  The eigenvalue problem on the circle for the non-self-adjoint operators
$L_{m}(V)=(-1)^{m}\frac{d^{2m}}{dx^{2m}}+V$, $m\in \mathbb{N}$ with singular
complex-valued 2-periodic distributions $V\in H_{per}^{-m}[-1,1]$ is studied.
Asymptotic formulae for the eigenvalues uniformly in $V$ in the space
$H_{per}^{m}[-1,1]$ and local uniformly in $V$ in the space
$H_{per}^{-m}[-1,1]$ are found.
