Asymptotic behavior of the density of the spectral measure of the singular Sturm-Liouville operator (Q2387919)

For the Sturm-Liouville operator \(Ly=-y''+qy\) with Robin boundary conditions at 0 in \(L^2[0,\infty)\) and with potential \(q\in C^0[0,\infty)\cap C^3(0,\infty)\), the leading terms of the asymptotics of the spectral measure for large \(\lambda\) are given under rather general hypotheses on the potential. The particular focus is that smoothness of \(q\) at 0 is not required, and this results in terms more complicated than powers of \(\sqrt{\lambda}\) in the expansion. Near~0, growth estimates \(x^\nu q^{(\nu)}(x)\to0\) are required for \(\nu\leq3\). Requirements near infinity are some monotonicity, growth estimates slightly weaker than \(-Cx^2\leq q\leq C\), and certain matching growth estimates for the derivatives. This generalizes work by \textit{M. S. P. Eastham} [Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 1, 37--45 (1998; Zbl 0896.34017)].