Analytic perturbation theory for a periodic potential

The operator $\mathbf H_\alpha=(-\Delta)^l+\alpha V$ is considered in $L_2(\mathbf R^n)$; here $4l>n+1$, $n\geqslant2$, $V$ is a periodic potential, and $\alpha$ is a perturbation parameter, $-1\leqslant\alpha\leqslant1$. An analytic perturbation theory with respect to $\alpha$ is constructed for Block eigenfunctions and the corresponding eigenvalues of $\mathbf H_\alpha$. It is proved that, for large energies, when the quasimomentum belongs to a sufficiently rich set they admit expansion in a Taylor series in the disk $|\alpha|\leqslant1$, and these series are asymptotic in the energy and infinitely differentiable with respect to the quasimomentum.
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