相关论文: Spectrum through pseudospectrum

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its…

We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…

This paper presents enhancement strategies for the Hermitian and skew-Hermitian splitting method based on gradient iterations. The spectral properties are exploited for the parameter estimation, often resulting in a better convergence. In…

There is widespread interest in calculating the energy spectrum of a Hamiltonian, for example to analyze optical spectra and energy deposition by ions in materials. In this study, we propose a quantum algorithm that samples the set of…

A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…

We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{\"o}dinger Hamiltonian: $H=p^2/2\mu+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$…

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not…

This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or…

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.

In data science, individual observations are often assumed to come independently from an underlying probability space. Kernel matrices formed from large sets of such observations arise frequently, for example during classification tasks. It…

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with…

Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been…

A procedure for counting the number of eigenvalues of a matrix in a region surrounded by a closed curve is presented. It is based on the application of the residual theorem. The quadrature is performed by evaluating the principal argument…

The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of…

Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…

We describe a general approach for computing generators for elimination ideals associated with matrix and hypermatrix spectral decomposition constraints. We derive from these generators iterative procedures for approximating the spectral…

For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit…

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic…