  We exploit the so called form-local subordination in the analysis of
non-symmetric perturbations of unbounded self-adjoint operators with isolated
simple positive eigenvalues. If the proper condition relating the size of gaps
between the unperturbed eigenvalues and the strength of perturbation, measured
by the form-local subordination, is satisfied, the root system of the perturbed
operator contains a Riesz basis and usual asymptotic formulas for perturbed
eigenvalues and eigenvectors hold. The power of the abstract perturbation
results is demonstrated particularly on Schr\"odinger operators with possibly
unbounded or singular complex potential perturbations.
