Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry

We consider an invariant quantum Hamiltonian H = − Δ LB + V in the L 2 space based on a Riemannian manifold M ̃ with a countable discrete symmetry group Γ . Typically, M ̃ is the universal covering space of a multiply connected Riemannian manifold M and Γ is the fundamental group of M . On the one h...

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Bibliographic Details
Published in:Journal of mathematical physics 2008-03, Vol.49 (3), p.033518-033518-16
Main Authors: KOCABOVA, P, STOVICEK, P
Format: Article
Language:English
Subjects:
BOUNDARY CONDITIONS
BOUNDARY-VALUE PROBLEMS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Combinatorics
Exact sciences and technology
FUNCTIONS
Geometry
HAMILTONIANS
HILBERT SPACE
INTEGRALS
Mathematical methods in physics
Mathematics
Physics
PROPAGATOR
QUANTUM MECHANICS
Quantum theory
RIEMANN SPACE
Sciences and techniques of general use
SYMMETRY
SYMMETRY GROUPS
Topological manifolds
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Summary:We consider an invariant quantum Hamiltonian H = − Δ LB + V in the L 2 space based on a Riemannian manifold M ̃ with a countable discrete symmetry group Γ . Typically, M ̃ is the universal covering space of a multiply connected Riemannian manifold M and Γ is the fundamental group of M . On the one hand, following the basic step of the Bloch analysis, one decomposes the L 2 space over M ̃ into a direct integral of Hilbert spaces formed by equivariant functions on M ̃ . The Hamiltonian H decomposes correspondingly, with each component H Λ being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations Λ of Γ . On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation Λ of Γ ) 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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.2898484