Upper bounds to the eigenvalues of the Dirac Hamiltonian

The eigenvalue problem of a matrix representation of the one-electron Dirac Hamiltonian in a finite-dimensional functional space is considered. Theorems which determine conditions under which the matrix eigenvalues are upper bounds to the exact bound state energies of the Dirac Hamiltonian are formulated. Asymptotic simplifications of the theorems are also mentioned. Numerical examples corresponding to a spherical Coulomb potential in the Slater-type and Gauss-type bases are presented.