On the Uniqueness Criterion for Solutions of the Sturm–Liouville Equation

We consider the Sturm–Liouville equation
$$ -y''+qy=\lambda^2y $$
in an annular domain $K$ from $\mathbb C$ and obtain necessary and sufficient conditions on the potential $q$ under which all solutions of the equation $-y''(z)+q(z)y(z)=\lambda^2y(z)$, $z\in\gamma$, where $\gamma$ is a certain curve, are unique in the domain $K$ for all values of the parameter $\lambda\in\mathbb C$.