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Inverse spectral results on even dimensional tori


Gordon, C; Guerini, P; Kappeler, T; Webb, D (2008). Inverse spectral results on even dimensional tori. Annales de l'institut Fourier, 58(7):2445-2501.

Abstract

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.

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.

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3 citations in Web of Science®
3 citations in Scopus®
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Additional indexing

Other titles:Résultats spectraux inverses sur les tores de dimension paire
Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2008
Deposited On:09 Nov 2009 00:31
Last Modified:05 Apr 2016 13:23
Publisher:Association des Annales de l'Institut Fourier
ISSN:0373-0956
Official URL:http://aif.cedram.org/aif-bin/fitem?id=AIF_2008__58_7_2445_0
Permanent URL: https://doi.org/10.5167/uzh-21420

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