to the scheme in question. In other words, $\psi_{\text{ref}}$ for (S2) numerical solutions is the (S2) FFT-based numerical solution associated to the time step $\tau = 0.001 \cdot 2^{-10}$, and so on. The same references are used later on in order to estimate the accuracy of the corresponding multiwavelet simulations.

In fact we can again see an obvious accuracy outperformance of the periodic spectral method in Figure 11. This time, however, we knew a priori that the best FFT calculations are achieved with the rough grid consisting of only 64 points. Indeed, refining the grid to 256 or 1024 one encounters poorer accuracy due to the Gibbs phenomenon leading to high oscillations of solutions close to the boundary points. It does not affect the multiresolution approach, of course. We do not include figures corresponding to refined grid FFT calculations here, as they exhibit behavior similar to Figure 11. However, these FFT results demonstrate poorer convergence, deviating from the theoretically predicted rate, before the MRA calculations approach stagnation. In other words, FFT-based spectral calculations can yield inferior results compared to MRA-based ones in terms of accuracy.

An interesting and noteworthy outcome is that the extensions (5.12) of Yoshida's schemes (5.8) for the time-dependent potential case degenerate to first order. According to [40] it may happen that some exponential integrators lose their theoretically predicted accuracy after the Suzuki's extension described in Subsection 5.2. It seems to us that Yoshida's schemes (5.8) cannot be efficiently extended to non-autonomous equations. The first author encountered a similar problem before [19].

## 7. CONCLUSION

We have developed a multiresolution analysis of the semigroup $\exp(it\partial_x^2)$ and introduced MRA-based numerical methods for solving the time-dependent Schrödinger equation. These methods offer several notable advantages. First of all, they can be seamlessly integrated with existing tools for solving stationary problems. Specifically, the application of singular integral operators arising from potential $V$, as utilized in Kohn-Sham and Hartree-Fock theories, is already well-established in multiwavelet bases. Consequently, these techniques can be readily adapted to address time evolution problems. Secondly, these methods offer spatial adaptivity, which is particularly crucial for quantum chemistry simulations. In these simulations, it is essential to handle Coulomb-type singularities and solutions with cusps. Therefore, an automatic grid refinement based on specified precision can be extremely beneficial.

There are some drawbacks to consider. Achieving sparsity in the multiresolution representation of $\exp(it\partial_x^2)$, which is obviously needed for efficient computations, requires large time steps $t$ and a high MRA order $\ell$. However, a use of a too large MRA order $\ell$ can slow down simulations, while overly large time steps $t$ may compromise the accuracy of the results obtained.

In the future, we plan to focus on developing artificial boundary conditions for the MRA-based methods introduced here. This approach will enable us to use larger time steps while maintaining accuracy, by confining the computational domain to a relatively compact size. Additionally, it will facilitate the use of reasonable values for $\ell$. The techniques developed will be applied to problems where FFT-based methods encounter challenges, specifically in modeling the attosecond dynamics of electrons in molecules exposed to intense laser pulses.

## DECLARATIONS

The CRediT taxonomy of contributor roles [5, 13] is applied. The “Investigation” role also includes the “Methodology”, “Software”, and “Validation” roles. The “Analysis” role also includes the “Formal analysis” and “Visualization” roles. The “Funding acquisition” role also includes the “Resources” role. The contributor roles are visualized in the following authorship attribution matrix, as suggested in Ref. [1].

**Acknowledgments.** The authors are grateful to S. R. Jensen and Ch. Tantardini for numerous helpful discussions. The research was supported by the Research Council of Norway through its Centres of Excellence scheme, Hylleraas-senteret, project number 262695.
<table><caption>TABLE 1. Levels of contribution: major, minor.</caption><thead><tr><th></th><th>ED</th><th>YZ</th><th>LF</th></tr></thead><tbody><tr><td>Conceptualization</td><td></td><td></td><td></td></tr><tr><td>Investigation</td><td></td><td></td><td></td></tr><tr><td>Data curation</td><td></td><td></td><td></td></tr><tr><td>Analysis</td><td></td><td></td><td></td></tr><tr><td>Supervision</td><td></td><td></td><td></td></tr><tr><td>Writing – original draft</td><td></td><td></td><td></td></tr><tr><td>Writing – revisions</td><td></td><td></td><td></td></tr><tr><td>Funding acquisition</td><td></td><td></td><td></td></tr><tr><td>Project administration</td><td></td><td></td><td></td></tr></tbody></table>

REFERENCES

[1] Researchers are embracing visual tools to give fair credit for work on papers. https://www.natureindex.com/news-blog/researchers-embracing-visual-tools-contribution-matrix-give-fair-credit-authors-scientific-papers. Accessed: 2021-5-3.

[2] Mrcpp repository, 2024. Accessed: 9 February 2024.

[3] Vampyr repository, 2024. Accessed: 9 February 2024.

[4] ABRAMOWITZ, M., Ed. Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 10. print., with corr ed. No. 55 in Applied mathematics series. U. S. Government Printing Office, Washington, DC, 1972.

[5] ALLEN, L., SCOTT, J., BRAND, A., HLAVA, M., AND ALTMAN, M. Publishing: Credit where credit is due. Nature 508 (2014), 312–313.

[6] ALPERT, B. A class of bases in $l^2$ for the sparse representation of integral operators. SIAM Journal on Mathematical Analysis 24, 1 (1993), 246–262.

[7] ALPERT, B., BEYLKIN, G., GINES, D., AND VOZOVOI, L. Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics 182, 1 (2002), 149–190.

[8] BEYLKIN, G., COIFMAN, R., AND ROKHLIN, V. Fast wavelet transforms and numerical algorithms i. Communications on Pure and Applied Mathematics 44, 2 (1991), 141–183.

[9] BEYLKIN, G., AND KEISER, J. M. On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. Journal of Computational Physics 132, 2 (1997), 233–259.

[10] BISCHOFF, F. A. Chapter one - computing accurate molecular properties in real space using multiresolution analysis. In State of The Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More, L. U. Ancarani and P. E. Hoggan, Eds., vol. 79 of Advances in Quantum Chemistry. Academic Press, 2019, pp. 3–52.

[11] BJORGVE, M., TANTARDINI, C., JENSEN, S. R., GEREZ S., G. A., WIND, P., DI REMIGIO EIKAS, R., DINVAY, E., AND FREDIANI, L. VAMPYR—A high-level Python library for mathematical operations in a multiwavelet representation, 04 2024.

[12] BLANES, S., AND MOAN, P. Splitting methods for the time-dependent schrödinger equation. Physics Letters A 265, 1 (2000), 35–42.

[13] BRAND, A., ALLEN, L., ALTMAN, M., HLAVA, M., AND SCOTT, J. Beyond authorship: attribution, contribution, collaboration, and credit. Learn. Publ. 28 (2015), 151–155.

[14] CHIN, S. A. Symplectic integrators from composite operator factorizations. Physics Letters A 226, 6 (1997), 344–348.

[15] CHIN, S. A., AND CHEN, C. R. Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation. The Journal of Chemical Physics 114, 17 (05 2001), 7338–7341.

[16] CHIN, S. A., AND CHEN, C. R. Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials. The Journal of Chemical Physics 117, 4 (07 2002), 1409–1415.

[17] COCCIA, E., AND LUPPI, E. Time-dependent ab initio approaches for high-harmonic generation spectroscopy. Journal of Physics: Condensed Matter 34, 7 (nov 2021), 073001.

[18] COHEN, A., DAUBECHIES, I., AND FEAUVEAU, J.-C. Biorthogonal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 45, 5 (1992), 485–560.

[19] DINVAY, E., KALISCH, H., AND PĂRĂU, E. I. Fully dispersive models for moving loads on ice sheets. Journal of Fluid Mechanics 876 (2019), 122–149.

[20] EVANS, L. C. Partial differential equations, 2nd ed ed. No. v. 19 in Graduate studies in mathematics. American Mathematical Society, Providence, R.I, 2010. OCLC: ocn465190110.
<table><caption>TABLE 1. Levels of contribution: major, minor.</caption><thead><tr><th></th><th>ED</th><th>YZ</th><th>LF</th></tr></thead><tbody><tr><td>Conceptualization</td><td></td><td></td><td></td></tr><tr><td>Investigation</td><td></td><td></td><td></td></tr><tr><td>Data curation</td><td></td><td></td><td></td></tr><tr><td>Analysis</td><td></td><td></td><td></td></tr><tr><td>Supervision</td><td></td><td></td><td></td></tr><tr><td>Writing – original draft</td><td></td><td></td><td></td></tr><tr><td>Writing – revisions</td><td></td><td></td><td></td></tr><tr><td>Funding acquisition</td><td></td><td></td><td></td></tr><tr><td>Project administration</td><td></td><td></td><td></td></tr></tbody></table>

REFERENCES

[1] Researchers are embracing visual tools to give fair credit for work on papers. https://www.natureindex.com/news-blog/researchers-embracing-visual-tools-contribution-matrix-give-fair-credit-authors-scientific-papers. Accessed: 2021-5-3.

[2] Mrcpp repository, 2024. Accessed: 9 February 2024.

[3] Vampyr repository, 2024. Accessed: 9 February 2024.

[4] ABRAMOWITZ, M., Ed. Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 10. print., with corr ed. No. 55 in Applied mathematics series. U. S. Government Printing Office, Washington, DC, 1972.

[5] ALLEN, L., SCOTT, J., BRAND, A., HLAVA, M., AND ALTMAN, M. Publishing: Credit where credit is due. Nature 508 (2014), 312–313.

[6] ALPERT, B. A class of bases in $l^2$ for the sparse representation of integral operators. SIAM Journal on Mathematical Analysis 24, 1 (1993), 246–262.

[7] ALPERT, B., BEYLKIN, G., GINES, D., AND VOZOVOI, L. Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics 182, 1 (2002), 149–190.

[8] BEYLKIN, G., COIFMAN, R., AND ROKHLIN, V. Fast wavelet transforms and numerical algorithms i. Communications on Pure and Applied Mathematics 44, 2 (1991), 141–183.

[9] BEYLKIN, G., AND KEISER, J. M. On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. Journal of Computational Physics 132, 2 (1997), 233–259.

[10] BISCHOFF, F. A. Chapter one - computing accurate molecular properties in real space using multiresolution analysis. In State of The Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More, L. U. Ancarani and P. E. Hoggan, Eds., vol. 79 of Advances in Quantum Chemistry. Academic Press, 2019, pp. 3–52.

[11] BJORGVE, M., TANTARDINI, C., JENSEN, S. R., GEREZ S., G. A., WIND, P., DI REMIGIO EIKAS, R., DINVAY, E., AND FREDIANI, L. VAMPYR—A high-level Python library for mathematical operations in a multiwavelet representation, 04 2024.

[12] BLANES, S., AND MOAN, P. Splitting methods for the time-dependent schrödinger equation. Physics Letters A 265, 1 (2000), 35–42.

[13] BRAND, A., ALLEN, L., ALTMAN, M., HLAVA, M., AND SCOTT, J. Beyond authorship: attribution, contribution, collaboration, and credit. Learn. Publ. 28 (2015), 151–155.

[14] CHIN, S. A. Symplectic integrators from composite operator factorizations. Physics Letters A 226, 6 (1997), 344–348.

[15] CHIN, S. A., AND CHEN, C. R. Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation. The Journal of Chemical Physics 114, 17 (05 2001), 7338–7341.

[16] CHIN, S. A., AND CHEN, C. R. Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials. The Journal of Chemical Physics 117, 4 (07 2002), 1409–1415.

[17] COCCIA, E., AND LUPPI, E. Time-dependent ab initio approaches for high-harmonic generation spectroscopy. Journal of Physics: Condensed Matter 34, 7 (nov 2021), 073001.

[18] COHEN, A., DAUBECHIES, I., AND FEAUVEAU, J.-C. Biorthogonal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 45, 5 (1992), 485–560.

[19] DINVAY, E., KALISCH, H., AND PĂRĂU, E. I. Fully dispersive models for moving loads on ice sheets. Journal of Fluid Mechanics 876 (2019), 122–149.

[20] EVANS, L. C. Partial differential equations, 2nd ed ed. No. v. 19 in Graduate studies in mathematics. American Mathematical Society, Providence, R.I, 2010. OCLC: ocn465190110.
[21] FANN, G., BEYLKIN, G., HARRISON, R. J., AND JORDAN, K. E. Singular operators in multiwavelet bases. *IBM Journal of Research and Development* 48, 2 (2004), 161–171.

[22] GRAY, S. K., AND VEROSKY, J. M. Classical Hamiltonian structures in wave packet dynamics. *The Journal of Chemical Physics* 100, 7 (04 1994), 5011–5022.

[23] HARRISON, R. J., FANN, G. I., YANAI, T., GAN, Z., AND BEYLKIN, G. Multiresolution quantum chemistry: Basic theory and initial applications. *The Journal of Chemical Physics* 121, 23 (11 2004), 11587–11598.

[24] JENSEN, S. R., SAHA, S., FLORES-LIVAS, J. A., HUHN, W., BLUM, V., GOEDECKER, S., AND FREDIANI, L. The elephant in the room of density functional theory calculations. *The Journal of Physical Chemistry Letters* 8, 7 (2017), 1449–1457. PMID: 28291362.

[25] KAYE, J., BARNETT, A., AND GREENGARD, L. A high-order integral equation-based solver for the time-dependent schrödinger equation. *Communications on Pure and Applied Mathematics* 75, 8 (2022), 1657–1712.

[26] LUCA FREDIANI, EIRIK FOSSGAARD, T. F., AND RUUD, K. Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the poisson and bound-state helmholtz kernels in three dimensions. *Molecular Physics* 111, 9-11 (2013), 1143–1160.

[27] MALLAT, S. G. Multiresolution approximations and wavelet orthonormal bases of $l^2(r)$. *Transactions of the American mathematical society* 315, 1 (1989), 69–87.

[28] MALLAT, S. G. A theory for multiresolution signal decomposition: the wavelet representation. *IEEE transactions on pattern analysis and machine intelligence* 11, 7 (1989), 674–693.

[29] MOUROU, G. Nobel lecture: Extreme light physics and application. *Rev. Mod. Phys.* 91 (Jul 2019), 030501.

[30] NISOLI, M., DECLEVA, P., CALEGARI, F., PALACIOS, A., AND MARTÍN, F. Attosecond electron dynamics in molecules. *Chemical Reviews* 117, 16 (2017), 10760–10825. PMID: 28488433.

[31] PESKIN, U., KOSLOFF, R., AND MOISEYEV, N. The solution of the time dependent Schrödinger equation by the $(t, t')$ method: The use of global polynomial propagators for time dependent Hamiltonians. *The Journal of Chemical Physics* 100, 12 (06 1994), 8849–8855.

[32] SANZ-SERNA, J. M., AND PORTILLO, A. Classical numerical integrators for wave-packet dynamics. *The Journal of Chemical Physics* 104, 6 (02 1996), 2349–2355.

[33] STRICKLAND, D. Nobel lecture: Generating high-intensity ultrashort optical pulses. *Rev. Mod. Phys.* 91 (Jul 2019), 030502.

[34] SUZUKI, M. General theory of higher-order decomposition of exponential operators and symplectic integrators. *Physics Letters A* 165, 5 (1992), 387–395.

[35] SUZUKI, M. General decomposition theory of ordered exponentials. *Proceedings of the Japan Academy, Series B* 69, 7 (1993), 161–166.

[36] SUZUKI, M. New scheme of hybrid exponential product formulas with applications to quantum monte-carlo simulations. In *Computer Simulation Studies in Condensed-Matter Physics VIII* (Berlin, Heidelberg, 1995), D. P. Landau, K. K. Mon, and H.-B. Schüttler, Eds., Springer Berlin Heidelberg, pp. 169–174.

[37] TANTARDINI, C., DINVAY, E., PITTELOUD, Q., GEREZ S., G. A., JENSEN, S. R., WIND, P., REMIGIO, R. D., AND FREDIANI, L. Advancements in quantum chemistry using multiwavelets: Theory, implementation, and applications. In preparation, 2024.

[38] VENCE, N., HARRISON, R., AND KRSTIĆ, P. Attosecond electron dynamics: A multiresolution approach. *Phys. Rev. A* 85 (Mar 2012), 033403.

[39] WALKER, R. B., AND PRESTON, R. K. Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator. *The Journal of Chemical Physics* 67, 5 (09 1977), 2017–2028.

[40] WIEBE, N., BERRY, D., HÖYER, P., AND SANDERS, B. C. Higher order decompositions of ordered operator exponentials. *Journal of Physics A: Mathematical and Theoretical* 43, 6 (jan 2010), 065203.

[41] YIN, J. A fourth-order compact time-splitting method for the dirac equation with time-dependent potentials. *Journal of Computational Physics* 430 (2021), 110109.

[42] YOSHIDA, H. Construction of higher order symplectic integrators. *Physics Letters A* 150, 5 (1990), 262–268.

Email address: evgueni.dinvay@uit.no

Email address: jl.zabelina@gmail.com

Email address: luca.frediani@uit.no
[21] FANN, G., BEYLKIN, G., HARRISON, R. J., AND JORDAN, K. E. Singular operators in multiwavelet bases. *IBM Journal of Research and Development* 48, 2 (2004), 161–171.

[22] GRAY, S. K., AND VEROSKY, J. M. Classical Hamiltonian structures in wave packet dynamics. *The Journal of Chemical Physics* 100, 7 (04 1994), 5011–5022.

[23] HARRISON, R. J., FANN, G. I., YANAI, T., GAN, Z., AND BEYLKIN, G. Multiresolution quantum chemistry: Basic theory and initial applications. *The Journal of Chemical Physics* 121, 23 (11 2004), 11587–11598.

[24] JENSEN, S. R., SAHA, S., FLORES-LIVAS, J. A., HUHN, W., BLUM, V., GOEDECKER, S., AND FREDIANI, L. The elephant in the room of density functional theory calculations. *The Journal of Physical Chemistry Letters* 8, 7 (2017), 1449–1457. PMID: 28291362.

[25] KAYE, J., BARNETT, A., AND GREENGARD, L. A high-order integral equation-based solver for the time-dependent schrödinger equation. *Communications on Pure and Applied Mathematics* 75, 8 (2022), 1657–1712.

[26] LUCA FREDIANI, EIRIK FOSSGAARD, T. F., AND RUUD, K. Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the poisson and bound-state helmholtz kernels in three dimensions. *Molecular Physics* 111, 9-11 (2013), 1143–1160.

[27] MALLAT, S. G. Multiresolution approximations and wavelet orthonormal bases of $l^2(r)$. *Transactions of the American mathematical society* 315, 1 (1989), 69–87.

[28] MALLAT, S. G. A theory for multiresolution signal decomposition: the wavelet representation. *IEEE transactions on pattern analysis and machine intelligence* 11, 7 (1989), 674–693.

[29] MOUROU, G. Nobel lecture: Extreme light physics and application. *Rev. Mod. Phys.* 91 (Jul 2019), 030501.

[30] NISOLI, M., DECLEVA, P., CALEGARI, F., PALACIOS, A., AND MARTÍN, F. Attosecond electron dynamics in molecules. *Chemical Reviews* 117, 16 (2017), 10760–10825. PMID: 28488433.

[31] PESKIN, U., KOSLOFF, R., AND MOISEYEV, N. The solution of the time dependent Schrödinger equation by the $(t, t')$ method: The use of global polynomial propagators for time dependent Hamiltonians. *The Journal of Chemical Physics* 100, 12 (06 1994), 8849–8855.

[32] SANZ-SERNA, J. M., AND PORTILLO, A. Classical numerical integrators for wave-packet dynamics. *The Journal of Chemical Physics* 104, 6 (02 1996), 2349–2355.

[33] STRICKLAND, D. Nobel lecture: Generating high-intensity ultrashort optical pulses. *Rev. Mod. Phys.* 91 (Jul 2019), 030502.

[34] SUZUKI, M. General theory of higher-order decomposition of exponential operators and symplectic integrators. *Physics Letters A* 165, 5 (1992), 387–395.

[35] SUZUKI, M. General decomposition theory of ordered exponentials. *Proceedings of the Japan Academy, Series B* 69, 7 (1993), 161–166.

[36] SUZUKI, M. New scheme of hybrid exponential product formulas with applications to quantum monte-carlo simulations. In *Computer Simulation Studies in Condensed-Matter Physics VIII* (Berlin, Heidelberg, 1995), D. P. Landau, K. K. Mon, and H.-B. Schüttler, Eds., Springer Berlin Heidelberg, pp. 169–174.

[37] TANTARDINI, C., DINVAY, E., PITTELOUD, Q., GEREZ S., G. A., JENSEN, S. R., WIND, P., REMIGIO, R. D., AND FREDIANI, L. Advancements in quantum chemistry using multiwavelets: Theory, implementation, and applications. In preparation, 2024.

[38] VENCE, N., HARRISON, R., AND KRSTIĆ, P. Attosecond electron dynamics: A multiresolution approach. *Phys. Rev. A* 85 (Mar 2012), 033403.

[39] WALKER, R. B., AND PRESTON, R. K. Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator. *The Journal of Chemical Physics* 67, 5 (09 1977), 2017–2028.

[40] WIEBE, N., BERRY, D., HÖYER, P., AND SANDERS, B. C. Higher order decompositions of ordered operator exponentials. *Journal of Physics A: Mathematical and Theoretical* 43, 6 (jan 2010), 065203.

[41] YIN, J. A fourth-order compact time-splitting method for the dirac equation with time-dependent potentials. *Journal of Computational Physics* 430 (2021), 110109.

[42] YOSHIDA, H. Construction of higher order symplectic integrators. *Physics Letters A* 150, 5 (1990), 262–268.

Email address: evgueni.dinvay@uit.no

Email address: jl.zabelina@gmail.com

Email address: luca.frediani@uit.no

DEPARTMENT OF CHEMISTRY, UIT THE ARCTIC UNIVERSITY OF NORWAY, PO BOX 6050 LANGNES, N-9037 TROMSØ, NORWAY