Some properties of root functions of ordinary second-order differential operators in the absence of the Carleman condition (Q946122)

Let the differential expression \[ \ell (y)(x):=-y''(x)+q(x)y(x)\quad \text{for}\quad x\in (a,b),\quad q\in L^1(a,b)\quad\text{and}\quad q(x)\in \mathbb{C}. \] The root functions are regular solutions \(Lu_{k}+\mu _{k}^{2}u_{k}=\theta _{k}u_{k-1}\) where \(\theta _{k}=0\) if \(u_{k}\) is an eigenfunction, and otherwise \(\theta _{k}=1\) with \(\mu _{k}=\mu _{k-1}.\) We say that \(\left\{ \mu _{k}\right\} \) satisfy Carlerman's condition if \(\sup \left| \text{Im}\mu _{k}\right| <\infty.\) The author obtains a sufficient condition for the root functions \(\left\{ u_{k}\right\} \) to satisfy the Bessel property \[ \sum_{k\geq 0}\left| \left( f,u_{k}\right) \right| ^{2}\leq \beta \left| \left| f\right| \right| ^{2} \] in case Calerman's condition does not hold.