Analysis of quantum decay law: is quantum tunneling really exponential?

The exponential decay law is well established since its first derivation in 1928; however, it not exact but only an approximate description. In recent years some experimental and theoretical indications for non-exponential have been documented. First we solve analytically the time-dependent Schrödinger equation one dimension a potential consisting of infinite wall plus rectangular barrier with finite width also cut harmonic oscillator by considering as sequence square potentials. Then using staggered Leap-Frog method, potential. both methods, time dependence survival probability particle parameter \( \lambda \) are analyzed. results exhibit behavior at short intermediate times.