While the Sauter-Schwinger effect describes nonperturbative electron-positron pair creation from vacuum by a strong and slowly varying electric field E_strong via tunneling, the dynamically assisted Sauter-Schwinger effect corresponds to a strong (exponential) enhancement of the pair-creation probability by an additional weak and fast electric or electromagnetic pulse E_weak. Using the WKB and worldline instanton method, we find that this enhancement mechanism strongly depends on the shape of

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... e fast pulse. For the Sauter profile 1/^2(ω t) considered previously, the threshold frequency ω_crit (where the enhancement mechanism sets in) is basically independent of the magnitude E_weak of the weak pulse---whereas for a Gaussian pulse (-ω^2t^2), an oscillating profile cos(ω t) or a standing wave cos(ω t)cos(kx), the value of ω_crit does depend (logarithmically) on E_weak/E_strong.