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.
Asymptotics of the Spectral Density for Radial Dirac Operators with Divergent Potentials
We study the asymptotics of the spectral density of one-dimensional Dirac systems on the half-line with an angular momentum term and a potential tending to infinity at infinity. The problem has two singular end-points; however, as the spectrum is simple, the derivative of the spectral matrix has only one non-zero eigenvalue which we take to be the spectral density. Our main result shows that, assuming sufficient regularity of the potential, there are no points of spectral concentration for large values of the spectral parameter outside a neighbourhood of a discrete set of exceptional points.
Weyl's theory and the method of complex rotation
The spectral properties of a singular second order differential operator L is analysed in terms of Weyl's theory. A spectral function is derived and the results are briefly compared with scattering theory. The analytic properties of L, its associated Green's function and related quantities of modern scattering theory are analysed. It is shown that the analytic continuation of an appropriately dilated operator L allows for an analytic extension of the singular case to a higher order Riemann sheet. The present theory is thus applicable to the study of the entire bound and continuous spectrum of L. A numerical realization of the theory is presented and applied to the scattering potential of Moiseyev et al. (1978, Molec. Phys., 36, 1613). New resonance structures are thereby found. Their asymptotic behaviour is discussed and an attempt to classify this type of structure is made. Implications of the present theory for predissociations in diatomic molecules, analysis of scattering experiments and ab initio stud...