Evolution of electric-field-induced quasibound states and resonances in   one-dimensional open quantum systems
  A comparative analysis of three different time-independent approaches to studying open quantum structures in uniform electric field $\mathscr{E}$ was performed using the example of one-dimensional attractive or repulsive $\delta$-potential and surface that supports the Robin boundary condition. The three considered methods exploit different properties of the scattering matrix $S(\mathscr{E};E)$ as a function of energy $E$: its poles, real values, and zeros of the second derivative of its phase. The essential feature of the method of zeroing the resolvent, which produces complex energies, is the unlimited growth of the wave function at infinity, which is, however, eliminated by the time-dependent interpretation. The real energies at which the unitary scattering matrix becomes real correspond to the largest possible distortion, $S=+1$, or its absence at $S=-1$ which in either case leads to the formation of quasibound states. Depending on their response to increasing electric intensity, two types of field-induced positive energy quasibound levels are identified: electron- and hole-like states. Their evolution and interaction in enlarging field lead ultimately to the coalescence of pairs of opposite states, with concomitant divergence of the associated dipole moments in what is construed as an electric breakdown of the structure. The characteristic features of the coalescence fields and energies are calculated and the behavior of the levels in their vicinity is analyzed. Similarities between the different approaches and their peculiarities are highlighted; in particular, for zero-field bound state in limit of vanishing $\mathscr{E}$, all three methods produce same results, with their outcomes deviating from each other according to growing electric intensity. 
O. Olendski
cond-mat.mes-hall math-ph math.MP physics.atom-ph quant-ph