Nonexponential decay and autoionising states

Complex QDT has been used to compute autoionising Rydberg series, most notably for the two-electron molecule H2 which, like He, exhibits autoionising series [279]. 

It was implicit in the last section and indeed in all the discussion so far that the decay of an autoionising resonance should be exponential in time. This may seem to be obvious, and is indeed verified with good accuracy in experiments, but there are also fundamental reasons for believing that it is not strictly correct [280] according to quantum mechanics, the exponential decay law is violated for very long times, where the probability of nonexponential decay eventually prevails, a fact which has been recognised in nuclear and particle theory [282], but has not so far been verified experimentally. 

The conditions under which nonexponential decay of autoionising states might appear have been considered phenomenologically [281], and it was suggested that the crucial issue is how close a resonance lies above the ionisation limit. In fact, it is the ratio T/Eo, where Eq the resonance energy is measured from the threhold, which must be significant for effects to appear.4 

Ab initio calculations by numerical integration of the Schrodinger equar tion suggest that nonexponential decay, resulting in a slight apparent lengthening of the lifetime, may be observable in some cases [280]. 

3 The easiest way to see this is simply to substitute a complex energy into the time-dependent Schrodinger equation the real part of the energy can then be associated with an oscillatory eigenfunction, while the imaginary part is associated with exponential decay. 

---