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Applying splitting methods with complex coefficients to the numerical integration of unitary problems

  • *Corresponding author: Sergio Blanes

    *Corresponding author: Sergio Blanes 

Work supported by Ministerio de Ciencia e Innovación (Spain) through project PID2019-104927GB-C21/AEI/10.13039/501100011033. A.E.-T. has been additionally funded by the predoctoral contract BES-2017-079697 (Spain)

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  • We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group $ \mathrm{SU}(2) $. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.

    Mathematics Subject Classification: Primary: 65L05, 37M15; Secondary: 65P10, 65M22.

    Citation:

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  • Figure 1.  Left: 2-norm error vs. computational cost (number of exponentials) for $ \Psi_{SC,c}^{[3]} $ (dotted line), $ \mathcal{S}^{[4]} $ (real coefficients, solid line), $ \Psi_{P,c}^{[4]} $ (complex coefficients, dash-dotted line) and $ \Psi_{SC,c}^{[4]} $ (dashed line). Right: Error in unitarity for $ \Psi_{P,c}^{[4]} $ (solid line) and the symmetric-conjugate methods $ \Psi_{SC,c}^{[3]} $ (dotted line) and $ \Psi_{SC,c}^{[4]} $ (dashed line)

    Figure 2.  Absolute value of the eigenvalues of the approximate solution matrix obtained with $ \Psi_{P,c}^{[4]} $ with complex coefficients ($ k = 1 $, black dashed line), $ \Psi_{SC,c}^{[3]} $ (blue dotted line) and $ \Psi_{SC,c}^{[4]} $ (red, solid line)

    Figure 3.  Error in norm of the approximate solution (left) and error in energy (37) (right) for the quartic potential (36) obtained by the palindromic schemes $ \Psi_{P,r}^{[4]} $ (magenta, dashed line), $ \Psi_{P,c}^{[4]} $ (blue dotted line) and the symmetric-conjugate method $ \Psi_{SC,c}^{[3]} $ (black solid line) along the integration interval. The step size is chosen so that all methods have the same computational cost

    Figure 4.  Error in norm of the approximate solution (left) and error in energy (37) (right) for the quartic potential (36) obtained by the palindromic scheme $ \Xi_{P,r}^{[4]} $ (blue dotted line), and the symmetric-conjugate schemes $ \Psi_{SC,r}^{[3]} $ (black solid line) and $ \Xi_{SC,r}^{[4]} $ (magenta dashed line) along the integration interval. The step size is chosen so that all methods have the same computational cost

    Figure 5.  Error in norm of the approximate solution (left) and error in energy (37) (right) for the Pöschl–Teller potential (38) obtained by the palindromic scheme $ \Psi_{P,c}^{[4]} $ (blue dotted line) and the symmetric-conjugate method $ \Psi_{SC,c}^{[3]} $ (black solid line) along the integration interval. The result achieved by $ \Psi_{P,r}^{[4]} $ is out of the scale

    Figure 6.  Error in norm of the approximate solution (left) and error in energy (37) (right) for the Pöschl–Teller potential (38) obtained by the palindromic scheme $ \Xi_{P,r}^{[4]} $ (blue dotted line), and the symmetric-conjugate schemes $ \Psi_{SC,r}^{[3]} $ (black solid line) and $ \Xi_{SC,r}^{[4]} $ (magenta dashed line) along the integration interval

    Figure 7.  Maximum of error in the expected value of the energy in the interval $ t \in [0,100] $ as a function of the time step (left) and the computational cost (number of FFTs, right) for several splitting schemes. Pöschl–Teller potential

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