On the Spectral Theory of Operator Pencils in a Hilbert Space

Abstract

Consider the operator pencil L _λ = A ? λB ? λ ² C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L _λ are investigated in the case when A, B and C are K-positive and K-symmetric operators in H, and a method to bracket the eigenvalues of L _λ is developed by using a variational characterization of the problem (i) L _λ u = 0. The method generates a sequence of lower and upper bounds converging to the eigenvalues of L _λ and can be considered an extension of the Temple-Lehman method to quadratic eigenvalue problems (i).     

Eigenvectors of the recursion operator and a symmetry structure for the coupled KdV hierarchy

Abstract

It is shown that eigenvectors of the recursion operator L with the eigenvalue λ _i and the inverse of the recursion operator L _i ≡ L?λ_i for the coupled KdV hierarchy (CKdVH) can be obtained in terms of squared eigenfunctions of the associated linear problem. The symmetry structure and corresponding infinite dimensional Lie algebras of CKdVH are also given. Using both the local and nonlocal symmetries of CKdVH, one can obtain some exact group invariant solutions and various new infinite-dimensional and finite-dimensional integrable models.     

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and S_L(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.     

Thickness and angular dependence of the L‐edge X‐ray absorption of nickel thin films

We report on the near‐edge X‐ray absorption fine structure spectroscopy of the L₃ (2p_3/2) and L₂ (2p_1/2) edges for ferromagnetic pure nickel transition metal and show that the L_2,3 edge peak intensity and satellite feature at ~6 eV above the L₃ edge in nickel increase with increasing nickel film thickness both in the total electron yield and transmission modes. The absorption spectra of nickel metal, however, exhibit strong angular‐dependent effects when measured in total electron yield mode. In addition, we calculated the mean electron escape depth of the emitted electrons (λ_e), which was found for pure nickel metal to be λ_e=25 ± 2 Å. We point out the advantages of the total electron yield technique for the study of the L‐edge of 3d transition metals. Copyright © 2011 John Wiley & Sons, Ltd.     

光纤四波混频的零色散波长涨落效应和相位敏感放大

本文讨论了光纤四波混频零色散波长的随机涨落,运用随机微分方程(SDE)推导出平均参量增益、相位转换效率及增益带宽的理论表达式.理论与实验有很好的一致性.进一步分析了这一效应对相位敏感放大器的噪音系数可能产生的限制.     

Frequency reference for atomic transitions of Rb D<Subscript>2</Subscript>-line based on the effect of selective reflection

On an example of the D₂-line of the Rb atoms the work of the frequency reference of atomic transitions is demonstrated, based on the application of the spectrum of a selective reflectance (SR) from the boundary of atom vapors with the use of nano-cell (NC) with the thickness L ~ λ/2, where λ is the laser wavelength equal to 780 nm. When changing the thickness of the nano-cell near the thickness L ~ λ/2, we observe the inversion of sign of the SR slope profile which is positive when L < λ/2 and negative when L > λ/2. In the case when the incidence angle of the laser beam on the surface of the nano-cell is close to the normal, in real-time it is possible to form the derivative of the SR which represents a resonance peak with ~35 MHz spectral linewidth and located at the atomic transition. The phenomenon of oscillation of the sign of slope while changing the nano-cell thickness from L ~ λ/2 up to L ~ 3/2λ is demonstrated. The practical application of the SR is noted.     

半柔性网络剪切模量的计算

采用ANSYS软件计算了二维半柔性网络的剪切变形,得到了纤维绕交点可自由旋转和不可自由旋转的剪切模量G_fix和G_free,分析了纤维交点数、长度、纤维力学参数对半柔性网络剪切模量的影响,对比发现G_fix/G_free存在极小值,G_fix/G_free≈1.05. 关键词： 半柔性网络 复杂网络动力学 软凝聚态物质 多尺度     

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT

Three classes of reciprocal graphs, viz. monocycle (G^C_n), linear chain (G^L_n) and star (G^K_n) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (G^C_n) and linear chain (G^L_n) are obtained by putting a pendant vertex to each vertex of simple monocycle (C_n) and simple linear chain (L_n), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n ? 1) peripheral vertices of the star graph (K_{1, (n?1)}). An n-fold rotational axis of symmetry for G^C_n and (n ? 1)-fold rotational axis of symmetry for G^K_n have been exploited for obtaining their respective condensed graphs. The condensed graph for G^L_n has been generated from that of G^C_n incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.     

On the theory of the diffraction of light by two parallel ultrasonic waves,one being then th harmonic of the other

Assuming the validity ofRaman andNath's preliminary theory, the amplitudes of the light diffracted by two parallel ultrasonic waves consisting of a fundamental tone and its n^th harmonic are compared in the following cases:

1. (1)

   the sound waves are superposed;

2. (2)

   the sound waves are situated side by side.

The diffraction patterns are approximately the same if the condition πnL/λ^*¦sin ?_s¦?1 is fulfilled, where sin?_s=?sλ/μ₀λ^*(s being an integer);λ is the wave-lenght of the incident light in vacuum, λ^* is the wave-lenght of the fundamental tone of the sound waves, μ₀ is the refractive index of the medium andL the width of the sound field. The good agreement between recent experiments ofMurty andRao in the arrangement of case (2) and theoretical results calculated in case (1) is thus justified. The theoretical confirmation of older experiments byPande, Pancholy andParthasarathy is thereby established at the same time.

     

Is there a hadron spin-flip contribution to the Coulomb-hadron interference at small momentum transfer and high energies?

The analysing power A_N is examined in the range of the Coulomb-hadron interference on the basis of the experimental data from p_L = 6 GeV/c up to 200 GeV/c taking account of a phenomenological analysis at p_L = 6 GeV/c and a dynamic high-energy spin model. The results are compared with the new RHIC data at p_L = 100 GeV/c. The new experimental data obtained at RHIC indicate small contributions of the hadron spin-flip amplitude.     

The Effect of Rotating Cylindrical Langmuir Probes in Magnetron Plasmas

Using a cylindrical Langmuir probe, the plasma properties (ion density, electron temperature, floating and plasma potentials) in a magnetron sputter source have been investigated, along one particular line‐of‐sight, but for different probe‐orientations with respect to the B‐field. The plasma in the region hosen for observation is haracterised by electrons, which are magnetised (Larmor radius r_le < both the electron mean‐free‐path λ_e, and plasma extension L) and ions, which are not (their Larmor radius r_I > L, λ_I). Through the development of a simple expression for the electron saturation current at different probe angles relative to the local B‐field, it is possible to correct for the diminished electron currents due to their restricted transport across the field. The results indicate that the measured ion density, electron temperature, floating and plasma potentials are unaffected by the probe orientation, however the electron saturation current is attenuated when the probe is aligned along the B‐field. A simple model for the collection of electrons indicates that classical electron diffusion may not operate in the magnetron, with cross‐field electron transport dominated by anomalous, possibly Bohm, diffusion.     

Symmetry Classification of the One–Dimensional Second Order Equation of a Hydrodynamic Type

Abstract

The paper contains a symmetry classification of the one–dimensional second order equation of a hydrodynamical type L(Lu)+λLu=F (u), where L ≡ ? _t+u? _x. Some classes of exact solutions of this equation are given.     

Dependence of the cesium absorption D1 linewidth on the thickness of atomic vapor column

A new nanocell has been elaborated, where the thickness of the atomic vapor column varies smoothly in the range L = 350–5100 nm. The cell allows studying the behavior of the resonance absorption at the D₁ line of cesium atoms by varying the thickness from L = λ / 2 to L = 5 λ with the step λ / 2 (λ being the resonant wavelength of the laser, 894 nm) and the laser intensity. It is shown that at low laser intensities a narrowing of the resonance absorption spectrum is observed for L = (2n + 1)λ/2 (with an integer n) up to L = (7/2)λ, whereas for L = nλ the spectrum broadens. The developed theoretical model well describes the experiment.     

基于金银合金薄膜的近红外表面等离子体共振传感器研究

利用淀积在玻璃衬底上的金银合金薄膜作为表面等离子体共振(SPR)芯片, 构建了Kretschmann结构的近红外波长检测型SPR传感器. 采用不同浓度的葡萄糖水溶液测试了金银合金薄膜SPR传感器的折射率灵敏度. 实验结果表明随着入射角从7.5°增大到 9.5°, SPR吸收峰的半高峰宽从292.8 nm 减小到 131.4 nm, 共振波长从 1215 nm蓝移到 767.7 nm, 折射率灵敏度从35648.3 nm/RIU 减小到 9363.6 nm/RIU.在相同的初始共振波长(λ_R)下获得的金银合金薄膜SPR折射率灵敏度高于纯金膜(纯金膜在λ_R=1215 nm下的折射率灵敏度为29793.9 nm/RIU). 利用1 μmol/L的牛血清蛋白(BSA)水溶液测试了传感器对蛋白质吸附的响应.结果表明, BSA分子吸附使得金银合金薄膜SPR吸收峰红移了12.1 nm而纯金膜SPR吸收峰仅红移了9.5 nm. 实验结果还表明, 在相同λ_R下, 金银合金薄膜SPR吸收峰的半高峰宽大于纯金膜的半高峰宽, 因此其光谱分辨率比纯金膜SPR传感器低. 关键词： 金银合金薄膜 表面等离子体共振 波长检测型 高灵敏度     

Innere Konversion der 161 keV- und 223 keV-Übergänge in133Cs

Using a Ge(Li)-Si (Li)-spectrometer the 161 keV and 223 keV transitions of¹³³Cs were investigated. TheK-conversion coefficients were deduced to beα _K =0.205±0.007 andK-conversion coefficients were deduced to beα _K =0.0743±0.0043, respectively. TheK/L ratio for the 161 keV transition is 4.72±0.37. From theK-conversion coefficients were deduced to beα _K -values,M1 penetration factors ofλ=8.6±3.4 andλ=7.8 _?2.8 ^3.0 were deduced which are inconsistent with known data from angular correlation experiments (λ=40±10 andλ=90±13). The penetration factor fromK/L-measurement for the 161 keV-transition isλ=52±19. A theoretical reinvestigation of the formulas used in the literature appears recommendable. Furthermore, theK/L ratio of the 1770 keV transition of²⁰⁷Pb was determined experimentally to be 5.52±0.54.     

确定等价电子杨盘基的等概率比对方法

给出了等价电子正则杨盘T^［λ］_ig的基本对称算子、完全对称算子概念,同时给出了这些对称算子作用于任一Slater函数_i所产生的根态、生成态概念.由正交归一化杨盘T^［λ］_ie的纵置换算子A^［λ］_ie的构造规则,给出了A^［λ］_ie中存在的对称算子和确定T^［λ］_ie的等概率比对方法,从而基本避免了牵涉到许多算子的极其复杂的代数,给出了求解N值较大的电子系统杨盘基问题的新方法. 关键词： 正则杨盘 对称算子 根态 等概率比对方法     

A new theoretical development of the limiting electric conductivity of ions in solution

The physical interpretation of the limiting equivalent conductivity of binary electrolytes in solution, Λ⁰ (=Σ_i?=?+,– λ⁰_i), has been a challenge for almost a century. Here, as a follow-up on a preliminary communication [D. Fraenkel, Mol. Phys. 115, 2944 (2017)], a quantitative theoretical treatment is offered and fully developed for the limiting ionic equivalent conductivity, λ⁰_i, based on assuming that around any singled-out ion i in solution, there is an electrostatic atmosphere that includes oriented solvent dipoles and the counterion as ‘charge density’. This leads to a simple mathematical expression for λ⁰_i as the result of the tandem operation of the dipole viscosity and dipole relaxation forces on the drifting ion. Contrary to literature teaching, λ⁰_i is found proportional to the effective ionic radius, r_i. An ‘electric ion-radius scale’ derived from experimental λ⁰_i’s for aqueous solutions, compares reasonably well with known ion-size scales. A newly defined solvent-specific dipole random-walk correction factor, ξ, is used for nonaqueous solvents as an adjustable parameter, to attain a parallel electric ion-size scale. The new theory provides overall a comprehensive view of the limiting ion mobility in solution, based on the ionic properties, solvent variables and universal constants.     

AlGaN/GaN场板结构高电子迁移率晶体管的场板尺寸优化分析

对不同场板尺寸的AlGaN/GaN 场板结构高电子迁移率晶体管进行了研究,建立简化模型分析场板长度对沟道电场分布的影响.结果表明,调整钝化层厚度和场板长度都可以调制沟道电场的分布形状,当场板长度较小时,随着长度的增大器件击穿电压随之增加,而当长度增大到一定程度后器件击穿电压不再增加.通过优化场板长度,器件击穿电压提高了64%,且实验结果与模拟结果相符. 关键词： AlGaN/GaN 击穿电压 场板长度     

Quantum diffusion and scaling of localized interacting electrons

A new numerical method is introduced that enables a reliable study of disorder‐induced localization of interacting particles. It is based on a quantum mechanical time evolution calculation combined with a finite size scaling analysis. The time evolution of up to four particles in one dimension is studied and localization lengths are defined via the long‐time saturation values of the mean radius, the inverse participation ratio and the center of mass extension. A systematic study of finite size effects using the finite size scaling method is performed in order to extract the localization lengths in the limit of an infinite system size. For a single particle, the well‐known scaling of the localization length λ₁ with disorder strength W is observed, λ₁ ∝ W^—2. For two particles, an interaction‐induced delocalization is found, confirming previous results obtained by numerically calculating matrix elements of the two‐particle Green's function: in the limit of small disorder, the localization length increases with decreasing disorder as λ₂ ∝ W^—4 and can be much larger than <$>\mitlambda λ₁. For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n‐particle localization length λ_n is derived and shown to be in agreement with the numerical data, λ_n ∝ λ₁. Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ₁ and n. Hence, no extendedstates exist in one dimension, at least for spinless fermions.     

E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.