Splitting methods for the time-dependent Schrödinger equation

Blanes, Sergio; Moan, P. C.

doi:10.1016/s0375-9601(99)00866-x

Cited by 66 publications

(64 citation statements)

References 16 publications

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“…This method is mainly used in its second-order variant in the time of propagation for time-dependent potentials, where no change is required in comparison with the time-independent case. However, also higher order split-operator schemes have been derived, including fourth [32][33][34][35] and higher-order expansions [36][37][38][39]. We also mention important counterexamples, like world-line Monte Carlo [40][41][42], where discretization errors are completely removed [43], or open systems where the main contribution to the systematic error is due to retardation effects which cannot be dressed into the form of simple potentials [44].…”

Section: Introductionmentioning

confidence: 99%

Fast converging path integrals for time-dependent potentials: I. Recursive calculation of short-time expansion of the propagator

Balaž¹,

Vidanović²,

Bogojević³

et al. 2011

J. Stat. Mech.

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In this and subsequent paper [1] we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniques to study properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models.

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Section: Introductionmentioning

confidence: 99%

Fast converging path integrals for time-dependent potentials: I. Recursive calculation of short-time expansion of the propagator

Balaž¹,

Vidanović²,

Bogojević³

et al. 2011

J. Stat. Mech.

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“…As the discretization of an unbounded operator, H(t) can be of arbitrarily large norm. Magnus integrators are an interesting class of numerical methods for such problems [3,12]. Though the error behavior of such methods is well understood in the case of moderately bounded H(t) [6,7], no results are so far available when H(t) becomes large.…”

mentioning

confidence: 99%

On Magnus Integrators for Time-Dependent Schrödinger Equations

Hochbruck

Lubich

2003

SIAM J. Numer. Anal.

101

110

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Abstract. Numerical methods based on the Magnus expansion are an efficient class of integrators for Schrödinger equations with time-dependent Hamiltonian. Though their derivation assumes an unreasonably small time step size as would be required for a standard explicit integrator, the methods perform well even for much larger step sizes. This favorable behavior is explained, and optimal-order error bounds are derived which require no or only mild restrictions of the step size. In contrast to standard integrators, the error does not depend on higher time derivatives of the solution, which is in general highly oscillatory.

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“…This symmetry imposes that r 3 = r 4 , which is a useful property in some cases. For example, in [6] these schemes were applied to a particular representation of the Schrödinger equation where unitarity was not exactly preserved (as was the case for other symplectic integrators) but was retained at higher order than the order of the method because r 3 = r 4 . The most efficient fourth-order methods of [13], denoted by (m = 4, n = 4) and (m = 6, n = 4), have E f = 0.342 and E f = 0.322, respectively.…”

Section: Rkn2 This Corresponds To the Case [Y [Y [Y Xmentioning

confidence: 99%

“…However, this particular RKN2 method preserves unitarity to fifth order, as can be seen from the slope of its curve. This method has been used in [6] in a time-dependent problem. Finally, we have to say that the RKN2 method works this efficiently only if one uses the error in energy as a measure of accuracy, while the error in position still behaves as predicted by the effective error analysis.…”

Section: Autonomous Systemsmentioning

confidence: 99%