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Levinson's theorem and Titchmarsh-Weyl m(λ) theory for Dirac systems*

Published online by Cambridge University Press:  14 November 2011

D. B. Hinton ,
M. Klaus  and
J. K. Shaw
D. B. Hinton
Affiliation:
Mathematics Department, University of Tennessee, Knoxville, TN 37996, U.S.A.
M. Klaus
Affiliation:
Mathematics Department, Virginia Tech, Blacksburg, VA 24061, U.S.A.
J. K. Shaw
Affiliation:
Mathematics Department, Virginia Tech, Blacksburg, VA 24061, U.S.A.
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Synopsis

A Levinson theorem is proved for a Dirac system with one singular endpoint. The number ofbound state is expressed in terms of the change in asymptotic phase of an appropriate solution and in terms of factors whose values depend on the presence of half-bound states. The behaviour of the asymptotic phase is used to determine the asymptotic behaviour of the Titchmarsh-Weyl m-function.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 173 - 186
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Barthélémy, M. C.. Contribution à l'étude de la diffusion par un potentiel central dans la théorie de l'électron de Dirac II. Ann. Inst. H. Poincaré A 7 (1967), 115–143.Google Scholar
2Coppel, W. A.. Stability and Asymptotic Behavior of Differential Equations (Boston: Heath, 1965).Google Scholar
3Evgrafov, M. A.. Analytic Functions (Philadelphia: Saunders, 1966).Google Scholar
4Hinton, D. B. and Shaw, J. K.. Absolutely continuous spectra of Dirac systems with long range, short range, and oscillating potentials. Quart. J. Math. Oxford Ser. (2) 36 (1985), 183–213.CrossRefGoogle Scholar
5Hinton, D. B., Mingarelli, A. B., Read, T. T. and Shaw, J. K.. On the number of eigenvaluesin the spectral gap of a Dirac system. Proc. Edinburgh Math. Soc. 29 (1986), 367–378.CrossRefGoogle Scholar
6Klaus, M.. On the variation-diminishing property of Schrodinger operators. Proceedings of the Canadian Mathematical Society: 1986 Seminar on Oscillation, Bifurcation, and Chaos (Providence: American Mathematical Society, 1987).Google Scholar
7Levinson, N.. On the uniqueness of the potential in a Schrodinger equation for a given asymptotic phase: Mat.-Fys. Medd. Danske Vid. Selsk. 25 (1949), 1–29.Google Scholar
8Levitan, B. M. and Sargsjan, I.. Introduction to Spectral Theory: Self adjoint Ordinary Differential Operators. Translations of Mathematical Monographs 39 (Providence, R.I.: American Mathematical Society, 1975).CrossRefGoogle Scholar
9Marchenko, V. A.. Sturm-Liouville Operators and Applications. Operator theory: Advances and Applications 22 (Basel: Birkhäuser, 1986).CrossRefGoogle Scholar
10Ma, Z. and Ni, G.. Levinson theorem for Dirac particles. Phys. Rev. D 31 (1985), 1482–1488.CrossRefGoogle ScholarPubMed
11Martinez, J. C.. Levinson's theorem for a fermion-monopole system. J. Phys. A 20 (1987), 2903–2908.CrossRefGoogle Scholar
12Newton, R. G.. Scattering Theory of Waves and Particles (New York: McGraw-Hill, 1966).Google Scholar
13Reed, M. and Simon, B.. Methods of Modern Mathematical Physics III: Scattering Theory (New York: Academic Press, 1979).Google Scholar
14Weidmann, J.. Spectral Theory of Ordinary Differential Operators. Lecture notes in mathematics 1258 (Berlin: Springer, 1987).CrossRefGoogle Scholar