Transfer Matrix for the Cantor Staircase Fractal Potential

The precision analytical method developed previously for studying nonrelativistic particle tunneling through the self-similar fractal potential is used to derive the transfer matrix for the Cantor staircase fractal potential. A generalized functional equation for the transfer matrix is derived and a solution is retrieved for the case in which the fractal dimensionality of the corresponding Cantor set is close to unity. The tunneling parameters are calculated including the transmission coefficient and phases of scattered waves.     

Bispectrum Analysis of One-Dimensional Regular Fractals with Additive Random Noise

The bispectrum of the Cantor set that is a typical regular fractal is calculated and its fractalities are shown. A relation between the bispectrum and the fractal dimension of the object is elucidated. Effects of additive random noise on the scaling property of the bispectrum are compared with those of the corresponding power spectrum.     

Propagation of Waves Through one-Dimensional Cantor-Like Fractal Media

We consider the resonant tunneling of electrons through one-dimensional Cantor-like fractal barriers. By means of a transfer matrix method, we present a general formalism to calculate the transmission and give some numerical examples. It is found that the transmission spectrum shows rich fractal patterns due to the self-similar geometry of the Cantor set. The scaling behaviour of the transmission spectrum is explored. By plotting the amplitudes of the wave functions, we also investigate the quasi-localization properties of the electrons.     

Transmission of quantum particles through a one-dimensional fractal potential barrier

We consider scattering of a nonrelativistic quantum particle by a one-dimensional fractal potential barrier carried by a generalized Cantor set. We obtain recurrence relations for the reflection coefficient and examine the scaling properties as functions of the wave number.V. D. Kuznetsov Siberian Physicotechnical Institute, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–127, July, 1993.     

Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives

In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.     

Cumulative Diminuations with Fibonacci Approach,Golden Section and Physics

In this study, physical quantities of a nonequilibrium system in the stages of its orientation towards equilibrium has been formulated by a simple cumulative diminuation mechanism and Fibonacci recursion approximation. Fibonacci p-numbers are obtained in power law forms and generalized diminuation sections are related to diminuation percents. The consequences of the fractal structure of space and the memory effects are concretely established by a simple mechanism. Thus, the reality why nature prefers power laws rather than exponentials ones is explained. It has been introduced that, Fibonacci p-numbers are elements of a Generalized Cantor set. The fractal dimensions of the Generalized Cantor sets have been obtained by different methods. The generalized golden section which was used by M.S. El Naschie in his works on high energy physics is evaluated in this frame.     

Fractal model of the atom and some properties of the matter through an extended model of scale relativity

The scale relativity model was extended for the motions on fractal curves of fractal dimension D _F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D _F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D _F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.     

Phase Times of Electron Passage through the Self-Similar Fractal Potential

Within the framework of the method developed previously for studying electron tunneling through the self-similar fractal potential (SFP), functional equations are derived for phase tunneling times (PTT). Their analysis shows that in the case of the SFP, the PTT cannot be interpreted as a dwell time of the particle in the barrier region. In particular, it turns out that the PTT through SFP barriers is negative for the short-wavelength region. This is also the case for SFP wells and electrons with arbitrary energy. It is shown that for the limiting SFP, when the fractal dimensionality is equal to unity, the PTT coincide with those for the -potential of the same power. Thus, the PTT are independent of the barrier width for the limiting SFP. This extraordinary result is caused by the fact that, as demonstrated previously, the probability that the electron with exactly preset energy is found inside the limiting SFP is equal to zero.     

Diffusion dynamics in external noise-activated non-equilibrium open system-reservoir coupling environment

The diffusion process in an external noise-activated non-equilibrium open system-reservoir coupling environment is studied by analytically solving the generalized Langevin equation. The dynamical property of the system near the barrier top is investigated in detail by numerically calculating the quantities such as mean diffusion path, invariance, barrier passing probability, and so on. It is found that, comparing with the unfavorable effect of internal fluctuations, the external noise activation is sometimes beneficial to the diffusion process. An optimal strength of external activation or correlation time of the internal fluctuation is expected for the diffusing particle to have a maximal probability to escape from the potential well.     

High-Dimensional Integrable Models with Conformal Invariance

Using the (2 1)-dimensional Schwartz dcrivative, the usual (2 1)-dimensional Schwartz Kadomtsev-Petviashvili (KP) equation is extended to (n 1)-dimensional conformal invariance equation. The extension possesses Painlcvc property. Some (3 1)-dimensional examples are given and some single three-dimensional camber soliton and two spatial-plane solitons solutions of a (3 1)-dimensional equation are obtained.     

关于推广的组态混合法(Ⅰ)

本文提出了一个借助无规位相近似法的解,应用推广的组态混合法以求得更精确解的方法,导出了有关矩阵元的一组明显表达式。此外并证明了推广的组态混合法是自洽的,给出了有关的格林函数所满足的运动方程及展开式,根据后者对推广的组态混合法的解的性质进行了讨论。     

On the possibility of an analytic solution of the three-body problem

Three-body Faddeev equations in the Noyes-Fiedeldey form are rewritten as a matrix analog of a one-dimensional nonrelativistic Schrödinger equation. Unlike the method of K-harmonics, where a similar equation was obtained by expansion of a three-body Schrödinger equation wavefunction into the orthogonal set of functions of two variables (K-harmonics), the use of the Noyes-Fiedeldey form of Faddeev equations allows us to limit ourselves to the expansion in functions of one variable only. The solutions of the above mentioned matrix equation are obtained. These solutions converge uniformly within every interval of continuity of the matrix, which corresponds to the potential of that equation. Their asymptotic behavior for large interparticle distances is discussed. The solutions for the harmonic oscillator, inverse-square, and Coulomb-Kepler potentials are found. It is shown that energy levels in the last case may be calculated from a simple formula which is very similar to the corresponding formula for the two-body Coulomb-Kepler problem. This formula can be easily generalized to the case of n particles interacting with inverse distance potentials.     

On Asymptotic Nonlocal Symmetry of Nonlinear Schrödinger Equations

Abstract

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schrödinger equations. An important class of solutions of the free Schrödinger equation with improved smoothing properties is obtained.     

含超导材料Cantor序列一维光子晶体的传输特性

利用传输矩阵法研究了按分形三分Cantor序列构成的一维超导光子晶体的传输特性。数值结果表明,与传统电介质材料构成的光子晶体的频谱不同,这种光子晶体频谱没有展现出自相似性。此外,当构成光子晶体的两种材料的光学厚度相同时,光子晶体的频谱具有明显的周期性,并且在光子带隙中间成对地出现缺陷模。这种特性可以用于设计通道滤波器。     

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

In this article, we study the (2+1)-extension of Burgers equation and the KP equation. At first, based on a known Bäcklund transformation and corresponding Lax pair, an invariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation is worked out. Given a known solution and using the invariance, we can find solutions of the (2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgers equation which cannot be directly obtained by constraining the invariance of the (2+1)-extension of Burgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgers equation can help us find the solutions of KP equation. At last, based on the invariance of Burgers equation, the corresponding recursion formulae for finding solutions of KP equation are digged out. As the application of our theory, some examples have been put forward in this article and some solutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.     

Lagrange formalism for particles moving in a space of fractal dimension

Analogs of the Lagrange equation for particles evolving in a space of fractal dimension are obtained. Two cases are considered: 1) when the space is formed by a set of material points (a so-called fractal continuum), and 2) when the space is a true fractal. In the latter case the fractional integrodifferential formalism is utilized, and a new principle for devising a fractal theory, viz., a generalized principle of least action, is proposed and used to obtain the corresponding Lagrange equation. The Lagrangians for a free particle and a closed system of interacting particles moving in a fractal continuum are derived. Zh. Tekh. Fiz. 68, 7–11 (February 1990)     

A note on open‐chain transfer matrices from q‐deformed su(2|2)S‐matrices

In this note, we perform Sklyanin's construction of commuting open‐chain/boundary transfer matrices to the q‐deformed SU(2|2) bulk S‐matrix of Beisert and Koroteev and a corresponding boundary S‐matrix. This also includes a corresponding commuting transfer matrix using the graded version of the q‐deformed bulk S‐matrix. Utilizing the crossing property for the bulk S‐matrix, we argue that the transfer matrix for both graded and non‐graded versions contains a crucial factor which is essential for commutativity.     

Partial diagonalization of Bethe-Salpeter type equations

We consider an equation of the Bethe-Salpeter type, with arbitrary potential and kernel, respectively, for space-like momentum transfer. The invariance group of the equation is then the Lorentz-group in three dimensions, the O(1, 2) group. The standard procedure for the diagonalization of such equations (valid for square integrable solutions only) is generalized to include the case of power bounded solutions, by means of a generalized O(1, 2) expansion formalism. The result is a two-dimensional integral equation for the O(1, 2) expansion coefficients. The right-most l-plane singularities of these determine the asymptotic behaviour of the amplitudes as in ordinary Regge theory. The formalism can be applied to other dynamical equations possessing O(1, 2) symmetry.