# Cascade Calculations with Schematic Interactions

## TL;DR

This paper investigates cascade calculations in schematic Hamiltonians, demonstrating exponential decay of transition strengths and providing an analytic proof for this behavior in the weak coupling limit.

## Contribution

It introduces a reverse cascade calculation approach and analytically proves exponential transition strength decay in the weak coupling limit.

## Key findings

- Transition strengths decrease exponentially with excitation energy.
- Analytic proof of exponential behavior in the weak coupling limit.
- Reverse cascade calculation method demonstrated.

## Abstract

In previous works we considered schematic Hamiltonians represented by simplified matrices. We defined 2 transition operators and calculated transition strengths from the ground state to all exited states. In many cases the strengths decreased nearly exponentially with excitation energy. Now we do the reverse We start with the highest energy state and calculate the cascade of transitions until the ground states 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 T2 transition operator.