Fourier transforms of orthogonal polynomials of singular continuous spectral measures

We discuss algorithms for the solution of the Schrodinger time-dependent equation, based on orthogonal polynomial decomposition of the exponential function. After reviewing the classical Chebyshev series approach and its iterated version, we show their inefficiency when applied to operators with singular continuous spectral measures. We then introduce new decompositions based on the spectral measure of the problem under consideration, which are especially suited to deal with this case. A fast version of these algorithms is also developed and shown to achieve the theoretical maximum performance.