INVERSE SPECTRAL RESULTS ON EVEN DIMENSIONAL TORI

Given a Hermitian line bundle L over a flat torus M, a connection del on L, and a function Q on M, one associates a Schrodinger operator acting on sect ions of L; its spectrum is denoted Spec(Q; L, del). Motivated by work of V. Guillemin in dimension two, we consider lino bundles over tori of arbitrary even dimension with "translation invariant" connections del, and we address the extent to which the spectrum Spec(Q; L, del) determines the potential Q. With a genericity condition, we show that if the connection is invariant under the isometry of M defined by the map x -> -x, then the spectrum determines the even part of the potential. In dimension two, we also obtain information about the odd part of the potential. We obtain counterexamples showing that the genericity condition is needed even in the cast, of two-dimensional tori. Examples also show that the spectrum of the Laplacian defined by a connection on it line bundle over a flat torus determines neither the isometry class of the torus nor the Chern class of the line bundle. In arbitrary dimensions, we show that the collection of all the spectra Spec(Q; L, del), as del ranges over the translation invariant connections, uniquely determines the potential. This collection of spectra is a natural generalization to line bundles of the classical Bloch spectrum of the torus.