An analytic study of the ionization from an ultrathin quantum well in a weak electrostatic field

We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction \psi_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, \psi_F(0,t) is a key element to their evaluation. We obtain a closed expression for \psi_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |\psi_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.