Evolution of quantum system in order domain of Chebyshev operator

Abstract

A cosine transform between the order and angle of the Chebyshev operator is identified. Because the order and angle form a conjugate pair similar to energy and time, the Chebyshev state can be considered as a cosine-type evolution state in the order domain, analogous to a time-dependent wave packet. The order/angle formulation is analytically equivalent to the time/energy formulation, but the former may have some numerical advantages in certain applications. This is illustrated by examining the spectral method and the filter-diagonalization method in both formulations.