稠密点谱本征值的扰动：一般Floquet哈密顿量
Perturbation of an eigenvalue from a dense point spectrum: a general Floquet Hamiltonian
摘要：
考虑Hilbert空间$L^2([0,T],E,dt)$中的扰动Floquet Hamilton算子$-i\partial_t + H + βV(ωt)$.这里$H$是$E$中的自伴算子，其离散谱服从增长间隙条件，$V(t)$是$E$中的对称有界算子，其依赖于$t$ $2π$ -周期性地，$ω= 2π/T$是频率，$β$是耦合常数。对于几乎所有的$ω$，未扰动部分的谱$Spec(-i\partial_t + H)$是纯点的，并且在$R$中是稠密的。这一事实排除了正则微扰理论的应用。然而，我们表明，几乎所有的$ω$和提供$V(t)$是足够光滑的，微扰理论仍然是有意义的，但是，有两个修改。首先，耦合常数被限制到一个集合$I$，它不需要是一个区间，但0仍然是$I$的密度点。第二，瑞利-薛定谔级数对扰动本征值和扰动本征向量是渐近的。
英文摘要：
We consider a perturbed Floquet Hamiltonian $-i\partial_t + H + βV(ωt)$ in the Hilbert space $L^2([0,T],E,dt)$. Here $H$ is a self-adjoint operator in $E$ with a discrete spectrum obeying a growing gap condition, $V(t)$ is a symmetric bounded operator in $E$ depending on $t$ $2π$-periodically, $ω= 2π/T$ is a frequency and $β$ is a coupling constant. The spectrum $Spec(-i\partial_t + H)$ of the unperturbed part is pure point and dense in $R$ for almost every $ω$. This fact excludes application of the regular perturbation theory. Nevertheless we show, for almost all $ω$ and provided $V(t)$ is sufficiently smooth, that the perturbation theory still makes sense, however, with two modifications. First, the coupling constant is restricted to a set $I$ which need not be an interval but 0 is still a point of density of $I$. Second, the Rayleigh-Schrodinger series are asymptotic to the perturbed eigen-value and the perturbed eigen-vector.
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