Cascade calculation with schematic interactions

In previous works we considered schematic Hamiltonians represented by simplified matrices. We defined two transition operators and calculated transition strengths from the ground state to all excited states. In many cases the strengths decreased nearly exponentially with the excitation energy. Now we do the reverse. We start with the highest energy state and calculate the cascade of transitions until the ground state is reached. On a log plot we show the average transition strength as a function of the number of energy intervals that were crossed. We give an analytic proof of exponential behavior for transition strength in the weak coupling limit for the [Formula: see text] transition operator.