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Perturbation of the Continuous Spectrum of Even Order Differential Operators

Published online by Cambridge University Press:  20 November 2018

John B. Butler Jr.
John B. Butler Jr.*
Affiliation:
University of Arizona
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Let L0 be a differential operator of even order n = 2v on the half open interval 0 ≤ t < ∞ which is formally self adjoint and satisfies the conditions of Kodaira (5, p. 503). We consider a perturbed operator of the form L∈ = Lo + ∈q where q(t) is a real-valued bounded function and ∈ is a real parameter. The object of this paper is to set up conditions on the operator L0 and the function q(t) such that L∈ determines a self-adjoint operator H∈ and such that the spectral resolution operator E∈(Δ) corresponding to H∈ is analytic in a neighbourhood of ∈ = 0, where Δ is a closed bounded interval.

Our conditions are a natural generalization of conditions considered by Moser for the case n = 2(6). Moser has given a number of examples showing that when his conditions do not hold E∈(Δ) need not be analytic.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 309 - 323
Copyright
Copyright © Canadian Mathematical Society 1960

References

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