caa a22 4500 44539126X CHVBK 20180317143047.0 cr unu---uuuuu 170323e20110801xx s 000 0 eng 10.1007/s10955-011-0276-x doi (NATIONALLICENCE)springer-10.1007/s10955-011-0276-x Gamow Vectors and Borel Summability in a Class of Quantum Systems [Elektronische Daten] [O. Costin, M. Huang] 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, ψis 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. Springer Science+Business Media, LLC, 2011 Gamow vectors nationallicence Schrödinger equation nationallicence Borel summability nationallicence Transseries nationallicence Costin O. Mathematics Department, The Ohio State University, 43210, Columbus, OH, USA aut Huang M. Department of Mathematics, The University of Chicago, 5734 S. University Avenue, 60637, Chicago, IL, USA aut Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/4(2011-08-01), 846-871 0022-4715 144:4<846 2011 144 10955 https://doi.org/10.1007/s10955-011-0276-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-011-0276-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Costin O. Mathematics Department, The Ohio State University, 43210, Columbus, OH, USA aut NATIONALLICENCE 700 1- Huang M. Department of Mathematics, The University of Chicago, 5734 S. University Avenue, 60637, Chicago, IL, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/4(2011-08-01), 846-871 0022-4715 144:4<846 2011 144 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer