Transplantation and isospectrality. I (Q1813033)

This paper gives new combinatorial geometric proofs of Sunada's theorem concerning isospectral manifolds (both for the eigenvalue spectrum and for the length spectrum). The proofs also apply to non-smooth metrics and to natural operators. The paper includes examples of non-isometric complete non-compact manifolds that are isospectral in the following sense: they have the same discrete part of the spectrum and their scattering matrices are unitarily conjugate (in this sense, one can say that these manifolds have the same continuous spectra).