Imaginary potential as a counter of delay time for wave reflection from a one-dimensional random potential

We show that the delay time distribution for wave reflection from a one-dimensional (one-channel) random potential is related directly to that of the reflection coefficient, derived with an arbitrarily small but uniform imaginary part added to the random potential. Physically, the reflection coefficient, being exponential in the time dwelt in the presence of the imaginary part, provides a natural counter for it. The delay time distribution then follows straightforwardly from our earlier results for the reflection coefficient, and coincides with the distribution obtained recently by Texier and Comtet [C. Texier and A. Comtet, Phys. Rev. Lett. 82, 4220 (1999)], with all moments infinite. The delay time distribution for a random amplifying medium is then derived. In this case, however, all moments work out to be finite.