Related papers: Eigenvalue rigidity of hyperbolic surfaces in the …

In this paper, we propose a new trace finite element method for the {Laplace-Beltrami} eigenvalue problem. The method is proposed directly on a smooth manifold which is implicitly given by a level-set function and require high order…

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this…

For $\Gamma={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius $\cosh^{-1}(X/2)$. We show that, by averaging…

Let $M$ be an oriented geometrically finite hyperbolic manifold of infinite volume with dimension at least $3$. For all $k \geq 0$, we provide a lower bound on the $k$th eigenvalue of the Laplace-Beltrami operator of $M$ by the $k$th…

Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral method. Significantly improved eigenvalues, eigenfunctions are obtained efficiently for arbitrary $n, \ell$ quantum states by solving the relevant…

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\log\lambda)^{-\frac12}$ gain for…

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…

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus…

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the…

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 convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show…

Given a conformal action of a discrete group on a Riemann surface, we study the maximization of Laplace and Steklov eigenvalues within a conformal class, considering metrics invariant under the group action. We establish natural conditions…

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at…

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are…

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma\backslash \mathbb{H}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a…

We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a…

We provide lower estimates for the eigenvalues of the laplacian for hypersurfaces of the round sphere.

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound…

Let $\lambda_{1}(G)$ and $\mu_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $\lambda_{1}(G)$ based on…

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(\lambda\_k(\tau,\alpha))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $\tau$ is a positive number that defines the…