  We obtained analytically eigenvalues of a multidimensional Ising Hamiltonian
on a hypercube lattice and expressed them in terms of spin-spin interaction
constants and the eigenvalues of the one-dimensional Ising Hamiltonian (the
latter are well known). To do this we wrote down the multidimensional
Hamiltonian eigenvectors as the Kronecker products of the eigenvectors of the
one-dimensional Ising Hamiltonian. For periodic boundary conditions, it is
possible to obtain exact results taking into account interactions with an
unlimited number of neighboring spins. In this paper, we present exact
expressions for the eigenvalues for the planar and cubic Ising systems
accounting for the first five coordination spheres (that is interactions with
the nearest neighbors, the next neighbors, the next-next neighbors, the
next-next-next neighbors and the next-next-next-next neighbors). In the case of
free-boundary systems, we showed that in the two- and three-dimensions the
exact expressions could be obtained only if we account for interactions with
spins of first two coordination spheres and first three coordination spheres,
respectively.
