Gamow vectors and Borel summability

TL;DR

This paper demonstrates that the dispersive part of the wave function for certain quantum Hamiltonians is Borel summable and decomposes it into a sum involving Gamow vectors and resonances, providing a rigorous nonperturbative analysis of quantum resonances.

Contribution

It introduces a rigorous decomposition of wave functions into Borel summable series and Gamow vectors, clarifying the role of resonances in quantum dynamics beyond perturbation theory.

Findings

Dispersive wave function series is Borel summable.

Wave functions can be decomposed into Gamow vectors and resonances.

Wave functions are smooth but not analytic in time, with a natural boundary.

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.