Time asymptotics of \(e^{-ith(\kappa)}\) (Q2929389)

The mathematical problem on the time decay of certain resonance states in quantum mechanics is reduced to that of finite dimension, say, \(N\), of studying the time evolution of \(e^{-it h(\kappa)}\) of an analytic family of non-selfadjoint \(N\times N\)-matrices \(\{h(\kappa)\}\) where \(\kappa\) is a parameter in a small neighborhood of \(0\) in the complex plane. In the paper under review the authors elaborately use analytic perturbation theory to derive an asymptotic expansion for \(e^{-it h(\kappa)}\) in the simultaneous limits of \(\kappa\rightarrow 0\) and \(t\rightarrow \infty\).NEWLINENEWLINEThe main result is as follows. If \(\{h(\kappa)\}\) is a family of \(N\times N\)-matrices analytic in \(\kappa\) with \(h(0) = \lambda_0 I_{N\times N}\) for some real \(\lambda_0\) with \(I_{N\times N}\) the identity matrix, and if it holds for some real \(\alpha\) and complex \(\beta\) that \(h'(0)=\alpha I_{N\times N}\) and all the eigenvalues \(\lambda(\kappa)\) of \(h(\kappa)\) have the form \( \lambda(\kappa) = \lambda_0+\alpha\kappa+\beta\kappa^2 + O(|\kappa|^{2+1/N})\) as \(\kappa \rightarrow 0\), then \(e^{-it h(\kappa)}\) enjoys the asymptotic expansion NEWLINE\[NEWLINE e^{-it h(\kappa)} = e^{-it (\lambda_0+\alpha\kappa+\beta\kappa^2)} \big[I_{N\times N} + R(\kappa,t)\big], NEWLINE\]NEWLINE as \(\kappa \rightarrow 0\), uniformly in complex variable \(t\). Here \(R(\kappa,t)\) is analytic in both \(\kappa\) and \(t\), and has uniform bounds in the one complex variable \(t\kappa^2\).NEWLINENEWLINEThis yields, in particular, when all the eigenvalues of \(h(\kappa)\) have the form \( \lambda(\kappa) = \lambda_0 + \alpha\kappa +\beta\kappa^2 +o(|\kappa|^2)\) as \(\kappa\rightarrow 0\), with some real \(\alpha\) and complex \(\beta\) with \(\text{Im} \beta <0\), the following asymptotics: for some \(p \geq 2+1/N\) and for every \(\tilde{N} \geq (p-2)(N-1)N\), NEWLINE\[NEWLINE \big\|e^{-it h(\kappa)}\big\| = e^{-t\kappa^2|\text{Im} \beta|} \big[1+\sum_{k=1}^{N+\tilde{N}-2} O((t\kappa^2)^k) + e^{ct|\kappa|^{2+1/N}}\sum_{k=0}^{N-1} O((t\kappa^2)^{k+\tilde{N}})\big], NEWLINE\]NEWLINE as real \(\kappa \rightarrow 0\), uniformly \(t\geq 0\), where \(c\) is a positive constant. In fact, the correction \(R(\kappa,t)\) above consists of two terms \(F_{\tilde{N}}(\kappa,t)\) and \(R_{\tilde{N}}(\kappa,t)\), to be given pretty explicitly in the paper. This formula shows exponential decay in time of \(e^{-it h(\kappa)}\), modified by polynomial growth and the exponential factor \(e^{ct|\kappa|^{2+1/N}}\) coming from branching of eigenvalues. So it is interpreting the physical picture of exponential time decay of a resonance state. The result improves some of the precededing ones by \textit{W. Hunziker} [Commun. Math. Phys. 132, No. 1, 177--188 (1990; Zbl 0721.35047)] and by the authors [Ann. Henri Poincaré 11, No. 3, 499--537 (2010; Zbl 1208.81092)].