  Besides perturbation theory, which requires, of course, the knowledge of the
exact unperturbed solution, variational techniques represent the main tool for
any investigation of the eigenvalue problem of some semibounded operator H in
quantum theory. For a reasonable choice of the employed trial subspace of the
domain of H, the lowest eigenvalues of H usually can be located with acceptable
precision whereas the trial-subspace vectors corresponding to these eigenvalues
approximate, in general, the exact eigenstates of H with much less accuracy.
Accordingly, various measures for the accuracy of the approximate eigenstates
derived by variational techniques are scrutinized. In particular, the matrix
elements of the commutator of the operator H and (suitably chosen) different
operators, with respect to degenerate approximate eigenstates of H obtained by
some variational method, are proposed here as new criteria for the accuracy of
variational eigenstates. These considerations are applied to that Hamiltonian
the eigenvalue problem of which defines the "spinless Salpeter equation." This
(bound-state) wave equation may be regarded as the most straightforward
relativistic generalization of the usual nonrelativistic Schroedinger
formalism, and is frequently used to describe, e.g., spin-averaged mass spectra
of bound states of quarks.
