We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order $N^{-1}$ (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N \to \infty$. Our results confirm Bogoliubov’s predictions.

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of $\mathbb{R}^N$ with prescribed measure $m$ attains its maximum on the union of two disjoint balls of measure $m/2$. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

The rank of tensors is not additive with respect to the direct sum.