Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, p. 2403-2419

We construct pairs of compact Kähler-Einstein manifolds (M i ,g i ,ω i )(i=1,2) of complex dimension n with the following properties: The canonical line bundle L i = n T * M i has Chern class [ω i /2π], and for each positive integer k the tensor powers L 1 k and L 2 k are isospectral for the bundle Laplacian associated with the canonical connection, while M 1 and M 2 – and hence T * M 1 and T * M 2 – 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 L, pairs of potentials Q 1 , Q 2 on the base manifold, and pairs of connections 1 , 2 on L such that for each positive integer k the associated Schrödinger operators on L k are isospectral.

On construit des couples de variétés de Kähler-Einstein compactes (M i ,g i ,ω i ) (i=1,2) de dimension complexe n avec les propriétés suivantes : la première classe de Chern associée au fibré en droites canonique L i = n T * M i est ω i /2π, et pour tout entier positif k, les puissances tensorielles L 1 k et L 2 k sont isospectrales pour le Laplacien associé à la connexion canonique, mais M 1 et M 2 – et, en conséquence, T * M 1 et T * M 2 – 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 L, de couples de potentiels Q 1 , Q 2 sur la variété de base et de couples de connexions 1 , 2 telles que, pour tout entier positif k, les opérateurs de Schrödinger associés sur L k soient isospectraux.

DOI : https://doi.org/10.5802/aif.2612
Classification:  58J53,  53C20
Keywords: Geometric quantization, tensor powers of line bundles, Laplacian, isospectral line bundles
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     pages = {2403-2419},
     doi = {10.5802/aif.2612},
     mrnumber = {2849266},
     zbl = {1230.53084},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 60 (2010) no. 7, pp. 2403-2419. doi : 10.5802/aif.2612. http://www.numdam.org/item/AIF_2010__60_7_2403_0/

[1] Ballmann, Werner Lectures on Kähler manifolds, European Mathematical Society (EMS), Zürich, ESI Lectures in Mathematics and Physics (2006) | MR 2243012 | Zbl 1101.53042

[2] Bates, Sean; Weinstein, Alan Lectures on the geometry of quantization, American Mathematical Society, Providence, RI, Berkeley Mathematics Lecture Notes, Tome 8 (1997) | MR 1806388 | Zbl 1049.53061

[3] Bérard, Pierre Transplantation et isospectralité. I, Math. Ann., Tome 292 (1992) no. 3, pp. 547-559 | Article | MR 1152950 | Zbl 0735.58008

[4] Berezin, F. A.; Shubin, M. A. The Schrödinger equation, Kluwer Academic Publishers Group, Dordrecht, Mathematics and its Applications (Soviet Series), Tome 66 (1991) (Translated from the 1983 Russian edition by Yu. Rajabov, D. A. Leĭtes and N. A. Sakharova and revised by Shubin, With contributions by G. L. Litvinov and Leĭtes) | MR 1186643 | Zbl 0749.35001

[5] Brooks, Robert On manifolds of negative curvature with isospectral potentials, Topology, Tome 26 (1987) no. 1, pp. 63-66 | Article | MR 880508 | Zbl 0617.53048

[6] Brooks, Robert; Gornet, Ruth; Gustafson, William H. Mutually isospectral Riemann surfaces, Adv. Math., Tome 138 (1998) no. 2, pp. 306-322 | Article | MR 1645582 | Zbl 0997.53031

[7] Buser, Peter Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), Tome 36 (1986) no. 2, pp. 167-192 | Article | Numdam | MR 850750 | Zbl 0579.53036

[8] Gordon, Carolyn; Makover, Eran; Webb, David Transplantation and Jacobians of Sunada isospectral Riemann surfaces, Adv. Math., Tome 197 (2005) no. 1, pp. 86-119 | Article | MR 2166178 | Zbl 1083.58031

[9] Guillemin, Victor; Sternberg, Shlomo Geometric asymptotics, American Mathematical Society, Providence, R.I. (1977) (Mathematical Surveys, No. 14) | MR 516965 | Zbl 0364.53011

[10] Guillemin, Victor; Sternberg, Shlomo Symplectic techniques in physics, Cambridge University Press, Cambridge (1990) | MR 1066693 | Zbl 0734.58005

[11] Kuwabara, Ruishi Isospectral connections on line bundles, Math. Z., Tome 204 (1990) no. 4, pp. 465-473 | Article | MR 1062129 | Zbl 0728.53025

[12] Kuwabara, Ruishi Spectral geometry for Schrödinger operators in a magnetic field focusing on geometrical and dynamical structures on manifolds [translation of Sūgaku 54 (2002), no. 1, 37–57; MR1918697], Selected papers on analysis and differential equations, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 211 (2003), pp. 25-46 | MR 1918697 | Zbl 1122.58303

[13] Mcreynolds, D. B. Isospectral locally symmetric manifolds (Preprint, arXiv:math/0606540v2)

[14] Pesce, Hubert Variétés hyperboliques et elliptiques fortement isospectrales, J. Funct. Anal., Tome 134 (1995) no. 2, pp. 363-391 | Article | MR 1363805 | Zbl 0847.58077

[15] Rajan, C. S. On isospectral arithmetical spaces, Amer. J. Math., Tome 129 (2007) no. 3, pp. 791-806 | MR 2325104 | Zbl 1131.58023

[16] Reed, Michael; Simon, Barry Methods of modern mathematical physics. I-IV, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980) (Functional analysis) | MR 751959 | Zbl 0459.46001

[17] Śniatycki, Jędrzej Geometric quantization and quantum mechanics, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 30 (1980) | MR 554085 | Zbl 0429.58007

[18] Sunada, Toshikazu Riemannian coverings and isospectral manifolds, Ann. of Math. (2), Tome 121 (1985) no. 1, pp. 169-186 | Article | MR 782558 | Zbl 0585.58047

[19] Vignéras, Marie-France Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2), Tome 112 (1980) no. 1, pp. 21-32 | Article | MR 584073 | Zbl 0445.53026

[20] Wells, R. O. Jr. Differential analysis on complex manifolds, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 65 (1980) | MR 608414 | Zbl 0435.32004

[21] Woodhouse, N. M. J. Geometric quantization, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1992) (Oxford Science Publications) | MR 1183739 | Zbl 0747.58004

[22] Zelditch, Steven Isospectrality in the FIO category, J. Differential Geom., Tome 35 (1992) no. 3, pp. 689-710 | MR 1163455 | Zbl 0769.53026