Statistics of real eigenvalues in Ginibre's ensemble of random real matrices

The integrable structure of Ginibre's orthogonal ensemble of random matrices is looked at through the prism of the probability p(n,k) to find exactly k real eigenvalues in the spectrum of an n x n real asymmetric Gaussian random matrix. The exact solution for the probability function p(n,k) is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.     

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.     

Large deviations of extreme eigenvalues of random matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.     

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions , with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p _N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p _N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k ^th moment of p _N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p _N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.     

Numerical experiment of anharmonic oscillators by using the symplectic scheme-shooting method

Symplectic scheme-shooting method (SSSM) is applied to solve the energy eigenvalues of anharmonic oscillators characterized by the potentials V(x)=λx4 and V(x)=(1/2)x2+λx2α with α=2,3,4 and doubly anharmonic oscillators characterized by the potentials V(x)=(1/2)x2+λ1x4+λ2x6, and a high order symplectic scheme tailored to the "time"-dependent Hamiltonian function is presented. The numerical results illustrate that the energy eigenvalues of anharmonic oscillators with the symplectic scheme-shooting method are in good agreement with the numerical accurate ones obtained from the non-perturbative method by using an appropriately scaled basis for the expansion of each eigenfunction; and the energy eigenvalues of doubly anharmonic oscillators with the sympolectic scheme-shooting method are in good agreement with the exact ones and are better than the results obtained from the four-term asymptotic series. Therefore, the symplectic scheme-shooting method, which is very simple and is easy to grasp, is a good numerical algorithm.     

The Kochen-Specker theorem and Bell's theorem: An algebraic approach

In this paper we present a systematic formulation of some recent results concerning the algebraic demonstration of the two major no-hidden-variables theorems for N spin-1/2 particles. We derive explicitly the GHZ states involved and their associated eigenvalues. These eigenvalues turn out to be undefined for N=, this fact providing a new proof showing that the nonlocality argument breaks down in the limit of a truly infinite number of particles.     

Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices

We consider an open XXX spin chain with two general boundary matrices whose entries obey a relation, which is equivalent to the possibility to put simultaneously the two matrices in a upper-triangular form. We construct Bethe vectors by means of a generalized algebraic Bethe ansatz. As usual, the method uses Bethe equations and provides transfer matrix eigenvalues.     

Eigenvalues of the Zakharov-Shabat scattering problem for real symmetric pulses

The classical problem of determining the solitons generated from symmetric real initial conditions in the nonlinear Schr?dinger equation is revisited. The corresponding Zakharov-Shabat scattering problem is solved for real and symmetric double-humped rectangular initial pulse forms. It is found that such real symmetric pulses may generate eigenvalues with nonzero real parts corresponding to separating soliton pulse pairs. Moreover, it is found that the classical formula relating the number of eigenvalues to the area of the pulse is not always correct.     

Spectral Theory of the Klein–Gordon Equation in Pontryagin Spaces

In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in ; they include potentials consisting of a Coulomb part and an L _p -part with n ≤ p < ∞.Branko Najman: Deceased     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.     

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

It has been found that in the integration method for calculating the eigenvalues for propagation in two-dimensional ducts with flow care must be taken not to exclude roots which in the hardwall case are at infinity. When the real part of the wall admittance is positive starting values for the integration procedure to obtain these extra eigenvalues can be obtained by a limiting case analysis. When the imaginary part of the admittance is negative it appears that all the roots can be accounted for by the usual hardwall initial values κ_nb = nπ.     

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p _n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.     

Eigenvalue density of the non-Hermitian Wilson Dirac operator

We find the lattice spacing dependence of the eigenvalue density of the non-Hermitian Wilson Dirac operator in the ? domain. The starting point is the joint probability density of the corresponding random matrix theory. In addition to the density of the complex eigenvalues we also obtain the density of the real eigenvalues separately for positive and negative chiralities as well as an explicit analytical expression for the number of additional real modes.     

Higher-order semiclassical energy expansions for potentials with non-integer powers

In this paper, we present a semiclassical eigenenergy expansion for the potential |x|^α when α is a positive rational number of the form2n/m (n is a positive integer and m is an odd positive integer). Remarkably, this expansion is found to be identical to the WKB expansion obtained for the potentialx ^N(N-even), if2n/m is replaced byN. Taking the limitm → 2 of the above expansion, we obtain an explicit asymptotic energy expansion of symmetric odd power potentials |x|^2j+1 (j- positive integer). We then show how to develop approximate semiclassical expansions for potentials |x|^α when α is any positive real number.     

Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations. II

This is our second paper devoted to the study of some non-linear Schrödinger equations with random potential. We study the non-linear eigenvalue problems corresponding to these equations. We exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails. Contrary to our results in the first paper, the results established in the present paper hold for an arbitrary strength of the non-linear (cubic) term in the non-linear Schrödinger equation.     

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the 0 limit.Research partially supported by U.S.N.S.F. under Grant MCS-78-01885.     

New Quantization Method for Evaluation of Eigenenergies

A quantum dynamical equation is constructed as the limit of a sequence of functions (called Semiquantum momentum functions or SQMF). The quantum action variable J is defined as the limit of the sequence of contour integrals of SQMFs such that the quantization condition is J = n, where n is a nonnegative integer for eigenvalues and a noninteger for off eigenvalues. This quantization condition is exact and J is an analytic function of energy. Based on new definitions, an accurate numerical method is developed for obtaining eigenenergies. The method can be applied to both real and PT symmetric complex potentials. The validity and the accuracy of this new method is demonstrated with three illustrations.     

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ ^∞-norm of the corresponding eigenvectors is of order O(N ^−1/2), modulo logarithmic corrections. The upper bound O(N ^−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.     

A Hamiltonian method for finding broadband modal eigenvalues

For shallow water waveguides over a layered elastic bottom, modal eigenvalues can be determined by searching the locations in the complex plane of the horizontal wave number at which the complex phase function is a multiple of π [C. T. Tindle and N. R. Chapman, J. Acoust. Soc. Am. 96, 1777-1782 (1994)]. In this paper, a Hamiltonian method is introduced for tracing the path in the complex plane along which the phase function keeps real. The Hamiltonian method can also be extended to compute the broadband modal eigenvalues or the modal dispersion curves in the Pekeris waveguide with fluid/elastic bottoms. For each proper or leaky normal mode, a different Hamiltonian is constructed in the complex plane and used to trace automatically the complex dispersion curve with the eigenvalue in a reference frequency as the initial value. In contrast to the usual methods, the dispersion curve for each mode is determined individually. The Hamiltonian method shows good performance by comparing with KRAKEN.     

化学发光抑制法测定抗坏血酸

利用抗坏血酸对DTMC H2 O2 化学发光体系的抑制作用 ,研究了对抗坏血酸进行间接测定的可能性。试验发现 ,化学发光猝灭率 (R)与抗坏血酸浓度在 1 0× 10 - 7～ 8 0× 10 - 6 mol·L- 1 范围内呈线性关系 ,检出限达到 8 0× 10 - 8mol·L- 1 (S N =3)。对 1 0× 10 - 6 mol·L- 1 抗坏血酸进行 10次平行测定 ,其化学发光强度猝灭率相对标准偏差为 4 6 %。该猝灭体系不需要额外的掩蔽剂 ,方法简单、选择性好 ,可直接应用于一些食品中微量抗坏血酸的测定。