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

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

2002-09-27

arxiv.org/abs/math/0209380v2

J. Geom. Analysis, Vol. 13, no. 2 (2003), 279 - 306

Mathematics

Differential Geometry

34 pages, AMS-TeX; revised subsection 5.1

Scientific paper

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.

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