Vol. 25, No. 1, 1968

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 299: 1
Vol. 298: 1  2
Vol. 297: 1  2
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Other MSP Journals
On relatively bounded perturbations of ordinary differential operators

Colin W. Clark

Vol. 25 (1968), No. 1, 59–70
Abstract

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
Milestones
Received: 29 November 1966
Published: 1 April 1968
Authors
Colin W. Clark