Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T15:06:16.123Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Singular Behaviour of the Titchmarsh-Weyl m-Function for the Perturbed Hill’s Equation on the Line

Published online by Cambridge University Press:  20 November 2018

Dominic P. Clemence
Dominic P. Clemence*
Affiliation:
Department of Mathematics, NCA&T State University, Greensboro, North Carolina 27411, U.S.A., e-mail: clemence@athena.ncat.edu
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.

For the perturbed Hill’s equation on the line,

we study the behaviour of the matrix m-function at the spectral gap endpoints. In particular, we extend the result of Hinton, Klaus and Shaw that En, a gap endpoint, is a half-bound state (HBS) if and only if (E − En)½m(E) approaches a nonzero constant as E → En, to the present case.

Keywords

34L0534B2034B24
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 4 , 01 December 1997 , pp. 416 - 421
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Clemence, D. P., On the Titchmarsh-Weyl M(ï)-coefficient and spectral density for a Dirac system. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 259–277.Google Scholar
2. Clemence, D. P.,M-function behaviour for a periodic Dirac System. Proc.Roy. Soc. Edinburgh Sect.A124 (1994), 149–159.Google Scholar
3. Clemence, D. P., Continuity of the S-matrix for perturbed periodic Hamiltonian systems. preprint.Google Scholar
4. Clemence, D. P. and Klaus, M., Continuity of the S-matrix for the perturbed Hill's equation. J. Math. Phys. 35 (1994), 3285–3300.Google Scholar
5. Coppel, W. A., Stability and Asymptotic Behaviour of Differential Equations. Boston, D. C. Heath and Co, 1965.Google Scholar
6. Firsova, N. E., Riemann surface of quasimomentum and scatter theory for the perturbed Hill operator. J. Soviet Math. 11 (1979), 487–497.Google Scholar
7. Hinton, D. B., Klaus, M. and Shaw, J. K., On the Titchmarsh-Weyl function for the half line perturbed periodic Hill's equation. Quart. J. Math. Oxford 41 (1990), 189–224.Google Scholar
8. Hinton, D. B., Levinson's theorem and Titchmarsh-Weyl theory for Dirac systems. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 173–186.Google Scholar
9. Hinton, D. B. and Shaw, J. K., On the absolutely continuous spectrum of the perturbed Hill's equation. Proc. London Math. Soc. 50 (1985), 175–182.Google Scholar
10. Zheludev, V. A., Eigenvalues of the perturbed Schrödinger operator with a periodic potential. In: Topics in Mathematical Physics, (ed. M. Sh. Birman), Consultants Bureau, New York, 1968, Vol. 2, 87–101.Google Scholar