Investigation of Solutions of a Quasipotential Equation for Complex Values of Energy with Piecewise Constant Potentials

Solutions of the Schrödinger equation for complex values of the energy describe the quasistationary state. The energy spectrum of quasistationary states is quasidiscrete and consists of a series of broadened levels, width of which determines the lifetimes of the respective states. The introduction of quasistationary states only makes sense if the width of the respective quasidiscrete levels is small compared with the distances between levels. An investigation of the solutions of quasistationary states is carried out for the quasipotential equation with piecewise-constant potentials at various values of the parameter of the equation , included in the equation and the potential parameters. A comparative analysis of the solutions of the quasipotential equation for the different values of with the solutions of the Schrödinger equation is performed. It was found that at → 0 the solutions of quasipotential equation tend to the solutions of the Schrödinger equation. With the increasing of the parameter the lifetime of quasilevels for the quasipotential equation increases as compared with the results obtained for the Schrödinger equation except the level that is close to the edge of the barrier. For comparison, the wave functions for the Schrödinger equation and the quasipotential equation for fixed values of the potential parameters are shown.

quasistationary states, the quasipotential equation, the shift operator, piecewise constant potentials, lifetime, potential width