Imaginary parts of Stark–Wannier resonances

We consider a one-dimensional Stark–Wannier Hamiltonian, H=−d 2 /dx 2 +p(x)−εx, x∈ R , where p is a smooth periodic, finite-gap potential, and ε>0 is small enough. We compute rigorously the imaginary parts of the spectral resonances. For this purpose we develop some related elements of the adiabatic approach to the equations of the form −ψ ″ +p(x)ψ+q(εx)ψ=Eψ, ε→0.