Spectral analysis of Friedrichs systems

Many physical and engineering phenomenasuch as wave propagation, fluid flow, and electromagnetismare described by complex mathematical equations. Friedrichs systems provide a powerful unified mathematical framework to analyse such equations (within one framework), especially those that change their nature from one region to another (e.g., from elliptic to hyperbolic). One classical example is the Tricomi equation, which appears in modeling transonic airflowaroundaircraft. This project aims to develop new mathematical tools to better understand and solve these types of systems. Specifically, we will use modern methods from operator theory, a field of mathematics that provides key tools to look at differential equations as operators and develope the theories. A key concept in this project is the use of boundary triples which help describe the behavior of solutions at the boundary of a prescribed domain. It is helpful in describing useful tools like Weyl functions. These tools will enable us to analyse the spectrum of a systemessentially the possible behaviors or outcomes that it can produce. One major focus of the project is the development of a general method for treating systems where different parts of the domain follow different physical laws. For example, we aim to provide a new mathematical framework for coupling different types of equations together, which is crucial for modeling real-world phenomena such as mixed media or changing materials. We will also apply the theory to specific examples, including the Dirac system (important in quantum physics), time-harmonic Maxwell systems (used in electromagnetics), and semi-linear problems where nonlinearity plays a role. The project combines abstract mathematical thinking with practical applications, opening up new ways to understand complex physical systems. Ultimately, this research will contribute to both pure mathematics and applied sciences, offering new techniques for tackling some of the most challenging problems in modern physics and engineering.