Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T09:51:13.488Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Brownell, F.H., Spectrum of the static potential Schrödinger equation over En, Ann. Math., 54 (1951), 554–594.Google Scholar
2. Brownell, F.H., Perturbation of the n-dimensional Schrödinger differential operator, Abstract no. 430t, Bull. Amer. Math. Soc, 60 (1954). 346.Google Scholar
3. Coddington, E.A., On self-adjoint ordinary differential operators, Math. Scand., 4 (1956), 9–21.Google Scholar
4. Coddington, E.A., The spectral matrix and Green's functions for singular self-adjoint boundary value problems, Can. J. Math., 6 (1954), 169–185.Google Scholar
5. Kodaira, K., On ordinary differential equations of any even order and the corresponding eigenfunclion expansions, Amer. J. Math., 72 (3) (1950), 502–543.Google Scholar
6. Moser, J., Stôrungstheorie des kontinuierlichen Spektrums, Math. Ann. 125 (1953), 366–393.Google Scholar
7. Rosenblum, M., Perturbation of the continuous spectrum and unitary equivalence, Pac. J. Math., 7 (1957), 997–1010.Google Scholar
8. Stone, M.H., Linear transformations in Hilbert space, Amer. Math. Soc. Coll. Pub., 15 (1932).Google Scholar
9. Titchmarsh, E.C., Rigenfunction expansions, Part II, (London: Oxford University Press, 1958).Google Scholar
10. Windau, W., Ueber lineare differentialgleichungen vierter ordnung mit singularitaten und die dazugehôrigen darstellungen willkurlicher functionen, Math. Ann., 83 (1921), 256–279.Google Scholar