Distribution of the spectrum of a Sturm-Liouville operator perturbed by a delta interaction (Q6640559)

In this paper the author considers the operator $H_{a,b}$ in $L^2[0,+\infty)$ generated by\N\[\Nl_{a,b}y:=-\frac{d^2y}{dx^2}+q(x)y+a\delta(x-b)y,\ a,b>0,\N\]\Nunder the conditions $y'(b+)-y'(b-)=ay(b)$, $y(0)=0$, where $\delta$ is the Dirac-delta function, $y\in AC[0,\infty)\cap L^2[0,\infty)$, $y'\in AC([0,\infty)\setminus\{b\})$, $l_{a,b}y\in L^2[0,\infty)$, $q(x)\in C^2[0,\infty)$ is a real-valued and monotone increasing function such that $q\to\infty$ as $x\to\infty$, $q'>0$ and $q''\geq 0$ on $[0,\infty)$. The author proves that if $q'''$ exists on $[0,\infty)$ with\N\[\N\frac{q^{(k)}(x)}{q^{(k-1)}(x)}=O(x^{-1}),\text{ as }x\to\infty,\N\]\N$k=1,2,3$, then\N\[\N\lambda_0^0<\lambda_0<\lambda_1^0,\ \lambda_{n}^0\leq\lambda_{n}<\lambda_{n+1}^0,\ n\in \mathbb{N},\N\]\Nwhere $\{\lambda_{n}\}$ and $\{\lambda_{n}^0\}$ are the spectrum of $H$ and $H_0=H_{0,b}$, respectively.