Absolute continuity and spectral concentration for slowly 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 the boundary condition y(0) = 0 and where q has slow decay O(x −α ) (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333-348) to deal with the new theoretical and computational complexities which arise for slow decay.

Efficient calculation of spectral density functions for specific classes of singular Sturm–Liouville problems

“…We point out that for very slowly decaying exponentials, where values of are close to zero, this may not hold true, depending on the accuracy sought. The error bound of F 3, for example, is proportional to the fourth derivative of q, so it contains an 4 factor that is small. The second example is Bessel's equation of order one on [1, ∞) with a Neumann boundary condition at x = 1.…”

“…In [7] we improved on this by deriving a sequence of more sophisticated formulas for P , Q, R in terms of the general potential q and its derivatives; these usually allowed smaller values of x to be used for a given accuracy compared to the methods in [6], but did not take advantage of special forms of q. In this paper we assume q takes on one of the following special forms: The latter class includes many examples that occur in the study of resonances and gives rise to points of spectral concentration [4].…”

“…These types of potentials are the objects of continuing research, especially for applications dealing with resonance phenomena and spectral concentration; see Brown et al [2][3][4] for examples. A further paper [8] makes use of the present methods for the purpose of computing points of spectral concentration.…”

Absence of high-energy spectral concentration for Dirac systems with divergent potentials

“…In the case of the Schrödinger operator (1.1) considered in, for example, [1,3,7,8,11,24, § § 5.7 and 5.10], an integral formula for the spectral density is derived as a consequence of the asymptotic form of the solutions of the equation ly = λy as x → ∞ (see [6]). Here λ is the complex spectral parameter.…”

“…On the face of it, (2.9) presents an additional difficulty for large µ because, in contrast to the formula used in [1,3,7,8], it does not contain inverse powers of µ. However, in the next two sections we develop a new and efficient method of dealing with (2.9).…”