Spectral concentration and perturbed discrete spectra
The authors consider the Sturm-Liouville equation \[ -y''+ q(x)y= \lambda y\;(0\leq x<\infty), \quad u(0)=0, \] where the potential is either \[ q(x)= -cx^2+x \] or \[ q(x)= -cx-(1+x)^{-1} \] with a parameter \(c\geq 0\). It is known that the spectrum for \(c>0\) is continuous, whereas for \(c=0\) it is discrete in \((-\infty, \infty)\) and \((-\infty,0)\), respectively. The influence of the eigenvalues \(\lambda_n\) for \(c=0\) on the spectral function \(\rho_c\) for small \(c>0\) is investigated. It turns out that, as \(c\downarrow 0\), ``spectral concentration'' occurs successively near \(\lambda_0, \lambda_1, \lambda_2 \dots\), which is reflected by local maxima of the density \(\rho_c'\). The main technical ingredient is a new representation formula for \(\rho_c'\) in terms of a solution to a first order ODE.
scientific article; zbMATH DE number 1115946