Crossings and Anticrossings of Unbound States

We investigate the characteristic crossings and anticrossings of energies and widths of a doublet of resonances, observed in the vicinity of, and at a degeneracy of unbound states, when the control parameters of the system are varied. This characteristic behavior is explained in terms of the local, topological structure of the surfaces that represent the complex energy eigenvalues in parameter space in the vicinity of a degeneracy point. In the simple but illustrative case of the scattering of a beam of particles by a double barrier potential well with two regions of trapping, we solved numerically the implicit, transcendental equation that defines the eigenwave numbers of a degenerate isolated doublet of resonances as functions of the real, control parameters of the system. We found that, at a degeneracy of unbound states, the surface representing the resonance eigenwave numbers as functions of the control parameters has an algebraic branch point of rank one. Unfolding the degeneracy point, crossings and anticrossings of energies and widths are obtained as projections of sections of the eigenwave number surfaces.     

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer’s vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer’s. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer’s method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.     

Resonance energy level selection in symmetrical rectangular quadruple-barrier structures

Resonant tunneling is studied theoretically in symmetrical rectangular quadruple-barrier structures by taking into account mass difference between the well layer and the barrier layer. Analytical expressions for the transmission coefficient and the resonance condition are derived. Especially, it is obtained that two independent resonance energies can be determined analytically as a function of the outer and the central well widths, which may be very useful for resonant tunneling device fabrications. Furthermore, the variation of the resonance energy with the outer barrier width is investigated. It is found that the resonance level is rarely affected with a small change of the outer barrier width. It is derived analytically that the transmission peak is decreased to less than unity when the outer barrier width becomes thinner than a critical value.     

Impedance conditions for resonance propagation and resonance localization of waves in barrier structures

Impedance conditions for the resonance propagation and resonance localization of waves in barrier structures are formulated. Analytical expressions for the eigenvalues of various barrier structures are derived.     

非完整超晶格中电子透射问题的计算机模拟

采用转移矩阵方法,模拟研究了垒高无序和阱宽无序非完整超晶格的电子态问题.计算了垒高无序有限超晶格的透射谱和其局域态波函数以及阱宽无序有限超晶格的透射谱和本征值,直观地给出了垒高无序和阱宽无序非完整有限超晶格其电子态行为的物理图像.模拟结果表明:垒高无序和阱宽无序这两种常见非完整一维有限超晶格的子带带隙间均存在强烈的电子运动定域化,且电子波的布喇格散射对周期性势场更敏感;这两种非完整性引起的局域,通过计算电子局域态波函数和有限系统的本征值得到了证实;对本文讨论的这种类型和周期的超晶格,如果控制阱宽在9.1～10.9nm间随机变化,即阱宽的值最大相差1.8岫时,计算机模拟的结果是,阱宽的这种非周期性开始使子带的带隙消失.     

Manifestation of exchange effects in heavy-ion interactions

Three different approaches to taking into account exchange effects in heavy-ion collisions are studied. Within the first of them, the lowest eigenstates of the Hamiltonian are treated as forbidden states. In the second approach, the eigenstates of the normalization kernel of the resonating-group model that correspond to zero eigenvalues are treated as forbidden states. The third approach takes additionally into account semiforbidden states. The ¹⁶O + ¹⁶O system is considered. A hybrid approach that combines the methods of discrete and continuous mathematics is developed for calculating the widths of narrow resonance states. The resonance width calculated within the approach that takes into account semiforbidden states proves to be sharply different from the widths obtained within traditional approaches.     

A note on perfect revivals in finite waveguide arrays

We propose a simple and algorithmic method for designing finite waveguide arrays capable of diffractionless transmission of arbitrary discrete beams by virtue of perfect revivals. Our approach utilises an inverse matrix eigenvalue theorem published by Hochstadt in 1974, which states that the Jacobi matrix, describing the system's discrete evolution equations, is uniquely determined by its eigenvalues and the eigenvalues of its largest leading principal submatrix, as long as the two sets of eigenvalues interlace. It is further shown that, by arranging the two sets of eigenvalues symmetrically with respect to zero, the resulting Jacobi matrix has zero diagonal elements. Therefore, arrays with arbitrary revival lengths can be obtained by engineering only the inter-waveguide couplings.     

A modified characteristic equation for stability analysis of periodic reciprocal systems

A linear reciprocal system with periodic coefficients is stable if the system monodromy matrix has simple structure and eigenvalues all of modulus unity. Under the assumption that the former condition is true, it is proved in this paper that this criterion is equivalent to the condition that the roots, α, of an algebraic equation all lie in the interval −1 ⩽ α ⩽ 1, and an explicit scheme is presented for the derivation of this algebraic equation in terms of the coefficients of the characteristic equation of the system monodromy matrix. The modified criterion provides considerable computational advantage over the usual form of the criterion.     

On the width of resonances

A method to get both upper and lower bounds on real and imaginary parts of resonance eigenvalues is extended to Schrödinger operators with exterior dilation analytic potential. We apply it to a simple model potential where the bound states and resonances are exactly known.     

Probability density function of the sum of N partially correlated speckle patterns

A method is developed which allows exact calculation of the probability density function of the sum of N correlated speckle patterns. To find the density function, it is only necessary to first find the eigenvalues of an N × N coherence matrix. When the eigenvalues are distinct, the density function can be expressed as a simple sum of N exponential terms.     

Statistical properties of resonance widths for open quantum graphs

We connect quantum compact graphs with infinite leads, and turn them into scattering systems. We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph. The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads. Finally, we analyse the statistical properties of the resonance widths and compare our results with the predictions of random matrix theory. Deviations appearing due to the dynamical nature of the system are pointed out and explained.     

Statistical properties of resonance widths for open quantum graphs

We connect quantum compact graphs with infinite leads, and turn them into scattering systems. We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph. The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads. Finally, we analyse the statistical properties of the resonance widths and compare our results with the predictions of random matrix theory. Deviations appearing due to the dynamical nature of the system are pointed out and explained.     

Random, but not so much a parameterization for the returns and correlation matrix of financial time series

A parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied. This parameterization allows easy introduction of non-stationarity and it shows several of the characteristics of the true, observed realizations, such as fat tails, volatility clustering, and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of Random Matrix Theory results. The predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of Brazilian assets and it is shown that those predictions fit the data better than Random Matrix Theory.     

Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality

We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the nonorthogonality of resonance wave functions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum.     

Modified Hill determinant approach to the eigenvalues of the anharmonic oscillator

The unperturbed Hamiltonian of quantum anharmonic oscillator is modified by introducing a simple variational scale parameter. A suitable choice of this parameter makes the eigenvalues rapidly convergent for small size of the determinant in the method of infinite Hill determinant. Simple analytic expressions for the eigenvalues are obtained by matrix diagonalization method.     

Ar+17和 Ar+16谱线的电子碰撞展宽

利用修改后伦敦理工大学的UCL扭曲波程序, 本文计算了自由电子与原子或离子的碰撞反应矩阵, 以此得到它的散射矩阵和碰撞强度.利用碰撞强度, 研究电子碰撞对辐射谱线影响.具体地, 以Ar⁺¹⁷和Ar⁺¹⁶的α线和β线为例, 计算了不同电子温度和密度下谱线的展宽和位移.     

Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

The rich phenomenology of crossings and anticrossings of energies and widths, observed in an isolated doublet of resonances when one control parameter is varied, is fully explained in terms of the topological properties of the energy hypersurfaces close to the degeneracy point. The hypersurface representing the complex resonance eigenvalues, as functions of the control parameters, has an algebraic branch point of rank one, and branch cuts in its real and imaginary parts, in parameter space. Associated with this singularity in parameter space, the scattering matrix, S _ℓ (E), and the Green’s function, G _ℓ ⁽⁺⁾(k; r,r'), have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of any degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence.     

Level Compressibility of Certain Random Unitary Matrices

The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of this quantity for different random unitary matrices describing models with intermediate spectral statistics. The computations are based on the approach developed by G. Tanner for chaotic systems. The main ingredient of the method is the determination of eigenvalues of a transition matrix whose matrix elements equal the squared moduli of matrix elements of the initial unitary matrix. The principal result of the paper is the proof that the level compressibility of random unitary matrices derived from the exact quantisation of barrier billiards and consequently of barrier billiards themselves is equal to 1/2 irrespective of the height and the position of the barrier.