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

The periodic eigenvalue problem for the differential operator $(-1)^{m}d^{2m}/dx^{2m}+V$ is studied for complex-valued distribution V in the Sobolev 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 $(\lambda_{k})_{k\geq0}$ of complex 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.