On the spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator $$\hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3,$$ is proved given that either $A\in C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)$, 2q > n-2, or the Fourier series of the vector potential $A:R^n\to R^n$ is absolutely convergent. Here, $\hat V^{(s)}=(\hat V^{(s)})^*$ are continuous matrix functions and \hat V^{(s)}\hat \alpha_j=(-1}^s\hat \alpha_j\hat V^{(s)} for all anticommuting Hermitian matrices $\hat \alpha_j$, $\hat \alpha_j^2=\hat I$, s=0,1.Comment: 17 page

Motivic Milnor fibre for nondegenerate function germs on toric singularities

We study function germs on toric varieties which are nondegenerate for their Newton diagram. We express their motivic Milnor fibre in terms of their Newton diagram. We extend a formula for the motivic nearby fibre to the case of a toroidal degeneration. We illustrate this by some examples.Comment: 14 page

Correlations, spectra, and instability of phase-space density fluctuations in open-cluster models

The dynamical evolution of six open star cluster models is analyzed using the correlation and spectral analysis of phase-space density fluctuations. The two-time and mutual correlation functions are computed for the fluctuations of the phase-space density of cluster models. The data for two-time and two-particle correlations are used to determine the correlation time for phase-space density fluctuations ((0.1-1) τ v.r., where τ v.r. is the violent relaxation time of the model) and the average phase velocities of the propagation of such fluctuations in cluster models. These velocities are 2-20 times smaller than the root mean square velocities of the stars in the cluster core. The power spectra and dispersion curves of phase-space density fluctuations are computed using the Fourier transform of mutual correlation functions. The results confirm the presence of known unstable phase-space density fluctuations due to homologous fluctuations of the cluster cores. The models are found to exhibit a number of new unstable phase-space density fluctuations (up to 32-41 pairs of fluctuations with different complex conjugate frequencies in each model; the e-folding time of the amplitude growth of such fluctuations is (0.4-10) τ v.r. and their phases are distributed rather uniformly). Astrophysical applications of the obtained results (irregular structure of open star clusters, formation and decay of quasi-stationary states in such clusters) are discussed. © 2013 Pleiades Publishing, Ltd