التحليل الطيفي لمعادلة كوشي الخطية في فضاءات هيلبرت (Comprehensive Journal of Science, 2026)
This research aims to study the spectroscopy of the Cauchy linear equation within Hilbert spaces, by integrating the concepts of spectroscopy and operator theory within the framework of partial differential equations. The Cauchy linear equation is an essential mathematical model for understanding the evolution of dynamic systems over time, as it is widely used in quantum physics, systems theory, and mathematical engineering. This paper examines the impact of spectrum on the evolution and stability of solutions in systems with infinite dimensions, focusing on specific and non-specific actuators, and analyzing the eigenvalues and associated eigenvalues. The study provides an integrated mathematical framework for interpreting the relationship between spectrum and solution stability, while demonstrating practical applications in the analysis of physical systems, heat and wave equations. It also discusses advanced theories in spectroscopy such as Friedman's theory, extending concepts to multidimensional spaces. The results conclude that spectrum properties play a pivotal role in determining the behavior of time solutions and the stability of mathematical and physical systems.
Keywords: Spectroscopy, Linear Cauchy equation, Hilbert spaces, Indeterminate operators, Stability of solutions, Systems theory