Spectral Analysis of Multidimensional Fractal Spaces Utilizing the Theory of Broken Laplacians

Abstract

The paper contains the spectral study of Laplacians that are broken on multidimensional fractal domains with a particular focus on the convergence, stability, and localization phenomena. Self-similar structures in fractal geometry were built in two and three dimensions by iterating a system of functions, contraction mappings, and by means of self-similarity parameters that are carefully determined. In the study, the broken Laplacian operator, a Laplacian operator obtained by piecewise Laplacians and domain decomposition methods, has been analyzed analytically and numerically using graph approximations and discrete energy forms. The spectral problem was posed in both the classical and variational form, and the eigenvalues and eigenfunctions were calculated at the refinement levels, showing smooth convergence towards relative errors of less than 0.3 percent to the high-resolution approximations. The fractal domain was broken through controlled discontinuities, and the eigenvalues shifted measurably with increasingly large fracture parameters of 12% and larger eigenvalues changing up to 31.4%. Eigenfunction localization was noted as well, where 27 percent of the total L^2 -mass was found to be concentrated in less than 6 percent of the domain, which confirms the sensitivity of high-frequency modes to geometric irregularities. The reduction of spectral dimensions by 8.55% and lifting of degeneracies caused by symmetry under anisotropic fragmentation was found through comparative analysis with classical Laplacians. These findings support the methodological scheme and indicate that broken Laplacians are an effective and mathematically consistent way to describe spectral properties of complex fractal spaces, and that they can be applied in the modeling of diffusion, wave propagation, and energy localization in heterogeneous media.