Statistical theory of nuclear reactions and the Gaussian orthogonal ensemble

Using methods developed in field theory and statistical mechanics, especially in the context of the Anderson model as generalised by Wegner, a novel approach to the statistical theory of nuclear reactions is developed. A finite set of N bound states, coupled to each other by an ensemble of Gaussian orthogonal matrices, is considered and coupled to a set of channels via fixed coupling matrix elements. The ensemble average and the variance of the elements of the nuclear scattering matrix are evaluated, using the method of a generating function combined with the replica trick, followed by the Hubbard-Stratonovitch transformation and a modified loop expansion. In the limit N → ∞, it is shown quite generally that, aside from a trivial dependence on average S-matrix elements, the variance depends only on the transmission coefficients, and that the correlation width of a pair of S-matrix elements is given by a universal function of the transmission coefficients. A modified loop expansion yields an asymptotic series valid for strong absorption. The terms in this series are partly novel, and partly coincide with results obtained earlier in the framework of a model which did not take account of the GOE eigenvalue fluctuations. This suggests that average cross sections are mainly sensitive to the stiffness of the GOE spectrum. Fluctuation properties are also derived, and the link to Ericson fluctuation theory is established.     

Isoscalar Giant Monopole Resonance in Relativistic Continuum Random Phase Approximation

The isoscalar giant monopole resonance （ISGMR） in nuclei is studied in the framework of a fully consistent relativistic continuum random phase approximation （RCRPA）. In this method the contribution of the continuum spectrum to nuclear excitations is treated exactly by the single particle Green＇s function technique. The negative energy states in the Dirac sea are also included in the single particle Green＇s function in the no-sea approximation. The single particle Green＇s function is calculated numerically by a proper product of the regular and irregular solutions of the Dirac equation. The strength distributions in the RCRPA calculations, the inverse energyweighted sum rule m-1 and the centroid energy of the ISGMR in ^120Sn and ^208Pb are analysed. Numerical results of the RCRPA are checked with the constrained relativistic mean field model and relativistic random phase approximation with a discretized spectrum in the continuum. Good agreement between them is achieved.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Distribution of eigenfrequencies for vibrating plates

Acoustic spectra of free plates with a chaotic billiard shape have been measured, and all resonance frequencies in the range 0-500 kHz have been identified. The spectral fluctuations are analyzed and compared to predictions of the Gaussian Orthogonal Ensemble (GOE) of random matrices. The best agreement is found with a superposition of two independent GOE spectra with equal density which indicates that two types of eigenmodes contribute to the same extent. To explain and predict these results a detailed theoretical analysis is carried out below the first cut-off frequency where only flexural and in-plane vibrations are possible. Using three-dimensional plate dispersion relations and two-dimensional models for flexural and in-plane vibrations we obtained two first terms of the asymptotic expansion of the counting function of these eigenmodes. The contribution of edge modes is also discussed. The results are in a very good agreement with the experimentally measured number of modes. The analysis shows that the two types of modes have almost equal level density in the measured frequency interval, and this explains the observed spectral statistics. For a plate with broken symmetry in the up-down direction (where flexural and in-plane modes are strongly coupled) experimentally observed spectral fluctuations correspond to a single GOE spectrum. Above the first cut-off frequency a greater complexity of the spectral fluctuations is expected since a larger number of types of modes will contribute to the spectrum. Received 5 January 1999 and Received in final form 5 September 1999     

Eigenvalue density of cross-correlations in Sri Lankan financial market

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.     

Fluctuation properties of states in26Al

The fluctuation properties of the energy levels of²⁶Al from the ground state to excitation energies of about 8 MeV have been examined. Several statistics have been employed, and particular attention has been given to the suitability of these statistics for small sample sizes. The results show, for a variety of states, behavior intermediate between GOE and Poisson but apparently favoring GOE. The results are consistent with the fluctuations' being independent of isospin.     

Inverse problem for the Schrödinger equation with a repulsive potential

We study the dependence of the repulsive potentials on the discrete spectral characteristics of the Schrödinger equation. The behavior of the regular solutions and the corrections to the potential for various changes to the spectrum are analyzed. It is shown that for a change in the number of bound states, the asymptotic correction to the potential is related to the period of classical vibrations in the field of the reference potential.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 12–16, August, 1984.     

Local impurity phase pinning and pinning force in charge density waves

Starting from the static Fukuyama-Lee-Rice equation for a three-dimensional incommensurate charge density wave (CDW) in quasi one-dimensional conductors a solvable model for local phase pinning by impurities is defined and studied. We find that average CDW energy and average pinning force show critical behaviour with respect to the pinning parameter h. Specifically the pinning force exhibits a threshold at h=1 with exponent . Our model exemplifies a general concept of local impurity pinning in which the force exerted by the impurity on the periodic CDW structure becomes multivalued and metastable states appear beyond a threshold. It is found that local impurity pinning becomes less effective at low temperatures and may eventually cease completely. These results are independent of spatial dimensionality as expected for local impurity pinning. Comparison with Larkin's model is also made. Received 8 July 1998     

GRID lifetime measurements in59, 61, 63Ni following thermal neutron capture

The GRID-method has been used to measure the lifetimes of ten excited states in^{59, 61, 63}Ni following thermal neutron capture in Ni targets of natural isotopic composition. Four of the lifetimes have been determined for the first time, the other six lifetimes can be compared with the results of conventional DSA-measurements following charged particle induced reactions. Cascade feeding effects have been included in the analysis. Level energies and electromagnetic properties of negative parity states in⁵⁹Ni have been compared with the results of shell model calculations in 3p0h and 4p 1h model spaces. Statistical model estimates of the lifetimes as function of excitation energy and spin are also given.Work partially supported by Deutsches BMFT under contract 06GOE141     

Velocity fluctuations of a column of water streaming down a wire: the possibility of one-dimensional turbulence

In this paper we study the influence of the magneto-coupling effect between the longitudinal motion component and the transverse Landau orbits of an electron on transmission features in single barrier structures. Within the parabolic conduction-band approach, a modified one-dimensional effective-mass Schr?dinger equation, including the magneto-coupling effect generated from the position-dependent effective mass of the electron, is strictly derived. Numerical calculations for single barrier structures show that the magneto-coupling effect brings about a series of the important changes for the transmission probability, the above-barrier quasi-bound states, and the tunneling time. Through examining the variation of the above-barrier resonant-transmission spectrum with the barrier width and observing the well-defined Lorentzian line-shape of the above-barrier resonant peaks, we convincingly show that the above-barrier resonant transmission in single barrier structures is delivered by the above-barrier quasibound states in the barrier region, just as the below-barrier resonant tunneling in double barrier structures is mediated by the below-barrier quasi-bound states in the well. Furthermore, we come to the conclusion that the magneto-coupling effect brings about not only the splitting of the above-barrier quasi-bound levels but also the striking reduction of the level-width of the quasi-bound states, correspondingly, the substantial increase of the density of the quasi-bound states. We suggest that magneto-coupling effects may be observed by the measurements of the optical absorption spectrum associated with the above-barrier quasi-bound states in the single barrier structures. Received: 26 September 1997 / Revised: 26 November 1997 / Accepted: 15 December 1997     

New rational extensions of solvable potentials with finite bound state spectrum

Using the disconjugacy properties of the Schrödinger equation, we develop a new type of generalized SUSY QM partnership which allows generating new solvable rational extensions for translationally shape invariant potentials having a finite bound state spectrum. For this we prolong the dispersion relation relating the energy to the quantum number out of the physical domain until a disconjugacy sector. By Darboux–Bäcklund Transformations built on these prolonged states we obtain new regular isospectral extensions of the initial potential. We give the spectra of these extensions in terms of new orthogonal polynomials and study their shape invariance properties.     

Destruction of order in nuclear spectra by a residual GOE interaction

We consider hamiltonians of the type H = H₀ + λV_GOE where H₀ is a fixed N × N matrix and V_GOE represents a gaussian orthogonal ensemble. The change as a function of λ of the average level density and of the eigenvector correlations is studied, and related to the distribution of branch points of H. It is shown that the GOE interaction completely dominates the spectral properties of H when its spectrum covers the spectrum of H₀.     

Level repulsion in the ground-state region of nuclei

Using the nuclei in the nuclear table as members of an ensemble of hamiltonianas we show that, once nuclei with a systematic behaviour are eliminated, the nearest-neighbour spacings between states with the same spin and parity follow, in the ground state region, a distribution which is roughly consistent with that predicted by the gaussian orthogonal ensemble (GOE). These is some indication, which however is not conclusive, that the two-body random hamiltonian ensemble fits the data somewhat better than the GOE.     

Levelstatistics and stochasticity in a driven quantum system

We investigate quantum properties of one anisotropic spin driven by an external time-dependent magnetic field which shows a transition from regular to irregular dynamics with increasing field strength in the classical limit. In particular we study the statistical properties of the quasi-spectrum. Our results support the conjecture that Poisson- and GOE-statistics are to be associated with integrable and nonintegrable systems resp. in the semiclassical limit. Approaching the quantum case we observe significant deviations from GOE statistics.     

Resonant Continuum in the Relativistic Mean-Field Theory

Energies, widths and wave functions of the single-particle resonant continuum are determined by solvingscattering states of the Dirac equation with proper asymptotic conditions for the continuous spectrum in the relativisticmean-field theory. The relativistic regular and irregular Coulomb wave functions are calculated numerically. Theresonance states in the continuum for some closed- or sub-closed-shell nucleus in Sn-isotopes, such as 1 14Sn, 1 16Sn, 1 18Sn,and 120Sn are calculated. Results show that the S-matrix method is a reliable and straightforward way in determiningenergies and widths of resonant states.     

RVB description of the low-energy singlets of the spin 1/2 kagomé antiferromagnet

Extensive calculations in the short-range RVB (Resonating valence bond) subspace on both the trimerized and the regular (non-trimerized) Heisenberg model on the kagomé lattice show that short-range dimer singlets capture the specific low-energy features of both models. In the trimerized case the singlet spectrum splits into bands in which the average number of dimers lying on one type of bonds is fixed. These results are in good agreement with the mean field solution of an effective model recently introduced. For the regular model one gets a continuous, gapless spectrum, in qualitative agreement with exact diagonalization results. Received 7 March 2000     

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.     

A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

Theoretical Investigation of Femtosecond-Resolved Photoelectron Spectrum of Na2 Molecules

Effect of laser fields on Na2 interaction potentials is studied by calculating the time-resolved photoelectron spectrum （TRPES） with the time-dependent wave-packet method. It is shown that the photoelectron spectrum at different delay times reflects the population in different electronic states. We inspect the periodicity of vibrational motion in neutral states, and map the vibrational wave-packet propagation in corresponding internuclear coordinate.