Asymptotic behavior on \((-\infty , 0)\) of the spectral measures of a family of singular Sturm-Liouville operators (Q731548)

Let \(L_{\varepsilon q,\alpha}\) be the Sturm-Liouville operator in \(L^2(0,+\infty)\) generated by the differential expression \(-y''+\varepsilon q(x)y\), \(\varepsilon>0\), and the boundary condition \(y(0)\cos\alpha+y'(0)\sin\alpha=0\). If the real-valued potential \(q\in C[0,+\infty)\) satisfies the conditions \[ \lim_{x\to+\infty}q(x)=-\infty\text{ and } \int_{x_0}^{+\infty} |q(x)|^{-1/2}\,dx=+\infty, \] where \(x_0\) is a number such that \(q(x)<0\) for \(x\geq x_0\), it is known that the spectrum of \(L_{\varepsilon q,\alpha}\) fills the entire real axis, and its spectral measure is unique. Under some additional condition this measure has continuous density \(\rho'_{\varepsilon q,\alpha}\) on \(\mathbb{R}\). The aim of this paper is to determine the asymptotic behavior of this density when \(\lambda/\varepsilon\to-\infty\). Let \(\mathcal L\) be the class of all positive functions \(l\in W_{\infty}^3[c,+\infty)\) (the number \(c\) depends on \(l\in\mathcal L\)) whose first three derivatives admit the asymptotic estimate \(l^{(k)}=o(x^{-k}l(x))\), \(x\to+\infty\), \(k=1,2,3\). Suppose that the potential \(q\) admits the representation \(q(x)=-x^bl(x)\), \(l\in\mathcal L\), \(0<b\leq2\) for \(x>x_0\) and satisfies the estimate \(q''(x)=o(x^{-2})\) as \(x\to0+\). The authors prove that, under these conditions, for \(\lambda<0\) and \(\varepsilon>0\), \(\varepsilon=O(1)\), the following asymptotics holds: \[ \rho'_{\varepsilon q,\alpha} \sim \frac1\pi \frac{\sqrt{|\lambda|}}{(\sqrt{|\lambda|}\sin\alpha-\cos\alpha)^2}\exp\left(-2\int_0^{p(\eta)} \sqrt{\varepsilon q(t)-\lambda}\,dt\right), \] \(\eta=-\lambda/\varepsilon\to+\infty\), where \(p\) stands for the inverse function of \(-q\). Certain intervals for \(\lambda\) are excluded.