Numerical Approximation of Unitary Exponential Functions

Absorbing potentials are mathematical tools used to analyze complex systems, like those found in physics and engineering. These potentials can help us understand how these systems behave and make predictions about their future behavior. In this paper, we explore the use of complex absorbing potentials and present new algorithms for approximating them.
Imagine you’re trying to understand a complex electrical circuit. You could break it down into smaller parts and study each one separately, but that might not give you the full picture. A better approach would be to use an absorbing potential, which allows you to analyze the entire circuit as a single entity. This makes it easier to understand how the different parts of the circuit interact with each other and how they behave over time.
The algorithms we present in this paper are like a toolkit for creating these absorbing potentials. They allow us to approximate the complex functions that describe the behavior of the system, making it easier to analyze and predict its behavior. We demonstrate the effectiveness of our algorithms through numerical examples, showing how they can be used to study a wide range of systems, from simple circuits to more complex phenomena like chaos and pattern formation.
Overall, this paper provides a valuable tool for anyone working with complex systems, helping them to better understand these systems and make more accurate predictions about their behavior. By demystifying complex concepts through engaging analogies and metaphors, we hope to make this important research accessible to a wider audience of scientists and engineers.