Vol. 25, No. 1, 1968

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On relatively bounded perturbations of ordinary differential operators

Colin W. Clark

Vol. 25 (1968), No. 1, 59–70

This paper studies ordinary differential operators of the form

(− 1)mD2m + Q2m− 1D2m −1 + ⋅⋅⋅+ Q0,

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 Q2m1 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 Qk0. Proofs are based on the method of contour integration of resolvent operators.

Mathematical Subject Classification
Primary: 47.60
Secondary: 34.00
Received: 29 November 1966
Published: 1 April 1968
Colin W. Clark