The discreteness of the spectrum of self-adjoint, even order, one-term, differential operators

Lewis, Roger T.

doi:10.1090/S0002-9939-1974-0330608-8

The discreteness of the spectrum of self-adjoint, even order, one-term, differential operators
HTML articles powered by AMS MathViewer

by Roger T. Lewis PDF

Proc. Amer. Math. Soc. 42 (1974), 480-482 Request permission

Abstract:

An open question which was asked by I. M. Glazman is answered. It is well known that the condition \[ \lim \limits _{x \to \infty } {x^{2n - 1}}\int _x^\infty {{r^{ - 1}} = 0} \] is sufficient for the discreteness and boundedness from below of the spectrum of selfadjoint extensions of ${( - 1)^n}{(r{y^{(n)}})^{(n)}}$. This paper shows that the condition is also necessary.

References

Calvin D. Ahlbrandt, Equivalent boundary value problems for self-adjoint differential systems, J. Differential Equations 9 (1971), 420–435. MR 284636, DOI 10.1016/0022-0396(71)90015-5
I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey & Co., Inc., New York, 1966. Translated from the Russian by the IPST staff. MR 0190800
Roger T. Lewis, Oscillation and nonoscillation criteria for some self-adjoint even order linear differential operators, Pacific J. Math. 51 (1974), 221–234. MR 350112