  The problem of approximating the discrete spectra of families of self-adjoint
operators that are merely strongly continuous is addressed. It is well-known
that the spectrum need not vary continuously (as a set) under strong
perturbations. However, it is shown that under an additional compactness
assumption the spectrum does vary continuously, and a family of symmetric
finite-dimensional approximations is constructed. An important feature of these
approximations is that they are valid for the entire family uniformly. An
application of this result to the study of plasma instabilities is illustrated.
