  We discuss the problems arising when computing eigenvalues of self-adjoint
operators which lie in a gap between two parts of the essential spectrum.
Spectral pollution, i.e. the apparent existence of eigenvalues in numerical
computations, when no such eigenvalues actually exist, is commonplace in
problems arising in applied mathematics. We describe a geometrically inspired
method which avoids this difficulty, and show that it yields the same results
as an algorithm of Zimmermann and Mertins.
