How Placement of Delta Barriers in an Infinite Square Well Affects the Level Spacing Distribution

Recent research suggests that systems with chaotic classical dynamics have random-matrix-type quantum level spacing distributions, while those with regular classical dynamics have a Poissonian level spacing distribution. However, an infinite square well (ISW) of length L has uniformly-spaced unfolded energy eigenvalues. This spacing distribution is modified by the presence of Dirac delta barriers inside the well, with effects depending on the location of the barriers and the probability T that a particle in the well will transmit through each barrier. Previous work has shown that when the barriers are placed irregularly, the level spacing distribution transitions from Poisson to random matrix to Gaussian as T goes from 0 to 1, but uniform placement of the barriers results in a non-generic level spacing distribution. We show that if the barriers are placed at x = iL/j, where i and j are natural numbers with 0 < i < j for fixed j, there can be at most j unique spacings. We then examine cases where barriers are placed at x = rL, for irrational r < 1. We study how the level spacing distribution for this irrational barrier placement is approached by using increasingly accurate rational approximations for the r values at various values of T. In some cases, using rational approximations gives a better match to the limiting distribution than using the irrational values.