Beurling-Malliavin theory for Toeplitz kernels

We consider the family of Toeplitz operators $T_{J\bar S^{a}}$ acting in the Hardy space $H^2$ in the upper halfplane; $J$ and $S$ are given meromorphic inner functions, and $a$ is a real parameter. In the case where the argument of $S$ has a power law type behavior on the real line, we compute the critical value $$c(J,S)=\inf\left\{a: \ker T_{J\bar S^{a}}\ne0\right\}.$$ The formula for $c(J,S)$ generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.