Stark resonances in a quantum waveguide with analytic curvature

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide $\mathbf{\Omega }$ i.e. bound states of the corresponding Laplace operator $-\mathrm{\Delta }\mathrm{\_}\mathbf{\Omega }$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(\mathrm{\mid }s{\mathrm{\mid }}^{-\epsilon }),\epsilon >3$ at infinity. We show that under conditions on the electric field $\mathbf{F}$, ${\bf H}(F):= -\Delta_{{\bf \Omega}} + {\bf F}. {\bf x} $ has resonances near the discrete eigenvalues of $-\mathrm{\Delta }\mathrm{\_}\mathbf{\Omega }$.