Related papers: Eigenvalue asymptotics for a class of multi-variab…

This paper investigates the asymptotic behavior of the principal eigenvalue $\lambda(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -\Delta_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c…

We introduce the notion of Drinfeld modular forms with $A$-expansions, where instead of the usual Fourier expansion in $t^n$ ($t$ being the uniformizer at `infinity'), parametrized by $n \in \mathbb{N}$, we look at expansions in $t_a$,…

We study matrix integrals of the form $$\int_{\mathrm{USp(2n)}}\prod_{j=1}^k\mathrm{tr}(U^j)^{a_j}\mathrm d U,$$ where $a_1,\ldots,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact…

Any sequence of uniformly bounded $N\times N$ Hermitian Toeplitz matrices $\{\boldsymbol{H}_N\}$ is asymptotically equivalent to a certain sequence of $N\times N$ circulant matrices $\{\boldsymbol{C}_N\}$ derived from the Toeplitz matrices…

We study asymptotic spectral properties of the generalized Hilbert $L$-matrix \[ L_{n}(\nu)=\left(\frac{1}{\max(i,j)+\nu}\right)_{i,j=0}^{n-1}, \] for large order $n$. First, for general $\nu\neq0,-1,-2,\dots$, we deduce the asymptotic…

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced…

A generalized eigenvector of a hypermatrix, called the universal (U-) eigenvector, is proposed, which extended the notion of diagonal (D-) eigenvectors in the literature. Using the semi-tensor product, the homogeneous U-eigenequation can be…

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and…

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that \be \mathbb{E}[a_{j}]=0, \ \ \mathbb{E}[|a_{j}|^{2}]=1 \ \ \textrm{for}\,\…

Let $\Gamma\subset\mathbb{R}^2$ be a piecewise smooth closed curve with corners. We discuss the asymptotic behavior of the individual eigenvalues of the two-dimensional Schr\"odinger operator $-\Delta-\alpha\delta_\Gamma$ for…

The spectral problem for the high order differential operator with singular weight is considered. If the weight is a generalized derivative of self-similar function with zero spectral degree the asymptotics of eigenvalues is obtained. They…

The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the…

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(\alpha) = \{\alpha(nm)\}_{n,m=1}^\infty$, where $\alpha$ is a sequence of complex numbers. Helson matrices are considered as…

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[…

If $A$ is a $2n \times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^TAM = \left [ \begin{array}{cc} D & O \\ O & D \end{array} \right ]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal…

For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator…

We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our…

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct…