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

2002/09/30 by Carolyn S. Gordon, Dorothee Schueth
Mathematics · #math.DG #math.SP #msc:58J53 #msc:58J50

paper · pdf

published as J. Geom. Analysis, Vol. 13, no. 2 (2003), 279 - 306 · 34 pages, AMS-TeX; revised subsection 5.1

arxiv created 2002/11/18 · arxiv updated 2009/11/30

Abstract

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.

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