Showing papers on "Spectrum of a matrix published in 1997"
TL;DR: In this article, the correlation functions of the distribution of the eigenvalues of random matrices have been studied in two classes of random Hermitian matrix models: the one-matrix model and the two-dimensional model, and it has been observed that the correlation function of this distribution possesses universal properties, independent of the probability law of the stochastic matrix.
89 citations
TL;DR: In this article, an analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions and the associated eigen functions are L 2 integrable.
Abstract: Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L2 integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.
24 citations
01 Jan 1997
TL;DR: In this article, the n(≥ 2)th order difference equation is considered and the boundary conditions (28.5) and 28.4) for α, β, γ, δ and δ are satisfied.
Abstract: Here, we shall consider the n(≥ 2)th order difference equation
$${\Delta ^n}y(k) + \lambda Q(k,y(k)), \cdots ,{\Delta ^{n - 2}}y(k))\, = \,\lambda P(k,y(k),\Delta y(k), \cdots ,{\Delta ^{n - 1}}y(k)),\,k \in N(0,J - 1)$$
(29.1)
together with the boundary conditions (28.2) – (28.4), where the constants α, β, γ and δ satisfy the conditions (28.5) and (28.6). In (29.1), λ > 0, Q: N(0, J − 1) × ℝn−1 → ℝ, and P : N(0, J − 1) × ℝn → ℝ.
13 citations
TL;DR: The minimal eigenvalues of a class of block-tridiagonal matrices from telecommunication system analysis are studied and an eigenvalue analysis for two- user systems and efficient estimates for m-user systems are presented.
Abstract: In this correspondence, we study the minimal eigenvalues of a class of block-tridiagonal matrices from telecommunication system analysis. We present an eigenvalue analysis for two-user systems and efficient estimates for m-user systems.
11 citations
TL;DR: In this paper, the collision operator of the linear Boltzmann equation for active atoms submerged in the much denser perturber gas contains a collision rate and a kernel and these two quantities are combined into a single entity, called collision operator.
Abstract: The linear Boltzmann equation for active atoms submerged in the much denser perturber gas contains a collision rate and a kernel. These two quantities are combined into a single entity\char22{}the collision operator. The collision operator possesses several interesting properties, the most important being that it is Hermitian. The eigenvalues are negative with the exception of one eigenvalue, which is zero and corresponds to the Maxwellian (steady-state) velocity distribution. A set of functions, closely related to the eigenfunctions of the quantum-mechanical harmonic oscillator, is postulated to approximate the true eigenfunctions. This assumption was a basis of the method of modeling various physical phenomena occurring in the gaseous mixtures, subjected to a radiation field. The eigenvalues of the collision operator were treated as free parameters. In this paper we establish a direct relationship between the eigenvalues and the collision integrals, or transport coefficients, known from the kinetic theory of gases. The generating function approach is employed to derive expressions yielding the eigenvalues. The obtained results form a bridge between kinetic theory, atomic physics, and quantum optics.
8 citations
01 Feb 1997
TL;DR: Subspace iterations are used to minimise as generalised Ritz functional of a large, sparse Hermitean matrix in this paper, which shows that the computational cost does not increase substantially with m.
Abstract: Subspace iterations are used to minimise as generalised Ritz functional of a large, sparse Hermitean matrix. In this way, the lowest m eigenvalues are determined. Tests with 1 ≤ m ≤ 32 demonstrate that the computational cost (no. of matrix multiplies) does not increase substantially with m . This implies that, as compared to the case of a m = 1, the additional eigenvalues are obtained for free.
6 citations
01 Feb 1997
TL;DR: In this article, an exploratory study of the low-lying eigenvalues of the Wilson-Dirac operator and their corresponding eigenvectors is presented, with respect to their localization properties in the qurenched approximation for SU(2) and SU(3).
Abstract: An exploratory study of the low-lying eigenvalues of the Wilson-Dirac operator and their corresponding eigenvectors is presented. Results for the eigenvalues from quenched and unquenched simulations are discussed. The eigenvectors are studied with respect to their localization properties in the qurenched approximation for the cases of SU(2) and SU(3).
6 citations
TL;DR: In this article, the eigenvalue problem was considered in real Hilbert spaces and it was shown that λ 0 e ℝ is a bifurcation point for (1.1) if every neighbourhood of (λ 0, 0) in Ω × H contains solutions of (1).
Abstract: Let H be a real Hilbert space and let A: H→H be a nonlinear operator such that A(0) = 0. We consider the eigenvalue problemRecall that λ0 e ℝ is said to be a bifurcation point for (1.1) if every neighbourhood of (λ0, 0) in ℝ × H contains solutions of (1.1).
5 citations
TL;DR: In this article, a criterion for high-codimensional bifurcations with several pairs of purely imaginary eigenvalues in terms of the properties of coefficients of characteristic polynomials instead of those of eigen values is presented.
4 citations
TL;DR: In this paper, the spectrum and asymptotic distribution of singular Sturm-Liouville eigenvalue problems on the half-line has been investigated, where the eigenvalues have a weight function that changes sign.
Abstract: We investigate spectrum and asymptotic distribution of the eigenvalues of certain singular Sturm-Liouville eigenvalue problems on the half-line having a weight function that changes sign.
3 citations
TL;DR: In this article, the nonlinear part is linear at infinity but behaves like a function intersecting many of the eigenvalues of the linear operator, and the asymptotic linear operator can have a null space of any finite dimension.
Abstract: We study semilinear problems in which the nonlinear part is linear at infinity but behaves like a function intersecting many of the eigenvalues of the linear operator. The asymptotic linear operator can have a null space of any finite dimension.
TL;DR: It is shown that the majority of the present eigenvalue bounds, expressed in concise forms, are less restrictive and sharper than existing results.
Abstract: New upper bounds for the solution eigenvalues of the continuous algebraic matrix Riccati equation are developed They include bounds of the extreme eigenvalues, the summation and product of eigenvalues, the trace, and the determinant It is shown that the majority of the present eigenvalue bounds, expressed in concise forms, are less restrictive and sharper than existing results
10 Dec 1997
TL;DR: In this article, the eigenvalues problem of matrix A+B has been studied and several relationships among the Eigenvalues of matrices A, B, A and B are proposed.
Abstract: This paper deals with the eigenvalues problem of matrix A+B. Several relationships among the eigenvalues of matrices A, B and A+B are proposed. Some simplified formulas for the eigenvalues of low order matrices are also proposed. These results are applied to the robustness analysis of uncertain state space systems.
TL;DR: In this article, the complex eigenvalue problem of a mono-energetic neutron transport operator is studied in a homogeneous sphere with spherically symmetric scattering, and it is shown that the spectrum involves a countable infinity of complex Eigenvalues.
Abstract: The complex eigenvalue problem of a mono-energetic neutron transport operator is studied in a homogeneous sphere with spherically symmetric scattering. It is shown that the spectrum involves a countable infinity of complex eigenvalues.
TL;DR: In this paper, the authors generalize classical interlacing by using singular values of off-diagonal blocks of A to construct extended intervals that capture a larger number of eigenvalues of A.
TL;DR: In this paper, the point spectrum lying on the essential spectrum is investigated for the one-dimensional Schrodinger operator -$ + q(z) with decaying potential q, and weakly perturbed Stark-like operator −$ - lzlPsignz + q (z).
Abstract: The point spectrum lying on the essential spectrum is investigated for the one- dimensional Schrodinger operator -$ + q(z) with decaying potential q, and weakly perturbed Stark-like operator -$ - lzlPsignz + q(z). An elementary constructive technique is developed to obtain various results concerning embedded eigenvalues of Schrodinger operators. In Section 3 a constructive example of the Stark-like operator with the potential q decaying slightly slowlier than O(1/1z11-a/2) and dense point spectrum on the whole real axis is presented.
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TL;DR: In this paper, the spectral properties of the Laplacian with multiple point interactions in two-dimensional bounded regions were discussed and a mathematically sound formulation for the problem was given within the framework of the self-adjoint extension of a symmetric (Hermitian) operator in functional analysis.
Abstract: We discuss spectral properties of the Laplacian with multiple ($N$) point interactions in two-dimensional bounded regions. A mathematically sound formulation for the problem is given within the framework of the self-adjoint extension of a symmetric (Hermitian) operator in functional analysis. The eigenvalues of this system are obtained as the poles of a transition matrix which has size $N$. Closely examining a generic behavior of the eigenvalues of the transition matrix as a function of the energy, we deduce the general condition under which point interactions have a substantial effect on statistical properties of the spectrum.