Quantum Equivalent Magnetic Fields that are not Classically Equivalent

Authors

Carolyn Gordon, Dartmouth CollegeFollow
William Kirwin, Instituto Superior Tecnico
Dorothee Schueth, Humboldt-Universität zu Berlin
David Webb, Dartmouth College

Document Type

Article

Publication Date

10-15-2010

Publication Title

Annales de l'Institut Fourier

Department

Department of Mathematics

Abstract

We construct pairs of compact Kähler-Einstein manifolds (Mi,gi,ωi)(i=1,2) of complex dimension n with the following properties: The canonical line bundle Li=⋀ⁿT∗M_i has Chern class [ω_i/2π], and for each positive integer k the tensor powers L^⊗k₁ and L^⊗k₂ are isospectral for the bundle Laplacian associated with the canonical connection, while M₁ and M₂ – and hence T∗M₁ and T∗M₂ – 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 Q₁, Q₂ 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.

DOI

10.5802/aif.2612

Dartmouth Digital Commons Citation

Gordon, Carolyn; Kirwin, William; Schueth, Dorothee; and Webb, David, "Quantum Equivalent Magnetic Fields that are not Classically Equivalent" (2010). Dartmouth Scholarship. 3140.
https://digitalcommons.dartmouth.edu/facoa/3140