  There are classical theorems of analysis which, given certain conditions on a
perturbation, assert stability of the essential and absolutely continuous
components of the spectrum of a self-adjoint operator. Whereas the singular
component is known to be highly volatile under the weakest of all possible
perturbations, -- rank one.
  In this note I announce a theorem which asserts that, nevertheless, the
singular component of spectrum in an open interval is in a sense
\emph{relatively stable} provided the limiting absorption principle (LAP) holds
in the interval. One of the benefits of this result is the provision of a
method for disproving LAP where it is suspected to fail.
