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It has been reported in the literature that the survival probability $P(t)$near an exceptional point where two eigenstates coalesce should generallyexhibit an evolution $P(t) \sim t^2 e^{-\Gamma t}$, in which $\Gamma$ is thedecay rate of the coalesced eigenstate; this has been verified in a microwavebilliard experiment [B. Dietz, et al, Phys. Rev. E 75, 027201 (2007)]. However,the heuristic effective Hamiltonian that is usually employed to obtain thisresult ignores the possible influence of the continuum threshold on thedynamics. By contrast, in this work we employ an analytical approach startingfrom the microscopic Hamiltonian representing two simple models in order toshow that the continuum threshold has a strong influence on the dynamics nearexceptional points in a variety of circumstances. To report our results, wedivide the exceptional points in Hermitian open quantum systems into two cases:at an EP2A two virtual bound states coalesce before forming a resonance,anti-resonance pair with complex conjugate eigenvalues, while at an EP2B tworesonances coalesce before forming two different resonances. For the EP2B,which is the case studied in the microwave billiard experiment, we verify thesurvival probability exhibits the previously reported modified exponentialdecay on intermediate timescales, but this is replaced with an inverse powerlaw on very long timescales. Meanwhile, for the EP2A the influence from thecontinuum threshold is so strong that the evolution is non-exponential on alltimescales and the heuristic approach fails completely. When the EP2A appearsvery near the threshold we obtain the novel evolution $P(t) \sim 1 - C_1\sqrt{t} + D_1 t$ on intermediate timescales, while further away the parabolicdecay (Zeno dynamics) on short timescales is enhanced. We also discuss theparametric encirclement of the EP2A in an appendix.

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects