On construit des couples de variétés de Kähler-Einstein compactes () de dimension complexe avec les propriétés suivantes : la première classe de Chern associée au fibré en droites canonique est , et pour tout entier positif , les puissances tensorielles et sont isospectrales pour le Laplacien associé à la connexion canonique, mais et – et, en conséquence, et – ne sont pas homéomorphes. Dans le contexte de la quantification géométrique, nous interprétons ces exemples comme des champs magnétiques qui sont équivalents au sens quantique mais pas au sens classique. En plus, on construit beaucoup d’exemples de fibrés en droites , de couples de potentiels , sur la variété de base et de couples de connexions , telles que, pour tout entier positif , les opérateurs de Schrödinger associés sur soient isospectraux.
We construct pairs of compact Kähler-Einstein manifolds of complex dimension with the following properties: The canonical line bundle has Chern class , and for each positive integer the tensor powers and are isospectral for the bundle Laplacian associated with the canonical connection, while and – and hence and – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent. Moreover, we construct many examples of line bundles , pairs of potentials , on the base manifold, and pairs of connections , on such that for each positive integer the associated Schrödinger operators on are isospectral.
Keywords: Geometric quantization, tensor powers of line bundles, Laplacian, isospectral line bundles
Mot clés : quantification géométrique, puissances tensorielles des fibrés en droites, Laplacien, fibrés en droites isospectraux
@article{AIF_2010__60_7_2403_0, author = {Gordon, Carolyn and Kirwin, William and Schueth, Dorothee and Webb, David}, title = {Quantum {Equivalent} {Magnetic} {Fields} that {Are} {Not} {Classically} {Equivalent}}, journal = {Annales de l'Institut Fourier}, pages = {2403--2419}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {7}, year = {2010}, doi = {10.5802/aif.2612}, zbl = {1230.53084}, mrnumber = {2849266}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2612/} }
TY - JOUR AU - Gordon, Carolyn AU - Kirwin, William AU - Schueth, Dorothee AU - Webb, David TI - Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent JO - Annales de l'Institut Fourier PY - 2010 SP - 2403 EP - 2419 VL - 60 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2612/ DO - 10.5802/aif.2612 LA - en ID - AIF_2010__60_7_2403_0 ER -
%0 Journal Article %A Gordon, Carolyn %A Kirwin, William %A Schueth, Dorothee %A Webb, David %T Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent %J Annales de l'Institut Fourier %D 2010 %P 2403-2419 %V 60 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2612/ %R 10.5802/aif.2612 %G en %F AIF_2010__60_7_2403_0
Gordon, Carolyn; Kirwin, William; Schueth, Dorothee; Webb, David. Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2403-2419. doi : 10.5802/aif.2612. http://www.numdam.org/articles/10.5802/aif.2612/
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