On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential

We study the spectral properties of the operator $L_\beta$ associated with the quadratic form $\mathcal{L}_\beta=\int\limits_{0}^{\infty}(|y'|^2-\beta x^{-\gamma}|y|^2)dx$ with the domain ${Q_0=\{y\in W_2^1(0,+\infty): y(0)=0\}}$, $0<\gamma<2$, $\beta\in \mathbf{C}$, as well as of the perturbed operator $M_\beta=L_\beta+W$. Under the assumption $(1+x^{\gamma/2})W\in L^1(0,+\infty)$ we prove the existence of finite quantum defect of the discrete spectrum that was established earlier by L. A. Sakhnovich as $\beta>0$, $\gamma=1$ and for real $W$ satisfying a more strict decaying condition at infinity. The main result of the paper is the proof of necessity (with some reservations) of the sufficient conditions for $W(x)$ obtained earlier by Kh. Kh. Murtazin under which the Weyl function of the operator $M_\beta$ possesses an analytic continuation on some angle from non-physical sheet.