Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry
- Department of Mathematics, Faculty of Nuclear Science, Czech Technical University, Trojanova 13, 120 00 Prague (Czech Republic)
We consider an invariant quantum Hamiltonian H=-{delta}{sub LB}+V in the L{sup 2} space based on a Riemannian manifold M-tilde with a countable discrete symmetry group {gamma}. Typically, M-tilde is the universal covering space of a multiply connected Riemannian manifold M and {gamma} is the fundamental group of M. On the one hand, following the basic step of the Bloch analysis, one decomposes the L{sup 2} space over M-tilde into a direct integral of Hilbert spaces formed by equivariant functions on M-tilde. The Hamiltonian H decomposes correspondingly, with each component H{sub {lambda}} being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations {lambda} of {gamma}. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation {lambda} of {gamma}) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian H. We discuss these procedures in detail and show that in a sense they are mutually inverse.
- OSTI ID:
- 21100248
- Journal Information:
- Journal of Mathematical Physics, Vol. 49, Issue 3; Other Information: DOI: 10.1063/1.2898484; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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