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#### Detecting localized eigenstates of linear operators

(2017-11-30)

We describe a way of detecting the location of localized eigenvectors of a linear
system $Ax = \lambda x$ for eigenvalues $\lambda$ with $|\lambda|$ comparatively large.
We define the family of functions $f_{\alpha}: \left\{1.2. ...

#### Riesz Energy on the Torus: Regularity of Minimizers

(2017-11-30)

We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing
interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i -
x_j)}. \] The main result states that for a class of functions ...

#### A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue

(2017-04-23)

The purpose of this short note is to give a variation on the classical Donsker-Varadhan
inequality, which bounds the first eigenvalue of a second-order elliptic operator
on a bounded domain $\Omega$ by the largest mean first ...