Asymptotic behavior of the spectral measure density of a singular Sturm-Liouville operator as \(\lambda \rightarrow -\infty\) (Q960742)

It is a classical result that the Sturm Liouville differential expression \(-y'' +q y\) on \([0, \infty)\) equipped with selfadjoint boundary conditions induces a generalized Fourier transformation. This is a certain isometric mapping from \(L^2[0,\infty)\) into a Hilbert space \(L^2_\rho (-\infty,\infty)\) where \(\rho\) is a nondecreasing function. If the real potential \(q\) is bounded from below then also the spectrum and hence the support of the so-called spectral measure induced by \(\rho\) are bounded from below. Then \(\rho\) is constant on the left of a lower bound and the asymptotic behavior of \(\rho\) at \(+\infty\) is well studied. However, \(\rho\) need not be constant near \(-\infty\) if e.g. \(q(x) \longrightarrow -\infty\) as \(x \longrightarrow \infty\). Then also the asymptotic behavior of \(\rho\) at \(-\infty\) is of interest. There were already some approaches to this problem before. In particular, the asymptotic behavior at \(-\infty\) of the density \(\rho'\) of the spectral measure was estimated for the two cases \(q(x)=-bx\) and \(q(x)=-bx^2\) with \(b>0\) in previous papers by the same authors. In the present paper, this result is generalized to a wider class of potentials satisfying certain conditions at \(\infty\) and \(0\). At least this class contains the potentials \(q(x)=-bx^p\) with \(0 < p \leq 2, \; b > 0\) and \(q(x)=-b \ln^p(x + x_1)\) with \(p > 0, \; b > 0, \; x_1 \geq 1\).