  We construct pairs of conformally equivalent isospectral Riemannian metrics
$\phi_1 g$ and $\phi_2 g$ 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 $\phi_1$ and $\phi_2$ are isospectral potentials
for the Schr\"odinger operator $\hbar^2\Delta +\phi$. To our knowledge, these
are the first examples of isospectral potentials and of isospectral conformally
equivalent metrics on simply connected closed manifolds.
