arXiv 0802.07552008-02-06math-phmath.MP

与周期哈密顿量相关的非线性算子：具有两个涡旋的Aharonov-Bohm哈密顿量的例子

Propagators associated to periodic Hamiltonians: an example of the Aharonov-Bohm Hamiltonian with two vortices

P. Kocabova, P. Stovicek

摘要：

本文考虑了基于具有离散对称群$Γ$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，$\tilde{M}$是多连通流形$M$的泛覆盖空间，而$Γ$是$M$的基本群。对于$Γ$的任何酉表示$Λ$，人们可以将$M=\tilde{M}/Γ$上的另一个算子联系起来，称为$H_Λ$，它在形式上对应于与$H$相同的微分算子，但它由准周期边界条件确定。我们给出了一个简短的审查的Bloch分解$H$和一个公式有关的传播相关的哈密顿$H_Λ$和$H$。然后，我们集中在Aharonov-Bohm效应与两个涡的例子。我们详细解释了这种情况下传播器的构造，并指出了所有必要的中间步骤。

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 discrete symmetry group $Γ$. Typically, $\tilde{M}$ is the universal covering space of a multiply connected manifold $M$ and $Γ$ is the fundamental group of $M$. To any unitary representation $Λ$ of $Γ$ one can relate another operator on $M=\tilde{M}/Γ$, called $H_Λ$, which formally corresponds to the same differential operator as $H$ but which is determined by quasi-periodic boundary conditions. We give a brief review of the Bloch decomposition of $H$ and of a formula relating the propagators associated to the Hamiltonians $H_Λ$ and $H$. Then we concentrate on the example of the Aharonov-Bohm effect with two vortices. We explain in detail the construction of the propagator in this case and indicate all essential intermediate steps.

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摘要：

英文摘要：

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