Related papers: Eigenvalue Ratios for vibrating string equations w…

The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1) where V(n,d) stands for the volume of a Hamming ball of radius d. Recently Jiang and…

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has…

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$),…

We consider the problem of maximizing the first eigenvalue of the $p$-laplacian (possibly with non-constant coefficients) over a fixed domain $\Omega$, with Dirichlet conditions along $\partial\Omega$ and along a supplementary set $\Sigma$,…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in…

We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential…

For the Dirichlet realization of $-d^2/dx^2-\lambda^2V$ on a bounded interval, with $V$ a positive $C^2$ potential bounded away from $0$ and $\lambda>0$ a large parameter, we prove an asymptotic law for the values $\lambda_n$ of $\lambda$…

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(\Omega) =…

We characterize the potential-energy functions $V(x)$ that minimize the gap $\Gamma$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u := -\frac{d}{dx} \left(p(x)\frac{du}{dx}\right)+V(x) u = \lambda u, \quad\quad x\in…

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $\Omega$ which is composed of smooth surfaces joined along a line $\gamma$, the junction. Through this junction we impose the Kirchhoff-type vertex…

We consider eigenfunctions of a semiclassical Schr{\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform…

We provide bounds for the sequence of eigenvalues $\{\lambda_i(\Omega)\}_i$ of the Dirichlet problem $$ L_\Delta u=\lambda u\ \ {\rm in}\ \, \Omega,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus \Omega,$$ where $L_\Delta$ is the…

We introduce a \emph{spectral Dehn function} \[ \Lambda_{\mathcal{P}}(n):=\inf \lambda_1(\Delta), \] where $\lambda_1(\Delta)$ is the first Dirichlet eigenvalue of the random-walk Laplacian on a van Kampen diagram $\Delta$, and the infimum…

In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the…

Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $\lambda=\lambda_{k}=\dots = \lambda_{k+m-1}$, we characterise the lowest and highest orders in the set…

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we…

We prove that the optimal constant in the Lieb--Thirring inequality on a star graph with $N$ edges coincides with that on $\mathbb R$ if $N$ is even. For odd $N$ we show that this property holds when restricting to radial potentials and we…

We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a…

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the…