The convergence of the lowest eigenvalue of a real nearly-symmetric matrix

The lowest eigenvalue of a real nearly-symmetric matrix is expressed as a perturbation series in terms of the eigenvalues of the symmetric part and the matrix elements of the skew-symmetric part. It is shown that the resulting series is closely related to the perturbation series for the lowest eigenvalue of a related hermitian matrix. This enables the behaviour of the lowest eigenvalue of a nearly symmetric matrix as the dimension of the matrix is increased to be deduced from the behaviour of the lowest eigenvalue of a hermitian matrix. This is of considerable importance as the behaviour of the lowest eigenvalue of a hermitian matrix as the dimension of the matrix is increased can be much more readily established. A possible application to Boys' transcorrelated method of calculating atomic and molecular energies is suggested.     

Accurate non-perturbative solution of eigenvalue problems with application to anharmonic oscillator

A method for eigenvalue problems is presented. As an example, we have obtained very accurate eigenvalues and eigenfunctions of the quartic anharmonic oscillator.The method is non-perturbative and involves the use of an appropriately scaled set of basis functions for the determination of each eigenvalue. The claimed accuracy for all eigenvalues is 15 significant figures. The method does not deteriorate for higher eigenvalues.     

Iterative approach for the eigenvalue problems

An approximation method based on the iterative technique is developed within the framework of linear delta expansion (LDE) technique for the eigenvalues and eigenfunctions of the one-dimensional and three-dimensional realistic physical problems. This technique allows us to obtain the coefficient in the perturbation series for the eigenfunctions and the eigenvalues directly by knowing the eigenfunctions and the eigenvalues of the unperturbed problems in quantum mechanics. Examples are presented to support this. Hence, the LDE technique can be used for nonperturbative as well as perturbative systems to find approximate solutions of eigenvalue problems.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

Stationary perturbation theory in the probability representation of quantum mechanics

In the probability representation of quantum mechanics, the eigenvalue problems in Hilbert space appear as *-genvalue equations. We show the possibility of employing the nondegenerate stationary perturbation method in the probability representation of quantum mechanics. The perturbed eigentomograms and the eigenvalues of energy are shown to be computed ab initio in terms of tomographic symbols of the operators involved.     

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.     

Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides

A surface integral method is presented to calculate the eigenmodes of an uniaxially anisotropic embedded channel waveguide. The electromagnetic field components are expressed with electric and magnetic vector potentials which are parallel with the optic axis and for which the scalar Helmholtz-equations hold using surface integral representation and the Green's functions of the vector potentials. The single component vector potentials are expanded in Fourier-series on the internal side of the dielectric interface. The Fourier-coefficients and the corresponding eigenvalues are obtained by minimizing the quadratic difference of the longitudinal field components in the cladding and core along the dielectric interface. The convergence of the series expansion is examined numerically and the eigenvalues obtained with the surface integral method agree with those obtained by Finite Element Method up to 5 significant digits.     

On the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods

Recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators. The standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed. They are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds. Further, the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues. A new method is presented which combines multigrid techniques with the Lanczos process. The resulting scheme requires O(mn) arithmetic operations and O(n) storage requirement, where n is the number of unknowns and m, the number of needed eigenvalues. The discretization of the considered differential operators is realized by means of p-finite elements and is applicable on general geometries. Numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods, at least on current workstations. The ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a PC shows the potential of this method. Practical applications are found, e.g. in the numerical simulation of quantum billiards.     

Eigenvalues of structures with uncertain elastic boundary restraints

The eigenvalue problems of structures with random elastic boundary supports are studied in this paper. Using the perturbation method, the differential equations including stochastic distributed parameters and random boundary conditions that govern the eigenproblems are transformed to a series of deterministic differential equations and boundary conditions. Then the differential equations and boundary conditions are discretized utilizing the finite element method (FEM). This process is easy to be implemented since the resulting perturbation equations with different orders own the same FEM meshes. The first-order and second-order sensitivities of eigenvalues are derived through solving the deterministic algebraic equations. Upon determining these sensitivities of eigenvalues, the approximate statistic expressions of random eigenvalues considering uncertain elastic boundary supports are established. At the end, several numerical examples are given to illustrate the application and effectiveness of the present method. Comparison of the present numerical results with those from the Monte-Carlo simulation method verifies the accuracy of the developed method.     

Formation of Superheavy Nuclei in Massive Fusion Reactions

We recently performed a series of improvement on evaluation of eigenvalues without complicated iterations.In this work we first discuss evaluation of the lowest eigenvalue for given systems,by which one conveniently obtains the value of the lowest eigenvalue based on the dimension and width of given matrix.We also discuss a strong correlation between eigenvalues and diagonal matrix elements for large matrices,by which one is able to predict eigenvalues approximately without iterations.     

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.     

Padé approximants on a lattice

The linear harmonic oscillator on a lattice is solved analytically. Wave functions and energy eigenvalues are expressed in terms of Mathieu functions and characteristic values of the Mathieu equation respectively. The Padé-approximant method for calculating the continuous limit of energy eigenvalues is tested. It is found that the values of approximants at infinity do not converge. A modification of the Padé method is proposed which leads to convergent series. Implications for more complicated systems are discussed.     

Radiative temperature dissipation in a finite atmosphere—I. The homogeneous case

We have developed a new approach toward solving problems of linear radiative relaxation of LTE temperature perturbations in a plane-parallel atmosphere of finite extent. We show that the mathematical problem is one of solving an integral eigenvalue equation, for which non-trivial solutions exist only for discrete values of the radiative relaxation time. The solutions for the spatial part of the perturbation constitute a complete and orthogonal set of basis functions, making it possible to solve more general problems of temperature relaxation. In applying this method to radiative relaxation in the middle atmosphere of earth, we show how the additional influences of photochemical coupling, advection by winds, and eddy diffusion by small-scale turbulence may be easily included using matrix perturbation techniques. We have solved the homogeneous integral equation for a wide variety of vertical thicknesses in an idealized homogeneous slab medium. Adopting a number of different analytic line profiles (rectangular, Doupler, Voigt, and Lorentz) we have obtained numerical solutions using an exponential-kernel method for solving the integral equation. The discrete eigenvalue “spectrum” is presented for vertical optical depths (0–10³) at line-center, and is used in solving several initial-value problems for a decaying temperature perturbation. We find that the eigenvalue spectrum is bounded from above by the lowest-order eigenvalue, and bounded from below by the familiar transparent approximation. The dependence of the lowest even eigenvalue on optical depth and the relative separation of the higher eigenvalues are found to depend sensitively on the line profile.     

Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms,a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters.In the proposed method,interval variables are used to quantitatively describe all the uncertain parameters.Different order perturbations in both eigenvalues and eigenvectors are fully considered.By retaining higher order terms,the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors.The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach.Compared with the Monte Carlo method,two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.     

The multiple scattering theory of phonons in mass disordered lattices

A version of the multiple scattering theory previously applied to the electronic problem of substitutional alloys is completed and modified in order to handle the problem of vibrations of general mass-disordered crystal lattices. The Born series is expressed in terms describing the scattering corrections to the appropriate effective medium. A new physical interpretation of the effective medium condition is given and the total partial summation of the Born series for the full scattering T-matrix is completed to the infinite order. The T-matrix is thus re-expressed in terms corresponding to resonant “ping-pong” scattering of renormalized phonons on all possible clusters of scattering centres. Confining to clusters of only limited number of scattering centres yields a set of successive approximations. The one-centre approximation for cubic crystals complies with the Taylor's selfconsistent procedure. A simple illustrative example in the two-centre approximation is solved and discussed.     

Eigenvalues for the extremely underdamped Brownian motion in an inclined periodic potential

The eigenvalues for the Brownian motion in a periodic potential with an additive constant force are investigated in the low friction limit. First the Fokker-Planck equation for the distribution function in velocity and position space is transformed to energy and position coordinates. By a proper averaging process over the position coordinate a differential equation for the distribution function depending on the energy only is obtained. Next the eigenvalues and eigenfunctions are calculated from this equation by a Runge-Kutta method. Finally the problem is formulated in terms of an integral equation from which the lowest non-zero eigenvalue is obtained analytically in the bistability region in the zero temperature limit.     

Krein Signatures for the Faddeev-Takhtajan Eigenvalue Problem

One of the difficulties in analyzing eigenvalue problems that arise in connection with integrable systems is that they are frequently non-self-adjoint, making it difficult to determine where the spectrum lies. In this paper, we consider the problem of locating and counting the discrete eigenvalues associated with the Faddeev-Takhtajan eigenvalue problem, for which the sine-Gordon equation is the isospectral flow. In particular we show that for potentials having either zero topological charge or topological charge ± 1, and satisfying certain monotonicity conditions, the point spectrum lies on the unit circle and is simple. Furthermore, we give an exact count of the number of eigenvalues. This result is an analog of that of Klaus and Shaw for the Zakharov-Shabat eigenvalue problem. We also relate our results, as well as those of Klaus and Shaw, to the Krein stability theory for J-unitary matrices. In particular we show that the eigenvalue problem associated to the sine-Gordon equation has a J-unitary structure, and under the above conditions the point eigenvalues have a definite Krein signature, and are thus simple and lie on the unit circle.     

The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.     

Accurate summation of the perturbation series for periodic eigenvalue problems

We show that algebraic approximants prove suitable for the summation of the perturbation series for the eigenvalues of periodic problems. Appropriate algebraic approximants constructed from the perturbation series for a given eigenvalue provide information about other eigenvalues connected with the chosen one by branch points in the complex plane. Such approximants also give those branch points with remarkable accuracy. We choose Mathieu's equation as illustrative example. Received 6 December 2000     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.