No Arabic abstract
In this paper, we discuss duality about components of invariant variety of periodic points(IVPP) and fundamental domain of recurrence equation, and present an algorithm for the derivation of all components of IVPPs of any rational maps. It is based on the study of two examples of a 2 dimensional map and a 3 dimensional map. In particular, all components of IVPPs of the 2 dimensional example are completely determined by means of the cyclotomic polynomials.
Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau equation in bounded domains are considered. We investigate the effect of the boundaries on spiral motion under homogeneous Neumann boundary conditions, for small values of the twist parameter $q$. We derive explicit laws of motion for rectangular domains and we show that the motion of spirals becomes exponentially slow when the twist parameter exceeds a critical value depending on the size of the domain. The oscillation frequency of multiple-spiral patterns is also analytically obtained.
We present the fundamental solutions for the spin-1/2 fields propagating in the spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach.
The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, ${bf x}={bf x}(tau)$, $t=t(tau)$ instead of ${bf x}={bf x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $tilde H=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $tilde H$ vanishes for any solution, $tilde H=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $hat{tilde H}Psi=0$, which is precisely the Schrodinger equation $ihbarpartial_tPsi=hat HPsi$. As an illustration, we discuss quantum mechanics of the relativistic particle.
We study the spectrum of the linear operator $L = - partial_{theta} - epsilon partial_{theta} (sin theta partial_{theta})$ subject to the periodic boundary conditions on $theta in [-pi,pi]$. We prove that the operator is closed in $L^2([-pi,pi])$ with the domain in $H^1_{rm per}([-pi,pi])$ for $|epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{rm per}([-pi,pi])$.
Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this paper, we investigate the existence of such points in the spectrum of the 3-dimensional Schr{o}dinger operator $H = - Delta +V(textbf{x})$ with $V(textbf{x})$ being in a large class of periodic potentials. Specifically, we give very general conditions on the potentials which ensure the existence of 3-fold Weyl points on the associated energy bands. Different from 2-dimensional honeycomb structures which possess Dirac points where two adjacent band surfaces touch each other conically, the 3-fold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. To ensure the 3-fold and 3-dimensional conical structures, more delicate, new symmetries are required. As a consequence, new techniques combining more symmetries are used to justify the existence of such conical points under the conditions proposed. This paper provides comprehensive proof of such 3-fold Weyl points. In particular, the role of each symmetry endowed to the potential is carefully analyzed. Our proof extends the analysis on the conical spectral points to a higher dimension and higher multiplicities. We also provide some numerical simulations on typical potentials to demonstrate our analysis.