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We consider a quantity $\kappa(\Omega)$ -- the distance to the origin from the null variety of the Fourier transform of the characteristic function of $\Omega$. We conjecture, firstly, that $\kappa(\Omega)$ is maximized, among all convex balanced domains $\Omega\subset\Rbb^d$ of a fixed volume, by a ball, and also that $\kappa(\Omega)$ is bounded above by the square root of the second Dirichlet eigenvalue of $\Omega$. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between $\kappa(\Omega)$ and the eigenvalues of the Laplacians., Comment: pdflatex; 4 figures; revised and extended

We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability., Comment: 8 pages, LaTex

Let $\Omega$ be some domain in the hyperbolic space $\Hn$ (with $n\ge 2$) and $S_1$ the geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$. We prove the Payne-P\'olya-Weinberger conjecture for $\Hn$, i.e., that the second Dirichlet eigenvalue on $\Omega$ is smaller or equal than the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\lambda_2(B_R, V) / \lambda_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.

A new and elementary proof of a recent result of Laptev and Weidl is given. It is a sharp Lieb-Thirring inequality for one dimensional Schroedinger operators with matrix valued potentials., Comment: Replaces the version of June 28, 1999. A technical error in the proof has been corrected