Inverse spectral results on even dimensional tori
Annales de l'Institut Fourier, Volume 58 (2008) no. 7, p. 2445-2501

Given a Hermitian line bundle L over a flat torus M, a connection on L, and a function Q on M, one associates a Schrödinger operator acting on sections of L; its spectrum is denoted Spec(Q;L,). Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections , and we address the extent to which the spectrum Spec(Q;L,) 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 case of two-dimensional tori. Examples also show that the spectrum of the Laplacian defined by a connection on a 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,), as 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.

À un fibré en droites hermitien sur un tore plat M, une connexion sur L et une fonction Q sur M, on associe un opérateur de Schrödinger agissant sur les sections de L ; on note Spec(Q;L,) son spectre. À la suite du travail de V. Guillemin en dimension deux, on considère des fibrés en droites complexes au dessus de tores de dimension paire quelconque ainsi qu’une connexion «  invariante par translation  » fixée et on se demande dans quelle mesure Spec(Q;L,) détermine le potentiel Q. Sous une condition générique, on montre que le spectre détermine la partie paire du potentiel, à condition que la connexion soit invariante par l’isométrie du tore définie par l’application x-x. En dimension deux, on obtient également des informations sur sa partie impaire. On obtient des contre-exemples qui montrent que la condition générique utilisée est nécessaire même dans le cas des tores de dimension deux. Ces exemples montrent aussi que le spectre du laplacien défini par une connexion sur un fibré en droites sur un tore plat ne détermine ni la classe d’isométrie du tore ni la classe de Chern du fibré.

En dimension quelconque, on montre que la collection de tous les spectres Spec(Q;L,), lorsque parcourt l’ensemble des connexions invariantes, détermine le potentiel de manière unique. Cette collection de spectres est une généralisation naturelle aux fibrés en droites du spectre classique de Bloch sur le tore.

DOI : https://doi.org/10.5802/aif.2420
Classification:  58J50,  58J53
Keywords: Schrödinger operator, spectrum, line bundles over tori
@article{AIF_2008__58_7_2445_0,
     author = {Gordon, Carolyn S. and Guerini, Pierre and Kappeler, Thomas and Webb, David L.},
     title = {Inverse spectral results on even dimensional tori},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {7},
     year = {2008},
     pages = {2445-2501},
     doi = {10.5802/aif.2420},
     mrnumber = {2498357},
     zbl = {1159.58015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_7_2445_0}
}
Gordon, Carolyn S.; Guerini, Pierre; Kappeler, Thomas; Webb, David L. Inverse spectral results on even dimensional tori. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2445-2501. doi : 10.5802/aif.2420. http://www.numdam.org/item/AIF_2008__58_7_2445_0/

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