EIGENFUNCTION EXPANSIONS OF SECOND-ORDER ELLIPTIC OPERATORS

This study explores the theory of spectral representations for second-order elliptic operators. The existence and properties of eigenvalues and eigenfunctions in bounded regions are examined, and solutions are formulated through orthogonal basis functions in Hilbert spaces. The research further applies the obtained theoretical results to boundary value problems, investigates convergence behavior, and highlights their importance in numerical approximations. In addition, the effects of regularity conditions, parameter variations, and illustrative examples are discussed. The findings contribute to a deeper understanding of elliptic operator theory and open new directions for future investigations.