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Minimax principle for the Dirac equation.

The problem of calculating the eigenvalues of the Dirac equation by the finite-basis expansion method is studied. Bounds for the eigenvalues are obtained explaining the numerical results on the spectrum that have been observed previously. It is argued that the problem of variational collapse can be avoided by finding the minimum over the wave-function large component of the maximum over the wave-function small component of the energy functional. A numerical example is discussed.