Persistence of Eigenvalues and Multiplicity in the Dirichlet Problem for the Laplace Operator on Nonsmooth Domains

We consider the Dirichlet eigenvalue problem for the Laplace operator on a variable nonsmooth domain. We extend a result of Lupo and Micheletti concerning the structure of the set of domain perturbations which leave the multiplicity of an eigenvalue unchanged, and we study the set of perturbations which leave a certain eigenvalue unchanged.     

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(εz + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω_i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.     

处理偏心圆柱形气缝导热的一个微扰方法

本文讨论固体导热问题中偏心圆柱形气缝所引起的影响。我们利用二维拉普拉斯方程线性迭加解与复变函数的级数展开之间有对应关系,把热传导方程连同边界条件的定解问题变化成线性代数的一个本征值问题,它的本征值表示气缝内外界面沿圆周方向平均温度的差值,它的本征矢量则表达气缝内外界面上温度的角分布。由于气体导热系数比固体导热系数小很多,所以用微扰方法容易把本征值算到一级近似,零级近似的本征矢量可以从等温圆柱形界面气缝导热问题的熟知的保角变换解得到,这样便简单地给出偏心对气缝内外界面沿圆周方向平均温度的差值的影响。如果气缝的偏心是随机的,结果如何统计处理,在本文末节也讨论到。     

Ein Eigenwertverfahren zur Lösung des Einteilchenmodells für Atomkerne in ihrer Sattelpunktsdeformation

To study saddle point states of fissioning nuclei in the single-particle model nucleon wave functions and energy-levels for highly deformed nuclei are needed. In this paper a method is developped to calculate single-particle wave functions and energy eigenvalues for potentials with any axially symmetric deformation. For that purpose the eigenvalue problem for the partial diefferential operatorH is replaced by a discrete approximation. A special method is worked out to solve the resulting algebraic eigenvalue problem.     

A numerical method for acoustic normal modes for shear flows

The normal modes and their propagation numbers for acoustic propagation in wave guides with flow are the eigenvectors and eigenvalues of a boundary value problem for a non-standard Sturm-Liouville problem. It is non-standard because it depends non-linearly on the eigenvalue parameter. (In the classical problem for ducts with no flow, the problem depends linearly on the eigenvalue parameter.) In this paper a method is presented for the fast numerical solution of this problem. It is a generalization of a method that was developed for the classical problem. A finite difference method is employed that combines well known numerical techniques and a generalization of the Sturm sequence method to solve the resulting algebraic eigenvalue problem. Then a modified Richardson extrapolation method is used that dramatically increases the accuracy of the computed eigenvalues. The method is then applied to two problems. They correspond to acoustic propagation in the ocean in the presence of a current, and to acoustic propagation in shear layers over flat plates.     

Unified Solutions of the Hard-Core Fermi-and Bose-Hubbard Models

A unified algebraic approach to both the hard-core Fermi- and Bose-Hubbard models is extended to boththe finite- and infinite-site with periodic condition cases. Excitation energies and the corresponding wavefunctions ofboth the models with nearest neighbor hopping are exactly derived by using a new and simple algebraic method. It isfound that spectra of both the models are determined simply by eigenvalue problem of N × N hopping matrix, where Nis the number of sites for finite system or the period of sites for infinite system.     

Two and k-Photon Jaynes–Cummings Models and Dirac Oscillator Problem in Bargmann–Segal Representation

In this work, we show that the Bargmann–Segal representation is a very simple approach to obtain the energy eigenvalues of some two-level quantum systems. It is shown that for 2-photon and k-photon Jaynes–Cummings models, the Bargmann–Segal realization gives the same energy eigenvalue which obtained by Lie algebraic and the matrix methods. We also study the Dirac oscillator problem in this representation.     

Phase space derivation of a variational principle for one-dimensional Hamiltonian systems

We consider the bifurcation problem u″ + λu = N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue λ. A new derivation of a variational principle for the lowest eigenvalue λ is given. This derivation makes use only of simple algebraic inequalities and leads directly to a more explicit expression for the eigenvalue than what had been given previously.     

Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp（4,R） Lie Algebra

Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4,R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4,R)\supset U(2).     

Quarkonium bound-state problem in momentum space revisited

A semi-spectral Chebyshev method for solving numerically singular integral equations is presented and applied in the quarkonium bound-state problem in momentum space. The integrals containing both, logarithmic and Cauchy singular kernels, can be evaluated without subtractions by dedicated automatic quadratures. By introducing a Chebyshev mesh and using the Nystrom algorithm the singular integral equation is converted into an algebraic eigenvalue problem that can be solved by standard methods. The proposed scheme is very simple to use, is easy in programming and highly accurate.     

The quantum inverse scattering method for Hubbard-like models

This work is concerned with various aspects of the formulation of the quantum inverse scattering method for the one-dimensional Hubbard model. We first establish the essential tools to solve the eigenvalue problem for the transfer matrix of the classical “covering” Hubbard model within the algebraic Bethe ansatz framework. The fundamental commutation rules exhibit a hidden 6-vertex symmetry which plays a crucial role in the whole algebraic construction. Next we apply this formalism to study the SU(2) highest weights properties of the eigenvectors and the solution of a related coupled spin model with twisted boundary conditions. The machinery developed in this paper is applicable to many other models, and as an example we present the algebraic solution of the Bariev XY coupled model.     

Complex FEM modal solver of optical waveguides with PML boundary conditions

A full-wave modal analysis of two-dimensional, lossy and anisotropic optical waveguides using the finite element method (FEM) is presented. In order to describe the behavior of radiating fields, anisotropic perfectly matched layer boundary conditions are applied for the first time in modal solvers. The approach has been implemented using high order edge elements. The resulting sparse eigenvalue algebraic problem is solved through the Arnoldi method. Application to an antiresonant reflecting optical waveguide is reported.     

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ? 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.     

An approximate solution scheme for the algebraic random eigenvalue problem

A reduced basis formulation is presented for the efficient solution of large-scale algebraic random eigenvalue problems. This formulation aims to improve the accuracy of the first order perturbation method, and also allow the efficient computation of higher order statistical moments of the eigenparameters. In the present method, the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for Ritz analysis of the governing random eigenvalue problem. This leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest. Since, only two basis vectors are used to represent each eigenvector, explicit expressions for the random eigenvalues and eigenvectors can readily be derived. This enables the statistics of the random eigenparameters and the forced response to be efficiently computed. Numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system. It is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method.     

Angular momenta and eigenvectors of the weakly coupled T ⊗ t Jahn‐Teller system

The eigenvalues of the weakly coupled T ? t Jahn‐Teller problem are known for several decades, and the same holds also true for the eigenstates. These, however, are only given in the traditional position representation, which proves inconvenient if one attempts to extend the weak‐coupling treatment into the region of stronger coupling. Here the solution of the T ? t eigenvalue problem at weak coupling is derived in terms of creation and annihilation operators. This reformulation of the problem is nontrivial, since the algebraic form of the oscillator eigenvectors, being simultaneous angular‐momentum eigenstates, has been worked out only recently and is probably still widely unknown. The electronic and oscillator eigenstates are then coupled to form eigenvectors of the total angular momentum. Finally, in preparation for an extension of the weak‐coupling treatment, the action of the boson creation and annihilation operators on the oscillator eigenvectors is calculated, thus completing the algebraic approach to the weakly coupled T ? t system.     

Method for calculation of helical-waveguide eigenmodes on the basis of solving the equivalent two-dimensional problem by field expansion in circular-waveguide modes

We present a method for calculating the dispersion characteristics of eigenmodes of metal waveguides with helical corrugations on the inner surface, which is based on the transition to a new nonorthogonal system of coordinates. In the new coordinate system, the problem of finding helical-waveguide modes is rigorously equivalent to the problem of finding the modes of a circular unitradius waveguide which has an anisotropic filling and is homogenous in the longitudinal direction. To solve the equivalent problem, we expand the field of the desired eigenmode in the modes of an empty circular unit-radius waveguide. As a result, the problem is reduced to solving a generalized algebraic eigenvalue problem. The comparison with the results of earlier three-dimensional calculations shows that the developed method allows one to determine characteristics of helical-waveguides with sufficient accuracy for many important applications and requires much lower calculation costs.     

体积算符对任意价顶角的本征作用与本征值谱

利用体积算符所含抓三元组对自旋网任意价顶角中圈线作用的反称化和双元恒等式，证明了这种作用均为本征作用，本征值为-2.用代数方法给出了对任意价顶角求体积算符本征值的系统程式，得到了普遍情况下的3，4，5和6顶角体积本征值的具体代数表式. 关键词： 体积算符 任意价顶角 本征作用证明 本征值谱     

Integrability and exact solution of correlated hopping multi-chain electron systems

Exact quantum integrability is established for a class of multi-chain electron models with correlated hopping and spin models with interchain interactions, by constructing the related Lax operators and R-matrices through twisting and gauge transformations. Exact solution of the eigenvalue problem for commuting conserved quantities of such systems is achieved through algebraic Bethe ansatz, on the examples of Hubbard and t–J models with correlated hopping. Our systematic construction identifies the integrable subclass of such known solvable models and also generates new systems including the generalized t–J models. At the same time it makes proper correction to a well known model and resolves recent controversies regarding the equivalence and solvability of some known models.     

Spurious resonances in a version of the algebraic Resonating-Group Method

It is shown that the conditions claimed to transform the algebraic version of the Resonating-Group Model, originally invented for scattering problems, into a complex eigenvalue problem corresponding to resonant states are necessary but not sufficient. This can be concluded from the fact that false resonances are produced along with true ones. They can be distinguished and discarded by introducing an arbitrary non-linear parameter. The true solutions are invariant against this parameter but the false ones can be swept out even into non-physical regions of the energy.Devoted to Prof. E.W.Schmid on the occasion of his 60th birthday     

Soliton dynamics and elastic collisions in a spin chain with an external time-dependent magnetic field

In this paper, we investigate the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. Such dynamics can be described by a Landau–Lifshitz-type equation. We apply the Darboux transformation method to the linear eigenvalue problem associated with this equation, and obtain the multi-soliton solution with a purely algebraic iterative procedure. Through the analytical analysis and graphical illustrations for the solutions obtained, we discuss in detail the effects of an external magnetic field on the nonlinear wave. Under the action of an external field, although the amplitude, width and depth of soliton vary periodically with time and its symmetry property is changeable, the soliton can also propagate stably and it possesses particle-like behavior.