This note is on inverse spectral theory for the Schrödinger operator on a flat two-dimensional torus with electric and magnetic potentials. This problem can be remarkably rigid. For generic flat tori, if the variation of the magnetic field is strictly less than its mean, and the total magnetic flux on the torus is ±2π, then the spectrum of the Schrödinger operator determines both the electric and magnetic fields. This is in marked contrast to both the Schrödinger operator without a magnetic

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... d (see [ERT]) and the case of a magnetic field of mean zero (see [E]). In both those problems there are large families of isospectral fields, and rigidity results are much more difficult to obtain (see also [ER]). The observation that there can be spectral rigidity when the total flux is ±2π is due to Guillemin ([Gu]). Here we give a short proof of the slightly stronger result stated above. Instead of thinking of the Hamiltonian as acting on functions with values in a line bundle over the torus R 2 /L, we think of the Hamiltonian as acting on functions on R 2 which are invariant with respect to the "magnetic translations" associated to L. However, these two settings are completely equivalent. Our assumption that the variation of the magnetic field B(x) is strictly less than its mean b 0 takes the simple form The spectrum of the Laplacian plus lower order perturbations on flat tori has the feature that there are large families of spectral invariants corresponding to sets of geodesics with a fixed length. In analogy with results on S 2 Guillemin proposed the name "band invariants" for these families. The nice feature of the problem discussed here is that only the simplest of the band invariants are needed to prove rigidity. The first complete solution of an inverse spectral problem was Mark Krein's definitive analysis of the "weighted string", [K1,2]. Since that time many other inverse spectral problems in one space dimension have been solved (see [M]). In higher dimensions it is widely believed that, modulo natural symmetries and deformations like gauge transformation, most problems will be spectrally rigid. However, so far there have been relatively few settings where this has been proven (for instance those in [GuK] and [Ze]) and many interesting examples where it fails (see [S] and [G]). This should remain an active field of research for many years to come, and one can reasonably say that it began with the work of Mark Grigor'evich Krein.