Spectral concentration and rapidly decaying potentials

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)cosα + y′(0)sinα = 0 and where q ∈ L(0, ∞). A point μ0 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 Prüfer 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.