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Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential

Published online by Cambridge University Press:  14 November 2011

Karl Michael Schmidt
Karl Michael Schmidt
Affiliation:
Mathematisches Institut der Universität, Theresienstraße 39, D-80333 München, Germany
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Abstract

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 5 , 1996 , pp. 1087 - 1096
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Eastham, M. S. P. and McLeod, J. B.. The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 25–34.CrossRefGoogle Scholar
2Erdélyi, A.. Note on a paper by Titchmarsh. Quart. J. Math. Oxford (2) 14 (1963), 147–52.CrossRefGoogle Scholar
3Evans, W. D. and Harris, B. J.. Bounds for the point spectra of separated Dirac operators. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 1–15.CrossRefGoogle 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. and Shaw, J. K.. Dirac systems with discrete spectra. Canad. J. Math. 39 (1987), 100–22.CrossRefGoogle Scholar
6Kalf, H.. Non-existence of eigenvalues of Dirac operators. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 309–17.CrossRefGoogle Scholar
7Naboko, S. N.. Dense point spectra of Schrodinger and Dirac operators. Theoret. and Math. Phys. 68 (1986), 646–53.CrossRefGoogle Scholar
8Roos, B. W. and Sangren, W. C.. Spectra for a pair of singular first order differential equations. Proc. Amer. Math. Soc. 12 (1961), 468–76.CrossRefGoogle Scholar
9Titchmarsh, E. C.. On the nature of the spectrum in problems of relativistic quantum mechanics. Quart. J. Math. Oxford Ser. (2) 12 (1961), 227–40.CrossRefGoogle Scholar
10Vogelsang, V.. Absence of embedded eigenvalues of the Dirac equation for long range potentials. Analysis 7 (1987), 259–74.CrossRefGoogle Scholar
11Weidmann, J.. Absolutstetiges Spektrum bei Sturm-Liouville-Operatoren und Diracsystemen. Math. Z. 180 (1982), 423–7.CrossRefGoogle Scholar
12Weidmann, J.. Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258 (Berlin: Springer, 1987).CrossRefGoogle Scholar