Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics ${\varphi }_{1}g$ and ${\varphi }_{2}g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions ${\varphi }_1$ and ${\varphi }_2$ are isospectral potentials for the Schr"odinger operator ${\mathrm{\hslash }}^2\mathrm{\Delta }+\varphi$. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.