We consider m spinless Fermions in l > m degenerate single-particle levels interacting via a kbody random interaction with Gaussian probability distribution and k ≤ m in the limit l → ∞ (the embedded k-body random ensembles). We address the cases of orthogonal and unitary symmetry. We derive a novel eigenvalue expansion for the second moment of the Hilbert-space matrix elements of these ensembles. Using properties of the expansion and the supersymmetry technique, we show that for 2k > m, the average spectrum has the shape of a semicircle, and the spectral fluctuations are of Wigner-Dyson type. Using a generalization of the binary correlation approximation, we show that for k m l, the spectral fluctuations are Poissonian. This is consistent with the case k = 1 which can be solved explicitly. We construct limiting ensembles which are either fully integrable or fully chaotic and show that the k-body random ensembles lie between these two extremes. Combining all these results we find that the spectral correlations for the embedded ensembles gradually change from Wigner-Dyson for 2k > m to Poissonian for k m l. C
Using a novel approach, we investigate the shape of the average spectrum and the spectral fluctuations of the k-body embedded unitary ensemble in the limit of large matrix dimension. We identify the transition point between semicircle and Gaussian shape. The transition also affects the spectral fluctuations which deviate from Wigner-Dyson form and become Poissonian in the limit k ≪ m ≪ l. Here m is the number of Fermions and l the number of degenerate singleparticle states.
We investigate the shape of the spectrum and the spectral fluctuations of the k-body Embedded Gaussian Ensemble for Bosons in the dense limit, where the number of Bosons m → ∞ while both k, the rank of the interaction, and l, the number of single-particle states, are kept fixed. We show that the relative fluctuations of the low spectral moments do not vanish in this limit, proving that the ensemble is non-ergodic. Numerical simulations yield spectra which display a strong tendency towards picket-fence type. The wave functions also deviate from canonical random-matrix behaviour.Introduction. -Random-matrix theory (RMT) successfully describes the statistical behaviour of spectra and wave functions of a large variety of systems such as atoms, molecules, atomic nuclei and quantum dots [1,2]. However, this RMT modeling is not completely realistic since all many-body systems are effectively governed by one-and two-body forces. This fact led to work on the two-body random ensembles for Fermions [3][4][5][6] and to the introduction of the k-body embedded ensembles by Mon and French [7]. In the embedded ensembles, many-body states are constructed by distributing m particles over l degenerate single-particle levels. The matrix of the k-body interaction with k ≤ m is taken in this basis. For k < m, the m-body matrix elements of the random k-body interaction are correlated: The number of independent random variables is smaller than in RMT. Do these more realistic embedded ensembles yield the same results as RMT? Early numerical simulations for interacting Fermion systems [3,4] of rather small matrix dimension have shown that the spectral fluctuation properties of the embedded ensembles agree with those of RMT. Similar results were obtained in numerical simulations for Bosonic systems [8,9]. Moreover, the Fermionic ensembles were shown to be ergodic. As far as we know, there are no results on the spectral ergodicity for Bosons.Recently, three of the present authors introduced a novel analytical approach to the Fermionic embedded ensembles in the limit of infinite matrix dimension (l → ∞) [10]. The main results of this approach are: (i) For 2k > m, the average spectrum has semicircle shape, and the spectral fluctuation properties coincide with those of RMT; (ii) the spectral density changes shape at or near 2k = m and becomes Gaussian; (iii) in the dilute limit (k ≪ m ≪ l) c EDP Sciences * * * We are grateful to O. Bohigas and T.H. Seligman for stimulating discussions and useful suggestions. T.A. acknowledges support from the Japan Society for the Promotion of
We consider the scattering motion of the planar restricted three-body problem with two equal masses on a circular orbit. Using the methods of chaotic scattering we present results on the structure of scattering functions. Their connection with primitive periodic orbits and the underlying chaotic saddle are studied. Numerical evidence is presented which suggests that in some intervals of the Jacobi integral the system is hyperbolic. The Smale horseshoe found there is built from a countable infinite number of primitive periodic orbits, where the parabolic orbits play a fundamental role.
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