### abstract

- ~Received 21 October 1996! It is demonstrated in the context of the simple one-dimensional example of a barrier in an infinite well that highly complex behavior of the time evolution of a wave function is associated with the near degeneracy of levels in the process of tunneling. Degenerate conditions are obtained by shifting the position of the barrier. The complexity strength depends on the number of almost degenerate levels which depend on geometrical symmetry. The presence of complex behavior is studied to establish correlation with spectral degeneracy. @S1063-651X~97!09902-9# Tunneling processes have become of considerable interest as one of the possible mechanisms for creating highly complex behavior in the structure of the quantum wave function. Tomsovic and Ullmo @1# found that there is an interesting correlation between classical chaotic behavior and the rate of tunneling in the corresponding quantum system. The conclusion of their study is that chaos facilitates tunneling. On the other hand, Pattanayak and Schieve @2,3# found chaotic behavior in the semiclassical phase space ~defined by expectation values! of a one-dimensional time-independent Duffing oscillator where new variables, associated with dispersion of the quantum states, are defined and included in the description of the system. They concluded that quantum tunneling plays a crucial role for the chaotic behavior in the corresponding semiclassical maps. They have argued @3# that the spectrum becomes more complicated in the neighborhood of the separatrix. In a recent study, we have considered a model in which tunneling leads to highly complex behavior of the quantum wave function and its time dependence. The spectrum, as anticipated by Pattanayak and Schieve @2,3#, indeed makes a transition to more complex behavior in the presence of a classical separatrix @4#. It is clear that the near degeneracy of levels is necessary for the existence of significant tunneling. We directly investigate, in this work, the effect of near degeneracy in the presence of tunneling on the complexity and the behavior of the development of the wave function. This criterion is, in fact, closely analogous to the criterion of overlapping resonances for the onset of classical chaos @5#. The model we shall use is related to the one we previously explored @6#, i.e., a barrier embedded in an infinite well. By displacing the barrier in the double well system ~to the right or to the left!, certain positions are passed where the system becomes strongly near-degenerate. These positions occur at almost commensurate intervals. It is exactly for those positions that one may find significant tunneling accompanied by complex behavior. We show, moreover, that in the cases of very high degeneracy, tunneling from left to right has exponential decay, on a significant interval of time, but at other positions, where near degenerate conditions are somewhat weaker, the transition curve develops strong oscil