On the Asymptotics of the Spectral Density of Radial Dirac Operators with Divergent Potential

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.

Dirac Operators with Operator Data of Wigner-von Neumann Type

“…In order to prove boundedness of solutions, we perform a Prüfer transformation to an arbitrary solution U (x, η). Prüfer variables have been used many times in the spectral theory of Schrödinger and Dirac operators (e.g., [13,14,16,17,18,19,20,21,26,27,32]). We set…”