Sum rules for zeros and intersections of Bessel functions from quantum mechanical perturbation theory

Bessel functions play an important role for quantum states in spherical and cylindrical geometries. In cases of perfect confinement, the energy of Schrödinger and massless Dirac fermions is determined by the zeros and intersections of Bessel functions, respectively. In an external electric field, standard perturbation theory therefore expresses the polarizability as a sum over these zeros or intersections. Both non-relativistic and relativistic polarizabilities can be calculated analytically, however. Hence, by equating analytical expressions to perturbation expansions, several sum rules for the zeros and intersections of Bessel functions emerge.     

Mechanical model of round window membrane under reverse excitation

According to the vibration characteristics of the round window, a mechanical model of a round window membrane is established. The Euler equation of the round window and the complementary boundary conditions are derived by the variational principle. Combined with the Bessel function, an analytical solution of the round window displacement is obtained by MATHEMATICA. Combined with clinical characteristics of round window membrane lesion, the effect of sound transmission due to thickening of the round window membrane caused by the otitis media, shrinkage of the round window membrane area caused by otosclerosis, and hardening of the round window membrane itself is analyzed. The results show that with thickening of the round window membrane, the displacement of the round window membrane is decreased. In the meantime, with hardening of the round window membrane and shrinkage of the membrane area, the maximum displacement of the round window membrane is gradually reduced, leading to a decrease in sound transmission. Thus, the analytical analysis can avoid interference of environment and the technical level of personnel, and it can evaluate transmission performance of the round window membrane efficiently, providing a theoretical basis for the reverse excitation of artificial prosthesis.     

Spectral behavior of pulsed Bessel beams focused by a dispersive lens

Starting from the Huygens–Fresnel diffraction integral, the analytical expression for the power spectrum of pulsed Bessel beams focused by a dispersive aperture lens is derived and used to study the spectral anomalies of pulsed Bessel beams in the focused field. Numerical calculation results are given to illustrate the dependence of spectral anomalous behavior on the pulse parameters, truncation parameter and material dispersion of the lens. It is shown that near the phase singularities the spectral anomalies may take place. The potential applications of spectral anomalies of ultrashort pulsed beams in information encoding and information transmission are considered.     

Light-induced frequency shift of stimulated Raman scattering in Bessel light beams

The medium susceptibility related to stimulated Raman scattering (SRS) is known to be a complex value whose imaginary part determines the SRS gain coefficient whereas the real part constitutes a small fraction of the refractive index at the frequencies of the scattered emission that is proportional to the intensity of the exciting beam. Strong transverse nonuniformity of this beam causes focusing of high-frequency components of the scattered light and defocusing of low-frequency ones that inevitably affect their amplification efficiency. The effect for Gaussian pumping is responsible for a slight shift of the SRS gain spectrum towards higher frequencies. The present work analyzes theoretically SRS excited by a Bessel light beam and demonstrates that the axial component of the scattered emission experiences a frequency shift that is much greater than that for the case of Gaussian beam pumping. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 3, pp. 336–341, May–June, 2008.     

k-途径正则有向图

In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004.     

三线摆摆线质量对转动惯量精确测量的影响

三线摆法是测量转动惯量的常用方法, 理论上通常都是把摆线作理想处理(即摆线质量趋于0), 但在工程测量中, 被测工件往往较重, 摆线不再是轻质细线, 摆线质量势必会对空载悬盘的转动惯量测量产生影响. 本文应用哈密顿原理和变量替换, 导出三线摆摆线的变张力弦振动方程, 通过对摆线与悬盘连接点(x=0)的运动状态讨论, 推得摆线质量不能忽略情形下的对称式三线摆的转动惯量计算式, 并借助于MATLAB 软件快速便利地进行计算.     

Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics

In Phys. Lett. A 313 (2003) 343 we have found that the self-reciprocal Hankel transformation (HT) is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations' transformation in quantum mechanics.     

Vertical motion of a spherical body with decreasing mass

The paper studies the fall and ascent in air of a spherical body with radius linearly decreasing with time. The first integral of the nonlinear equation of motion is expressed in closed form in terms of Bessel functions on the assumption that the aerodynamic drag depends on the velocity as a quadratic polynomial. The second integral is evaluated by an approximate formula. The reliability of the analytic solution is confirmed by comparing it with a numerical solution of the Cauchy problem __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 118–125, June 2008.     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Frames of Periodic Shift-Invariant Spaces

This note studies Bessel sequences and frames of shift-invariant spaces generated by a countable set of periodic functions. We give characterizations under which the set of translations of the countable set is a Bessel sequence or a frame in terms of spectral decompositions of some self-adjoint operators.