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.
Spectral concentration and rapidly decaying potentials
Abstract We consider the spectral function ϱ ( μ ) ( μ ⩾0) for the Sturm-Liouville equation y ″ + ( λ − q ) y = 0 on [0, ∞) with a boundary condition y (0)cos α + y ′(0)sin α = 0 and where q ∈ L (0, ∞). A point μ 0 is called a point of spectral concentration if ϱ′ has a local maximum there. We give a new formula for ϱ′ in terms of the Prufer transformation angle θ and we identify a rapid transitional property of θ in the neighbourhood of points of spectral concentration. This property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by, for example, SLEDGE. Our approach also leads to the possibility of a theoretical investigation.
The Invariance of the Vacuum is the Invariance of the World
The following theorem is demonstrated: If the vacuum is invariant under the group generated by the space integral of the time component of a local vector current, then the Hamiltonian is invariant also. A similar theorem holds for the group generated by the space integral of a space component of a local axial-vector current.