# Inverse spectral results on even dimensional tori

Carolyn S. Gordon^{[1]}; Pierre Guerini^{[2]}; Thomas Kappeler^{[3]}; David L. Webb^{[1]}

- [1] Dartmouth College Department of Mathematics 6188 Kemeny Hall Hanover, NH 03755-3551 (USA)
- [2] CPGE Dumont d’Urville 83056 Toulon cedex (France)
- [3] Institut für Mathematik Universität Zürich-Irchel Winterthurerstrasse 190 8057 Zürich (Switzerland)

Annales de l’institut Fourier (2008)

- Volume: 58, Issue: 7, page 2445-2501
- ISSN: 0373-0956

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topGordon, Carolyn S., et al. "Inverse spectral results on even dimensional tori." Annales de l’institut Fourier 58.7 (2008): 2445-2501. <http://eudml.org/doc/10384>.

@article{Gordon2008,

abstract = {Given a Hermitian line bundle $L$ over a flat torus $M$, a connection $\nabla $ on $L$, and a function $Q$ on $M$, one associates a Schrödinger operator acting on sections of $L$; its spectrum is denoted $Spec(Q; L,\nabla )$. Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections $\nabla $, and we address the extent to which the spectrum $Spec(Q;L,\nabla )$ 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 $x\rightarrow -x$, 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 $Spec(Q; L,\nabla )$, as $\nabla $ 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.},

affiliation = {Dartmouth College Department of Mathematics 6188 Kemeny Hall Hanover, NH 03755-3551 (USA); CPGE Dumont d’Urville 83056 Toulon cedex (France); Institut für Mathematik Universität Zürich-Irchel Winterthurerstrasse 190 8057 Zürich (Switzerland); Dartmouth College Department of Mathematics 6188 Kemeny Hall Hanover, NH 03755-3551 (USA)},

author = {Gordon, Carolyn S., Guerini, Pierre, Kappeler, Thomas, Webb, David L.},

journal = {Annales de l’institut Fourier},

keywords = {Schrödinger operator; spectrum; line bundles over tori},

language = {eng},

number = {7},

pages = {2445-2501},

publisher = {Association des Annales de l’institut Fourier},

title = {Inverse spectral results on even dimensional tori},

url = {http://eudml.org/doc/10384},

volume = {58},

year = {2008},

}

TY - JOUR

AU - Gordon, Carolyn S.

AU - Guerini, Pierre

AU - Kappeler, Thomas

AU - Webb, David L.

TI - Inverse spectral results on even dimensional tori

JO - Annales de l’institut Fourier

PY - 2008

PB - Association des Annales de l’institut Fourier

VL - 58

IS - 7

SP - 2445

EP - 2501

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

LA - eng

KW - Schrödinger operator; spectrum; line bundles over tori

UR - http://eudml.org/doc/10384

ER -

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