A simple model of the decaying quantum state composed of a one-dimensional electron moving in the potential with the quantum dot separated from the infinite half-axis by the δ-function is studied. The quasibound quantum dot state properties calculated by means of approximate methods (complex energy technique and local density of states method) are compared with results obtained with the aid of an exact analytical non-stationary wave function. The physical meaning of the emitted electron wave function peculiarities is discussed.