Eigenfunction expansions associated with 1d periodic differential operators of order $2n$

Abstract

We prove an explicit formula for the spectral expansions in $L^2(\mathbb{R})$ generated by selfadjoint differential operators $$(-1{)}^n\frac{d^{2n}}{d{x}^{2n}}+\sum _{j=0}^{n-1}\frac{d^j}{d{x}^j}{p}_j(x)\frac{d^j}{d{x}^j},p_j(x+\pi )=p_j(x),x\in \mathbb{R}.$$

25 pages