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The periodic eigenvalue problem for the differential operator$(-1)^{m}d^{2m}/dx^{2m}+V$ is studied for complex-valued distribution V in theSobolev space $H^{-m\alpha}_{per}[-1,1]\;(m\in\mathbb{N},\; 0\leq\alpha<1)$.The following result is shown:  The periodic spectrum consists of a sequence $(\lambda_{k})_{k\geq0}$ ofcomplex eigenvalues satisfying the asymptotics (for any $\varepsilon>0$) $$  \lambda_{2n-1},\lambda_{2n}=n^{2m}\pi^{2m}+\hat{V}(0)\pm  \sqrt{\hat{V}(-2n)\hat{V}(2n)}+o(n^{m(2\alpha-1+\varepsilon)}), $$ where$\hat{V}(k)$ denote the Fourier coefficients of V.

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