The time-dependent Schroedinger equation, TDSE, is a partial differential equation (PDE) first order in real time, reducible to a diffusion equation in imaginary time. Exponential integrators or propagators with quantum Hamiltonians in the exponent are natural methods to describe evolution in time both for time independent and time dependent potentials [1].Accurate numerical solutions of multidimentional TDSE,s are the basis for the understanding of quantum phenomena in chemical reactions [2], energy transfer in photosynthesis [3], and new algorithms for quantum computing[4]. High order splitting methods of noncommuting operators in quantum Hamiltonians based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracy in the time step dt. Fourth and higher order accuracy propagators involve unphysical negative time steps which can be avoided by the use of complex (a+ib) time steps [1]. Extending these propagators to pure imaginary time (a=0) allows to calculate eigenvalues and eigenfunctions of time-independent TDSE,s [5]. Higher order splitting schemes with complex time steps avoid negative time steps and result in higher accuracy in simulations in mathematical finance [6],classical celestial mechanics [7] and nonlinear TDSE,s [8]. These various integration schemes of TDSE,s , their variations, "vices and virtues" will be reported and discussed in this presentation.

Co-author: Hui Zhong Lu

References:

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[8] W Bao,Q Du, SIAM J Sci Comput 25,1674 (2004)