具有离散对称的黎曼流形上的广义Bloch分析和传播子
Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry
摘要：
本文考虑了基于具有可数离散对称群$Γ$的黎曼流形$\tilde{M}$的$L^{2}$空间中的不变量子哈密顿量$H=-Δ_{LB}+V$.通常，$\tilde{M}$是多连通黎曼流形$M$的泛覆盖空间，而$Γ$是$M$的基本群。一方面，遵循Bloch分析的基本步骤，将$\tilde{M}$上的$L^{2}$空间分解为由$\tilde{M}$上的等变函数形成的Hilbert空间的直接积分。哈密顿量$H$相应地分解，每个分量$H_Λ$由准周期边界条件定义。准周期边界条件又由$Γ$的不可约酉表示$Λ$确定。另一方面，固定准周期边界条件（即，$Γ$的酉表示$Λ$），可以根据与哈密顿量$H$相关联的传播子来表示相应的传播子。我们详细讨论了这些程序，并表明，在某种意义上，他们是相互逆的。
英文摘要：
We consider an invariant quantum Hamiltonian $H=-Δ_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a countable discrete symmetry group $Γ$. Typically, $\tilde{M}$ 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 $\tilde{M}$ into a direct integral of Hilbert spaces formed by equivariant functions on $\tilde{M}$. The Hamiltonian $H$ decomposes correspondingly, with each component $H_Λ$ being defined by a quasi-periodic boundary condition. The quasi-periodic boundary conditions are in turn determined by irreducible unitary representations $Λ$ of $Γ$. On the other hand, fixing a quasi-periodic boundary condition (i.e., a unitary representation $Λ$ of $Γ$) one can express the corresponding propagator in terms of the propagator associated to the Hamiltonian $H$. We discuss these procedures in detail and show that in a sense they are mutually inverse.
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