  The first two eigenvalues of the Robin Laplacian are investigated along with
their gap and ratio. Conjectures by various authors for arbitrary domains are
supported here by new results for rectangular boxes.
  Results for rectangular domains include that: the square minimizes the first
eigenvalue among rectangles under area normalization, when the Robin parameter
$\alpha \in \mathbb{R}$ is scaled by perimeter; that the square maximizes the
second eigenvalue for a sharp range of $\alpha$-values; that the line segment
minimizes the Robin spectral gap under diameter normalization for each $\alpha
\in \mathbb{R}$; and the square maximizes the spectral ratio among rectangles
when $\alpha>0$. Further, the spectral gap of each rectangle is shown to be an
increasing function of the Robin parameter, and the second eigenvalue is
concave with respect to $\alpha$.
  Lastly, the shape of a Robin rectangle can be heard from just its first two
frequencies, except in the Neumann case.

Spectral Theory;Analysis of PDEs