Moments Of Eigenvalues Research Articles

Sharp Semiclassical Bounds for the Moments of Eigenvalues for Some Schrödinger Type Operators with Unbounded Potentials

We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we obtain sharp lower bounds for the moments of positive powers not exceeding one for such eigenvalues. When considering a Schrodinger operator with the relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the sharp Lieb-Thirring estimate for the moments of some negative powers of its eigenvalues.

Moments of the transmission eigenvalues, proper delay times and random matrix theory II

We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-Büttiker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of random matrix theory. The starting points are the finite-n formulae that we recently discovered [F. Mezzadri and N. J. Simm, “Moments of the transmission eigenvalues, proper delay times and random matrix theory,” J. Math. Phys. 52, 103511 (2011)]10.1063/1.3644378. Our analysis includes all the symmetry classes β ∈ {1, 2, 4}; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer [“Riemannian symmetric superspaces and their origin in random-matrix theory,” J. Math. Phys. 37(10), 4986 (1996)]10.1063/1.531675 and Altland and Zirnbauer [“Random matrix theory of a chaotic Andreev quantum dot,” Phys. Rev. Lett. 76(18), 3420 (1996)10.1103/PhysRevLett.76.3420; Altland and Zirnbauer “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B 55(2), 1142 (1997)]10.1103/PhysRevB.55.1142. Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. [“Full counting statistics of chaotic cavities from classical action correlations,” J. Phys. A: Math. Theor. 41(36), 365102 (2008)]10.1088/1751-8113/41/36/365102 and Berkolaiko and Kuipers [“Moments of the Wigner delay times,” J. Phys. A: Math. Theor. 43(3), 035101 (2010)10.1088/1751-8113/43/3/035101; Berkolaiko and Kuipers “Transport moments beyond the leading order,” New J. Phys. 13(6), 063020 (2011)]10.1088/1367-2630/13/6/063020. Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.

Universality in chaotic quantum transport: The concordance between random-matrix and semiclassical theories

Electronic transport through chaotic quantum dots exhibits universal, system-independent properties, consistent with random-matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semiclassical diagrams, through a connection to factorizations of permutations. We show agreement between the semiclassical and random matrix approaches to the moments of the transmission eigenvalues. The result is valid for all moments to all orders of the expansion in inverse channel number for all three main symmetry classes (with and without time-reversal symmetry and spin-orbit interaction) and extends to nonlinear statistics. This finally explains the applicability of random-matrix theory to chaotic quantum transport in terms of the underlying dynamics as well as providing semiclassical access to the probability density of the transmission eigenvalues.

Source enumeration in large arrays using moments of eigenvalues and relatively few samples

This study presents a method based on minimum description length criterion to enumerate the incident waves impinging on a large array using a relatively small number of samples. The proposed scheme exploits the statistical properties of eigenvalues of the sample covariance matrix (SCM) of Gaussian processes. The authors use a number of moments of noise eigenvalues of the SCM in order to separate noise and signal subspaces more accurately. In particular, the authors assume a Marcenko–Pastur probability density function (pdf) for the eigenvalues of SCM associated with the noise subspace. We also use an enhanced noise variance estimator to reduce the bias leakage between the subspaces. Numerical simulations demonstrate that the proposed method estimates the true number of signals for large arrays and a relatively small number of snapshots. In particular, the proposed method requires less number of samples to achieve the same correct enumeration probability compared to the state-of-the-art methods. The authors evaluated the assumed pdf in order to justify the limitation and the behaviour of the proposed method for small number of snapshots and array sizes.

Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I

We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre, and Jacobi ensembles for all the symmetry classes β ∈ {1, 2, 4} and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n → ∞.

Trace identities for commutators, with applications to the distribution of eigenvalues

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrodinger operators and Schrodinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda(N+1) in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrodinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue lambda(N+1) of the Laplace-Beltrami operator by a universal constant times vertical bar vertical bar h vertical bar N-2(infinity)2/d.

Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities

We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{\lambda_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $\tau_n=<\sum_{i=1}^N\lambda_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

Universal monotonicity of eigenvalue moments and sharp Lieb–Thirring inequalities

We show that phase space bounds on the eigenvalues of Schrödinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb–Thirring inequalities.

On the condition number distribution of complex wishart matrices

This paper investigates the distribution of the condition number of complex Wishart matrices. Two closely related measures are considered: the standard condition number (SCN) and the Demmel condition number (DCN), both of which have important applications in the context of multiple-input multiple-output (MIMO) communication systems, as well as in various branches of mathematics. We first present a novel generic framework for the SCN distribution which accounts for both central and non-central Wishart matrices of arbitrary dimension. This result is a simple unified expression which involves only a single scalar integral, and therefore allows for fast and efficient computation. For the case of dual Wishart matrices, we derive new exact polynomial expressions for both the SCN and DCN distributions. We also formulate a new closed-form expression for the tail SCN distribution which applies for correlated central Wishart matrices of arbitrary dimension and demonstrates an interesting connection to the maximum eigenvalue moments of Wishart matrices of smaller dimension. Based on our analytical results, we gain valuable insights into the statistical behavior of the channel conditioning for various MIMO fading scenarios, such as uncorrelated/semi-correlated Rayleigh fading and Ricean fading.

Universal Bounds and Semiclassical Estimates for Eigenvalues of Abstract Schrödinger Operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space, which extend those known previously for Laplacians and Schrodinger operators, freeing them from restrictive assumptions on the nature of the spectrum and allowing operators of much more general form. In particular, we allow for the presence of continuous spectrum, which is a necessary underpinning for new proofs of Lieb-Thirring inequalities. We both sharpen and extend universal bounds on spectral gaps and moments of eigenvalues {lambda(k)} of familiar types, and in addition we produce novel kinds of inequalities that are new even in the model cases. These include a family of differential inequalities for generalized Riesz means and a theorem stating that arithmetic means of {lambda(p)(k)}(k=1)(n) with p 8.

A Simple Proof of Hardy-Lieb-Thirring Inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).

Large System Spectral Analysis of Covariance Matrix Estimation

Eigendecomposition of estimated covariance matrices is a basic signal processing technique arising in a number of applications, including direction-of-arrival estimation, power allocation in multiple-input/multiple-output (MIMO) transmission systems, and adaptive multiuser detection. This paper uses the theory of non-crossing partitions to develop explicit asymptotic expressions for the moments of the eigenvalues of estimated covariance matrices, in the large system asymptote as the vector dimension and the dimension of signal space both increase without bound, while their ratio remains finite and nonzero. The asymptotic eigenvalue distribution is also obtained from these eigenvalue moments and the Stieltjes transform, and is extended to first-order approximation in the large sample-size limit. Numerical simulations are used to demonstrate that these asymptotic results provide good approximations for finite systems of moderate size.

Nonlinear statistics of quantum transport in chaotic cavities

In the framework of the random matrix approach, we apply the theory of Selberg's integral to problems of quantum transport in chaotic cavities. All the moments of transmission eigenvalues are calculated analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance and transmitted charge as well as for the variance of the shot-noise power in chaotic cavities. The obtained results are generally valid at arbitrary numbers of propagating channels in the two attached leads. In the particular limit of large (and equal) channel numbers, the shot-noise variance attends the universal value $1∕64\ensuremath{\beta}$ that determines a universal Gaussian statistics of shot-noise fluctuations in this case.

Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrödinger-like operators remain true, with possibly different constants, when the critical Hardy-weight C | x | − 2 C |x|^{-2} is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schrödinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z α = 2 / π Z\alpha =2/\pi , for α \alpha less than some critical value.

Estimating Eigenvalue Moments via Schatten Norm Bounds on Semigroup Differences

We derive new bounds on the moments of the negative eigenvalues of a selfadjoint operator B. The moments of order \(0<\gamma \leqslant 1\) are estimated in terms of Schatten-norm bounds on the difference of the semigroups generated by B and a reference operator A which is assumed to be nonnegative and selfadjoint. The estimate in the case γ = 1 is sharp.

Linking Confinement to Spectral Properties of the Dirac Operator

We represent Polyakov loops and their correlators as spectral sums of eigenvalues and eigenmodes of the lattice Dirac operator. The deconfinement transition of pure gauge theory is characterized as a change in the response of moments of eigenvalues to varying the boundary conditions of the Dirac operator. We argue that the potential between static quarks is linked to spatial correlations of Dirac eigenvectors.

Design of Reduced-Rank MMSE Multiuser Detectors Using Random Matrix Methods

Reduced-rank minimum mean-squared error (MMSE) multiuser detectors using asymptotic weights have been shown to reduce receiver complexity while maintaining good performance in long-sequence code-division multiple-access (CDMA) systems. In this paper, we consider the design of reduced-rank MMSE receivers in a general framework which includes fading, single and multiantenna receivers, as well as direct-sequence CDMA (DS-CDMA) and multicarrier CDMA (both uplink and downlink). In all these cases, random matrix results are used to obtain explicit expressions for the asymptotic eigenvalue moments of the interference autocorrelation matrix and for the asymptotic weights used in the reduced-rank receiver.

A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the nth Landau level an nth eigenvalue cluster, and study the distribution of eigenvalues in the nth cluster as n→∞. A complete asymptotic expansion for the eigenvalue moments in the nth cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the eigenvalue moments is obtained.

Approximation of some moments in eigenvalue problem including some random variables and its application

The approximation of some moments of eigenvalues and eigenvectors in eigenvalue problem including random variables using Taylor expansion technique is considered. This method is applied to the numerical calculation of the elastic wave equation assuming the stratified layer media with some uncertainties of the values of structural parameters. The accuracy of the approximation by comparing with the values taken from Monte Carlo simulation is observed.

On the Discrete Spectrum of a Pseudo-Relativistic Two-Body Pair Operator

We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.