Yi Wei, Yan V. Fyodorov
Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U(N)}$ according to the Haar measure.
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In this work we build a certain machine that allows to construct almost periodic Jacobi matrices with singularly continuous spectrum a prescribed p-adic hull.
The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solu…
For a quantum state represented as an $n\times n$ density matrix $σ\in M_n$, let ${\cal S}(σ)$ be the compact convex set of quantum states $ρ= (ρ_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $σ$, i.e., ${\rm tr}_1(ρ) = ρ_{11} + \cdots + ρ_{mm} = σ$. It is known that if $m \ge n$ then there is a rank one matrix $ρ\in {\cal S}(σ)$ satisfying ${\rm tr}_1(ρ) = σ$. If $m < n$, there may not be rank one matrix in ${\cal S}(σ)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set ${\cal S}$ We also determine $ρ^* \in {\cal S}(σ)$ with rank…
The paper deals with singular Schrödinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }γ_k δ(\cdot-z_k),\quad γ_k\in\mathbb{R}, \end{gather*} in $\mathsf{L}^2(\ell_-,\ell_+)$, where $(\ell_-,\ell_+)$ is a bounded interval, and $ δ(\cdot-z_k)$ is the Dirac delta-function supported at $z_k\in (\ell_-,\ell_+)$. It will be shown that the interaction strengths $γ_k$ and the points $z_k$ can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets o…
We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbations. The general results presented here includes weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so-called additive strongly singular perturbations.
An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_α(x;t)=x^α{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~α>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $λ_{N}$, of the Hankel matrix generated by the weight $w_α(x;t)$.
Random matrices with prescribed eigenvalues and expectation values for random quantum states
本文研究具有指定特征值的随机矩阵，证明在大维度下其元素线性泛函近似服从联合高斯分布，并应用于随机量子态可观测量期望值、子矩阵谱分布及 Schur–Horn 定理的概率版本。
We study eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrodinger operators with Holder regular potential, obtaining a version of Minami's inequality and Poisson statistics for the local eigenvalue spacings. The main additional new input are regular properties of the Furstenberg measures and the density of states obtained in some of the author's earlier work.
In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.
We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.
The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential V(x) = x^2 + lambda/x^alpha, alpha > 0, lambda > 0, and is valid for all discrete eigenvalues, arbitrary angular momentum ell, and spatial dimension N.
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.
Correlation functions between singular values and eigenvalues
研究通过生物正交复随机矩阵有限维集合中奇异值密度与特征值密度的显式双射关系，推导出j个特征值和k个奇异值诱导的概率测度，提出j,k点相关测度，并在奇异值遵循多项式族时得到闭式表达。
This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.
In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $δ$- and $δ'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.
Statistical field theory of random graphs with prescribed degrees
本研究针对指定度序列随机图难以应用统计场论的问题，基于随机正则图计数成果构建邻接矩阵谱的统计场论描述，推导正则图McKay分布，提出混合正则图特征值密度的Hammerstein型积分方程，数值求解结果与经验分布高度吻合。
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