Resonances in open systems are described by eigenvalue problems with radiation conditions at infinity and are relevant in fields such as acoustics, classical mechanics, quantum mechanics, and x-ray physics. This thesis focuses on optimizing resonances for multilayer x-ray resonators, which consist of multiple layers that support specific resonant states. These states can be excited by x-ray beams at particular grazing angles of incidence, corresponding to the system's resonant frequencies, resulting in significant field enhancement within the structure compared to the incident field. X-ray resonators or waveguides facilitate the filtering, guiding, and concentration of x-rays, beneficial for nanoscale x-ray structure analysis and imaging. The multilayer structures are characterized by the refractive index n, and the goal is to identify a function n that maximizes field enhancement at a resonant angle of incidence while adhering to constraints on n. The optimization problem employs an objective function involving complex resonances and resonance functions. Derivatives of these functions with respect to n are derived using perturbation theory of linear operators. Additionally, approximation formulas for reflectivity are developed, providing a mathematical basis for the kinematic approximation. Higher-order Taylor and Padé approximations yield significant improvements, particularly near the critical angle. The optimization proble
