Estimates for Periodic Eigenvalues of the Differential Operator $(-\mathbf{1}{)}^{\mathbf{m}}{\mathbf{d}}^{\mathbf{2}\mathbf{m}}\mathbf{/}\mathbf{d}{\mathbf{x}}^{\mathbf{2}\mathbf{m}}+\mathbf{V}$ with V -- Distribution

The periodic eigenvalue problem for the differential operator $(-1{)}^m{d}^{2m}\mathrm{/}d{x}^{2m}+V$ is studied for complex-valued distribution V in the Sobolev space $H_{per}^{-m\alpha }[-1,1](m\in \mathbb{N},0\le \alpha <1)$. The following result is shown: The periodic spectrum consists of a sequence $({\lambda }_k{)}_{k\ge 0}$ of complex eigenvalues satisfying the asymptotics (for any $\epsilon >0$) $${\lambda }_{2n-1},{\lambda }_{2n}=n^{2m}{\pi }^{2m}+\widehat{V}(0)\pm \sqrt{\widehat{V}(-2n)\widehat{V}(2n)}+o(n^{m(2\alpha -1+\epsilon )}),$$ where $\widehat{V}(k)$ denote the Fourier coefficients of V.