Abstract
The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as \({\mathcal {O}}\!\left( M\log M\right) \), where \(M\) is the number of degrees of freedom in the discretisation.
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Notes
Unless a non-existing term is subtracted and thus newly introduced instead of removed.
Using a Fourier basis the cost is \({\mathcal {O}}\!\left( M\log M\right) \).
As before, a tilde denotes a discretisation.
All powers of \(\tau \) are odd because of the palindromy of the symmetric BCH formula. Since \(\tau ={\mathrm {i}}\Delta t\), this means that they always contribute a multiple of \(\pm {\mathrm {i}}\).
References
Blanes, S., Casas, F. & Murua, A. (2006), Symplectic splitting operator methods tailored for the time-dependent Schrödinger equation, J. Chem. Phys. 124, 234–105.
Bungartz, H.-J. & Griebel, M. (2004), Sparse grids, Acta Numer. 13, 147–269.
Casas, F. & Murua, A. (2009), An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications, J. Math. Phys. 50, (electronic).
Faou, E. (2012), Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, The European Mathematical Society, Zürich.
Gallopoulos, E. & Saad, Y. (1992), Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Stat. Comput. 13, 1236–1264.
Golub, G. H. & Van Loan, C. F. (1996), Matrix Computations, 3rd edn, Johns Hopkins University Press, Baltimore.
Griffiths, D. J. (2004), Introduction to Quantum Mechanics, 2nd edn, Prentice Hall, Upper Saddle River, NJ.
Hairer, E., Lubich, C. & Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn, Springer, Berlin.
Hesthaven, J. S., Gottlieb, S. & Gottlieb, D. (2007), Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge.
Hochbruck, M. & Lubich, C. (1997), On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34, 1911–1925.
Iserles, A. (2008), A First Course in the Numerical Analysis of Differential Equations, 2nd edn, Cambridge University Press, Cambridge.
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. & Zanna, A. (2000), Lie-group methods, Acta Numer. 9, 215–365.
Jin, S., Markowich, P. & Sparber, C. (2011), Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numer. 20, 121–210.
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, The European Mathematical Society , Zürich.
McLachlan, R. I. & Quispel, G. R. W. (2002), Splitting methods, Acta Numer. 11, 341–434.
McLachlan, R. I., Munthe-Kaas, H. Z., Quispel, G. R. W. & Zanna, A. (2008), Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Found. Comput. Math. 8, 335–3554.
Oteo, J. A. (1991), The Baker–Campbell–Hausdorff formula and nested commutator identities, J. Math. Phys. 32, 419–424.
Reutenauer, C. (1993), Free Lie Algebras, London Mathematical Society Monographs 7, Oxford University Press, Oxford.
Tal Ezer, H. & Kosloff, R. (1984), An accurate and efficient scheme for propagating the time dependent Schrödinger equation, J. Chem. Phys. 81, 3967–3976.
Yošida, H. (1990), Construction of higher order symplectic integrators, Phys. Lett. 150, 262–268.
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Communicated by Elizabeth Mansfield.
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Bader, P., Iserles, A., Kropielnicka, K. et al. Effective Approximation for the Semiclassical Schrödinger Equation. Found Comput Math 14, 689–720 (2014). https://doi.org/10.1007/s10208-013-9182-8
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DOI: https://doi.org/10.1007/s10208-013-9182-8
Keywords
- Semiclassical Schrödinger equation
- Exponential splittings
- Zassenhaus splitting
- Spectral collocation
- Krylov subspace methods