Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ ( x , t ) for one dimensional Hamiltonians 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 , 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⋅ k log 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 .