Related papers: On the multiplicity of the eigenvalues of discrete…

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $\lambda_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for…

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among…

We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \hookrightarrow \operatorname{GL}_r$. Given a positive integer $n$, there exists a…

We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $\lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For…

We provide a proof of the conjecture formulated in \cite{Jak97,JNT01} which states that on a $n$-dimensional flat torus $\T^{n}$, the Fourier transform of squares of the eigenfunctions $|\phi_\lambda|^2$ of the Laplacian have uniform $l^n$…

This paper is concerned with the maximisation of the k'th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension d as k goes to infinity. We show that in any dimension maximisers exist for any given k, but that any…

Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$…

We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is…

In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus…

We give uniform upper and lower bounds for the L^2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results…

In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than $1$ are simple and also the multiplicity of Laplacian eigenvalue $1$ has been well studied…

We prove the discrete restriction conjecture holds with no loss when $p>\frac{2d}{d-4}$ and $d\geq 5$. That is, we show optimal $L^p$ bounds for eigenfunctions of the Laplacian on the square torus for large values of $p$. This improves the…

We study the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on a two-dimensional rectangle or torus. We show that for a rectangle with the side length ratio $r$, $M=\mathbb{N}$, the set of all positive…

Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We…

We establish uniform upper and lower bounds on the restrictions of the eigenfunctions of the Laplacian on the 2- and 3-dimensional standard flat torus to smooth hyper-surfaces with non-vanishing curvature.

Our main goal is to determine, under certain restrictions, the maximal closed connected subgroups of simple algebraic groups containing a regular torus. We call a torus regular if its centralizer is abelian. We also obtain some results of…

Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the…

We compute an upper bound for the dimension of the tangent spaces at classical points of certain eigenvarieties associated with definite unitary groups, especially including the so-called critically refined cases. Our bound is given in…

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio…