Exact solution of one class of Maryland model

The Hamiltonian H of one-body Maryland model is defined as the sum of a linear unperturbed Hamiltonian H_0 and the interaction V, which is a Toeplitz matrix. Maryland model with a doubly infinite Hilbert space are exactly solved. Special cases of one-body Maryland model include the original Maryland model (Phys. Rev. Lett. 49, 833 (1982) and Physica 10D, 369 (1984)), which describes a quantum kickied linear rotator and single band Bloch oscillations. Maryland model and single band Bloch oscillations are the same Hamiltonian in two different representations. A special case of many-body Maryland model is Luttinger model.Comment: 5 pages, no figure

General Theory of the Quantum Kicked Rotator. I

This is the first of a series of two papers. We discuss some basic problems of the quantum kicked rotator (QKR) and review some important results in the literature. We point out the flaws in the inverse Cayley transform method to prove dynamic localization. When $\tau/2\pi$, where $\tau$ is the kick period, is very close to a rational number, the localization length is larger than the typical localization length. We analytically prove anomalous localization and confirm it by numerical calculations. We point out open problems that need further work.Comment: 10 pages, 6 figure

Some new results on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int. J. Inf. Coding Theory, 2010, 1, pp. 149-170).Comment: 21 pages. We have changed the title of our pape

Competition between phase coherence and correlation in a mixture of Bose-Einstein condensates

Two-species hard-core bosons trapped in a three-dimensional isotropic harmonic potential are studied with the path-integral quantum Monte Carlo simulation. The double condensates show two distinct structures depending on how the external potentials are set. Contrary to the mean-field results, we find that the heavier particles form an outer shell under an identical external potential whereas the lighter particles form an outer shell under the equal energy spacing condition. Phase separations in both the spatial and energy spaces are observed. We provide physical interpretations of these phase separations and suggest future experiment to confirm these findings.Comment: 4 pages, 4 figures, submitted to Physical Review Letter

Non-convex Penalty for Tensor Completion and Robust PCA

In this paper, we propose a novel non-convex tensor rank surrogate function and a novel non-convex sparsity measure for tensor. The basic idea is to sidestep the bias of $\ell_1-$norm by introducing concavity. Furthermore, we employ the proposed non-convex penalties in tensor recovery problems such as tensor completion and tensor robust principal component analysis, which has various real applications such as image inpainting and denoising. Due to the concavity, the models are difficult to solve. To tackle this problem, we devise majorization minimization algorithms, which optimize upper bounds of original functions in each iteration, and every sub-problem is solved by alternating direction multiplier method. Finally, experimental results on natural images and hyperspectral images demonstrate the effectiveness and efficiency of the proposed methods

Condensate-profile asymmetry of a boson mixture in a disk-shaped harmonic trap

A mixture of two types of hard-sphere bosons in a disk-shaped harmonic trap is studied through path-integral quantum Monte Carlo simulation at low temperature. We find that the system can undergo a phase transition to break the spatial symmetry of the model Hamiltonian when some of the model parameters are varied. The nature of such a phase transition is analyzed through the particle distributions and angular correlation functions. Comparisons are made between our calculations and the available mean-field results on similar models. Possible future experiments are suggested to verify our findings.Comment: 4 pages, 4 figure

Elliptic billiard - a non-trivial integrable system

We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable system. However, second order statistics exhibit a novel property of long-range oscillations. Classical simulation shows that all the periodic orbits except two are not isolated. In Fourier analysis of the spectrum, all the peaks correspond to periodic orbits. The two isolated periodic orbits have small contribution to the fluctuation of level density, while non-isolated periodic orbits have the main contribution. The heights of the majority of the peaks match our semiclassical theory except for type-O periodic orbits. Elliptic billiard is a nontrivial integrable system that will enrich our understanding of integrable systems.Comment: 5 pages, 6 figure

Global Level Number Variance in Integrable Systems

We study previously un-researched second order statistics - correlation function of spectral staircase and global level number variance - in generic integrable systems with no extra degeneracies. We show that the global level number variance oscillates persistently around the saturation spectral rigidity. Unlike other second order statistics - including correlation function of spectral staircase - which are calculated over energy scales much smaller than the running spectral energy, these oscillations cannot be explained within the diagonal approximation framework of the periodic orbit theory. We give detailed numerical illustration of our results using four integrable systems: rectangular billiard, modified Kepler problem, circular billiard and elliptic billiard.Comment: 5 pages, 3 figure

A Model for Stock Returns and Volatility

We prove that Student's t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further argue that a more accurate measure of the volatility may be possible based on the fact that stock returns can be understood as the product distribution of the volatility and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.Comment: 17 pages, 30 figures, 2 table

Spectral and Parametric Averaging for Integrable Systems

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos - spectral averaging and parametric averaging. For spectral averaging, we introduce a new procedure - rescaled spectral averaging. Unlike traditional spectral averaging, it can describe the correlation function of spectral staircase and produce persistent oscillations of the interval level number variance. Parametric averaging, while not as accurate as rescaled spectral averaging for the correlation function of spectral staircase and interval level number variance, can also produce persistent oscillations of the global level number variance and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.Comment: 7 pages, 7 figure