Gamow vectors and Borel summability

Abstract

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.