Permanent magnetic moment in mesoscopic metals with spin-orbit interaction

We argue that at zero temperature an isolated metal particle (or an AB ring) with spin-orbit interaction and odd number of electrons will have a permanent magnetic moment, even in zero magnetic field (flux). In a zero-field-cooled state both the direction and the magnitude of the moment varies from particle to particle and averages to zero. In a field-cooled state it averages to $\sim \mu_{B}(k_{F}\ell) ^{1/2}$. We argue that the permanent moment is due to an uncompensated electron in the last occupied (Fermi) level. We introduce an effective single-electron Hamiltonian which accounts for spin-orbit coupling.Comment: 4 page

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

Quantum absorption in small metal particles

We evaluate the electric dipole absorption in small metal particles in a longitudinal electric field taking into account the Fermi-Thomas screening. When either the level broadening or the frequency of the field are larger than the mean energy-level spacing, the main contribution to absorption is classical, with quantum corrections. When both the broadening and the frequency are smaller than the mean level spacing, the absorption is manifestly quantum and can be understood in terms of the two-level system.Comment: 5 pages revte

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

On absence of steady state in the Bouchaud-M\'ezard network model

In the limit of infinite number of nodes (agents), the It\^o-reduced Bouchaud-M\'ezard network model of economic exchange has a time-independent mean and a steady-state inverse gamma distribution. We show that for a finite number of nodes the mean is actually distributed as a time-dependent lognormal and inverse gamma is quasi-stationary, with the time-dependent scale parameter.Comment: 6 pages, 4 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

Classical absorption in small metal particles and thin films

We study the electric dipole absorption in small metal particles and thin films in a longitudnal electric field. In diffusive approximation, we give both the phenomenological and microscopic derivations with the account for Thomas-Fermi screening and Drude relaxation.Comment: 11 page

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

Implied and Realized Volatility: A Study of the Ratio Distribution

We analyze correlations between squared volatility indices, VIX and VXO, and realized variances -- the known one, for the current month, and the predicted one, for the following month. We show that the ratio of the two is best fitted by a Beta Prime distribution, whose shape parameters depend strongly on which of the two months is used.Comment: 9 pages, 10 figures, 9 table

The Absence Of Saturation Of The Level Number Variance In A Rectangular Box

The variance of the number of levels in an energy interval around a level with large quantum numbers (semiclassical quantization) is studied for a particle in a rectangular box. Sampling involves changing the ratio of the rectangle's sides while keeping the area constant. For sufficiently narrow intervals, one finds the usual linear growth with the width of the interval. For wider intervals, the variance undergoes large, non-decaying oscillations around what is expected to be the saturation value. These oscillations can be explained as a superposition of just a few harmonics that correspond to the shortest periodic orbits in the rectangle. The analytical and numerical results are in excellent agreement.Comment: 15 pages, 4 figure