Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent

Gordon, Carolyn, et al. "Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent." Annales de l’institut Fourier 60.7 (2010): 2403-2419. <http://eudml.org/doc/116339>.

@article{Gordon2010,
abstract = {We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,\omega _i) (i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i=\bigwedge ^n T^*M_i$ has Chern class $[\omega _i/2\pi ]$, and for each positive integer $k$ the tensor powers $L_1^\{\otimes k\}$ and $L_2^\{\otimes k\}$ 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 $\nabla _1$, $\nabla _2$ on $L$ such that for each positive integer $k$ the associated Schrödinger operators on $L^\{\otimes k\}$ are isospectral.},
affiliation = {Dartmouth College Department of Mathematics Hanover, NH 03755 (USA); 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); Humboldt-Universität zu Berlin Institut für Mathematik 10099 Berlin (Germany); Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)},
author = {Gordon, Carolyn, Kirwin, William, Schueth, Dorothee, Webb, David},
journal = {Annales de l’institut Fourier},
keywords = {Geometric quantization; tensor powers of line bundles; Laplacian; isospectral line bundles; geometric quantization},
language = {eng},
number = {7},
pages = {2403-2419},
publisher = {Association des Annales de l’institut Fourier},
title = {Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent},
url = {http://eudml.org/doc/116339},
volume = {60},
year = {2010},
}

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
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2403
EP - 2419
AB - We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,\omega _i) (i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i=\bigwedge ^n T^*M_i$ has Chern class $[\omega _i/2\pi ]$, and for each positive integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ 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 $\nabla _1$, $\nabla _2$ on $L$ such that for each positive integer $k$ the associated Schrödinger operators on $L^{\otimes k}$ are isospectral.
LA - eng
KW - Geometric quantization; tensor powers of line bundles; Laplacian; isospectral line bundles; geometric quantization
UR - http://eudml.org/doc/116339
ER -