = Df(g)D(g) = l . (2.1) The unitary transformation in Eq.(2.1) between the two representa- tions is also expressed as a transformation between two sets of creation and annihilation operators with the aid of one-body unitary operator defi- ned by U(g) = exp{iG(g)} . where G(g)=Igapc^cp . (2.2) ap Since g is the hermitian matrix satisfying g=gt by definition, there holds the relation G(g) = Gt(g) and Ut(g) = U'1(g) . (2.3) As is easily proved by using the anticommutation relations between fer- mion operators, the two kinds of single - particle operators (c^,ca} and cp(g)) are related with each other through 5pag 1 ca(g) = U(g)caU(gr = I£pDja(g). (2.4) which expresses a transformation identical with Eq.(2.1). The unitary transformation defined within the space spanned by the whole single-particle states [§a) is also represented in another way by dividing them into occupied- (hole-) states and unoccupied- (particle-) states. Since this division is achieved after having introduced a stable mean-field satisfying the Hartree-Fock (HF) condition, let us discuss essential points of the HF theory. 2.2 Hartree-Fock Theory With the representation { H = HI+HII. H! = Z Topc^cp . Hn=i S Vap 5c^c5cY . (2.5) A stable mean-field for the N-particle system described by the Hamilto- nian in Eq.(2.5) is determined by the HF theory. It defines the HF state lgo> which is expressed by the single Slater-determinant wave function given as |go)= Jlc^gollO) • where ca(go)|O) = O , (2.6) a=l satisfying the HF condition. What should be determined by the HF condi- tion are specific values of the matrix elements of hermitian matrix go. The HF condition is expressed as - I I - 5{g[H|g) = 0 . (2.7) Referring to the state lgo> defined by the HF condition (2.7), one may introduce the particle- and hole-operators through aj(go), N+l£a = jJ.S b,(g0), Ha = ISN = b1(go)|go) = O. (2.8) Here and hereafter, subscripts (i,v denote the particle-states whereas i,j denote the hole-states. The capital letters M and N express the number of particle- and hole-states for the system under discussion, respectively. With the aid of the unitary transformation in Eq.(2.2), one may introduce another single Slater-determinant state I s • U(go)| In the same way as Eq.(2.8), a new set of particle- and hole-operators for the state I d>o> are introduced by a*. 6,. (2.10) which is derived from Eq.(2.8) by means of Eqs.(2.4) and (2.9). Here it should be mentioned that the state IOo> does not necessarily satisfy the HF condition. A relation between the two single Slater-determinant states Kt>0 > and fg0 > is specified by U(go) which just describes the relation between two sets of single - particle states (c£,ca) and (c^(go) .ca(go))- When the particle-hole representation in Eq.(2.10) is employed, the the state Igo > obtained by operating U(g) on |<&0) is described by using much - l j - simpler operator U{k) whose generator F(k) containes only particle - hole (ph-) and hp-components [33]. Namely , F(k) = (2.11) Here the notation I g> <=> I k> means that the states I g> and I k> are equivalent with each other except for the phase factor which has no physical relevance. The (N+M)x(N+M) hermitian matrix k in Eq.(2.11) is expressed by using an NxM sub-matrix K expressing ph-arnplitudes as 0, K k = (2.12) K\ 0 Since there holds Eq.(2.11), the unitary matrix D(g) in Eqs.(2.1) and (2.4) can be expressed by the matrix k as = elg=eUcsD(k) . (2.13) By using Eq.(2.12), the unitary matrix D(k) is expressed as cosy KKT, iK D(k) = (2.14) As is easily recognized, the single Slater-determinants lg>and lk> are characterized by the one-body density matrix which is expressed by the unitary matrix D(k) as N (2.15) - 16- which has no effects caused by the phase factor. By using Eq.(2.14), the unitary transformation in Eq.(2.4) and its inverse relation are given by t b,(k) = lbJfVl-C c') - +ZbJ(k)C|(i J J Jh t > bi =Zbj(k/Vl-C C ) +Za^(k)CVJ , (2.17) j V J}1 v where matrices C and C* are defined by (218) Equations (2.16) and (2.17) demonstrate another representation of the unitary transformation by means of the new variables C and Ct. Since the new variables C and Ct will turn out in Eq.(2.34) to be the canonical vari- ables, this representation has special dynamical meaning and is called a canonical-variable representation. As will be discussed in §2.3, dynamical aspects of the unitary transformation D(g) between two complete sets of single-particle states are simply explored by using this representation. Let us consider the varlational equation (2.7) with the use of the new variables C and Ct. For this purpose, we introduce a set of infinitesimal one-body operators defined by (2.19) - 17 - As is justified by using Eq.(2.18), expectation values of these operators with respect to the state lk> satisfy the following relations. (k|6J1(k)|k) = (kj ac,, dK ] fCKf J + sin KK ac,, JKKJ (2.20) Taking derivatives for both sides of Eq.(2.20) with respect to CVj and C*vj, and subtracting one from the other, we get 6^(k), 6VJ(k)] |k) = o . (2.21) Equation (2.21) means that the infinitesimal operators defined in Eq. (2.19) satisfy weak-boson commutation relations at any value of K^i, K*^. With the aid of the infinitesimal operators, the HF condition (2.7) defin- ing the HF state I ko> is expressed as - (k|[H,6VJ(k)]|k) = 0 ,V(vj) (2.22) k = k0 - 18 - As is well known, the HF condition (2.7) is conventionally rewritten in a form of determining the unitary matrix D(ko) in Eq.(2.14) as DaY(k0) = EY(k0) D^lko) , (2.23) which is called the HF equation. Equation (2.23) is a nonlinear eigenvalue equation because the HF potential f(ko) is expressed by the resultant eigenvectors (D^ko); a=l,...,N+M) as aYp55Yo ; ppa(k0)= Z Dpi(k0)D[a(k0) . (2.24) This nonlinear relation between the HF mean-field T+f(ko) and Day(ko) represents a self-consistent relation which should be satisfied by the self- sustained and self-organizing system. Here the eigenvalue Ey(ko) means the single - particle energy of c*(k0). In Eq.(2.24) i runs the hole-states which are determined by the first N lowest energy eigenstates of Eq.(2.23). Even though the condition (2.22) does not necessarily means to regard the first N lowest eigenstates as the hole-states, it is reasonable to do so because the HF theory intends to define the most energetically stable mean-field. If one take another set of eigenstates as the hole states, one gets another local minimum. 2.3. Symplectic Structure of the TDHF-Manifold As a preliminary for discussing the "classical-level" theory of the nuclear collective dynamics, we start with recapitulating the structure of the time-dependent Hartree-Fock (TDHF) theory [29-32]. The basic equation of the TDHF theory is given by i^l4(t)>=0 . (2.25) at Here and hereafter, we use a convention h = l. In Eq. (2.25), l4/(t)> denotes a time-dependent single Slater-determinant state given by {mjp| ^PjJ (2.26) where I kQ> is the HF stationary state determined in the previous sub- section and d^ and pj mean the particle - and hole - operators with respect to lko>; 8(ko|H|ko) = O, and dj saj(k0) . pf sbj(ko) . (2.27) In the same way as we introduced the new variables C^t instead of K^i, we may introduce C^t) instead of JC^t) by (228) By means of {C^.C^i}, the TDHF equation (2.25) is expressed as l\iC J- + iC^^r -H|fe) = O , |fc) ^ |W)) . (2.29) H,i[ ^JU oC^J Let us introduce the following one-body infinitesimal operators (2.30) In the same way as Eqs.(2.20) and (2.21), these infinitesimal operators satisfy the following relations; <*.| [d^tfe). dj(«] |«.) = S^SJJ . («,| [d^w. dVJ(fe)] |&) = o . (2.3i) With the aid of Eq.(2.30), the TDHF equation (2.29) is expressed as H|fc) = O . (2.32) Taking the variations in Eq.(2.32) with respect to Cyj* and Cvj for \k>, we get - 20- H.iL By using the "weak" boson-like commutation relations in Eq.(2.31), Eq. (2.33) reduces to a set of classical Hamilton's equations of motion where H =H[C ,C*)s< &IHI& > and the symbol {A, B)P B denotes the Poisson bracket defined as. Equation (2.34) clearly shows the symplectic structure of the TDHF mani- fold M2MN : {£ ; 2.4. Constrained Hartree-Fock Theory The HF theory discussed in §2.1 defines the local stationary point with the lowest energy. In order to get global information of the mean-field, e.g. an increase of the potential energy of the system induced by the deformation of the mean-field, it is convenient to apply the CHF theory. In order to discuss the CHF theory as simple as possible, let us take the mass quadrupole operator as a constraining operator. 0 2 Q2M=0 = Xq^ ^ • q3Mdn£(?)r Y2M(e.9)*pff) . (2-36) ccp which is suggested by the phenomenological liquid drop model. This means that we are considering the mean-field which is deformable towa- rd the quadrupole direction alone and is stationary toward the other directions perpendicular to the quadrupole deformation. The basic equa- tions of the CHF theory are given by -21 - 5(k(q)|H'|k(q)}|k=k()=0. H'=-H4Q20 . (2.37) with a constraint condition (ko(q)|Q2o|ko(q)) = q • (2.38) Here H' is called an auxiliary Hamiltonian of the CHF theory and X denotes the Lagrange multiplier. In the CHF theory, the single Slater-determinant state lk> in Eq.(2.11) is denoted by lk(q)>, because the CHF state lko(q)> satisfying the stationary condition (2.37) is defined for each specified expectation value q. By taking variations with respect to the canonical variables C,ii and C^i*. Eq.(2.37) is expressed as [fl'. 6^(q»] |k(q))|k(q) = ko(q) = 0 . V(vj) . (2.39) Determination of the CHF state lko(q)> is equivalent to defining the unitary transformation Dap(ko(q)) in Eq.(2.4). The CHF condition (2.37) is conventionary expressed in a form of the eigenvalue equation given by {2A0) where the HF potential rap(ko(q)) is given by using the resultant eigen- vectors {DaY(ko(q)); a=l N+M) as rap(k0(q)) = Pp«(kO(q)) = I Dp^kotqHD^lkofq)) . (2.41) In Eq.(2.40), Ey(ko(q)) denotes the single-particle energy corresponding to the creation operator cj(ko(q))- The q - dependence of the single - particle energy Ey(ko(q)) is well known from the Nilsson model [19] and is very- similar to the nonlinear oscillation of a pendulum [4J. Since the eigen- value equation (2.40) contains the resultant eigenvectors from the begin- ning. Eq.(2.40) is a nonlinear eigenvalue equation which should be solved self-consistently under the restriction of satisfying the constraint condition (2.38), i.e. q = Zq^R • Z DmfkoCqMDjjtkotq)) • (2.42) ap p 1=1 An increase of the mean-field energy caused by the defomiation may be regarded as the collective potential energy V(q) for the quadrupole deformation of the mean-field, which is £iven by V(q)s(ko(q))|H|ko{q))>-(ko(q = 0))|H|k0(q = 0)i) , (2.43) where lko(q=O)> is the HF state defined by Eq.(2.22) without any const- raint condition. Here it should be emphasized that the density matrix pap(ko(q)) is defined in such a self-consistent way that the first N lowest energy single- particle eigenfunctions of Eq.(2.40) construct the single Slater-determi- nant state lko(q)>. Namely, the configuration of the occupied states is defined at each value of q under a requirement of getting the most ener- getically favorable single Slater-determinant state. Consequently, there may occur some singularity in |5ap(ko(q)). provided there exists a level- crossing where a sudden change of the configuration in defining the hole states takes place. This singularity will be further discussed in §2.5. For making clear the physical meaning of the CHF theory, let us introduce a collective momentum operator ie satisfying i^-lk(q)>=P(q)lk(q)i^ > . (2.44) oq oq Taking derivatives in both sides of Eq.(2.38) with respect to q, we get a weak canonical condition (k(q)) H*o| e^ [Q2M=o • = k0 (2.45) With the use of the collective operators Q20 and P(q). the variational equation (2.39) is expressed as H'. Q20] |k(q)} k(q) = ko(q) = {k(q)|[H,920 = 0 (2.46a) k(q) = ko(q) = (k(q)|[H.P(q)]|k(q)) (2.46b) k(q) = ko(q) k(q) = ko(q) = 0 (2.46c) Here 5jjq) collectivelly represents ncn - collective infinitesimal operators sati- sfying | [5j.(q) , Q20] | | [Sj.(q) , P(q)J (2.47) and its explicit expression in terms of 6^(k(q)) and 6^(k(q)) is easily obtained by using the orthogonality relation (2.21). According to the relations (2.45) and (2.47), the auxiliary Hamiltonian H' in Eqs.(2.46a) and (2.46c) is replaced by the original Hamiltonian H. Consequently, the variational equa - tions (2.46a) and (2.46c) toward the directions perpendicular to the coll- ective direction give the same condition as the HF condition (2.22). Taking derivatives in both sides of Eq.(2.43) with respect to q, we get (2.48) k(q) = ko(q) 2 1 From Eqs.(2.46b) and (2.48), we get an obvious physical meaning of the Lagrange multiplier k as (2.49) 2.5. Adiabatlc Time-Dependent Hartree-Fock Theory In the previous subsection, we have discussed how the self-sustained system acquires the potential energy when it deforms toward the direction indicated by the phenomenological model. However, a time- evolutional path of the self-organizing system has not been discussed. Before discussing a theory which extracts the dynamical path selected by the self-organizing system, it is quite instructive to start with discussing how collective motion along the collective path defined by the CHF theory is described. This description provides us with the first understanding of the nonlinear dynamics between the single-particle and collective modes of motion [21,22]. The single Slater-determinant wave function describing the collective motion with momentum p along the collective path specified by the CHF theory is expressed as |p. q)sexp{-ip92o] |ko(q)) • (2-50) where lko(q)> is defined by the CHF equation (2.37). The time evolution of the state I p,q> is described by the following variational equation; 8{p.q|(H-iA5|p,q) = o . (2.51) Since the time dependence of lp,q> is supposed to be described by the collective variables p and q alone, Eq.(2.51) is rewritten as S(p.q|{H-i(pi- + qiL}|p,q) = o . (2.52) dp dq Here, let us assume that the collective motion is slow enough so that the perturbative expansion of Eq.(2.52) with respect to p is meaningful (adiabatic assumption). By using Eqs.{2.44) and (2.50), and introducing the adiabatic assumption, Eq.(2.52) is reduced to S(ko(q)| H |ko(q)) = 0. (2.53) By using the expression in Eq.(2.46), the variations in Eq.(2.53) toward the collective directions are given by = (ko(q)| P(q) , |ko(q)) . f2 Q20, . (2.54) According to the time-reversal invariance property of the Hamiltonian, Eq.(2.54) results in 2 p = (ko(q)| [P(q) . {H~P [[H.92O].92O]}] |ko(q)> . k q = p(ko(q)| [920 • [H.Q20]] ( o(q)) • (2.55) Equation (2.55) is the basic equation of adiabatic (A-) TDHF theory based on the CHF theory. In order to get physical meaning of Eq.(2.55), let us define the collective Hamiltonian through H(p.q) = (p.q|H|p.q)- (p = O.q = Oj H |p = O.q = 0) • 2 2 = (ko(q)| H + ip[H.92oj + |j-p [[H.92o].92o] |kO(q)> = 0)|H|ko(q = 0)) 2 = (ko(q)| H-ip [[H,92o].92o] |kO(q)> — •>![ — -(ko(q = O)|H|ko(q = 0)). (2.56) With the aid of the collective potential V(q) defined in Eq.(2.43) and an inertia parameter defined by M-^q) S the collective Hamiltonian is finally expressed as which gives a microscopic foundation of the phenomenological Bohr- Mottelson Hamiltonian. Since there holds ko(q)) , ]}] |ko(q)> , (2.59) which is derived from Eq.(2.56), the basic equation (2.55) of ATDHF theory is reduced into the classical collective equation of motion (2.60) dp dq Aiming at showing the dynamical relation between the collective motion organized by Eq.(2.60) and the single-particle dynamics, it is instructive to discuss property of the inertia parameter. The inertia parameter defined in Eq.(2.57) is known to be expressed in a simple form where Mq)> = aJ(ko(q))bI(ko(q)]|ko(q)) . (2.61) provided effects coming from the residual interaction are negligibly small. Equation (2.61) is known as Inglis formula [22, 45-47]. From the denom- inator of Eq.(2.61), it is easily recognized that the inertia parameter M(q) becomes large in a case where the single-particle energies of particle(n) and hole(i) states degenerate with each other, i.e. the level-crossing point. From the numerator of Eq.(2.61), it is also seen that the inertia parameter M(q) takes a quite large value when the single-particle wave function strongly depends on q near the quasi level-crossing point. Therefore, the properties of the collective mass parameter strongly depends on the level-crossing and quasi level-crossing in the single- particle states An essential limitation of the ATDHF theory rests on its assumption that the collective motion should be slow enough in comparison with the single-particle motion. The most energetically favorable stable mean-field determined by the CHF theory {under a specific constraining condition for the collective motion) accompanies sudden changes of the single- particle configurations when the level-crossing of the single-particle states takes place. Because of the adiabatic assumption, however, one can never investigate the microscopic single-particle dynamics near the level- crossing regions associated with the large-amplitude collective motion, which is essentially outside of the scope of the CHF+ATDHF theory. As far as one sticks to the CHF+ATDHF theory, therefore, one can not obtain any dynamical information why the system under consideration should move along the valley of the collective potential V(q) in Eq.(2.43), and why there occurs a sudden change in the single-particle configurations. Another drawback of the ATDHF theory in describing the large-ampli- tude collective motion of the self-organizing system is its artificial intro- duction of the collective coordinates. The collective motion of the self- sustained system could not necessarily be confined within the constrain- ed phase space (p,q}, i.e. the single Slater-determinant states lp,q>, be- cause a finite system could not solely develop toward the large-amplitude region of the specific direction, say the direction specified by some collective operator Q, without inducing any effects on the other degrees of freedom. A range of an applicability of the ATDHF method will be dis- cussed in §3.2. In spite of the above stated defects and limitations, the ATDHF theory has clearly shown a decisive relation between collective and single-parti- cle dynamics through Eq.(2.61). 3. Classical-Level Theory of Large-Amplitude Collective Motion 3.1. Self-Consistent Collective Coordinate Method In §2.3, we have seen that the TDHF equation (2.25) is formally equivalent to the classical canonical equations of motion in Eq.(2.34) with MN-degrees of freedom. Based on this fact, it is easily expected to devel- op a "classical-level" nuclear collective dynamics under a guidance of what has been developed in the general theory of classical dynamical systems. As discussed in §2, however, an essential difficulty of describing the large-amplitude collective motion in a self-organizing system as the nucleus lies in how to properly define the collective coordinates deter- mined by the system itself, because the self-sustained system selects by itself its evolutional path following its microscopic equations of motion. In other words one does not know in advance the unperturbed integrable Hamiltonian of the self-organizing system, in contrast with a case of cou- pled nonlinear oscillators whose unperturbed integrable Hamiltonian is known from the beginning. Applying the static HF theory where the nonlinearity specific for the finite and self-sustained system is properly taken into account, one gets an integrable HF Hamiltonian where the single-particle excitation modes are decoupled with one another. Namely, the single-particle motion within the stable mean-field is regarded as an integrable motion. Howev- er, the HF theory only gives local information valid for a tiny region near the HF local minimum. In order to get a coordinate system which is ca- pable of describing time variation of the mean-field, and is valid in a fairly wide area of the TDHF manifold, one has to extend a basic idea of the static HF or CHF theories to a case of time-dependent problem. Aiming at exploring the nonlinear dynamics between the time dependent mean- field and the nucleonic motion inside the mean-field, one has to start with finding out the most optimum coordinate system of the TDHF mani- fold, where the collective degrees of freedom are approximately decou- pled from the rest. Here, it should be mentioned that there have been made many efforts [48-54] to dynamically determine the proper collec- tive variables within the ATDHF theory. However, it is not easy task to extract the optimal collective variables within the "coordinate space" under the adiabatic approximation. In what follows in this section, we discuss within the TDHF symplectic manifold how to dynamically define Representative Trajectory a) Collective Regime b) Chaotic Regime Figure 3.1. Schematic illustration representing collective and chaotic behavior of trajectories in the TDHF manifold M2K_ sheet in a) repre- sents the collective submanifold 2 the optimum coordinate system where the collective and single-particle degrees of freedom are decoupled maximally. For the dynamical system organized by Eq.(2.34), one may imagine two extreme cases. If an ensemble of TDHF trajectories with slightly differ- ent initial conditions develops in almost the same way, forming a bundle, the system is classified to be in collective regime as shown in Fig.3.1a. In this case, one could introduce a representative trajectory of the bundle as well as a collective submanifold I2L, i.e. the integral surface of the representative trajectory, to which many trajectories are accumulating. On the other hand, the system is classified to be in chaotic regime, if many trajectories with almost the same initial conditions evolve quite differently, having lost their initial focussing (Fig.3.1b). In the latter case, one could not introduce any submanifold within the TDHF manifold M2MN. From the above consideration, one may expect that transition mechanism between the "regular" and "chaotic" motions may be studied by, at first, extracting the collective submanifold Z2L out of M2MN. In the collective regime where many trajectories are accumulating on a certain submanifold I2L, there must be an optimal coordinate system called a dynamical canonical coordinate (DCC) system [34] where the minimum number of relevant (collective) variables denoted as {^a.^a*: a=l ...,L«MN} is required in describing Z2L. The remaining canonical vari- : ables (^H, ^a* a= L+l MN) in the DCC system are called the irrelevant (intrinsic) variables. The coordinate system [C^i.C^i*] used in describing Eq.(2.34) is called an initial canonical coordinate (ICC) system, because this coordinate system has nothing to do with a specific correlation in the Hamiltonian H{C,C*). The extraction of 22L is achieved by determining a canonical transfor- mation between the ICC and DCC systems. Since the transformation is canonical, the local infinitesimal generators defined by OLD have to satisfy the following canonical variable conditions with the same form as Eq.{2.31); i.e., a=L+1 K> {3-3) where we have used Eq.(2.31). In Eq.(3.3) and hereafter, the double indexes (ni) specifying the particle-hole states are simply expressed by the single indexes (j)- The total number of particle-hole states MN is also simply expressed as K. In Eqs.(3.2) and (3.3). S denotes a real generating function of the canonical transformation between the ICC and DCC sys- - :s i - • tems. According to the canonical transformation in Eq.(3.3). the equa- tions of motion in the DCC system are given by ^ for a = v (3.4) In order to fix the transformation, it is decisive to introduce the fol- lowing expansion scheme: Since faa-fTa) are supposed to describe large- 2L amplitude collective motions inside S , whereas (^a,^*a) describe small fluctuations around 22L, it is reasonable to introduce a power series ex- pansion with respect to the irrelevant variables. For example. (3.5a) T Figure 3.2. Diffeomorphic mapping defining the collective submanifold Z2L in the TDHF manifold M2K. H = (3.5b) s = (3.5c) where the symbol [g] for any function g(ii,T|*;^,^*) denotes a function on the expansion surface Z2L. i.e. [g] = g(r|,ii*;^=O,^*=O). In Eq.(3.5), n denot- es a power of the irrelevant variables contained in each term. Since the lowest order Hamiltonian #(°) = \H] contains only the relevant variables, it is identified to be the collective Hamiltonian Hcon= Hl°l Now, let us define I2L within the TDHF manifold M2K. This task is achieved by determining the functional forms of ([Cj],[C*j)} with respect to (ria.'H'a}- Then, we have the following diffeomorphic mapping after getting {[Cj],[C*j]}; Diffeotnorphic Tlapplruj ; embedded, in M2K, (3.6) which defines I2L in M2K as is shown in Fig.3.2. According to the self- consistent collective coordinate (SCC) method [33], the basic equations for defining I2L within M2K are given by the following two requirements; R.I: There should hold the lowest order canonical variable condition; m (3.7) which is derived from Eq.(3.3). Condition (3.7) certifies that the relevant variables {ria.i"l*a} should be canonical. R.2: There should hold the equation of collective submanifold; (II) (3.8) dC Equation (3.8) is originally derived under the name of "the invariance principle of the time-dependent Schrodinger equation"[55]. The physi- cal meaning of Eq.(3.8) is obtained by the following consideration: As is discussed in the beginning of this subsection, the representative trajec- tory of the bundle in Fig.3.1a should satisfy both the equation of motion in the ICC system given by i[Cjl = [dtt /3Cj*J , and C.C. (3.9a) and the equation of motion in the DCC system given by ifia =d[H]/ch\a , and C.C. . (3.9b) These equations are obtained from Eqs.(2.34) and (3.4), respectively, by using the perturbative expansion in Eq.(3.5), and contain the symbol [ ] because the representative trajectory should always be travelling on E2L. Eliminating the time derivative terms in Eq.(3.9a) by means of Eq.(3.9b), we get a time independent equation (3.8), which is suitable for prescrib- ing the integral surface Z2L of the representative trajectory and is called the equation of collective submanifold. Equations (3.7) and (3.8) are solved perturbatively by expressing [Cj] and [C*j] in power series expansion with respect to ri and TJ*. W,]= I C,(r,s)n*rTf • (3.10) r,s r+s>l This expansion method is called the TJ,TJ'-expansion method. An impor- tant task in this expansion method is to set up a "boundary condition of Z2L" which is derived from the initial condition of the representative trajectory under discussion. An explicit method of getting the diffeo- morphic mapping is found elsewhere [33]. 3.2. Self-Consistent Constrained Hartree-Fock Theory In the previous subsection, we have discussed how to extract the inte- gral surface Z2L of the representative trajectory out of the TDHF manifold M2K. For making clear what has been done by the SCC method in terms of the single-particle dynamics, we start with reformulating the SCC - :u - method within the original form of the TDHF theory. This reformulation clearly shows that the SCC method just corresponds to a self-consistent constrained Hartree-Fock theory, where the constraining operators are dynamically defined by following the microscopic dynamics of the system itself. In marked contrast with the conventional CHF theory where the constraining operators are put in by hand, the SCC method intends to dynamically extract the constraining operators whose instantaneous di- rections just coincide with the "phase-space vector" of the representative trajectory in the TDHF manifold. In this sense, the SCC method reformu- lated in terms of the single-particle dynamics provides us with a theoret- ical foundation for specifying the single-particle states in deformable potentials (such as in the Nilsson model, the two-center shell model and Routhian in the rotating shell model) without introducing any adiabatic assumptions for the collective motion. By using the canonical variables Cj and C*j, the TDHF equation (2.25) is expressed as i~ - H mC,C )) = 0 . (3.11) at J I / The equation of motion of the representative trajectory is described by - H| \v{[C].[c'l)) . (3.12) In Eq.(3.12), the canonical variables Cj and C*j contained in the single Slater-determinant state IH/([C],[C*])> are sandwiched by the symbol [ ], because the representative trajectory is always bounded on the collective submanifold L2L. Since the time dependence of [Cj] and [C*j] is supposed to be described by the collective variables {T|a. T|*a) alone, we get the last expression of Eq.(3.12). By using the notation in Eq.{2.26), the state in Eq.(3.12) is expressed as {+f ^J (3.13) Here we introduce the following one-body collective operators defined by (3.14) Substituting Eq.(3.14) for Eq.(3.12), we get *])| fi ! \') O. (3.15) Now, let us discuss how to express the canonical variable condition (3.7) in terms of the single-particle dynamics. The collective variables T|a. and r|a* are certified to be canonical, provided there hold the following relations; (3-16) Equation (3.16) represents the canonical variable condition for the colle- ctive variables and is equivalent to Eq.(3.7). From Eq.(3.16). we derive the following "weak" boson-like commutation relation (3.17) Here it should be noticed that these relations in Eq.(3.17) hold at any 2L value of r|a and r\a* on E , whereas the conventional RPA modes satisfy the boson-like commutation relation at only one point of the starting sta- tionary HF state IVo >• In marked contrast with the conventional RPA modes, the collective operators defined in Eq.(3.14) contain the particle- particle and hole-hole operators with respect to the state IVF([C],[C*])> in addition to the particle-hole operators (see Eq.(3.32)). This desirable property of the collective operators assures that {OI(T\,T\'), C>a(rj,ri*)J are global operators, whereas the RPA modes are local operators. Namely, an exis- tence of particle-particle and hole-hole terms makes it possible to describe structual change of the collective operators (Oa(T|,Ti*}, da[x\,j\')} depen- ding on the amplitudes (TJ, TJ*). Taking variations of I¥([£],[£*])> in Eq.(3.15) with respect to r\b and Tib*, we get t g UC*])| [Ob(T],T1')/il{f,a(?a (i1,Ti*)-fi;£Ja(T1,Ti )}- fill (3.18) With the use of Eq.(3.17), Eq.(3.18) is reduced to the collective equations of motion in the classical-level. if|a = Mi (3.19) which describe the representative trajectory of the bundle on the collec- tive submanifold X2k Substituting Eq.(3.19) for Eq.(3.15), we get (3.20) which represents the equation oj collective submanifold in the language of the single-particle dynamics and is equivalent to Eq.(3.8). By defini- tion, the variations in Eq.(3.20) toward the collective directions are identically zero. When one takes variations toward the irrelevant direc- tions perpendicular to collective directions, one has H = 0 (3.21) -37 - In Eq.(3.21), a symbol 5j_ means variations perpendicular to the collective directions. If we express Eq.(3.21) in terms of the classical-level theory in §3.1, it is easily shown (56] that Eq.(3.21) Just corresponds to the maximal-decoupling condition given by From Eq.(3.22), it is clearly seen that the collective submanifold E2L is defined so as to satisfy the stationary condition with respect to the varia- tions towards the irrelevant directions. At this point, it should be remembered that the HF state just corre- sponds to the stationary point of the total Hamiltonian (potential energy) in the K-dimensional coordinate space: (Re(Cj), Re(Cj*)}, satisfying the condition 9H(Re(C), Re(C')) _ 3Re(C) " ' ( 3) whereas the collective submanifold E2L corresponds to the stationary surface of the total Hamiltonian in the 2K dimensional phase space M2K :{Cj, e*j}. In this sense, the SCC method is a natural extension of the HF theory from the stationary point to the stationary 2L-dimensional surface E2L Here, it is worthwhile mentioning that the conventional CHF theory satisfies the stationary condition with respect to variations toward the di- rections perpendicular to those specified by the given constraining oper- ators. It is also the case in Eq.(3.21). Namely, Eqs.(3.16) and (3.20) are just a generalization of the CHF equation given by Eqs.(2.37) and (2.38). The first important generalization rests on the fact that the CHF theory contains the constraining operator Q with fixed microscopic structure put in by hand, whereas the SCC equation (3.20) includes many pairs of con- straining operators [Oa,ol; a = l,---.L) which describe the collective directions chosen by the self-organizing system itself and change their microscopic structure depending on the amplitudes of collective motion. Physical meaning of this generalization is also easily understood by the following fact: The Lagrange multiplier in Eq.(2.37) is expressed by the static quantity, i.e. the derivative of the collective potential \=-dV(q)/dq (see Eq.(2.49)), whereas the coefficients appearing in front of the collective operators {Oa.dl) in Eq.(3.20) are nothing but a "phase-space vector" [d[H]/ [6a +61)/ ~J2 and collective momenta i(0a -6\)l V2 as constraining operators, whereas Eq.(2.37) in the CHF theory has only one type of constraining operator, i.e. the collective coordinate Q2M- This is the second generaliza- tion of the SCC method. Since the CHF theory gives an information only on the collective potential V(q). as we have seen in §2, the time evolution of collective motion should be treated by introducing another framework, e.g. the ATDHF theory on top of the CHF theory. On the other hand, the SCC method is capable of treating the time evolution of collective motion within the single framework without introducing any adiabatic assump- tion. In the next subsection, we will discuss a dynamical condition on which an adiabatic assumption of the collective motion, or more generally a sep- arate treatment between the collective and single-particle modes of mo- tion is Justified. For a pedagogical aid on this problem, we shortly discuss a criterion which givov, a range of validity to use the CHF-ATDHF. Since the collective coordinate Q2M *s introduced by hand, there is no dynamical reason why one may treat the collective motion within the constraining collective phase space (p,q} spanned by the single Slater-determinant wave function lp,q> in Eq.(2.50), forgetting about effects coming from the other degrees of freedom. In the ATDHF theory, the basic equation (2.54) is derived from Eq.(2.53) by taking variations only toward the dire- ctions P(q) and Q2M- ASi s obvious from the above derivation of the ATDHF equation, the separate treatment of collective motion by Eq.(2.54) is jus- tified insofar as the variational equation (2.53) toward the non-collective directions perpendicular to the collective ones is also satisfied; Here 8j_(q) collectively represents the non-collective infinitesimal operators satisfying Eq.(2.47). Equation (3.24) just gives a criterion of introducing an adiabatic assumption and of using the CHF+ ATDHF theory. 3.3. Stability and Separability Conditions of Collective Submanifold The main purpose of this subsection is to discuss an applicability of the SCC method [56,57]. Since the SCC method intends to extract the collective submanifold Z2L for a given representative trajectory, the bor- der of applicability is directly related to a characteristic feature of the representative trajectory under discussion. In order to make clear the range of applicability of the SCC method, let us start with introducing a new set of particle operators with respect to IXF([C].[C*])>. With the aid of the unitary operator in Eq.(3.13). one may define a new operator i\r! iiff / d*(Ti.Tfd* f)) = e I Jc^e I = zeiDcjDRrtp(Ti.ri*a j (3.25) In the same way as we have derived Eq.{2.40) from Eq.(2.37), Eq.(3.20) is expressed as a nonlinear eigenvalue equation for determining the unitary matrix Dpa("n,T)*) in Eq.(3.25) as • DpY(rt,Ti*) = EY(n,T]*)DaY(Ti.ri*), (3.26) where the eigenvalue Ea(T|,Ti*) means a single-particle energy of new basis operator d^Ti.T)*) defined in Eq.(3.25). The collective Hamiltonian [H ] and HF potential rop(ri,Ti*) in Eq.(3.26) are given by 1 Z 1.T]*) . (3.27) The nonlinear eigenvalue equation (3.26) should be solved under the canonical variable condition given by - .HI- (3.28, which is derived from Eq.(3.16). Equations (3.26) and (3.28) give anoth- er representation of the basic equations of the SCC method. This repre- sentation clearly shows how the SCC method generalizes the basic idea of the HF theory in Eqs.(2.23) and (2.24). and the CHF theory given by Eqs.(2.40),(2.41) and (2.42). We are now in a position to discuss a range of applicability of the SCC method. This task is carried out by studying the stability of the extracted submanifold £2L under small variations toward the non-collective direc- tions perpendicular to the collective ones at each point of Z2L. To this end, let us introduce particle- and hole-operators with respect to (3.29) Pi(T|.ri*) . lSa = i^N . which is defined at each point of E2L, i.e. at each value of T] and TJ*. With the aid of particle- and hole- operators in Eq.(3.29), the auxiliary (const- rained) Hamiltonian in Eq.(3.20) is expressed as 1 4 cxPyS Here the symbol : : means the normal-product with respect to the state X is I F([C],[C*])> and faP/ySCn/n*) related with the original interaction Vapj75 in Eq.(2.5) through Dj(a.{Tl.Tl*)Djp,(Tl.T,*)Valj.iYVDry-(Tl.Tl*)D56.(Tl.Tl*) . (3.31) -•ll - Aiming at properly introducing the non-collective excitation modes, let us first notice that the collective operators in Eq.(3.14) are, without any loss of generality, expressed as + Z©*vdi(Ti.TT)av(Ti.TT) + ZS8p[(Ti.Ti»)pl(Ti.-n*) + const. . (3.32) The equation determining the non-collective excitation modes (at each point of E2L) expressed as {1l| Pp,(Ti,Tr)a(1 (Ti.r))] . (3.33) is then given by an RPA-type equation, -I'I'.V'J' * Lt Pt J • (3-34) Here the coefficients appearing in Eq.(3.34) are given by ^liv.ijfn.Tl*) • (3-35) L L ' _! " ^ " • (3.36) a=l' The matrix in Eq.(3.34) is a projection operator satisfying P2=P. This projection operator certifies that the non-collective excitation modes (in Eq.(3.33)) determined by Eq.(3.34) are orthogonal to the collective operators. Moreover it also certifies that the RPA-type equation (3.34) has 2L zero energy solutions which correspond to the collective excitation modes t (Oa,Oa ;a = l.-,L). Here, it should be noticed that the correlation amplitudes of the non- collective modes Wiiiaan ^i 4>iiia defined by Eq.(3.34) are directly related to the quantities [dC^i/d^a] and [dC^/dtfa] in Eq.(3.5a), and express the exci- tation modes described by the irrelevant degrees of freedom %a, Z,*a under the small-amplitude approximation. By using the language of clas- sical-level theory in §3.1, the RPA-type equation (3.34) is directly derived from the equation of motion for the irrelevant degrees of freedom id^a/dt =dH/d£,*a in Eq.(3.4). This derivation is much more rigorous than the present derivation, because Eq.(3.34) is obtained from the basic dynami- cal equations of motion of the Hamiltonian system by applying the pertur- bative expansion in Eq.(3.5). In this derivation, it is shown [58] that we get Eq. (3.34) provided that there holds the following condition; ,3,3, a Since the time dependence of correlation amplitudes vj/"^ and cp Mi only comes from that of the collective amplitudes r\ and TI*. Eq.(3.38) simply means that the non-collective motion should not strongly depend on TJ.TI* and/or the collective motion should be quite slow in comparison with the non-collective excitation modes. Therefore, Eq.(3.38) is called the sepa- rability condition between collective and non-collective degrees of free- dom. Namely, the separate treatment between the relevant and irrele- vant degrees of freedom, i.e. the extraction of collective submanifold Z2L described by the relevant variables alone and the description of irrelevant excitation modes by the RPA-type equation (3.34), is justified under the condition (3.38). The condition (3.38) is regarded as a generalization of the adiabatic assumption which is supposed in the CHF+ATDHF theory from the outset. - 1 3 - An Ti,T]*-dependence of the eigenvalue WaOl/n*) and that of the correla- tion amplitudes yani and q>a^i tell us how the non-collective excitation modes change as the collective motion develops toward the large-ampli- tude region. It also indicates what kinds of non-collective modes of motion (irrelevant degrees of freedom) are responsible in changing the microscopic structure of the collective operators as the collective motion evolves into the large-amplitude region. As is easily recognized from the general discussion on the stability matrix of the RPA theory, Eq.(3.34) gives an information on the stability of the collective submanifold £2L against the specific non-collective excitation mode a. If there holds the stability condition. co2(r|,ri*)>>O : Va and Vft.Tf) , (3.39) the subspace L2L remains stable within the TDHF manifold M2K, being well decoupled from the rest space of M2K. With the aid of both the stability and separability conditions, one may divide the collective submanifold 22L into three characteristic regions. Collective Region Dissipative Region Chaotic Region Stability co2 » 0 to2 > 0 co2 <, 0 Condition Separability well separated weakly separated strongly mixed Condition In the collective region, a TDHF trajectory starting from the vicinity of 2L E^L iS expected to stay always in a tiny region near Z . In the dissipat- ive region, a TDHF trajectory found near Z2L at certain instant may oscil- late toward the irrelevant direction (perpendicular to the submanifold) with a fairly large amount of deviation from S2L. In the chaotic region, the trajectory occasionally travelling near X2L does not come back in its neighborhood. These typical situations are schematically illustrated in Fig.3.3. With the aid of a simple soluble model Hamiltonian with SU(3) symmetry whose explicit expression is given in Eq.(5.1), we have evaluated [59] the stability and separability condition and divided the collective submani- fold into the above stated three characteristic regions. By calculating the - M - a) Collective Regime b) Disspative Regime c) Chaotic Regime Figure 3.3. Schematic illustration of the TDHF trajectories starting near £2L with open circle in three different cases. The representative trajec- tory starting with the solid circle is also indicated. TDHF trajectory starting from the vicinity of the collective submanifold, we get three characteristic trajectories shown in Fig.3.4. As is seen from Fig.3.4, the stability and separability conditions for the collective sub- manifold £2L (associated with the DCC system) provide us with a firm starting point for investigating the rich structure of the TDHF manifold near the collective submanifold £2L. When Z2L satisfies both the separa- bility and stability conditions for ail the non-collective modes, the auxil- iary Hamiltonian defined in Eq.(3.20) has definite physical significance. In this case, the introduction of TI.T)'-dependent HF single-particle states £=8 £=5 E=l a) Collective Regime b) Dissipative Regime C) Chaotic Regime Figure 3.4. TDHF trajectories for three different cases starting from vicinity of the collective submanifold L2L. 1 ;) defined in Eq.(3.26) has its sound basis. If E2L does not satisfy the sepa- rability condition for a special non - collective mode X^(T|,TI*), the concept of T),T)#-dependent single-particle states becomes obscure, because the sepa- rate (adiabatic) treatment of the collective (relevant) degrees of freedom from the special non-collective (irrelevant) degree of freedom is not Jus- tified any more. In this case, we have to extend the collective submani- fold so as to explicitly include the corresponding irrelevant variables (%a, £ot*). in order to investigate the single-particle dynamics which is respo- nsible for violating the separability condition. The above discussion is useful for studying a range of applicability of many phenomenological stable mean-fields applied for the nucleus, like the Nilsson model, two center shell model, rotating shell model, etc. A border indicated by the stability condition is simply understood in terms of the single-particle dynamics as follows: If the effects coming from V(r|,T|*) in Eq.(3.35) are negligibly small, a point on Z2L satisfying &>aCn.n*)=O just corresponds to a level-crossing point Ell(r\,T]*)=Ei{r\,T]*), i.e. a singular point where the particle- and hole-energies degenerate with each other. Thus, it is clearly expected that a dynamical relation between a characteristic feature of the collective trajectory described by Eq.(3.19) and the r|,r|*-dependent single-particle energy Ey{r\,r\*) may be explored on the basis of the SCC method. Since the SCC method does not depend on the adiabatic assumption, these studies would give a fur- ther understanding on the nonlinear relation between a specific property of the collective mass parameter M(q) and an occurrence of the level- crossing, which has been discussed within the conventional CHF+ATDHF theory. The discussion of this subsection may be summarized by stating that the study on the collective dynamics within the classical-level theory (i.e. within the TDHF manifold] is still :n its very early stage. We have divided the collective submanifold E2L into three characteristic regimes, i.e. the collective, dissipative and stochastic regions. However, this classification is too coarse to examine an inexhaustibly rich structure of the TDHF manifold. In such a finite, self-sustained system as the nucleus, the sepa- rability and stability conditions for I2L have to be examined carefully for individual non - collective excitation modes X^(T),T)*J. Since one may naturally expect not only many level-crossing points on L2L indicated by the stabil- ity condition, but also many tiny regions where the separability condition does not hold, a detailed study of the single-particle dynamics on what - Hi - are happening in the "dissipative region" will be the central theme for further development of the nuclear collective dynamics. These problems are strongly related to exploring the dynamical relation between two stable mean-fields with different symmetry, like the phase transition in finite systems, fission process, shape coexistence phenomena, decay of superdeformed states, etc. Furthermore, there might be many other collective submanifolds {2 s] which are embedded in other parts of the TDHF manifold. An interrela- tionship between various collective submanifolds is another important subject to be explored in the classical-level theory of nuclear collective dynamics. 3.4 Resonant Interaction between Relevant and Irrelavant Degrees of Freedom At the end of this section, it is worthwhile to discuss an important problem which appears in defining the collective submanifold L2L. As discussed in §3.1, the basic equations (3.7) and (3.8) of the SCC method are usually solved perturbatively by using the T|,r|'-expansion method. In calculating the higher order expansion coefficients Cj(r.s) in Eq.(3.10), one often encounters with a difficulty of "small denominators" which pre- vents the perturbation series from converging. This difficulty is related to the resonant relation between the relevant and irrelevant degrees of freedom. Note that there also hold other kinds o[ resonant relations which are satisfied within the relevant variables and within the irrelevant variables alone. Since the small resonant denominators are known to be the source of the inexhaustible rich structure of the phase space, let us briefly discuss a general picture which has been established in the theory of classical non- linear systems [3,4,35-38]: The island structure is related to an existence of regular motion described by the action variables alone. The system still remains to be integrable, when there exists an isolated single-reso- nance. This is simply because of the fact that one may eliminate the single-resonance by using the canonical transformation to a newly intro- duced resonant action variable. A successive canonical transformations eliminating isolated primary resonances generates secondary and tertiary resonances, giving an explanation for an inexhaustible island structure of regular motion. Isolated resonance usually introduces new stable and unstable orbits (fixed points) in pairs. In between various islands of the regular motion, there are separate levels called separatrix which passes through the unstable periodic solutions. If the separatrices of two differ- ent resonances start to touch with each other, i.e. two different resonan- ce zones start to overlap, the overlapping zone changes into a stochastic sea (overlap criteria 14,38]) by breaking the celebrated KAM torus. The above stated extraordinary picture of the classical phase space, which had not been expected before the digital computer was invented, is also valid for the TDHF manifold. In order to explicitly examine the resonant structure of the TDHF manifold near the collective submanifold Z2L, the SCC method has been reformulated in Refs. 60a and 60b by using the action-angle variable rep- resentation (Ia, *3 • Sa = V^'93 • (3-40) Since the DCC system Is defined so as to satisfy the maximal-decoupling condition (3.22), i.e. H^=0, the Hamiltonian in the action-angle variable representation is obtained by substituting Eq.(3.40) for Eq.(3.5b), and is expressed as H = () Jf) »'=J? <°>(I, [2) (2) (2) % =JD (I,({.;J,cp) = «: (71(I, In order to carefully examine how the collective motion undergoes a change as it develops into the dissipative region, and in order to explore the effects of resonances on the collective motion, one may employ vari- ous classical perturbation methods which have been extensively devel- oped in the classical nonlinear dynamical theory (3]. Aiming at evaluating the resonance points, and dividing the total Hamiltonian into the inte- grable part and non-integrable part, it was proposed in Refs. 60a and 60b to use the Birkhoff-Gustavson normal form perturbation theory [61] in the Lie derivative formalism. In this method, one may extract an optimal - -IK - integral form of the Hamiltonian by making use of a successive canonical transformations, (I.4>; J.cp) -» M1; J'.cp'} -> [r.f; J",9"} -»••-. (3.42) Here a successive canonical transformations is determined in such a way that the angle variable dependent terms of the resultant Hamiltonian are eliminated as much as possible. In §4.5, we will treat the quantum cor- respondent of this problem in detail. By means of this framework which is free from the adiabatic assump- tion, one may proceed to explore the single-particle dynamics which are responsible for violating the separability condition and for generating the resonant points (fixed points) in the TDHF manifold. Insofar as the mean fields are well realized in the nucleus, it is clear from the above discus- sion that the nucleus provides us with a new uncultivated field suitable for studying the order-to-chaos transition dynamics which has been studied in the classical nonlinear systems. The quantum analogue of these prob- lems will be extensively discussed in §§4 and 5. With regards to this point, it should be mentioned that there are some efforts for getting the complete set of closed TDHF trajectories, i.e. the stable fixed points within the TDHF manifold [62]. Although these problems are very interesting and important for the further development of the classical-level theory of nuclear collective dynamics, they are beyond the scope of the present article. -49- 4. Quantum Theory of Nuclear Collective Dynamics 4.1 Concepts of Quantum "Integrability" and Quantum "Chaos" In contrast with the classical chaos, the investigation on the "quantum chaos" is seemed to be still in its immature stage. However, there are some attempts to properly define the concept of quantum "integrability", a counter concept of quantum "chaos", on the basis of both the quantum group approach [63] and group theoretical consideration [64]. On the other hand, there is a well establish method for describing statistical aspects of an ensemble of eigenstates in terms of the random matrix the- ory [65,66]. As is well known, the statistics of nearest neighbor level spacing of the neutron resonance states is well described by the gaussian orthogonal ensemble (GOE) [67]. As an argument relevant to the quan- tum chaos, the following empirical correspondence is frequently stated [68]: If the classical system is integrable, the corresponding quantized system shows the Poisson distribution in the statistics of the nearest level spacing. If it is non-integrable or chaotic, on the other hand, its quantum correspondent shows the Wigner distribution. On the basis of this empir- ical rule, it is widely accepted that the order-to-chaos transition mecha- nism in quantum systems can be studied in terms of a transition from the Poisson to the Wigner distributions [69]. Here, it should be noticed that the above discussion treats a statistical aspects of an ensemble of eigen- states, whereas the classical order-to-chaos transition is discussed by using the dynamical properties exhibited by individual trajectories. Moreover, the discussion based on the GOE introduces some probabilistic assumptions from the beginning. Without introducing probabilistic assumptions, there have been made many attempts which intend to study how a classically chaotic system should be quantized, and how the remnants of phase space trajectories are found in the corresponding quantum system [70,71]. In these inves- tigations, the relation between invariant phase space structure and prop- erties of individual quantum eigenfunctions has attracted many attentions after a spectacular observation of scarred eigenstates by Heller [72]. This observation strongly suggests that there might be an inexhaustible struc- ture in the Hilbert space like in the classical phase space of nonlinear system. In this section, we discuss another way which enables us to investigate the concepts of quantum integrability and quantum chaoticity, on the basis of a recent development of the quantum theory of nuclear collective dynamics [44]. In the previous section, we have discussed how to extract an approximate integral surface Z2L for a given trajectory in the TDHF manifold M2K. It has been shown that the stability and separability con- ditions of £2L provide us with a powerful method in understanding the microscopic dynamics of nuclear collective motion in terms of the single- particle dynamics. It has been also emphasized that the order-to-chaos transition mechanism investigated in the general theory of classical dynamical systems plays a decisive role in understanding dynamical behavior of the large-amplitude collective motion in the TDHF manifold. What we discuss in this section is how to extend the basic idea developed in §3 to a full quantum system by going beyond Jie TDHF (mean-field) theory. A clue toward such an extension is to recognize that there holds a clear-cut correspondence between the canonical variable representation of the TDHF theory and the boson expansion theory [39-43! for even- number many-fermion systems: According to the boson expansion theory, the fermion pair operators { d^dy, pfpk, d*p|, pjdv } defined in Eq.(2.27) are known to be mapped Into a boson space by where [Bn1,B^k] = 8^v8lk . [B^.Bvk] = O. (4.1a) The canonical variable representation of the TDHF theory discussed in §2.3 simply leads us to - :> i - (4.1b) which demonstrates that the canonical variables (C^i, C^i*) in the TDHF theory are c-number correspondents of the boson operators introduced In the boson expansion theory [43]. 4.2 Basic Idea of Quantum Theory of Nuclear Collective Dynamics In the classical-level theory of the TDHF manifold, an Introduction of the DCC system associated with Z2L plays a central role in characterizing the integrability of the system. In a full quantum case, we start with dis- cussing an importance of introducing a concept of dynamical representa- tion (DR) [44] associated with a collective subspace under discussion. In order to illustrate the basic idea as simple as possible, we first dis- cuss an ideal case. Let us suppose that, with the use of the boson expan- sion theory, the even-number many-fermion system under consideration has been transformed into a boson system described by K-kinds of boson operators {BJ.B];j = l K) . defined in Eq.(4.1), where the double indices (p.i) are simply expressed by the single index (j). The representation using (B.Bt) will be called an initial representation (IR). The IR is a kinematical representation becau- se it has nothing to do with a specific correlation characteristic to the collective motion under discussion, generated by the Hamiltonian H(Bt,B). The boson space T2K is spanned by B nj 4 2 h nK ) ^ ni==( j) I o ) • < - ) where 10) denotes the vacuum of the boson operators (Bt.B), i.e. BjlO) = 0. The dynamics of the system is governed by the Heisenberg equation of motion iBt(t)=FB}(t).H(Bt.B)l , (4.3) where the time-dependent operator Btj(t) is defined as eJt. (4 4) Let us consider an ideal situation where a set of collective excited states constitutes an invariant subspace (, i.e. an ideal collective subspace: K»L; 12K z> ) 12L of the Hamiltonian. In this case, there must be an optimum representation called the DR where the minimum number of relevant operators is required in describing 12L. Let (aj.a];j = l L} . (4.5a) be a set of relevant (collective) operators necessary for spanning 12L and let t {(3J,pJ ;j = L + l K} . (4.5b) be the remaining operators called the irrelevant (intrinsic) operators in the DR. In the DR, the time evolution is still governed by H(Bt,B). Namely, there holds idj(t) = (aj(t) ,H(Bt,B)J ; j = 1...L, ijjj(t) = [p](t) .HfB+.B)] ; j = L + 1....K. (4.6) where the time-dependent operators in the DR are given by at(t),eiH(Bt,B)tate-iH(Bt.B)t( pt(tJ „,e iH(B+.B)tpte-iH(Bt,B)t (4.7) In the DR, the Hamiltonian is expressed as F(at ; i ) t t F(at a: t ) t.B) = e -« P' -P H(a .a:p .p)e- - P 'P (4.8) Here an operator exp{F(Bt,B)}=exp{F(at,a,bt,b)) denotes a nonlinear uni- tary transformation between IR and DR; B)J = 1 L> pt =e-F(Bt.B)BteF(Bt.B) , j = L + l,...K. (4.9) This transformation is a quantum analogue of the Lie transformations [3, 60b,74,75] which have been developed in the classical-level theory. The dynamical condition specifying the invariant subspace is obviously given by the complete decoupling condition between the relevant and irrele- vant operators; With the aid of condition (4.10), the time-dependent operators in Eq. (4. 7) are reduced into aj{t) = e^cotttate-i«co[tt ^ pt(t) = eirtirrtpte-irtirrti (411) satisfying t t t iaJ (t) = [aJ (t).rtcott(a ,a)]; j = l L . (A-II) + t ip](t) = [pJ (t),rtirr(p ,p)] ; j = L + l K. (4.12) Namely, the relevant and irrelevant operators are governed separately by Hco[t and Hirr, which satisfy the following commutability condition t t (A-III) [Kcott(a ,a).Jiirr(p .p)l = 0. (4.13) Equations (A-I), (A-II) and (A-III) constitute a set of basic relations char- acterizing the invariant subspace 12L of the Hamiltonian. From the above discussion, it is clear that an introduction of the DR for a given invariant subspace 12L is decisive in manifesting the invariant property of the Hamiltonian. 4.3 Basic Equations for Determining the Dynamical Representation Let us discuss the general case where the complete decoupling condi- tion (4.10) does not necessarily hold. Namely, we Intend to extract a set of collective states which forms an approximate invariant subspace of the Hamiltonian. To this aim, we start with a resultant DR where the collec- tive subspace denoted as D2L: {lmi,m2,...niL; nL+i=...=nK=0>) is expressed as 0 |mllm2 mL;nL+i=-=nK = 0) = n-rrMPl ) • (4-14) j=i-y'mj!V J/ where IO>= exp(-F(Bt, B) JIO). In order to get the DR, i.e. to determine a generating function F(Bt,B)=-Ft(Bt,B) of the unitary transformation in Eq.(4.9), we have to discuss how to characterize the approximate invari- ant subspace. /x' ;;.ing at carrying out the above subject, we start with introducing the following notation for the vacuum state; 10 >= IO»R®IO»I ; OjlO»R=0 . pjl0»I=0 . (4.15) In this notation, the total Hilbert space is expressed as a product space whose basis states are represented as Im1,m2 niL;nL+1,nL+2 nK > = Im1,m2 mL »R ®lnL+1,nL+2 nK »! n^Jpi o »R n r J=lVmJ !^ ' j=L+lVnj With the aid of this notation, the Hamiltonian is decomposed into H(B'.B) = where f t Kco[i (a ,a) = j « 01 H(B ,B) 10 »! . (4.18) and Art in Eq.(4.17) are supposed to be expressed as the normal ordered form with respect to ocf.cc.pt and p. It is obvious that Hcocc satis- fies [«cott . Pj] = 0 • (4.19) In order to get the basic equations specifying the most optimum representation, we have to employ the following expansion scheme: Since the relevant operators are supposed to describe the large-amplitude col- lective motion within D2L whereas the irrelevant operators describe small-fluctuations around D2L, it is reasonable to introduce a power series expansion with respect to the Irrelevant boson operators. For example, is expanded as AH = £rt(n), n=l = I fp] -^(OKp,. H(Bt,B)]IO»i + h.c. } , -- I pt-I«OI[p,.[H(B,B). P^IHO))! pk J.k=L+l J J t + Z N-Jk{pMI«0l(pkpj,H(B .B)jl0»I + h.c.}.andetc. (4.20) j.k=L+l l J J Here, the normalization constant is given by (4.21) Within the first order expansion in AK, the time-dependent operators in Eq.(4.7) are expressed as o}(t) B (4.22a) pj(tj tJ. p]j . (4.22b) If there holds the maximal-decoupling condition (B-I) Ji(l) = O . (4.23) the relevant time-dependent operators in Eq.(4.22a) satisfy the following equation of motion given as (B-IIa) i^a](t) = [a](t), JicoCC] , (4.24) within the approximation of neglecting the second order effects with respect to the irrelevant operators. Further more, the irrelevant time- dependent operators in Eq.(4.22b) satisfy (B-IIb) i-j-pt(t) = [pf(U«[2)]. (4.25) at J J provided that there holds the separability condition expressed as (B-III) [«cott.«(2J]=.0. (4.26) By comparing Eqs.(B-I),(B-II) and (B-III) with Eqs.(A-I),(A-II) and (A-HI), it is easily seen that the maximal-decoupling condition (4.23) and the separability condition (4.26) just correspond to the complete-decoupling condition (4.10) and the commutability condition (4.13) in the ideal case, respectively. Equations (B-I), (B-IIj and (B-III) constitute the basic equa- tions of quantum theory of nuclear collective dynamics [44]. 4.4 Dynamical Representation and Extraction of Collective Subspace D21^ It has been shown elsewhere [44] that the maximal-decoupling condi- tion (B-I) gives necessary conditions for determining the functional form of F(Bt,B) which specifies the unitary transformation between IR and DR. Here It should be noticed that the canonical variable condition (3.7) has played an important role In the classical-level theory discussed in §3, together with the equation of collective submanifold in Eq.(3.8). In the quantum theory, however, there are no equations corresponding to the canonical variable condition, because the unitarity of the transformation is automatically imposed by Eq.(4.9). If we employ the Lie-derivative for- malism in the classical-level theory, the similarity between the classical- level theory and the quantum theory becomes more transparent [60b]. According to the quantum theory of nuclear collective dynamics , the generating function F(Bt,B) specifying the DR is determined by the maximal decoupling condition (B-I), under such a boundary condition that the relevant and irrelevant operators are reduced to lim(aj) = BJ . forj= 1 L; lim(pj) = BJ ,forj= L + l K . (4.27) in their small-amplitude region, respectively. This boundary condition is only one physical input of the theory in specifying the collectivity under discussion. A detailed discussion how to get the unitary transformation is found in Ref.44. The usefulness and feasibility of the DR has been demon- strated by applying it to the pairing Hamiltonian [76] and it has been shown that the dynamical structure of phase transition from normal to super phases is well described by this theory. We are now at the position to discuss a range of applicability of D2L spanned by the basis states in Eq.(4.14) which are described by the rele- vant operators alone. After getting an explicit expression of F(Bt,B), we have a concrete forms of Hcott and Mt. With the aid of an explicit expre- ssion of Hcoll. we have a set of collective excited states by diagonalizing KcoIX within D2L; rt E e 2L cott W>R = n'^»R- W>R D :{lm1,m2 mL»R}. (4.28) With the aid of the collective excited states I (I»R thus introduced, the separability condition (4.26) is expressed as L'))R=0. n*n'. (4.29) Equation (4.29) simply means that the collective states I H»R defined by Hcotl should not be disturbed under the effects of H(2), and it gives the first condition to evaluate the applicability of collective states. If there holds the separability condition (4.29), it is easily derived that K(2) has only diagonal terms with respect to the collective states, i.e., H(2) is expressed as . (4.30) Here. |H>= (4.31) Let us consider effects of diagonal components 3i^(2) on the collective states. Introducing a general coherent state for the irrelevant part of the wave function given by mR . (4.32) we have Zj7 Yj S^ > 0 , for each collective state |^»R , (4.35) which is the second condition to evaluate the applicability of collective states. Here, it is worthwhile mentioning that the stability condition is given for each collective state IJI»R, i.e. the stability of collective states depends on the quantum number (i characterizing the amplitude of col- lective motion. At the end of this subsection, it should be stated that the number of relevant operators 2L necessary for describing the collective states under consideration is determined in such a way that the corresponding sub- space D2L should be the smallest subspace where both the stability and separability conditions are satisfied. 4.5 How to Evaluate Stochasticlty of Individual Elgenfunctions In the previous subsections, we have discussed how to dynamically ex- tract the collective subspace D2L. and how to evaluate the range of appli- cability of D2L by introducing the separability condition (4.29) and the stability condition (4.35). These concepts just correspond to the collec- tive submanifold Z2L, separability condition (3.38) and stability condition (3.39) in the classical-level theory, respectively. It has been shown in §3 that an introduction of the DCC system (TI,T]*£4*) gives a powerful tool in analyzing various structure of the TDHF manifold near D2L. In order to carefully examine an exceedingly rich structure of the TDHF manifold in- duced by an existence of the resonances, however, it is convenient to introduce an action-angle variable representation {Ia.a: Ja. mn-Representation Aiming at extending the separability and stability conditions given by Eqs.(4.29) and (4.35) into the whole Hilbert space and introducing the most optimum basis states for a given set of collective excited states, we hereafter consider the simplest case with L=l and K=2, i.e. the system described by one relevant at and one irrelevant |lt boson operators. The extension to the general case is straightforward. In this case, the physi- cal boson space X corresponding to the fermion space is described as X: {|m,n) ; m + n The projection operator 1 onto 1 is expressed as 1 = I |m,n){m,n| . (4.38) m,n (m+nSN) The representation using the basis states (lm,n>) is called the inn-repre- sentation. As is easily recognized, the mn-representation has a sense in every part of the physical boson space, i.e. every eigenstates of the total Hamiltonian are properly described by one of the lm.n>-states, when the complete decoupling condition (4.10) is satisfied. Note that the boson number m and n are quantum analogue of the magnitudes of the relevant action Ia and irrelevant action Ja in Eq.(3.40), respectively. jj.n-Representa.tion Let us divide the physical boson space T into multi-layers of subspaces (Dn ; n=0,l N). Here the n-th subspace Dn consists of a set of basis states having the same irrelevant boson number n, i.e. Dn:{jm.n) ;m=0.1 N-n ) . (4.39) 2L Here, Dn=ojust corresponds to the collective subspace D discussed in the previous subsection. Since the collective excitation modes are supposed to be not strongly affected by an existence of small number of irrelevant bosons, a set of subspaces {Dn} has a sense at n«N. The projection operator P(n) onto Dn is given by P(n) = I? |m,n )< m,n|, 1 = £P(n) . (4.40) m=0 n=O By using the projection operator P(n). AJt in Eq.(4.17) is divided into two parts -62- = X P(n)AJi P(n'). (4.41) n,n' (n*n') Here, we briefly discuss characteristic roles of each part of the Hamilto- nian. As is clearly understood from Eq.(4.19). HCOU commutes with any irrelevant boson operators by definition. Namely, it does not violate the irrelevant quantum number n. It is also the case for the part AKdla in Eq.(4.41), because it is effective only between two states with the same irrelevant boson number n, and it expresses blocking effects of the irrele- vant bosons on the collective states. On the other hand, Mioii~diu plays a role to violate the irrelevant quantum number n. Let us extend the concept of collective quanta JI defined in Eq.(4.28) to the whole physical boson space. This task is achieved by diagonalizing H(Bt,B) within the subspace Dn with arbitrary n, f (fta P(n)H(B .B)P(n)|n.n) = P(n){Hcott +AH }P(n)|n.n) = E[T' IM . |n,n) e Dn :{ |m,n) ;m = O,--,N-n) . (4.42) Here, the quantum number \i coincides with the definition in Eq.(4.28) in a case with n=0 and specifies the relevant part of the states in order of their energy eigenvalues; 4 43 H = 0.1 N-n with EJJJ, & E[gx < ^jStt-n • < - ^ With the aid of a set of basis states ln,n> thus obtained, the subspace Dn is alternatively expressed as Dn:{||i.n) ;n = 0,l N-n). (4.44) The representation using the basis states I n,n> is called the |i.n-represen- tation. Here, it should be mentioned that this is the best representation one may choose in incorporating the dynamical correlations among the relevant bosons into the basis states, under the restriction of conserving the total number of good quantum numbers. The classical correspondent of the (^,n)-basis state may be a phase space point which is expressed as (I'(J), J}; Here, I'(J) means a new rele- vant action variable which is defined in the Birkhoff-Gustavson perturba- tion method by taking account of the effects coming from an existence of the irrelevant actions J, and by eliminating the angle-variable dependent terms in the Hamiltonian as much as possible. Hv-Representation Since the relevant parts of the basis states ln.n> and \y.,n'> with n*n' «N are expected to have almost the same structure, one may divide the physical boson space X into a set of vertical-layers of subspaces n^. {|n.n> ; n = 0,1 N-^} . (4.45) The projection operator P(n) onto Up. is expressed as P(H) = I |fi.n )< |i,n| .1=1 P(n). (4.46) n=0 H=l By means of P(n), the Hamiltonian is expressed as lHfB+.B) UIl" |i.n) E[.n) ( |l.n| + Mi°^~dia . (4.47) H=0n=0 ^ where AK°JJ "<*-"* splits into two parts ^oJJ-ctu* = H[stab] + ^fsepa] ^ rt[stabl = ^ H[stab] = ^ p(ilLmof H K = In.n'(n*n') Hn.ji' = 2n.n'(n^') P(U)AJi » P(H ) . (4.48) In the expression of Eq.(4.48), it is directly recognized that a part conserves the relevant quantum number JJ. with violation of the irrelevant quantum number n. For a case with n=0, rt'stabJ acts to induce an 2L instability of the collective subspace D = Dn=o (see Eq.(4.35) where the diagonal component of the second order Hamiltonian Ji^(2j is discussed). - (i-i - On the other hand, a part HlsePal violates both the relevant quantum number |i and irrelevant quantum number n. From the above discussion, it becomes clear that the dynamical effects coming from Jtlstabl can be incorporated into the basis vectors by diagonalizing H(Bt,B) within the subspace Rji, i.e. |,v) = E^v|^v) . IMeR^. (4.49) Since there holds f 10 abI P(n)H(B ,B)P(n) = H\i){HcM + Aft* +H[f }p(ji) . (4.50) the dynamical correlations contained in H(Bt,B) except for a part Jilsepa] are successfully incorporated into the basis states ln,v>. By using ln,v> basis states, the subspace R|i is expressed as %: {|n,v);v = 0.1 N-n}. (4.51) The representation using the basis states I |i,v> is called the nv-repre- sentation. As is easily understood from the above prescription, this is the best representation (the most optimum representation) one may choose in incorporating the correlations responsible for a set of collective excit- ed states under consideration, into the basis states as much as possible under a restriction of conserving the number of quantum numbers. The unperturbed Hamiltonian expressing regular quantum motions is given by Eq.(4.50), because the (n,v)-basis states thus introduced are still charac- terized by the same quantum numbers as that of the (m.n)-basis states. The perturbative interaction destroying the regularity is given by Jt'sePal. As is also clear from the above derivation, the range of an applicability of the |iv-representation is studied by only the separability condition given by rt^ = O.(n*n') . (4-52) because the nv-representation is defined so as to satisfy the stability condition (u.vj Jt[ftab] JM-V) = 0. for all n with v * v'. (4.53) Equations (4.52) and (4.53) are natural extensions of the separability and stability conditions for the collective subspace D2L given by Eqs.(4.29) and (4.35), respectively, to the whole Hilbert space. The classical correspondent of the (p.,v)-basis state may be expressed by a phase space point {I'(J),J'(r)): Here I'(J) denotes a renormalized rel- evant action variable defined in the (ja.,n)-representation and J'(l') denotes a renormalized irrelevant action variable which is defined per- turbatively by taking account of the effects caused by an existence of the renormalized relevant action I'(J), and by eliminating the remaining angle dependent terms in the Hamiltonian as much as possible. The classical- level correspondent of unperturbed Hamiltonian InP(^)H(Bt,B)P(|i) = £^v [H,v>-E|i,v- Definition of an Onset of Quantum Chaos Let {I i >} be a set of exact eigenstates of the total Hamiltonian H(B,Bt) lH(Bf.B) I Ji)= E, |i) , |i) € \ :{|m,n) ; m + n < N} . (4.54) From the above discussion, it is clear that the exact eigenstate I i > is no more specified by one of the (n.v)-basis states when the separability con- dition (4.52) is violated. Namely the exact eigenstate is not specified by K-kinds of quantum numbers any more but by smaller number of quantum numbers in accordance with the violation of the separability condition. In other words, there occurs a dissolution of quantum numbers when the separability condition does not hold. In the classical-level theory of the TDHF manifold, a dissolution of integrability simply means an onset of chaotic motion. In the quantum system, therefore, it is reasonable to state that a dissolution of quantum numbers may indicate an onset of "quantum chaos". In accordance with the above discussion on the quan- tum chaos, we may classify the exact eigenstates {I i >) into three charac- teristic cases with the aid of the (|i,v)-basis states; Quantum integrable motion ; If one finds one of the (|i,v)-basis states for a given eigenstate II > satisfying l<(i,vli >|2 =i, then li > is classified to be in "quantum integrable motion" because it is specified by K-kinds of quantum numbers (n,v) associated with the DR. Quantum "dissipative" motion; If li > is described perturbatively by starting from the most appropriate (^,v)-basis state, then it is specified to be in "quantum dissipative motion" because it is still characterized by K- kinds of asymptotic quantum numbers. Quantum chaotic motion ; If li > is not described perturbatively by starting from any {^,v)-basis states, then it is regarded to be in "quantum chaotic motion" because I i> is not characterized by K-kinds of quantum numbers any more. The above criterion of the quantum chaos is very similar to what has been proposed by Hose-Taylor |77]. Since their 50% criterion strongly depends on the employed representation, it does not enjoy its proper role until the most optimum representation is defined uniquely. In order to express the chaoticity of each eigenstate, let us expand the eigenstate I i > in terms of the (n,v)-basis states as ^ (4.55) Looking at the distribution of square of expansion coefficients la^y'l2, one may recognize whether the eigenstate I i > belongs to the quantum integrable, dissipative or chaotic regime. A figure expressing la^v1'2 is called the fractional parentage plot (FPP) [73] and will be further discuss- ed in §5. With the aid of the expansion coefficients, one may introduce a (ji,v)-entropy by (4.56) which clearly expresses a degree of chaoticity of the individual eigenstate li >. This subsection is ended by stating some comments. We have so far discussed on a set of global quantum numbers which are defined in every part of the Hilbert space. As is easily understood from the definition of the (n.v)-basis states, number operators counting the n,v quanta may have very complicated structure. This is simply because the (|i,v)-basis states is introduced by diagonalizing a part of the Hamiltonian under the restri- ction of not decreasing the number of good quantum numbers of the (m.n)-basis state. In other words, the \i,v quanta are state-dependent objects. The second comment is concerned with a special way of con- structing the (n,v)-basis states. As discussed in §4.4, the DR and associ- ated nv-representation are defined for a set of specific collective excited states one wants to describe. If one intends to describe another set of collective excited states, one has to define another DR and corresponding Hv-representation. Although the DR and its uv-representation are defined in the whole Hiibert space, their border of applicability are determined by the separability condition (4.52) and are bounded in a limited region of the Hilbert space. In the present case with two degrees of freedom, the (fi.v)-basis states has physical significance in a region with 0 4.6. Resonant Quantum Number Let us discuss what happens as the collective excited states develops into the large-amplitude region. Here, it should be noticed that the divi- sion of the Hilbert space into three characteristic regimes, i.e. the quan- tum integrable, quantum dissipative and quantum chaotic regions dis- cussed in the previous subsection may be too rough to explore an exceed- ingly rich structure of the Hilbert space. Since each DR is specified at the small-amplitude limit of the specific collective excited states (see Eq.(4.27)), this exploration may be performed by carefully examining what kinds of dynamical change takes place as the collective states approach to the border of applicability of the DR, i.e. the region where the quantum dissipative motion appears. In order to study the detailed structure of the Hilbert space, one should pay careful attention on the special roles of individual unperturbed quantum states in the (iv-repre- sentation. As discussed in §3.4 and is known in the classical dynamical system, an appearance of the resonant denominator is the source of an exceeding complexity of topology [3J. Let us discuss what happens when the quantum states satisfy the resonant condition. As is seen from Ref.44 where an explicit way of getting the generating function F(Bt.B) is discussed, there appears resonant denominator when we solve the maximal-decoupling condition (4.23) by using the perturbative expansion method. It also appears in evaluating the effects coming from 3iIsePal per- turbatively on the basis of the |iv-representation when the unperturbed energies of the |j.v-representation E^v in Eq.(4.49) degenerate acciden- tally. In the following, we discuss an effect of resonances by using the mn- representation. Although the resonance condition depends on the em- ploied representation, the following discussion is applicable for any rep- resentation. We restrict ourselves to a simple system with two degrees o[ freedom. The unperturbed harmonic Hamiltonian of the boson system with two degrees of freedom is given by Ho =CO1B|B1+G>2BJ,B2 , (4.57) and the mn-representation of the system is given by introducing the following basis states; Here m and n are the quantum numbers of the "oscillator quanta" which are eigenvalues of the following two commutable operators, N^BJBi . N2=BJ,B2 . (4.59) — i;(iiul - The resonant condition for the harmonic Hamiltonian is expressed as -k2w2 =0 . kx and k2; integer . (4.60) II the resonant condition (4.60) is satisfied, one may introduce new quan- tum number 7i called "resonant oscillator quanta" which is an eigenvalue of the following operator ^ = k2N1+k1N2 . (4.61) To make the following discussion simple, we hereafter treat a special case with kj=2 and k2=l. However the generalization for any integer values of ki and k2 is straightforward. By making use of the resonant oscillator quanta, one may introduce the following subspace; -2.1>.fflyC-4.2).—. ^ or (4.62) whose elements have the same quantum number 9(, It is also noted that the states in D^have the same unperturbed energy Eo=coiSV!; Now let us discuss another new quantum number specifying the states when the resonant condition (4.61) is satisfied. For this aim, we consi- der the simplest case with 5\£=2; (4.63) With the aid of the last notation in Eq.(4.63), the total Hamiltonian in the subspace D7^2 is generally represented as (4-64) The first term denotes the unperturbed harmonic Hamiltonian and a denotes a degree of violation from complete degeneracy. If the two unperturbed states are completely degenerate, there holds a resonance condition oc=0. The second term expresses an anharmonic interaction - 70- (465) within the subspace jD^=2. Apparently, a type of the anharmonic Hamil- tonian acting between two degenerate unperturbed states depends on kj and k2 in the resonant condition (4.60). Now let us consider the reso- nant case cc=O. In this case, one may introduce a set of new basis states which satisfy . t±=±l . (4.67) Here it is worthwhile to mention that there exists a new representation ft+ called "resonant representation" which diagonalizes the anharmonic Hamiltonian. Since the eigenstates R.± of the Hamiltonian in Eq.(4.64) with oc=O are still specified by two kinds of quantum number (j£ t±) called "local resonant quanta" (instead of the original global two oscillator quanta (m.n)), there does not occur any dissolution in the number of the good quantum numbers. Namely, there still exist new regular quantum states which incorporate such anharmonic effects that otherwise induce a well-known divergent difficulty in the perturbative treatment based on the mn-representation. An extremely important character of a pair of resonant quanta (JV)t±) rests on its local property. If there exists two states whose unperturbed energies are accidentally degenerate, one may always introduce a corresponding local subspace D^=2 spanned by these two states, irre- spective of the other part of the Hilbert space. Then one may further introduce local operator T expressing anharmonic coupling between these two state. Namely, the local isolated resonant quanta are defined in a tiny region of the Hilbert space. This local property is very similar to the isolated integral of the classical system. Note that the first integrals have global character, whereas the isolated integrals have local property. The above discussion can be extended to the non-degenerate case. If the subspace U^=2 is an approximate invariant subspace of the total Hamiltonian, the eigenstates of the Hamiltonian in Eq.(4.64) have a sense -71 — irrespective of the values of a and p. The eigenvalues and eigenstates of Eq.(4.64) are given by U± = Q±U± (4.68) As is easily justified, U± reduce to two states in Eq.(4.63) in a case with P=0, whereas they reduce to K± in a case with a=0. As discussed above, the global oscillator quanta (m.n) become good quantum numbers in a case with P=0 and the local resonant quanta (2£t) become good quantum numbers in a case with a=0. More generally, there always exist two good quantum numbers (9\£O), provided the subspace D^2 is an approximate invariant subspace of the total Hamiltonian. The above discussion may be summarized by stating that, as far as the subspace J5^=2 is an invariant subspace of the total Hamiltonian, there exist regular quantum states specified by a set of local quantum numbers {?{,&), even in a case where the global quantum numbers (m,n) does not remain to be good quantum numbers any more. Summarizing this section, it may be stated that we have acquired a new method of analyzing the complicated structure of the Hilbert space. The separability condition (4.52) gives a criterion for an applicability of the (iv-representation. It tells us to what extent a specific feature of the collective excited states characterized in the small-amplitude region persists or continues to survive. If the separability condition is violated due to an existence of the single-resonant denominator, there appear new states which are not described by the ^v-basis states any more. In order to describe new coming states which have quite different property from that of the collective states in small-amplitude region, we get a new representation where two quantum numbers out of K-kinds of quantum numbers specifying the nv-representation are replaced by a pair of local resonant quantum numbers. Since there does not occur any dissolution in number of the good quantum numbers, the new coming states are also classified to be in "quantum integrable motion" with other property. By means of the separability condition and a pair of local resonant quantum numbers, we may explore exceeding rich structure of the Hilbert space by starting the specific collective excited states in the small-amplitude region . - TA - 5. Application of Quantum Theory 5.1 Simple Model Hamiltonian In §4, we have discussed how to extract an approximate invariant sub- space D2L of the Hamiltonian out of the total Hilbert space 12K by intro- ducing the DR. We further defined the (n,v)-basis states which allow us to characterize the quantum integrabiliry and quantum chaoticity for indi- vidual eigenfunctions. It was shown that a range of applicability of the extracted subspace is examined by the separability and stability condi- tions, and an effect of resonant denominator plays an important role in generating a new set of ordered quantum states. In order to illustrate a feasibility of the quantum theory of nuclear collective dynamics in explor- ing various structure of the total Hilbert space, let us apply it to a simple soluble three-level SU(3) model Hamiltonian by Li, Klein and Dreizler [78] with slight modification given as - _ 2 . Vl . . V2 - - 1=0 '" 2 2 N . m=l There are three single-particle levels with energies EO f H(B,B ) = (Ei -eoJB^Bj + (E2 - - 7 1 - (5.2) The "physical" boson space corresponding to the fermion space of states is given by t m n lm,n>=-7Lr-7L.(B1 ) (Bj,) l0> . with 0 Sm + n <, N . (5.3) Vml Vnf 1 z In this paper, we consider a set of "collective" excited states whose small-amplitude motion near the stationary state 10> is described by a single kind of boson operators Bit and Bi. Since the maximal-decoupling condition (4.23) is already satisfied by the Hamiltonian itself in Eq.(5.2), the representation using the boson operators (Bit.Bi; 1=1,2) just corre- sponds to the DR for the present case. Namely, the set of basis states in Eq.(5.3) provides us with the (m.n)-basis states defined in Eq.(4.16). 5.2. Fractional Parentage Plot According to the discussion in §4.5, the chaoticity of each eigenstate is visualized by expanding it in terms of the (p.,v)-basis states. The jxv- representation of {li>} is shown in Fig.5.1, where the size of a solid circle is proportional to the square of expansion coefficients la^v1'2- The parameters used in calculating Fig.5.1 are N=10, £I-EO=1 and E2-eo=^2. According to the special symmetry of the interaction in Eq.(5.2), the Hamiltonian conserves the evenness or oddness of the population in the relevant and irrelevant boson numbers. In our numerical calculation, we consider the subspace with even number operators for both the relevant and irrelevant bosons. If the strength of the interaction is relatively weak, each eigenstate li> is characterized by one of the (|i,v)-basis states. In this case (Fig.5. la), the system may be classified to be in "quantum integrable motion", because each eigenstates are specified by two (K=2) quantum numbers. If the force strength becomes large (Fig.5.lc). the expansion coefficients are distributed over a wide range of the (|i,v)-basis states, indicating the dissolution of quantum numbers associated with the DR. In this case, the system may be classified to be in "quantum chaotic motion", because it Is hard to characterize individual eigenstates by two quantum numbers. In the intermediate case (Fig.5. lb), the expansion coefficients spread out over many (ji,v)-basis states. Since the diagonal components still remain the largest, the system in Fig.5. lb may be classi- fied to be in "quantum dissipative • motion". These figures are called the fractional parentage plots • (FPP). Since the FPP clearly • visualizes the chaoticity of indivi- • dual eigenfunctions, it may corre- • spond to the Poincare section map in the classical system which clearly shows the chaotic- * ity of individual trajectories. a) Quantum integrable motion. V1=V2=-0.022 ID-* I')-* • • • • • • • • Rp.a b) Quantum dissipative motion. c) Quantum chaotic motion. V^-0.11. V2=-0.33. V!=V2=-0.33 Figure 5.1. Fractional parentage plot for the case with N=10. (Taken from Ref. [73]) It has been shown [82] that specific feature of the FPP near the stable mean-field are well understood by means of a "classical-level" Hamiltonian h(a.P)H la,p>=exp[aBj-a*B1J • exp (pB^-p*B2} IO>=la»Rip»I . (5.5) The Hamiltonian h(oc,p) simply corresponds to the classical-level Hamil- tonian appearing in the TDHF theory discussed in §3. As is easily recog- nized, a classical analogue of the HlsepaJ jn Eq.(4.48) near an irrelevant local minimum Po is expressed as (5.6) where a^ is determined in such a way that R«CC I p.»R takes the maxi- mum value at a=ocJ1. In Eq.(5.6), a^'-ajx represents a distance between two states I H'»R and I |I»R , and A(3=p-po- Here Po denotes a local mini- mum point of hfa^.p). The quantum analogue of po for a given oc^ is expressed by the lowest energy eigenvalue state of Eq.(4.49). In this way, one may get a classical image of the nv-representation and structure of the perturbative interaction Ji'sePal. Further discussion is found else- where [82]. 5.3. Stochasticity of Ensemble of Eigenstates Let us discuss a relation between the present definition of quantum chaos for each eigenstate and the usual definition of quantum chaos for an ensemble of eigenstates. In order to get the nearest neighbor level spacing statistics, we calculated the Hamiltonian in Eq.(5.2) with N=60, ei-£o=l and £2-£o= 2. The final dimension of the Hamiltonian matrix to be diagonalized is Nd=496 which might be sufficiently large for discussing a variety of statistical aspects of an ensemble of eigenstates. In Fig.5.2, the FPP is shown for three different cases. In this figure, the squares of ex- pansion coefficients are shown by the lengths of vertical segments. As is easily seen from Fig.5.2, three different parameter sets for Vj and V2 used in our calculation just correspond to the quantum integrable, quan- tum dissipative and quantum stochastic cases, respectively. Correspond- ing to these three parameter sets, we have calculated the nearest neigh- bor level spacing statistics which are shown in Fig.5.3. In Case A (Fig.5.3a). the level statistics gives a strong peak at the origin, indicating an "integrability" of the system. With increased effects of Vi- and V2-interactions. the peak at the origin becomes less pronounced. Namely, the Poisson type distribu- tion in Case A is disturbed toward the Wigner type distribution when the system shows "quantum dissi- pative" character as in Case B. In Case C where almost all eigenstates are classified to be in quantum chaotic motion, the peak at the origin disappears. aj Case A. Vi=V2=-0.01 b) Case B. V^-0.01, V2=-0.05. c)CaseC. V^-0.10, V2=-0.13. Figure 5.2. Fractional parentage plot for the case with N=60. (Taken from Ref.82.) From these numerical calculations, one may conclude that the present definition of quantum chaos seems to be compatible with the usual dis- cussion based on the GOE. Here it should be remembered that the random matrix theory relies on some probabilistic assumptions: The matrix elements should be random and the resultant distribution should be representation indepen- P tx P tx) LOt - \ VI • -0 01 12 --0 01 ••n • Mf . n • 1 w X 1 n 1 u I. II I-« a) Case A. b) Case B. c) Case C Figure 5.3. Nearest neighbor level spacing statistics for all eigenstates. Solid and dashed curves show the Wigner and Poisson distribution. (Taken from Ref.82.) dent. In this treatment, the boundary effects specific for such a finite system as the nucleus are not properly considered. In order to eliminate the boundary effects from our calculation, we divided the total eigenstates Nd=496 into three groups specified by their energy eigenvalues: the low- est 200, the intermediate 200 and the highest 96. By extracting 200 intermediate eigenstates, we have calculated the nearest neighbor level spacing statistics shown in Fig.5.4 where the transition from the Poisson to Wigner distributions is much more pronounced than the case shown in Fig.5.3. From the above numerical calculation, it may be said that an appearance of the Wigner type distribution function does not directly correspond to (even though it is compatible with) the dissolution of quan- tum numbers associated with the DR. In other words, the dissolution of 1ft 1.90 a) Case A. b) Case B. c) Case C Figure 5.4. Nearest neighbor level spacing statistics for 200 intermediate eigenstates. (Taken from Ref.82.) - 79 - quantum numbers indicating the individual chaoticity does not necessar- ily mean an appearance of probabilistic feature which is the basic assumption of the random matrix theory. 5.4. Structure of Eigenfunctions Let us discuss how to get an experimental indication of an onset of quantum chaos besides the nearest neighbor level spacing. Detailed information on the wave functions is obtained by evaluating many kinds of transition probabilities between many eigenstates. In the present model, the collectivity of the system is evaluated by the following fermion opera- tors t!=Klo + Koi. (5.7) The boson representation of the collective operator is given by BTT*> t}T TD fC Q\ The matrix element between two eigenstates is expressed as Ti,f=| Until now we have concentrated ourselves to a set of (m,n)-basis states with even number of m and n. because of the special symmetry of the Hamiltonian in Eq.(5.2). In order to calculate the matrix elements Tij, we calculate the exact eigenstates in (m,n)=(even,odd). (odd,even) and (odd,odd) sectors in addition to the (even,even) sector. With the aid of numerical calculation for Tirf, we get interesting information how the collectivity associated with the relevant degrees of freedom undergoes a change as the stochasticity of individual eigenstates develops. However, it is not easy to deduce some general conclusion out of numerous data {Tj,f}, whose number is O(Nd2). Aiming at extracting some physical information from the numerous experimental data, there has been pro- posed the Ey- Ey plot 183,84] which has aroused many theoretical dis- cussions on the rotational damping mechanism [85,86]. Namely, one gets some statistical aspect of the rotational motion out of many sequential - 80- transitions. By means of two successive matrix elements and energy eigenvalues, one may introduce the following quantity: - (Ej - Ek)TJik , with Ej > Ej > Ek (5.10) Plotting a huge number of sequential values (Ijjk. Ijkl) as well as (Ijkl. Ijjk) in the 2-dimensional xy-plane. one may evaluate statistical features of the system in terms of local information obtained from two successive transition probabilities. From the above discussion, it is quite interesting to study whether or not the stochasticity occurring in the individual eigenfunctions induces some irregularity in the (Iyk. Ijy)-plot. We start with a specific situation where all the eigenstates are classified to be in "ideal integrable" motion, i.e. a case with no interaction Vi= V2 =0. In this case, we get two straight dotted lines in the xy-plane shown in Fig.5.5. By switching on the effects coming from the Vi- and V2-interactions, we get various (Iyk. Ijkl)-plots illustrated in Fig.5.6. As is seen from Fig.5.6a which corresponds to Case A in Fig.5.2, there exists a pronounced regularity when all the eigenstates are classified to be Jn quantum regular motion. On the other hand, it is hard to see any regularity in Fig.5.6c which represents Case C in Fig.5.2. In between these two extreme cases, we find remnants of regularity in Fig.5.6b. It is striking that there still remains information on the stochasticity of individual eigenfunctions after accumulating numerous local information between two eigenstates connect- to.o ed by the collective operator. As far as the present model Hamil- 34.0 - y-y tonian is concerned, it is clearly yy'' y-y seen that the (Iijk.Ijkl)-plot gives • 1.0 • important information on the dis- solution of quantum numbers. -u.o - y'y' Here it should be noticed that the •y motional narrowing mechanism »-0 - y-y' [85, 86] was studied numerically y'y' to.o - by calculating a strength function • lO'.O -JI.0 -|J'.» HO J*0 MO for the system with two degrees of Figure 5.5 Simulated Ey-EY plot for freedom very similar to the pre- integrable case. (Taken from Ref.82) sent model Hamiltonian. It is a very interesting problem to explore whether the rotational damping Is induced by the dissolution of quan- tum numbers specifying the colle- ctive rotational motion or by the motional narrowing mechanism. - •' *• ^* • •'*' '' "'' -•o. o -it- o -ii- a 11. a )». o to. o a) Case A. •31.0 -12.0 IJ-0 31-0 10.0 -J*.O -lJ.0 1J-0 J».O tO.O b) Case B. c) Case C. Figure 5.6. Simulated Ey-Ey plot for three characteristic cases in Fig.5.2. (Taken from Ref.82.) 5.5, inexhaustible Rich Structure of the Hilbert Space In the preceding subsections, we have applied the quantum theory of nuclear collective dynamics summarized in §4 to a simple model Hamil- tonian with two degrees of freedom. It was illustrated by using numerical calculation that a violation of separability condition (4.52) seen in the FPP gives a reasonable criterion for an onset of quantum chaos. However, the analysis in the preceding subsections is too coarse to examine exceed- ingly rich structure of the Hilbert space. Here it is worthwhile to notice that a set of expansion coefficients a^y1 appearing in the FPP is directly related with the separability condition (4.52). As is easily recognized from the definition of nv-representation in §4.5, the exact eigenstate li> is approximated as H.v within the first order perturbation. Here, the zero-th order state I i>(°) is selected out of the (p.,v)-basis states by using the maximum overlap condi- tion with li>, i.e. U>(0)=lmvi> with max {< nvli >} . (5.12) Comparing Eq.(4.55) with Eq.(5.11), one gets a relation, P ^v,: foi*H). (5.13) insofar as the perturbative treatment starting from I nivi> is applicable for reproducing the exact eigenstate li>. It is clear from Eqs.(5.11) and (5. 13) that the violation of separability condition induces a mixture of the (^,v)-basis states. At this stage, one should remember an essential role of the resonant denominator discussed in §4.6. If the exact eigenstate li> is in regular regime, there is only one large expansion coefficient a^v/ in the FPP which expresses the zero-th order component of the wave function. If the effects coming from the perturbative interaction JiJsepa) start to act, there realizes physical situation where the discussion in §4.6 is applica- ble; If one additional and non negligible expansion coefficient denoted as a a ears in novo' fao* ^ PP the FPP besides the maximum coefficient aWvj' , one may introduce a local resonant subspace D^=2 and resonant Hamilto- nian ^V=2 expressed as : flnlVl). -83 - (5,4| where we use the notations in Eqs.(4.63) and (4.64). As discussed in §4.6, the resonant Hamiltonian Pffc2 generates a pairs of new eigenstates characterized by new quantum numbers (3^fi), even though the separabil- ity condition is violated. Namely, there occurs a local rearrangement of the quantum numbers from ((i,v) to (5\£Q), conserving the number of the quantum numbers. Since the above stated situation is expected to occur many times, there happen many local rearrangements of the quantum number sufficiently before the system reaches to the dissipative regime. In order to get geometrical and topological understanding what are happening in the Hilbert space under a presence of the resonance denominator, let us introduce the Husimi distribution function [81] ^(qi.pi: q2.P2) of the exact eigenstate li> by using the coherent state defined in Eq.(5.5); Pi = ^-V -a), ql = -^(a* + a) . • (5.15) A profile of the distribution on the qi-pi plane is given by ^'(qi.Pi) = Jdp2^'(qi.Pi;q2=0.P2) • (5.16) In Fig.5.7, we illustrate g'(qi.pi) for six typical eigenstates. The para- meters used in this calculation are N=30, ei-eo=l, £2-eo= 2 and Vi=V2 =0.01. Note that the system with these parameters is very near vicinity of the quantum integrable regime, because the interaction strength are sufficiently small in comparison with the level spacing. In the present case, the resonant condition is given by e2-e0H2(E1-e0) . (5.17) and the resonant oscillator quanta X. in Eq.(4.61) is approximately ex- pressed as -HI - . (5.18) Here it should be noticed from the structure of the interaction in Eq.(5.2) that the small resonant denominator first appears in the third order perturbation term, provided one evaluates the effects of the inter- action on the basis of the mn-representation. If one uses the fxv-repre- sentation, as is explicitly shown in Eq.(5.11), the resonant denominator appears in the first order perturbation term. This fact simply means a superiority of the (n,v)-basis states over the (m.n)-basis states. Now let us consider a resonant subspace D^=40, because the effects of the interaction become large near jy=40 region due to the boson enhancement factor. Since we are considering (m,n)=(even,even)-sector, the (n.v)-basis states contained in D^=40 are (^,v)=(20,10), (16,12), (12,14), (8,16), (4,18) and (0,20) whose unperturbative energies approx- imately degenerate around EnV=40. Our numerical calculation shows that there are large couplings between (16,12) and (12,14), and between (8,16) and (4,18). Figs.5.7a and 5.7f show the profiles of the distribution 0qi.Pi) for the exact eigenstates whose main components are described by a single (n,v)-basis state, i.e. (20,10) and (0,20), respectively. Figs.5.7b and 5.7c show the profiles of exact eigenstates which are mainly compos- ed of a pair of (n.v)-basis states;{(8,16),(4,18)}, whereas Figs.5.7d and 5.7e are the profiles of exact eigenstates whose main components are a pair of (n.v)-basis states;{(16,12).(12,14)). Namely V^=40 contains two small dimensional resonant subspaces D^=2 : {J8.16). 14.18)} and {) 16.12). |12,14)} , (5.19) to which one may apply the discussion in §4.6. In Fig.5.8, we illustrate the phase space structure of the correspond- ing classical system governed by the classical-level Hamiltonian in Eq.(5.4). It is the Poincare section map of various trajectories on pi-qi plane with condition q2=0. The parameters used in calculating Fig.5.8 are the same as those used in Fig.5.7 and the total energy of the classical system is E=40. In Fig.5.8, there appear five kinds of regular motions; The inner most, middle and outer most concentric circles. In between these three types of concentric circles, there exist additional two regular motions expressed by inner four-crescents and outer four-crescents structure. Although the separatrices are not explicitly shown in Fig.5.8, - 8:") - b) e) 1*1.*|l.|.t.tl.•#.•!! Ml.•|l*l>t.fl.••.»!! CICC"H**t- ' III * III(>II»U> " III ' ». If 1*1.»H«I.t.tl. ••• ill I.'1 'I I Figure 5.7. A profile of Husimi distribution of the exact eigenstate on qi plane. (Taken from Ref.87.) -86- each four-crescents struc- ture is surrounded by a corresponding separatrix. It is well known from the general theory of the non- linear classical system that a pair of stable and unsta- ble fixed points usually appears simultaneously, provided that there exists a resonant denominator. The stable fixed point just 01 corresponds to the center Figure 5.8. Phase space structure of the of the crescent and the classical system corresponding to Fig.5.7. separatrix is running (Taken from Ref.87.) through the unstable fixed point. Here it should be noticed that the situation in the classical system just stated above is very similar to what we encountered in the quantum syst- em discussed in §4.6; two pairs of quantum resonant states in Fig.5.7 (i.e. (Figs.5.7b and 5.7c) and {Figs.5.7d and 5.7e}) and two pairs of fixed points (one stable and one unstable) in Fig.5.8 are results of an existence of the resonant denominator. They always appear and disappear in pair irrespective of whether it is quantum or classical. A similarity between quantum resonant states and classical fixed points is spectacular; the inner pair of fixed points in Fig.5.8 shows almost the same topological structure as the profiles of exact eigenfunction in Figs.5.7b and 5.7c, and the outer pairs in Fig.5.8 shows the same as in Figs.5.7d and 5.7e. The outermost and inner most concentric circles in Fig.5.8 have almost the same geometrical structure as the profiles of eigenfunctions in Figs.5.7a and 5.7f. Figure 5.7 is the first result of our numerical calculation on the phase space structure of the exact eigenfunctions whose characteristic feature are well explained by the quantum theory of nuclear collective dynamics. It is astonishing that there also exists an exceedingly rich structure even in the Hilbert space near vicinity of the quantum Integrable regime specified by a certain local (jo.,v) -basis states. 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