# American Institute of Mathematical Sciences

April  2022, 9(2): 85-101. doi: 10.3934/jcd.2021022

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

 1 Universitat Politècnica de València, Instituto de Matemática Multidisciplinar, 46022-Valencia, Spain 2 Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071-Castellón, Spain 3 Departament de Matemàtiques, Universitat Jaume I, 12071-Castellón, Spain

*Corresponding author: Sergio Blanes

Received  March 2021 Revised  August 2021 Published  April 2022 Early access  December 2021

Fund Project: 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)

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.

Citation: Sergio Blanes, Fernando Casas, Alejandro Escorihuela-Tomàs. Applying splitting methods with complex coefficients to the numerical integration of unitary problems. Journal of Computational Dynamics, 2022, 9 (2) : 85-101. doi: 10.3934/jcd.2021022
##### References:
 [1] A. Bandrauk, E. Dehghanian and H. Lu, Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett., 419 (2006), 346-350. [2] S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math., 54 (2005), 23-37.  doi: 10.1016/j.apnum.2004.10.005. [3] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016. [4] S. Blanes, F. Casas, P. Chartier and A. Escorihuela-Tomàs, On symmetric-conjugate composition methods in the numerical integration of differential equations, arXiv: 2101.04100 (to appear in Math. Comput.). doi: 10.1090/mcom/3715. [5] S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comput., 82 (2013), 1559-1576.  doi: 10.1090/S0025-5718-2012-02657-3. [6] S. Blanes, F. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl., 45 (2008), 89-145. [7] S. Blanes, F. Casas and A. Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl., 50 (2010), 47-60.  doi: 10.1007/bf03322541. [8] S. Blanes and P. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7. [9] F. Casas, P. Chartier, A. Escorihuela-Tomàs and Y. Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math., 381 (2021), 113006.  doi: 10.1016/j.cam.2020.113006. [10] F. Castella, P. Chartier, S. Descombes and G. Vilmart, Splitting methods with complex times for parabolic equations, BIT Numer. Math., 49 (2009), 487-508.  doi: 10.1007/s10543-009-0235-y. [11] J. Chambers, Symplectic integrators with complex time steps, Astron. J., 126 (2003), 1119-1126.  doi: 10.1086/376844. [12] S. Flügge, Practical Quantum Mechanics, Springer, 1971. [13] A. Galindo and P. Pascual, Quantum Mechanics. I., Texts and Monographs in Physics. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-83854-5. [14] F. Goth, Higher order auxiliary field quantum Monte Carlo methods, arXiv: 2009.04491. [15] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, 2006. [16] E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496.  doi: 10.1090/S0025-5718-09-02213-3. [17] E. Hansen and A. Ostermann, High order splitting methods for analytic semigroups exist, BIT Numer. Math., 49 (2009), 527-542.  doi: 10.1007/s10543-009-0236-x. [18] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, 2008. doi: 10.4171/067. [19] R. McLachlan and R. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053. [20] T. Prosen and I. Pizorn, High order non-unitary split-step decomposition of unitary operators, J. Phys. A: Math. Gen., 39 (2006), 5957-5964.  doi: 10.1088/0305-4470/39/20/021. [21] Q. Sheng, Solving partial differential equations by exponential splitting, IMA J. Numer. Anal., 9 (1989), 199-212. doi: 10.1093/imanum/9.2.199. [22] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N. [23] M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys., 32 (1991), 400-407.  doi: 10.1063/1.529425.

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##### References:
 [1] A. Bandrauk, E. Dehghanian and H. Lu, Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett., 419 (2006), 346-350. [2] S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math., 54 (2005), 23-37.  doi: 10.1016/j.apnum.2004.10.005. [3] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016. [4] S. Blanes, F. Casas, P. Chartier and A. Escorihuela-Tomàs, On symmetric-conjugate composition methods in the numerical integration of differential equations, arXiv: 2101.04100 (to appear in Math. Comput.). doi: 10.1090/mcom/3715. [5] S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comput., 82 (2013), 1559-1576.  doi: 10.1090/S0025-5718-2012-02657-3. [6] S. Blanes, F. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl., 45 (2008), 89-145. [7] S. Blanes, F. Casas and A. Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl., 50 (2010), 47-60.  doi: 10.1007/bf03322541. [8] S. Blanes and P. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7. [9] F. Casas, P. Chartier, A. Escorihuela-Tomàs and Y. Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math., 381 (2021), 113006.  doi: 10.1016/j.cam.2020.113006. [10] F. Castella, P. Chartier, S. Descombes and G. Vilmart, Splitting methods with complex times for parabolic equations, BIT Numer. Math., 49 (2009), 487-508.  doi: 10.1007/s10543-009-0235-y. [11] J. Chambers, Symplectic integrators with complex time steps, Astron. J., 126 (2003), 1119-1126.  doi: 10.1086/376844. [12] S. Flügge, Practical Quantum Mechanics, Springer, 1971. [13] A. Galindo and P. Pascual, Quantum Mechanics. I., Texts and Monographs in Physics. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-83854-5. [14] F. Goth, Higher order auxiliary field quantum Monte Carlo methods, arXiv: 2009.04491. [15] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, 2006. [16] E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496.  doi: 10.1090/S0025-5718-09-02213-3. [17] E. Hansen and A. Ostermann, High order splitting methods for analytic semigroups exist, BIT Numer. Math., 49 (2009), 527-542.  doi: 10.1007/s10543-009-0236-x. [18] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, 2008. doi: 10.4171/067. [19] R. McLachlan and R. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053. [20] T. Prosen and I. Pizorn, High order non-unitary split-step decomposition of unitary operators, J. Phys. A: Math. Gen., 39 (2006), 5957-5964.  doi: 10.1088/0305-4470/39/20/021. [21] Q. Sheng, Solving partial differential equations by exponential splitting, IMA J. Numer. Anal., 9 (1989), 199-212. doi: 10.1093/imanum/9.2.199. [22] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N. [23] M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys., 32 (1991), 400-407.  doi: 10.1063/1.529425.
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)
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)
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
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
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
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
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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