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

Received:
Accepted:
DOI: 10.5802/aif.2612
Classification: 58J53,  53C20
Keywords: Geometric quantization, tensor powers of line bundles, Laplacian, isospectral line bundles
Gordon, Carolyn 1; Kirwin, William 2; Schueth, Dorothee 3; Webb, David 1

1 Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)
2 Instituto Superior Tecnico CAMGSD Departamento de Matematica Av. Rovisco Pais, 1049-001 Lisboa (Portugal) Current Address: University of Cologne Mathematisches Institut Weyertal 86 50931 Cologne (Germany)
3 Humboldt-Universität zu Berlin Institut für Mathematik 10099 Berlin (Germany)
@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},
     mrnumber = {2849266},
     zbl = {1230.53084},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2612/}
}
TY  - JOUR
TI  - Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent
JO  - Annales de l'Institut Fourier
PY  - 2010
DA  - 2010///
SP  - 2403
EP  - 2419
VL  - 60
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2612/
UR  - https://www.ams.org/mathscinet-getitem?mr=2849266
UR  - https://zbmath.org/?q=an%3A1230.53084
UR  - https://doi.org/10.5802/aif.2612
DO  - 10.5802/aif.2612
LA  - en
ID  - AIF_2010__60_7_2403_0
ER  - 
%0 Journal Article
%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 https://doi.org/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, Volume 60 (2010) no. 7, pp. 2403-2419. doi : 10.5802/aif.2612. https://aif.centre-mersenne.org/articles/10.5802/aif.2612/

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

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

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

[4] Berezin, F. A.; Shubin, M. A. The Schrödinger equation, Mathematics and its Applications (Soviet Series), Volume 66, Kluwer Academic Publishers Group, Dordrecht, 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 | Zbl

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

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

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

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

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

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

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

[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. Transl. Ser. 2) Volume 211, Amer. Math. Soc., Providence, RI, 2003, pp. 25-46 | MR | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

Cited by Sources: