We study the distribution of eigenvalues of the Schrödinger operator with a complex valued potential V . We prove that if |V | decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterised.The particular solution of Painlevé II that arises is a double shifted Bäckl...

In this paper, we study the inverse problem for Dirac differential operators with  discontinuity conditions in a compact interval. It is shown that the potential functions can be uniquely determined by the value of the potential on some interval and parts of two sets of eigenvalues. Also, it is shown that the potential function can be uniquely determined by a part of a set of values of eigenfun...

Let Xn be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ > 0 constant, and Rn an n×N random matrix independent of Xn. Assume, almost surely, as n →∞, the empirical distribution function (e.d.f.) of the eigenvalues of 1 N RnR ∗ n converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio n...

in this article we consider the sequences of sample and population covariance operators for a sequence of arrays of hilbertian random elements. then under the assumptions that sequences of the covariance operators norm are uniformly bounded and the sequences of the principal component scores are uniformly sumable, we prove that the convergence of the sequences of covariance operators would impl...

Takemura and Sheena (2002) derived the asymptotic joint distribution of the eigenvalues and the eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. They also showed necessary conditions for an estimator of the population covariance matrix to be minimax for typical loss functions by calculating the asymptotic risk of the estimator. In this paper, we furthe...

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyltype inequalities. The top eig...

The stability analysis of islanded inverter-based microgrids (IBMGs) is increasingly an important and challenging topic due to the nonlinearity of IBMGs. In this paper, a new linear model for such microgrids as well as an iterative method to correct the linear model is proposed. Using the linear model makes it easy to analyze the eigenvalues and stability of IBMGs due to the fact that it derive...