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.
Titchmarsh's asymptotic formula for periodic eigenvalues and an extension to the p-Laplacian
An asymptotic formula for periodic eigenvalues, due to Titchmarsh, is taken to a higher degree of accuracy using a Prufer transformation method, not depending on Floquet theory. With the idea of the rotation number, this method is then extended to the p-Laplacian.
Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential
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.