Spectral asymptotics of the Sturm-Liouville operator on the half-line with potential tending to \(-\infty \): II (Q1007007)

It is a classical result that the Sturm-Liouville differential expression \[ l(y):=-y'' +q y\text{ on } [0, \infty) \] equipped with selfadjoint boundary conditions induces a generalized Fourier transformation by \[ \widehat{f}(\lambda) := \int_0^\infty f(x) \varphi(x,\lambda) \; dx, \] where \(\varphi(\cdot,\lambda)\) denotes the solution of \(l(y)=\lambda y\) satisfying the boundary conditions at \(0\). Then \(f \longrightarrow \hat{f}\) defines an isometric mapping from \(L^2[0,\infty)\) into a Hilbert space \(L^2_\rho (-\infty,\infty)\) where \(\rho\) is a nondecreasing function. If \(q(x) \longrightarrow -\infty\) as \(x \longrightarrow \infty\) then the spectrum of the associated operator is not bounded from below and hence the asymptotic behavior at \(-\infty\) of \(\rho\) and of the density \(\rho'\) of the spectral measure are of interest. In the first part of the paper [cf. \textit{A.S. Pechentsov, A.Yu. Popov}, Differ. Equ. 44, No. 5, 659--667 (2008; Zbl 1182.34107)] such asymptotic results at \(-\infty\) were formulated for a class of potentials \(q\) satisfying certain conditions at \(\infty\) and \(0\) including at least the potentials \(q(x)=-bx^p\) with \(0 < p \leq 2, \; b > 0\). Moreover, in the first part it was shown how the asymptotics follow from asyptotic estimates for the solution \(\varphi(x,\lambda)\) and for \(\frac{\partial \varphi}{\partial x}(x,\lambda)\) as \(\lambda \longrightarrow -\infty\) and \(x \longrightarrow \infty\). Now, as a direct continuation of the first part the second part of the paper presents a proof of these asymptotic estimates for \(\varphi(x,\lambda)\) and \(\frac{\partial \varphi}{\partial x}(x,\lambda)\). This completes the proof of the main results of the whole paper.