Spectral concentration for perturbed equations of harmonic oscillator type

Abstract: Sturm–Liouville potentials of the form x^a f (εx) are considered, where a>0, f decays sufficiently rapidly at infinity, and ε is a small positive parameter. It is shown that there are a finite number N(ε) of spectral concentration points, and computational evidence is given to support the conjecture that N(ε) increases to infinity as ε decreases to zero.