  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.
