Extreme Values of the Fiedler Vector on Trees

Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $\lambda_{2}(G) > 0$, also known as the algebraic connectivity, as well as the associated eigenvector $\phi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $\phi_2$ are assumed at the endpoints of the longest path in $G$. Our results also apply to more general graphs that `behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for the eigenvector $\phi_k$

On localization of eigenfunctions of the magnetic Laplacian

Let $\Omega \subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:\Omega \rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) \psi = \lambda \psi$ that are highly localized in a small region of $\Omega$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0 \in \Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{\lambda}-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where \left|\mbox{curl} A \right| is small. The result is illustrated with numerical examples

A compactness principle for maximising smooth functions over toroidal geodesics

DOI: 10.1017/S000497271800163

Improved bounds for Hermite-Hadamard inequalities in higher dimensions

Let $Omega subset mathbb{R}^n$ be a convex domain and let $f:Omega ightarrow mathbb{R}$ be a positive, subharmonic function (i.e. $Delta f geq 0$). Then $$rac{1}{|Omega|} int_{Omega}{f dx} leq rac{c_n}{ |partial Omega| } int_{partial Omega}{ f dsigma},$$ where $c_n leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $Omega_2 subset Omega_1 subset mathbb{R}^n$: $$ rac{|partial Omega_1|}{|Omega_1|} rac{| Omega_2|}{|partial Omega_2|} leq n.$