Abstract
In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems.
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References
Bader, P., Iserles, A., Kropielnicka, K. & Singh, P. (2014), ‘Effective approximation for the semiclassical Schrödinger equation’, Found. Comput. Math. 14(4), 689–720.
Bransden, B. H. & Joachain, C. J. (1983), Physics of Atoms and Molecules, Prentice Hall, Englewood Cliffs, NJ.
Casas, F. & Murua, A. (2009), ‘An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications’, J. Math. Phys. 50(3), 033513, 23.
Gaim, W. & Lasser, C. (2014), ‘Corrections to Wigner type phase space methods’, Nonlinearity 27(12), 2951–2974.
Gottlieb, D., Hussaini, M. Y. & Orszag, S. A. (1984), Theory and applications of spectral methods, in ‘Spectral methods for partial differential equations’, SIAM, Philadelphia, PA, pp. 1–54.
Griffiths, J. D. (2004), Introduction to Quantum Mechanics, 2nd edn, Prentice Hall, Upper Saddle River, NJ.
Hagedorn, G. A. (1980), ‘Semiclassical quantum mechanics. I. The \(\hbar \rightarrow 0\) limit for coherent states’, Comm. Math. Phys. 71(1), 77–93.
Hairer, E. & Iserles, A. (2016), ‘Numerical stability in the presence of variable coefficients’, Found. Comput. Maths. DOI:10.1007/s10208-015-9263-y.
Hairer, E., Lubich, C. & Wanner, G. (2010), Geometric numerical integration, Vol. 31 of Springer Series in Computational Mathematics, Springer, Heidelberg. Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition.
Hesthaven, J. S., Gottlieb, S. & Gottlieb, D. (2007), Spectral methods for time-dependent problems, Vol. 21 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.
Hochbruck, M. & Lubich, C. (1997), ‘On Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 34(5), 1911–1925.
Iserles, A. (2009), A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics, second edn, Cambridge University Press, Cambridge.
Iserles, A. (2014), ‘On skew-symmetric differentiation matrices’, IMA J. Numer. Anal. 34(2), 435–451.
Iserles, A., Kropielnicka, K. & Singh, P. (2015), On the discretisation of the semiclassical Schrödinger equation with time-dependent potential, Technical Report 2015/NA02, DAMTP, University of Cambridge.
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. & Zanna, A. (2000), ‘Lie-group methods’, 9, 215–365.
Jin, S., Markowich, P. & Sparber, C. (2011), ‘Mathematical and computational methods for semiclassical Schrödinger equations’, Acta Numer. 20, 121–209.
Lubich, C. (2015), ‘Time integration in the multiconfiguration time-dependent Hartree method of molecular quantum dynamics’, Appl. Math. Res. Express. AMRX (2), 311–328.
McLachlan, R. I. & Quispel, G. R. W. (2002), ‘Splitting methods’, Acta Numer. 11, 341–434.
Meyer, H. D., Manthe, U. & Cederbaum, L. S. (1990), ‘The multi-configurational time-dependent Hartree approach’, Chem. Phys. Lett. 165, 73–78.
Oteo, J. A. (1991), ‘The Baker-Campbell-Hausdorff formula and nested commutator identities’, J. Math. Phys. 32(2), 419–424.
Shapiro, M. & Brumer, P. (2003), Principles of the Quantum Control of Molecular Processes, Wiley-Interscience, Hoboken, NJ.
Singh, P. (2016), Algebraic theory for higher-order methods in computational quantum mechanics, Technical report, DAMTP, University of Cambridge.
Teufel, S. (2012), ‘Semiclassical approximations for adiabatic slow-fast systems’, EPL. DOI:10.1209/0295-5075/98/50003.
Townsend, A. & Olver, S. (2015), ‘The automatic solution of partial differential equations using a global spectral method’, J. Comput. Phys. 299, 106–123.
Acknowledgments
Various parts of this paper are based on joint research with my colleagues Philipp Bader (La Trobe), Ernst Hairer (Geneva), Karolina Kropielnicka (Gdańsk) and Pranav Singh (Cambridge). I wish to acknowledge not just their mathematical contribution but also the great pleasure of collaborating with them. I also wish to thank a number of colleagues for very fruitful discussions, in particular Helge Dietert (Cambridge) and Caroline Lasser (Munich).
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Communicated by Albert Cohen.
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Iserles, A. The Joy and Pain of Skew Symmetry. Found Comput Math 16, 1607–1630 (2016). https://doi.org/10.1007/s10208-016-9321-0
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DOI: https://doi.org/10.1007/s10208-016-9321-0
Keywords
- Splitting methods
- Semiclassical equations
- Differentiation matrices
- Zassenhaus splitting
- Stable semidiscretization