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

In this paper we investigate the operator $H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}({\Bbb R}^{2})$, where $\beta>0$ and $\Gamma$ is a closed $C^{4}$ Jordan curve in ${\Bbb R}^{2}$. We obtain the asymptotic form of each…

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and…

Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q = 2n+2$. For $1 < q < \infty$, $\gamma > 0$ and an exponent function $p(\cdot)$ on $\mathbb{H}^n$, which satisfy log-H\"older conditions, with $0 < p_{-} \leq p_{+} < \infty$, we introduce…

We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2 ({\Bbb R}) $. As an example, we consider integral operators $H$ in the space $L^2 ({\Bbb…

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding…

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and…

In the article, we investigate the average behaviour of normalised Hecke eigenvalues over certain polynomials and establish an estimate for the power moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of weight $k…

For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schr\"{o}der paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schr\"{o}der paths of length $\ell$. For constants $\alpha, \beta$, in…

To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0…

We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2)^{\beta}(t^2-x^2)^\alpha h(x),~x\in(-1,1),$$ where $\beta>-1$, $\alpha+\beta>-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for…

We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\rightarrow x+\omega$. We assume that the Lyapunov exponent…

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how…

In this paper we introduce the class of matrix valued asymmetric truncated Hankel operators. By using characterizations of matrix valued asymmetric truncated Toeplitz operators, we characterize matrix valued asymmetric truncated Hankel…

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The…

Every diagonalmatrix D yields an endomorphism on the n-dimensional complex vectorspace. If one provides this space with Hoelder norms, we can compute the operator norm of D. We define homogeneous weighted spaces as a generalization of…

Let $M= \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We show that the eigenvalue counting function $N(\lambda)$ for any fixed element of a family…

We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $\omega \in \Omega$ by the law $[H_{\omega} \textbf{u}]_{n} := D(T^{n -…

Let $A$ be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, $V$, so that $AV>0$ and $A$ is Hankel with respect to $V$, i.e. $V^*A = AV$, if and only if $A$…

Let $(G,\kappa)$ be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let $\tilde{G}$ be the complexification of $G$. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain…