Gamow Vectors and Borel Summability in a Class of Quantum Systems

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 compactly supported.

We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^−1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form $\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}$, where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ∼const⋅klog k+k ² π 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 ^∞ in t but nowhere analytic on ℝ⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ℝ⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.