Transitions of electrons from a bound state into the continuous spectrum. Quadratic approximation in a model of a short-range potential

A study is made of the interaction of a discrete leveLwith a continuous spectrum in the model of a short-range potential with a logarithmic derivative that depends quadratically on the time. I t is possible to develop a unified approach which enables one to follow the transition between the well-known limiting cases and find a previously unknown universal numerical coefficient in the adiabatic theory for the continuous spectrum. The form of the spectrum is established for the emitted electrons and the behavior of the ionization probability is found in the case when the discrete level is near the boundary of the continuous spectrum. An approximate formula that holds for all values of the effective parameter of the problem is proposed forthe spectrum of the emitted electrons; this agrees well with the results of a numerical calculation.