Accuracy of approximate eigenstates

Besides perturbation theory, which requires the knowledge of the exact unpe rturbed solution, variational techniques represent the main tool for any in vestigation of the eigenvalue problem of some semibounded operator H in qua ntum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H can be located with acceptable pre cision whereas the trial-subspace vectors corresponding to these eigenvalue s approximate, in general, the exact eigenstates of H with much less accura cy. Accordingly, various measures for the accuracy of approximate eigenstat es derived by variational techniques are scrutinized. In particular, the ma trix elements of the commutator of the operator H and (suitably chosen) dif ferent operators with respect to degenerate approximate eigenstates of H ob tained by the variational methods are proposed as new criteria for the accu racy of variational eigenstates. These considerations are applied to that H amiltonian the eigenvalue problem of which defines the spinless Salpeter eq uation. This bound-state wave equation may be regarded as the most straight forward relativistic generalization of the usual nonrelativistic Schrodinge r formalism, and is frequently used to describe, e.g. spin-averaged mass sp ectra of bound states of quarks.