MiS Preprint Repository

The intertwining operator constructed in Annals of Mathematics, 154(2001), 437-475; and 161(2005), 343-395 does not appear in the right form. It is established there by using only the anticommutators $J_1$ and $J_1^{\prime }$. The correct operator involves all endomorphisms, $J_{\alpha }$, which are unified by the Z-Fourier transform.

Although some of the correct elements of the previous constructions are kept, this idea is established by a new technique which yields the various isospectrality theorems stated in the papers on a much larger scale. The new results include new isospectrality examples living on sphere$\times$ball- and sphere$\times$sphere-type manifolds. Among them, there are such discrete isospectrality families where one of the members is homogeneous while the others are locally inhomogeneous (striking examples). Furthermore, a large class of new isospectrality families are constructed by $\sigma$ deformations.