Absence of high-energy spectral concentration for Dirac systems with divergent potentials
It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.
On the Asymptotics of the Spectral Density of Radial Dirac Operators with Divergent Potential
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.
( 18 ) The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices
Recently E. C. Titchmarsh has treated the theory of expansion of arbitrary functions in terms of the eigenfunctions of a differential operator of the second order by a new method and obtained results of importance for applications.' The method of Titchmarsh is based solely on the calculus of residues. In the present paper we shall first give another proof of Titchmarsh's results based, on the general theory of linear operators in Hilbert space and, secondly, applying them to differential equations of Schrodinger type, show that a theorem of leisenberg2 concerning the S-matrix can be founded on these results.