A variation of Landau's eigenvalue theorem describing the phase transition of the eigenvalues of a time-frequency limiting, self adjoint operator is presented. The total number of degrees of freedom of square-integrable, multi-dimensional, bandlimited functions is defined in terms of Kolmogorov's $n$-width and computed in some limiting regimes where the original theorem cannot be directly applied. Results are used to characterize up to order the total amount of information that can be transported in time and space by multiple-scattered electromagnetic waves, rigorously addressing a question originally posed in the early works of Toraldo di Francia and Gabor. Applications in the context of wireless communication and electromagnetic sensing are discussed.