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

Abstract

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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 S p e c ( 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 S p e c ( 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 S p e c ( 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.

How to cite

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Gordon, 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 -

References

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  11. T. Kappeler, On isospectral periodic potentials on a discrete lattice II., Adv. in Appl. Math. 9 (1988), 428-438 Zbl0675.35023MR968676
  12. P. Kuchment, Floquet theory for partial differential equations, (1993), Birkhäuser Zbl0789.35002MR1232660
  13. P. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 624-646 Zbl0083.31801MR97628
  14. R. S. Palais, T. E. Stewart, Torus bundles over a torus, Proc. Amer. Math. Soc. 12 (1961), 26-29 Zbl0102.38702MR123638
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