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

We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within…

We study the family of compact operators $B_{\alpha} = V A_{\alpha} V$, $\alpha>0$ in $L^2(\mathbb R^d)$, $d\ge 1$, where $A_{\alpha}$ is the pseudo-differential operator with symbol $a_{\alpha}(\boldsymbol\xi) = a(\alpha\boldsymbol\xi)$,…

In this survey we show how to produce asymptotics of determinants of structured matrices using operator theory methods. We describe the asymptotics for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and generalizations of…

We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses…

We study the Schroedinger operator with a constant magnetic field in the exterior of a compact domain in $\mathbb{R}^{2d}$, $d\geq 1$. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give…

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with…

We provide eigenvalue asymptotics for a Dirac-type operator on $\mathbb Z^n$, $n\geq 2$, perturbed by multiplication operators that decay as $|\mu|^{-\gamma}$ with $\gamma<n$. We show that the eigenvalues accumulate near the value of the…

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite…

We study the eigenvalues and the eigenvectors of $N\times N$ structured random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and $W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry technique…

We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\alpha>0,\;\;t\geq 0. $$ If $t=0$, it is reduced to the classical symmetric…

We consider the smallest eigenvalue distributions of some Freud unitary ensembles, that is, the probabilities that all the eigenvalues of the Hermitian matrices from the ensembles lie in the interval $(t,\infty)$. This problem is related to…

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a rapidly decaying potential, as the magnetic field strength, $B$, tends to infinity. After a suitable rescaling, this becomes a semiclassical problem where…

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of…

We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities. This generalises two results: 1) a result of Berestycki, Webb and Wong [5] for root-type…

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $\alpha$ and the others of weight $1$. We denote by $n$ the order of the graph and suppose that $n$ tends to infinity. We notice…

Large H-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

In this article, we obtain some results in the direction of ``infinite dimensional symplectic spectral theory". We prove an inequality between the eigenvalues and symplectic eigenvalues of a special class of infinite dimensional operators.…

We give a construction of a class of magnetic Laplacian operators on finite directed graphs. We study some general combinatorial and algebraic properties of operators in this class before applying the Harrell-Stubbe Averaged Variational…

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size.…