空间绝热微扰理论
Space-Adiabatic Perturbation Theory
摘要：
我们研究了薛定谔方程$i\epsi\partialψ_t(x)/\partial t = H(x,-i\epsi\nabla_x) ψ_t(x)$的近似解，给出了作为符号$H(q,p)$的Weyl量子化的哈密顿量，该符号在具有快速"内部“自由度的希尔伯特空间$\Hi_{\rm f}$上的有界算子空间中取值。假设$H(q,p)$具有孤立的能带。使用Nenciu和Sordoni的方法\cite{NS}，我们证明了带间跃迁被抑制到任何顺序的$\epsi$。因此，与该能带相关联，存在$L^2(\mathbb{R}^d,\Hi _{\rm f})$的子空间，该子空间在酉时间演化下几乎不变。我们开发了一个系统的微扰计划的有效哈密顿量的计算约带内的时间演化。作为一般微扰方案的例子，我们讨论了狄拉克和玻恩-奥本海默型哈密顿量，我们也重新考虑时间绝热理论。
英文摘要：
We study approximate solutions to the Schrödinger equation $i\epsi\partialψ_t(x)/\partial t = H(x,-i\epsi\nabla_x) ψ_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\Hi_{\rm f}$ of fast ``internal'' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \cite{NS} we prove that interband transitions are suppressed to any order in $\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\mathbb{R}^d,\Hi _{\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.
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