Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-27T07:07:39.394Z Has data issue: false hasContentIssue false
  • English
  • Français

The convexity of the spectral function in Sturm–Liouville problems

Part of: Boundary value problems

Published online by Cambridge University Press:  26 February 2010

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
Department of Computer Science, Cardiff University, P.O. Box 916, Cardiff CF24 3XF.
Get access

Extract

§1. Introduction. The spectral function ρα(μ) (−∞<μ<∞) associated with the Sturm–Liouville equation

and a boundary condition

is a non-decreasing function of μ which is defined in terms of the Titchmarsh–Weyl function mα(λ) for (1.1) and (1.2).

MSC classification

Secondary: 34B24: Sturm-Liouville theory
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 319 - 326
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Brown, B. M., Eastham, M. S. P. and McCormack, D. K. R.. Spectral concentration and rapidly decaying potentials. J. Camp. Appl, Math., 81 (1997), 333–348.Google Scholar
2.Brown, B. M., Eastham, M. S. P. and McCormack, D. K. R.. Spectral concentration and perturbed discrete spectra. J. Comp. Appl. Math., 86 (1997), 415–425.Google Scholar
3.Brown, B. M., Eastham, M. S. P. and McCormack, D. K. R.. Absolute continuity and spectral concentration for slowly decaying potentials. J. Comp. Appl. Math., 94 (1998), 181–197.Google Scholar
4.Eastham, M. S. P.. The asymptotic form of the spectral function in Sturm-Liouville problems with a large potential like -xc(c≤2). Proc. Roy. Soc. Edinburgh 128A (1998), 37–45.CrossRefGoogle Scholar
5.Eastham, M. S. P.. On the location of spectral concentration for Sturm-Liouville problems with rapidly decaying potential. Mathematika, 45 (1998), 25–36.Google Scholar
6.Eastham, M. S. P.. On the location of spectral concentration for perturbed discrete spectra. Mathematika, 46 (1999), 145–154.Google Scholar
7.Eastham, M. S. P. and Kalf, H.. Schrödinger-type operators with continuous spectra. Research Notes in Mathematics 65 (Pitman, London, 1982).Google Scholar
8.Engdahl, E. and Brändas, E.. Resonance regions determined by a projection-operator formulation. Phys. Rev. A 37 (1988), 4145–4152.Google Scholar
9.Froese, R.. Asymptotic distribution of resonances in one dimension. J. Diff. Equations 137 (1997), 251–272.CrossRefGoogle Scholar
10.Harris, B. J.. The form of the spectral functions associated with Sturm-Liouville problems with continuous spectrum. Mathematika, 44 (1997), 149–161.Google Scholar
11.Hislop, P. D. and Sigal, I. M.. Introduction to spectral theory. Springer Applied Math. Sciences 113 (Springer, Berlin, 1996).Google Scholar
12.Lehr, H., Chatzidimitriou-Dreismann, C. A. and Brändas, E.. Pole strings of exponentially damped periodic potentials. Phys. Scripta, 49 (1994), 528–535.Google Scholar
13.Marletta, M.. The detection of quantum-mechanical resonances using the Pruess method, Technical Note ACM 90–14, Royal Military College of Science, 1990.Google Scholar
14.Pruess, S. A., Fulton, C. T. and Xie, Y.. Performance of the software package SLEDGE, Colorado School of Mines Department of Math, and Comp. Sci. Tech. Rep. MCS-91–19, 1991.Google Scholar
15.Reed, M. and Simon, B.. Methods of modern mathematical physics vol. 4: Analysis of operators, (Academic Press, New York, 1978).Google Scholar
16.Rittby, M., Elander, N. and Brändas, E.. Weyl's theory and the method of complex rotation: A synthesis for a description of the continuous spectrum. Molecular Phys., 11 (1982), 1–20.Google Scholar
17.Titchmarsh, E. C.. Eigenfunclion expansions part I, 2nd ed. (Clarendon Press, Oxford, 1962).Google Scholar