Asymptotics of the spectrum of the differential operator \(-y^{\prime\prime} + q(x)y\) with a boundary condition at zero and with rapidly growing potential (Q2577325)

Consider the operator \(\mathcal L_q\) given in \(L_2(\mathbb{R}^+)\) by the expression \(-y''+q(x)y\) and the boundary condition \(y(0)\cos\alpha+y'(0)\sin\alpha=0\). The potential \(q\) is such that \(q\in C[0,+\infty)\cap C^2(0,+\infty)\), \(q''(x)\geq0\) for \(x\) sufficiently large and \(\lim_{x\to+\infty}xq'(x)/q(x)=+\infty\). Such functions increase more rapidly at infinity than any power of~\(x\). Denote by \(p\) the function inverse to~\(q\). The operator \(\mathcal L_q\) has a discrete spectrum \(\{\lambda_n\}_{n=1}^{\infty}\), where the eigenvalues \(\lambda_n\) are assumed to be arranged in ascending order. The author proves the following asymptotic properties of the spectrum. 1)~If \(q''(x)\leq(q'(x))^{\gamma}\) for some \(\gamma\), \(1<\gamma<4/3\) and \(\log q(x)/\log^2x\uparrow+\infty\) as \(x\to+\infty\), then \(\lambda_n\sim(\pi n)^2p^{-2}((\pi n)^2)\) as \(n\to\infty\). 2)~If \(\log q(x)=\beta^{-2}\log^2x+o(\log x)\) as \(x\to+\infty\) with some \(\beta>0\), then \(\lambda_n\sim(\pi n)^2p^{-2}((\pi n)^2)\exp(2\beta^2)\) as \(n\to\infty\). 3)~For potentials sufficiently rapidly growing at infinity, two terms of the asymptotics of the spectrum are computed.