  We consider the eigenvalue problem of certain kind of non-compact linear
operators given as the sum of a multiplication and a kernel operator. A
degenerate kernel method is used to approximate isolated eigenvalues. It is
shown that entries of the corresponding matrix of this method can be evaluated
exactly. The convergence of the method is proved; it is proved that the
convergence rate is $O(h)$. By some numerical examples, we confirm the results.
