In this work, we study the bifurcation structures and the stability of multidimensional localized states within
coherently driven Kerr optical cavities with parabolic potentials in 1D, 2D, and 3D systems. Based on symmetric
considerations, we transform higher-dimensional models into a single 1D model with a dimension parameter.
This transformation not only yields a substantial reduction in computational complexity, but also enables an
efficient examination of how dimensionality impacts the system dynamics. In the absence of nonlinearity,
we analyze the eigenstates of the linear systems. This allows us to uncover a heightened concentration of
the eigenmodes at the center of the potential well, while witnessing a consistent equal spacing among their
eigenvalues, as the dimension parameter increases. In the presence of nonlinearity, our findings distinctly
reveal that the stability of the localized states diminishes with increasing dimensionality. This study offers an
approach to tackling high-dimensional problems, shedding light on the fundamental dimensional connections
among radially symmetric states across different dimensions, and providing valuable tools for analysis.