Abstract
This study investigates the computational implementation of spectral theory and its applications in analyzing complex mathematical problems. It explores the use of modern programming languages and scientific libraries for implementing and visualizing complex mathematical concepts, particularly focusing on spectral theory within quantum physics. The research employs computational methods to address the challenges in interpreting spectral properties, utilizing the Lanczos spectral method for eigenvalue calculation in large, sparse matrices. The results illustrate the effectiveness of these computational techniques in visualizing quantum states, demonstrating the potential of advanced programming in understanding and solving intricate problems in quantum physics and spectral graph theory. The study's findings are significant in bridging computational methods with theoretical spectral analysis, offering a new perspective on the application of computational techniques in scientific research.
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