Spectral Theory is a branch of mathematics that studies the spectrum of operators, particularly linear operators on function spaces. It focuses on understanding how these operators can be represented in terms of their eigenvalues and eigenvectors, which are fundamental in solving differential equations and analyzing physical systems. In the context of quantum mechanics, Spectral Theory helps describe the behavior of quantum systems by examining the energy levels of particles. The theory is essential for understanding Hermitian operators, which represent observable quantities, and it provides tools for analyzing stability and dynamics in various scientific fields, including engineering and signal processing.