Imaginary parts of Stark-Wannier resonances

We consider a one-dimensional Stark-Wannier Hamiltonian, H = d(2)/dx(2) + p(x) - epsilon x, x is an element of R, where p is a smooth periodic, finite-gap potential, and epsilon > 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 - psi" + p(x) psi + q(epsilon x) psi = E psi, epsilon --> 0. (C) 1998 American Institute of Physics.