SMATRIX

The Gamow vector description of resonances is compared with the S-matrix and the Green function descriptions using the example of the square barrier and similar potentials. By imposing different boundary conditions on the time independent Schrodinger equation, we get either eigenvectors corresponding to real eigenvalues (Dirac kets) and the real ``physical'' spectrum or we get eigenvectors corresponding to complex eigenvalues (Gamow vectors) and the resonance spectrum. We will show that the poles of the S-matrix are the same as the poles of the Green function and as the complex eigenvalues of the Schrodinger equation subject to a purely outgoing boundary condition. We also obtain the basis vector expansion generated by the Gamow vectors. The time asymmetry built into the purely outgoing boundary condition will be revealed. It will be also shown that the probability to detect the decay within a shell around the origin of the decaying state follows the exponential law if the Gamow vector (resonance) contribution to this probability is the only contribution that is taken into account.