Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics

 Access to full text

1. [1] J. Abhau and M. Thalhammer, A numerical study of adaptive space and time discretisations for Gross–Pitaevskii equations. J. Comput. Phys.231 (2012) 6665–6681. MR2965095
2. [2] R.A. Adams, Sobolev Spaces. Academic Press, Orlando, Fla. (1975). Zbl1098.46001MR450957
3. [3] W. Bao, D. Jaksch and P. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys.187 (2003) 318–342. Zbl1028.82501MR1977789
4. [4] W. Bao and J. Shen, A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput.26 (2005) 2010–2028. Zbl1084.35083MR2196586
5. [5] C. Bardos, I. Catto, N. Mauser and S. Trabelsi, Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree-Fock equations: A sufficient condition. Appl. Math. Lett.22 (2009) 147–152. Zbl1163.35464MR2482267
6. [6] C. Bardos, I. Catto, N. Mauser and S. Trabelsi, Setting and analysis of the multi-configuration time-dependent Hartree–Fock equations. Arch. Ration. Mech. Anal.198 (2010) 273–330. Zbl1229.35221MR2679373
7. [7] M.H. Beck, A. Jäckle, G.A. Worth, and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets. Phys. Rep.324 (2000) 1–105. 
8. [8] M.H. Beck and H.-D. Meyer, An efficient and robust integration scheme for the equations of the multiconfiguration time-dependent Hartree (MCTDH) method. Z. Phys. D42 (1997) 113–129. 
9. [9] S. Blanes and P.C. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math.142 (2002) 313–330. Zbl1001.65078MR1906732
10. [10] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer Verlag, New York, 2nd edition (2002). Zbl0804.65101MR1894376
11. [11] I. Burghardt, H.-D. Meyer and L.S. Cederbaum, Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method. J. Chem. Phys.111 (1999) 2927–2939. 
12. [12] J. Caillat, J. Zanghellini, M. Kitzler, W. Kreuzer, O. Koch and A. Scrinzi, Correlated multielectron systems in strong laser pulses – an MCTDHF approach. Phys. Rev. A 71 (2005) 012712. 
13. [13] M. Caliari, Ch. Neuhauser and M. Thalhammer, High-order time-splitting Hermite and Fourier spectral methods for the Gross–Pitaevskii equation. J. Comput. Phys.228 (2009) 822–832. Zbl1158.65340MR2477790
14. [14] S. Descombes and M. Thalhammer, An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT Numer. Math.50 (2010) 729–749. Zbl1205.65250MR2739463
15. [15] P.A.M. Dirac, Note on exchange phenomena in the Thomas atom. Proc. Cambridge Philos. Soc.26 (1930) 376–385. Zbl56.0751.04JFM56.0751.04
16. [16] J. Frenkel, Wave Mechanics, Advanced General Theory. Clarendon Press, Oxford (1934). Zbl0013.08702
17. [17] L. Gauckler, Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal.49 (2011) 1194–1209. Zbl1223.65079MR2813177
18. [18] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Springer Verlag, Berlin–Heidelberg–New York (2002). Zbl0994.65135MR1904823
19. [19] E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Springer Verlag, Berlin–Heidelberg–New York (1987). Zbl0638.65058MR868663
20. [20] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities. Cambridge Univ. Press, Cambridge (1934). Zbl0634.26008
21. [21] T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Berlin–Heidelberg–New York (1966). Zbl0435.47001MR203473
22. [22] T. Kato and H. Kono, time-dependent multiconfiguration theory for electronic dynamics of molecules in an intense laser field. Chem. Phys. Lett. 392 (2004) 533–540. 
23. [23] M. Kitzler, J. Zanghellini, Ch. Jungreuthmayer, M. Smits, A. Scrinzi and T. Brabec, Ionization dynamics of extended multielectron systems. Phys. Rev. A 70 (2004) 041401(R). 
24. [24] O. Koch, The variational splitting method for the multi-configuration time-dependent Hartree–Fock equations for atoms. To appear in J. Numer. Anal. Indust. Appl. Math.7 (2012) 1–13. MR2950041
25. [25] O. Koch, W. Kreuzer and A. Scrinzi, MCTDHF in ultrafast laser dynamics. AURORA TR-2003-29, Inst. Appl. Math. Numer. Anal., Vienna Univ. of Technology, Austria (2003). Available at http://www.othmar-koch.org/research.html. 
26. [26] O. Koch, W. Kreuzer and A. Scrinzi, Approximation of the time-dependent electronic Schrödinger equation by MCTDHF. Appl. Math. Comput.173 (2006) 960–976. Zbl1088.65092MR2207989
27. [27] O. Koch and C. Lubich, Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics. M2AN Math. Model. Numer. Anal. 41 (2007) 315–331. Zbl1135.81380MR2339631
28. [28] O. Koch and C. Lubich, Analysis and time integration of the multi-configuration time-dependent Hartree–Fock equations in electron dynamics. ASC Report 4/2008, Inst. Anal. Sci. Comput. Vienna Univ. of Technology (2008). Zbl1218.81040
29. [29] O. Koch and C. Lubich, Variational splitting time integration of the MCTDHF equations in electron dynamics. IMA J. Numer. Anal.31 (2011) 379–395. Zbl1218.81040MR2813176
30. [30] Y. Kwon, D.M. Ceperley and R.M. Martin, Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study. Phys. Rev. B58 (1998) 6800–6806. 
31. [31] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford–New York, 3rd edition (1977). Zbl0178.57901
32. [32] C. Lubich, A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math.48 (2004) 355–368. Zbl1037.81634MR2056923
33. [33] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Zurich Lect. Adv. Math. Europ. Math. Soc., Zurich (2008). Zbl1160.81001MR2474331
34. [34] C. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comp.77 (2008) 2141–2153. Zbl1198.65186MR2429878
35. [35] R. McLachlan and R. Quispel, Splitting methods. Acta Numer.11 (2002) 341–434. Zbl1105.65341MR2009376
36. [36] H.-D. Meyer, F. Gatti and G.A. Worth, editors. Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley-VCH, Weinheim, Berlin (2009). 
37. [37] H.-D. Meyer, U. Manthe and L.S. Cederbaum, The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett.165 (1990) 73–78. 
38. [38] H.-D. Meyer and G.A. Worth, Quantum molecular dynamics: Propagating wavepackets and density operators using the multi-configuration time-dependent Hartree (MCTDH) method. Theo. Chem. Acc.109 (2003) 251–267. 
39. [39] M. Miklavčič, Applied Functional Analysis and Partial Differential Equations. World Scientific, Singapore (1998). Zbl0913.35002
40. [40] I. Nagy, R. Diez Muiño, J.I. Juaristi and P.M. Echenique, Spin-resolved pair-distribution functions in an electron gas: A scattering approach based on consistent potentials. Phys. Rev. B 69 (2004) 233105. 
41. [41] M. Nest and T. Klamroth, Correlated many-electron dynamics: Application to inelastic electron scattering at a metal film. Phys. Rev. A 72 (2005) 012710. 
42. [42] M. Nest, T. Klamroth and P. Saalfrank, The multiconfiguration time-dependent Hartree–Fock method for quantum chemical calculations. J. Chem. Phys. 122 (2005) 124102. 
43. [43] C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numer. Math.49 (2009) 199–215. Zbl1162.65385MR2486135
44. [44] V.M. Perez-Garcia and X. Liu, Numerical methods for the simulation of trapped nonlinear Schrödinger systems. Appl. Math. Comput.144 (2003) 215–235. Zbl1024.65084MR1993427
45. [45] J.C. Slater, Quantum Theory of Molecules and Solids. McGraw–Hill, New York, Toronto, London 1 (1960). Zbl0045.28405
46. [46] G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal.5 (1968) 506–517. Zbl0184.38503MR235754
47. [47] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Appl. Math. Sci. Springer Verlag, New York (1999). Zbl0928.35157
48. [48] M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal.46 (2008) 2022–2038. Zbl1170.65061MR2399406
49. [49] M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal.50 (2012) 3231–3258. Zbl1267.65116MR3022261
50. [50] S. Trabelsi, Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions. C. R. Acad. Sci. Paris, Ser. I 345 (2007) 145–150. Zbl1120.35077MR2344813
51. [51] H.F. Trotter, On the product of semi-groups of operators. Proc. Amer. Math. Soc.10 (1959) 545–551. Zbl0099.10401MR108732
52. [52] J. Zanghellini, M. Kitzler, T. Brabec and A. Scrinzi, Testing the multi-configuration time-dependent Hartree–Fock method. J. Phys. B: At. Mol. Phys. 37 (2004) 763–773. 
53. [53] J. Zanghellini, M. Kitzler, C. Fabian, T. Brabec and A. Scrinzi, An MCTDHF approach to multi-electron dynamics in laser fields. Laser Phy.13 (2003) 1064–1068.