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

Abstract

The periodic eigenvalue problem for the differential operator $(-1{)}^m{d}^{2m}/d{x}^{2m}+V$ 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 $\epsilon >0$) $${\lambda }_{2n-1},{\lambda }_{2n}=n^{2m}{\pi }^{2m}+\hat{V}(0)\pm \sqrt{\hat{V}(-2n)\hat{V}(2n)}+o(n^{m(2\alpha -1+\epsilon )}),$$ where $\hat{V}(k)$ denote the Fourier coefficients of V.

arXiv comment13 pages

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