This paper studies ordinary
differential operators of the form
over a finite interval I. The coefficients Qj are bounded operators in L2(I). This
operator is treated as a perturbation T + A of the operator T, which is
generated by the leading term (−1)mD2m plus suitable boundary conditions. The
main hypothesis is that Q2m−1 can be written as the sum of a compact
operator and a bounded operator of sufficiently small norm. Given that
T is a discrete spectral operator, with eigenvalues {λn}, it is shown that
T + A is also a discrete spectral operator, with eigenvalues {λn′} satisfying
|λn′− λn| = O(|λn|k∕2m), where k is the largest integer ≦ 2m − 1 for which
Qk≠0. Proofs are based on the method of contour integration of resolvent
operators.
