  We establish sufficiency conditions in order to achieve approximation to
discrete eigenvalues of self-adjoint operators in the second-order projection
method suggested recently by Levitin and Shargorodsky, [math.SP/0212087]. We
find explicit estimates for the eigenvalue error and study in detail two
concrete model examples. Our results show that, unlike the majority of the
standard methods, second-order projection strategies combine non-pollution and
approximation at a very high level of generality.
