Singularly perturbed periodic and semiperiodic differential operators

Abstract

Qualitative and spectral properties of the form-sums S_{\pm}(V):=D_{\pm}^{2m}\dotplus V(x),\quad m\in \mathbb{N}, in the Hilbert space $L_2(0,1)$ are studied. Here the periodic $(D_{+})$ and the semiperiodic $(D_{-})$ differential operators are $D_{\pm }:u\mapsto -i{u}^{\prime }$, and $V(x)$ is a 1-periodic complex-valued distribution in the Sobolev spaces $H_{per}^{-m\alpha }$, $\alpha \in [0,1]$.

13 pages

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