Gamow vectors and Borel summability

Abstract

We analyze the detailed time dependence of the wave function $\psi (x,t)$ for one dimensional Hamiltonians $H=-{\partial }_x^2+V(x)$ where $V$ (for example modeling barriers or wells) and $\psi (x,0)$ are {\em compactly supported}. We show that the dispersive part of $\psi (x,t)$, its asymptotic series in powers of $t^{-1/2}$, is Borel summable. The remainder, the difference between $\psi$ and the Borel sum, is a convergent expansion of the form ${\sum }_{k=0}^{\infty }{g}_k{\Gamma }_k(x)e^{-{\gamma }_{k}t}$, where ${\Gamma }_k$ are the Gamow vectors of $H$, and ${\gamma }_k$ are the associated resonances; generically, all $g_k$ are nonzero. For large $k$, ${\gamma }_k\sim const\cdot k\log k+k^2{\pi }^{2}i/4$. 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 $\psi$ 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 $\psi$ is perhaps surprising: in general (even in simple examples such as square wells), $\psi (x,t)$ turns out to be $C^{\infty }$ in $t$ but nowhere analytic on $\RR^+$. In fact, $\psi$ is $t-$analytic in a sector in the lower half plane and has the whole of $\RR^+$ a natural boundary.