Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using Krylov–Schur method and Jacobi–Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov–Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization.     

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.     

Analytical solutions of coupled-mode equations for microring resonators

We present a study on analytical solutions of coupled-mode equations for microring resonators with an emphasis on occurrence of all-optical EIT phenomenon, obtained by using a cofactor. As concrete examples, analytical solutions for a 3×3 linearly distributed coupler and a circularly distributed coupler are obtained. The former corresponds to a non-degenerate eigenvalue problem and the latter corresponds to a degenerate eigenvalue problem. For comparison and without loss of generality, analytical solution for a 4×4 linearly distributed coupler is also obtained. This paper may be of interest to optical physics and integrated photonics communities.     

Orthogonal and Symplectic Matrix Models: Universality and Other Properties

We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures. For these models we find the bounds (similar to those for the hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also the universality of local eigenvalue statistics in the bulk.     

SUPER INTEGRABLE SYSTEMS AND ITS INFNITE CONSERVED CUKRENTS

The 3×3 super extended eigenvalue problem is considered. We deduce a class of completely integrable equations. Its reduced forms are givan and their infinite conserved currents have been discussed.     

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

We consider the ensemble of adjacency matrices of Erd?s-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption \({p N \gg N^{2/3}}\), we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd?s-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd?s-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.     

Rank One Non-Hermitian Perturbations of Hermitian $$\beta $$-Ensembles of Random Matrices

We provide a tridiagonal matrix model and compute the joint eigenvalue density of a rank one non-Hermitian perturbation of a random matrix from the Gaussian or Laguerre \(\beta \)-ensemble.     

A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

We develop a parallel Jacobi–Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi–Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc’s efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrödinger’s equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi–Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.     

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.     

A new Monte Carlo power method for the eigenvalue problem of transfer matrices

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Â^k are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.     

Estimates of the first two buckling eigenvalues on spherical domains

In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate of the second eigenvalue in terms of the first eigenvalue, which improves on a recent result obtained by Wang and Xia (2007) [1].     

Open Quantum Random Walks with Decoherence on Coins with n Degrees of Freedom

In this paper we define a new type of decoherent quantum random walk with parameter 0≤p≤1, which becomes a unitary quantum random walk (UQRW) when p=0 and an open quantum random walk (OQRW) when p=1, respectively. We call this process a partially open quantum random walk (POQRW). We study the limiting distribution of a POQRW on Z ¹ subject to decoherence on coins with n degrees of freedom. The limiting distribution of the POQRW converges to a convex combination of normal distributions, under an eigenvalue condition. A Perron-Frobenius type of theorem is established to determine whether or not a POQRW satisfies the eigenvalue condition. Moreover, we explicitly compute the limiting distributions of characteristic equations of the position probability functions when n=2 and 3.     

Many-body Hamiltonian with screening parameter and ionization energy

We prove the existence of a Hamiltonian with ionization energy as part of the eigenvalue, which can be used to study strongly correlated matter. This eigenvalue consists of total energy at zero temperature (E ₀) and the ionization energy (ξ). We show that the existence of this total energy eigenvalue, E ₀±ξ, does not violate the Coulombian atomic system. Since there is no equivalent known Hamilton operator that corresponds quantitatively to ξ, we employ the screened Coulomb potential operator (Yukawa-type), which is a function of this ionization energy to analytically calculate the screening parameter (σ) of a neutral helium atom in the ground state. In addition, we also show that the energy level splitting due to spin-orbit coupling is inversely proportional to ξ eigenvalue, which is also important in the field of spintronics.     

Asymptotics of eigenvalues of the Schrödinger operator with a strong δ-interaction on a loop

In this paper we investigate the operator H_β=−Δ−βδ(·−Γ) in , where β>0 and Γ is a closed C⁴ Jordan curve in . We obtain the asymptotic form of each eigenvalue of H_β as β tends to infinity. We also get the asymptotic form of the number of negative eigenvalues of H_β in the strong coupling asymptotic regime.     

Vacuum instability of massless electrodynamics and the Gell-Mann-Low eigenvalue condition for the bare coupling constant

We discuss a connection of the Gell-Mann-Low eigenvalue condition for the bare coupling α₀ with the normal phase vacuum rearrangement in massless electrodynamics which results in the fermion acquiring a physical mass. A new approach to determine the Gell-Mann-Low eigenvalue for α₀ is proposed.     

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.     

The Lie–Poisson representation of the nonlinearized eigenvalue problem of the Kac–van Moerbeke hierarchy

The Kac–van Moerbeke hierarchy is studied by a 3×3 discrete eigenvalue problem and the corresponding nonlinearized one an integrable Poisson map with a Lie–Poisson structure is also presented. Moreover, the 2×2 nonlinearized eigenvalue problem associated with the Kac–van Moerbeke hierarchy is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation.     

Eigenvector Localization in Real Networks and Its Implications for Epidemic Spreading

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum K-core in a K-core decomposition. The localization of the principal eigenvector is often strongly correlated with the value of the largest eigenvalue, which is given by the local eigenvalue of the corresponding localization subgraph, but different scenarios sometimes occur. We additionally show that simple targeted immunization strategies for epidemic spreading are extremely sensitive to the actual localization set.

     

One-factor model for the cross-correlation matrix in the Vietnamese stock market

Random matrix theory (RMT) has been applied to the analysis of the cross-correlation matrix of a financial time series. The most important findings of previous studies using this method are that the eigenvalue spectrum largely follows that of random matrices but the largest eigenvalue is at least one order of magnitude higher than the maximum eigenvalue predicted by RMT. In this work, we investigate the cross-correlation matrix in the Vietnamese stock market using RMT and find similar results to those of studies realized in developed markets (US, Europe, Japan) , , , , , , , ,  and  as well as in other emerging markets, ,  and . Importantly, we found that the largest eigenvalue could be approximated by the product of the average cross-correlation coefficient and the number of stocks studied. We demonstrate this dependence using a simple one-factor model. The model could be extended to describe other characteristics of the realistic data.