SPECTRAL EXPANSIONS FOR SECOND-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

This paper investigates the spectral expansion theory for second-order elliptic differential operators. The existence of eigenvalues and eigenfunctions in bounded domains is established, and solutions are expressed in terms of an orthonormal basis. The obtained results are applied to analyze boundary value problems, evaluate convergence properties, and provide a theoretical foundation for numerical methods. The influence of model parameters, regularity, and practical examples are briefly discussed. The results pave the way for future research opportunities.