Let $\Delta$ be the Dirichlet Laplacian on a bounded smooth open subset $\Omega \subset \mathbb{R}^N$, $N \geq 1$, (with domain $H^2 \cap H^1_0 \subset L^2$). Do we know necessary and/or necessary and sufficient conditions on $\Omega$ such that the eiganvalues $\{\lambda_k\}_{k \geq 1}$:
1) are simple.
2) satisfy the gap condition $\inf_{k \geq 1} |\lambda_{k+1}-\lambda_k|>0$.