Spectral fluctuation and 1/f(alpha) noise in the energy level statistics of interacting trapped bosons

It has been recently shown numerically that the transition from integrability to chaos in quantum systems and the corresponding spectral fluctuations are characterized by 1/f(alpha) noise with 1 <= alpha <= 2. The system of interacting trapped bosons is inhomogeneous and complex. The presence of an external harmonic trap makes it more interesting as, in the atomic trap, the bosons occupy partly degenerate single-particle states. Earlier theoretical and experimental results show that at zero temperature the low-lying levels are of a collective nature and high-lying excitations are of a single-particle nature. We observe that for few bosons, the P(s) distribution shows the Shnirelman peak, which exhibits a large number of quasidegenerate states. For a large number of bosons the low-lying levels are strongly affected by the interatomic interaction, and the corresponding level fluctuation shows a transition to a Wigner distribution with an increase in particle number. It does not follow Gaussian orthogonal ensemble random matrix predictions. For high-lying levels we observe the uncorrelated Poisson distribution. Thus it may be a very realistic system to prove that 1/f(alpha) noise is ubiquitous in nature.

Energy-level statistics of interacting trapped bosons

It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s = 0, which signifies the presence of a large number of quasidegenerate states.

On global linearization of planar involutions

Let phi: a"e(2) -> a"e(2) be an orientation-preserving C (1) involution such that phi(0) = 0. Let Spc(phi) = {Eigenvalues of D phi(p) | p a a"e(2)}. We prove that if Spc(phi) aS, a"e or Spc(phi) a (c) [1, 1 + epsilon) = a... for some epsilon > 0, then phi is globally C (1) conjugate to the linear involution D phi(0) via the conjugacy h = (I + D phi(0)phi)/2,where I: a"e(2) -> a"e(2) is the identity map. Similarly, we prove that if phi is an orientation-reversing C (1) involution such that phi(0) = 0 and Trace (D phi(0)D phi(p) > - 1 for all p a a"e(2), then phi is globally C (1) conjugate to the linear involution D phi(0) via the conjugacy h. Finally, we show that h may fail to be a global linearization of phi if the above conditions are not fulfilled.

TRANSLATION AND CULTURAL ADAPTATION OF THE GLOBAL APPRAISAL OF INDIVIDUAL NEEDS - INITIAL

The objective of this study was to perform the translation on and cultural adaptation of the Global Appraisal of Individual Needs - Initial instrument, and calculate its content validity index. This is a methodological study designed for the cultural adaptation of the instrument. The instrument was translated into Portuguese in two versions that originated the synthesis of the translations, which were then submitted to the evaluation of four judges, experts in the field of alcohol and other drugs. After the suggested changes were made, the instrument was back-translated and resubmitted to the judges and authors of the original instrument, resulting in the final version of the instrument, Avaliacao Global das Necessidades Individuais - Inicial. The content validity index of the instrument was 0.91, considered valid according to the literature. The instrument Avaliacao Global das Necessidades Individuais - Inicial was culturally adapted to the Portuguese language spoken in Brazil; however, it was not submitted to tests with the target population, which suggests further studies should be performed to test its reliability and validity.

Tradução e adaptação cultural do global appraisal of individual needs - initial

Este estudo objetivou traduzir e adaptar culturalmente o instrumento Global Appraisal of Individual Needs - Initial e calcular seu Índice de Validade de Conteúdo. Trata-se de estudo metodológico, de adaptação cultural do instrumento. O instrumento foi traduzido para o português em duas versões que deram origem à síntese das traduções, submetida à avaliação de quatro juízes experts na área de álcool e outras drogas. Após modificações, foi retraduzido e ressubmetido aos juízes e autores do instrumento original, resultando na versão final do instrumento, Avaliação Global das Necessidades Individuais - Inicial O Índice de Validade de Conteúdo do instrumento foi de 0,91, considerado válido pela literatura. O instrumento Avaliação Global das Necessidades Individuais - Inicial é um instrumento adaptado culturalmente para o português falado no Brasil; entretanto, não foi submetido a testes com a população-alvo, o que sugere que sejam realizados estudos que testem sua confiabilidade e validade.