Wannier-Stark effect on surface states

A recursive Green-function (RGF) technique is used to study the surface states of an electrified solid. The concatenating nature of the method leads to a continued-fraction form of the RGF, which can be expressed as a ratio of Bessel functions. The surface density of states (SDOS), derived from the RGF, reveals a quasi-Stark-ladder distribution of the energy levels at the surface atom. The dependence of the SDOS on the applied field and surface perturbation parameters is discussed.     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

The wavelet dimension function is the trace function of a shift-invariant system

In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

     

Mach's principle. Part 1. Initial state of the universe the universe

The integral formulation of equations of general relativity proposed earlier as a mathematical tool for Mach's principle forbids the conventional singular cosmologies but is compatible with the de Sitter initial space.     

Analysis of optical characteristics of holey photonic crystal devices with the extended coupled-mode formalism

Presented here is a new approach for analysis of the so-called holey photonic crystals—a class of electro-optical components, in which periodicity of air holes in dielectric media is used for confinement of light. This class includes several kinds of microstructured fibers, semiconductor lasers etc. Accurate evaluation of optical characteristics of those devices is usually a complicated problem due to the large dimensions and the fine structure of their refractive index distribution. Furthermore, usually, only numerical solutions for this class of optical components are available. The overwhelming majority of the physical models, suitable for analysis of holey photonic devices, proceed from the “natural” assumption: the devices are considered as arrays of air holes, surrounded by dielectric material. In this work we propose another model. Namely, we treat them as arrays of dielectric spots (waveguides), embedded in the air (cladding material). This model allows utilization of the extended coupled-mode theory (a relatively new approach designed for analysis of infinite arrays of coupled waveguides and previously considered inapplicable to holey optical components) for calculations of the latter. In this sense, we present a new method for analysis of holey photonic crystals. On the one hand, our method allows analytical evaluation of some optical characteristics of holey optical components (such as the number of photonic bands and bandwidth). On the other hand, accurate numerical computation of the photonic band structure of the holey photonic devices, incorporating a large number of holes, can be done with this technique on a timescale of several minutes.