Singular eigenvalue problems on the circle

The eigenvalue problem on the circle for the non-self-adjoint operators $L_m(V)=(-1{)}^m\frac{d^{2m}}{d{x}^{2m}}+V$, $m\in \mathbb{N}$ 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.