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Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups
We construct pairs of conformally equivalent isospectral Riemannian metrics
and on spheres and balls for certain
dimensions , the smallest of which is , and on certain compact simple
Lie groups. In the case of Lie groups, the metric is left-invariant. In the
case of spheres and balls, the metric is not the standard metric but may be
chosen arbitrarily close to the standard one. For the same manifolds we
also show that the functions and are isospectral potentials
for the Schr\"odinger operator . To our knowledge, these
are the first examples of isospectral potentials and of isospectral conformally
equivalent metrics on simply connected closed manifolds.Comment: 34 pages, AMS-TeX; revised subsection 5.
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