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Let $m\in \mathbb{N}$, $\alpha\in[0,1]$, and $V$ be a 1-periodiccomplex-valued distribution in the negative Sobolev space $H^{-m\alpha}[0,1]$.The singular non-self-adjoint eigenvalue problem $D^{2m}u+V u=\lambda u$, $D=-id/dx$, with semi-periodic boundary conditions is investigated. The uniform in$V$ asymptotic and non-asymptotic eigenvalue estimates are found and proved.The case of periodic boundary conditions was studied by authors earlier.

Uniform estimates for the semi-periodic eigenvalues of the singular differential operators