Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent

Gordon, Carolyn; Kirwin, William; Schueth, Dorothee; Webb, David

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2403-2419.

On construit des couples de variétés de Kähler-Einstein compactes $(M_i,g_i,{\omega }_i)$ ($i=1,2$) de dimension complexe $n$ avec les propriétés suivantes : la première classe de Chern associée au fibré en droites canonique $L_i={\bigwedge }^n{T}^{*}{M}_i$ est ${\omega }_i/2\pi$, et pour tout entier positif $k$, les puissances tensorielles $L_1^{\otimes k}$ et $L_2^{\otimes k}$ 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 ${\nabla }_1$, ${\nabla }_2$ telles que, pour tout entier positif $k$, les opérateurs de Schrödinger associés sur $L^{\otimes k}$ soient isospectraux.

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.

Classification : 58J53, 53C20
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

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