Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T19:00:09.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Subordinacy Analysis and Absolutely Continuous Spectra for Sturm-Liouville Equations with Two Singular Endpoints

Published online by Cambridge University Press:  20 November 2018

Dominic P. Clemence
Dominic P. Clemence*
Affiliation:
Department of Mathematics North Carolina Agricultural and Technical State University Greensboro, NC 27411 USA
Rights & Permissions [Opens in a new window]

Abstract

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.

The Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

Keywords

34L0534B2034B24
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 1 , 01 March 1998 , pp. 23 - 27
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Behncke, H., Absolute continuity of Hamiltonians with von Neumann-Wigner potentials. Proc. Amer.Math. Soc. 111 (1991), 373–384.Google Scholar
2. Behncke, H., Absolute continuity of Hamiltonians with von Neumann-Wigner potentials: II. Manuscripta Math. 71 (1991), 163–181.Google Scholar
3. Bennewitz, C., Spectral asymptotics for Sturm-Liouville equations. Proc. London. Math. Soc. (3) 59 (1989), 294–338.Google Scholar
4. Clark, S. L. and Hinton, D. B., Strong nonsubordinacy and absolutely continuous spectra for Sturm-Liouville equations. Differential Integral Equations (3) 6 (1993), 573–586.Google Scholar
5. del Rio Castillo, R. R., On conditions which guarantee stability of the absolutely continuous spectrum of ordinary differential operators. Quart. J. Math. Oxford (2) 39 (1988), 269–280.Google Scholar
6. Gilbert, D. J., On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints. Proc. Roy. Soc. Edinburgh Sect. A 112 (1987), 213–229.Google Scholar
7. Gilbert, D. J. and Pearson, D. B., On subordinacy and analysis of the spectrum of one dimensional operators. J. Math. Anal. Appl. 128 (1987), 30–56.Google Scholar
8. Stoltz, G., On the absolutely continuous spectrum of perturbed periodic Sturm-Liouville operators. J. Reine Angew. Math. 416 (1991), 1–23.Google Scholar