Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

作者：Kostakova P; Stovicek P^*

来源：Journal of Geometry and Physics, 2011, 61(3): 727-744.

DOI：10.1016/j.geomphys.2010.12.004

• Bloch analysis
• Periodic Hamiltonian
• Covering space
• Propagator
• Green function

Given an invariant gauge potential and a periodic scalar potential (V) over tilde on a Riemannian manifold (M) over tilde with a discrete symmetry group Gamma, consider a Gamma-periodic quantum Hamiltonian (H) over tilde = -(Delta) over tilde (B) + (V) over tilde where (Delta) over tilde (B) is the Bochner Laplacian. Both the gauge group and the symmetry group Gamma can be noncommutative, and the gauge field need not vanish. On the other hand, Gamma is supposed to be of type I. With any unitary representation Lambda of Gamma one associates a Hamiltonian H-Lambda = -Delta(Lambda)(B) + V on M = (M) over tilde/Gamma where V is the projection of (V) over tilde onto M. We describe a construction of the Bloch decomposition of (H) over tilde into a direct integral whose components are H-Lambda, with Lambda running over the dual space (Gamma) over cap. The evolution operator and the resolvent decompose correspondingly. Conversely, given Lambda is an element of (Gamma) over cap, one can express the propagator K-t(Lambda) (y(1), y(2)) (the kernel of exp(-itH(Lambda))) in terms of the propagator (K) over tilde (t) (y(1), y(2)) (the kernel of exp(-it (H) over tilde)) as a weighted sum over Gamma. Such a formula is known in theoretical physics for the case when the gauge field vanishes and (M) over tilde is a universal covering space of a multiply connected manifold M. We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.

• 出版日期2011-3