Abstract
The radial Dirac operator with a potential tending to infinity at infinity and satisfying a mild regularity condition is known to have a purely absolutely continuous spectrum covering the whole real line. Although having two singular end-points in the limit-point case, the operator has a simple spectrum and a generalised Fourier expansion in terms of a single solution. In the present paper, a simple formula for the corresponding spectral density is derived, and it is shown that, under certain conditions on the potential, the spectral function is convex for large values of the spectral parameter. This settles a question considered in earlier work by M. S. P. Eastham and the author.
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Brunnhuber R., Eckhardt J., Kostenko A., Teschl G.: Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. Monatsh. Math. 174, 515–547 (2014)
Coddington E.A., Levinson N.: Theory of ordinary differential equations. McGraw-Hill Book Company Inc, New York (1955)
Eastham M.S.P., Schmidt K.M.: Absence of high-energy spectral concentration for Dirac systems with divergent potentials. Proc. R. Soc. Edinb. 135A, 689–702 (2005)
Eastham M.S.P., Schmidt K.M.: Asymptotics of the spectral density for radial Dirac operators with divergent potentials. Publ. RIMS Kyoto Univ. 44, 107–129 (2008)
Eckhardt, J., Kostenko, A., Teschl, G.: Spectral Asymptotics for Canonical Systems. J. Reine Angew. Math. (2015) (to appear)
Erdélyi A.: Note on a paper by Titchmarsh. Quart. J. Math. 14, 147–152 (1963)
Fulton C., Langer H.: Sturm-Liouville oprators with singularities and generalized Nevanlinna functions. Compl. Anal. Oper. Theory 4, 179–243 (2010)
Fulton C., Langer H., Luger A.: Mark Krein’s method of directing functionals and singular potentials. Math. Nachr. 285, 1791–1798 (2012)
Gesztesy F., Zinchenko M.: On the spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279, 1041–1082 (2006)
Kac I.S.: On the existence of spectral functions of certain second-order singular differential systems. Dokl. Akad. Nauk SSSR (N.S.) 106, 15–18 (1956)
Kostenko A., Sakhnovich A., Teschl G.: Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials. Int. Math. Res. Not. 2012, 1699–1747 (2012)
Schmidt K.M.: Absolutely continuous spectrum of Dirac systems with potentials infinite at infinity. Math. Proc. Camb. Philos. Soc. 122, 377–384 (1997)
Schmidt K.M., Yamada O.: Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity. Publ. RIMS Kyoto Univ. 34, 211–227 (1998)
Weidmann, J.: Spectral theory of ordinary differential operators, Springer Lecture Notes in Mathematics 1258, Berlin (1987)
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Dedicated to the memory of Michael S. P. Eastham
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Schmidt, K.M. On the Asymptotics of the Spectral Density of Radial Dirac Operators with Divergent Potential. Integr. Equ. Oper. Theory 85, 137–149 (2016). https://doi.org/10.1007/s00020-015-2276-8
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DOI: https://doi.org/10.1007/s00020-015-2276-8