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

الأصل · EN

We construct pairs of conformally equivalent isospectral Riemannian metrics ϕ₁ g and ϕ₂ g on spheres Sⁿ and balls Bⁿ⁺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 ϕ₁ and ϕ₂ are isospectral potentials for the Schrödinger operator ℏ²Δ+ϕ. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.