Particle-Time Duality in the Kicked Ising Chain II: Applications to the   Spectrum
  Previously, we demonstrated that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for $N$ spins at time $T$ is related to one of a non-unitary evolution operator for $T$ spins at time $N$. Using this duality relation we obtain the oscillating part of the density of states for a large number of spins. Furthermore, the duality relation explains the anomalous short-time behavior of the spectral form factor previously observed in the literature. 
M. Akila, D. Waltner, B. Gutkin, T. Guhr
nlin.CD cond-mat.stat-mech quant-ph