Darboux transformations of coherent states of the time-dependent singular oscillator

Darboux transformation of both Barut-Girardelo and Perelomov coherent states for the time-dependent singular oscillator is studied. In either case, a measure that realizes the resolution of the identity operator in terms of coherent states is found and a corresponding holomorphic representation is constructed. For the particular case of the free particle moving with the fixed value of the square of angular momentum equal to 2 it is shown that the Barut-Giriardelo coherent states are more localized at the initial time moment while the Perelomov coherent states are more stable with respect to time evolution. It is also illustrated that Darboux transformation may keep unchanged this different time behaviour.