Spectral concentration and rapidly decaying potentials

Antibound States and Exponentially Decaying Sturm–Liouville Potentials

“…We note that resonances which are close to the real z-axis can, by (1.6), produce a local maximum of ρ (µ), that is, spectral concentration [6,7].…”

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).…”

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

“…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.…”