Gamow向量与Borel可和性
Gamow vectors and Borel summability
摘要：
我们分析了一维哈密顿量$H=-\partial_x^2+V(x)$的波函数$ψ(x,t)$的详细时间依赖性，其中$V$（例如建模势垒或阱）和$ψ(x,0)$是{ \em compensated}的。我们证明了$ψ(x,t)$的色散部分，它的渐近级数在$t^{-1/2}$的幂，是Borel可和的。其余部分，即$ψ$和Borel和之间的差，是$\sum_{k=0}^{\infty}g_k Γ_k(x)e^{-γ_k t}$形式的收敛展开，其中$Γ_k$是$H$的伽莫夫向量，$γ_k$是相关的共振;一般来说，所有$g_k$都不为零。对于大型$k$，使用$γ_{k}\sim const\cdot k\log k +k^2π^{2}i/4$。当时间不是很大时，伽莫夫矢量的效果是可见的，并且分解以物理相关的方式严格定义了非微扰机制中的共振和伽莫夫矢量。分解允许计算$ψ$的中型和大型$t$，以任何规定的指数精度，使用最佳截断的幂级数加上100多个伽莫夫向量的贡献。$ψ$的解析结构可能令人惊讶：一般来说（即使在简单的例子中，如方孔），$ψ(x,t)$在$t$中是$C^\infty$，但在$\RR^+$中没有解析结构。事实上，$ψ$是$t-$在下半平面的扇区中的解析，并且具有整个$\RR^+$的自然边界。
英文摘要：
We analyze the detailed time dependence of the wave function $ψ(x,t)$ for one dimensional Hamiltonians $H=-\partial_x^2+V(x)$ where $V$ (for example modeling barriers or wells) and $ψ(x,0)$ are {\em compactly supported}. We show that the dispersive part of $ψ(x,t)$, its asymptotic series in powers of $t^{-1/2}$, is Borel summable. The remainder, the difference between $ψ$ and the Borel sum, is a convergent expansion of the form $\sum_{k=0}^{\infty}g_k Γ_k(x)e^{-γ_k t}$, where $Γ_k$ are the Gamow vectors of $H$, and $γ_k$ are the associated resonances; generically, all $g_k$ are nonzero. For large $k$, $γ_{k}\sim const\cdot k\log k +k^2π^{2}i/4$. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way. The decomposition allows for calculating $ψ$ for moderate and large $t$, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions. The analytic structure of $ψ$ is perhaps surprising: in general (even in simple examples such as square wells), $ψ(x,t)$ turns out to be $C^\infty$ in $t$ but nowhere analytic on $\RR^+$. In fact, $ψ$ is $t-$analytic in a sector in the lower half plane and has the whole of $\RR^+$ a natural boundary.
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