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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 certaindimensions $n$, the smallest of which is $n=7$, and on certain compact simpleLie groups. In the case of Lie groups, the metric $g$ is left-invariant. In thecase of spheres and balls, the metric $g$ is not the standard metric but may bechosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ wealso show that the functions $\phi_1$ and $\phi_2$ are isospectral potentialsfor the Schr\"odinger operator $\hbar^2\Delta +\phi$. To our knowledge, theseare the first examples of isospectral potentials and of isospectral conformallyequivalent metrics on simply connected closed manifolds.

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