The aim of the project MA14-002 'Adaptive Time-Splitting for Many-Body Quantum Propagation' supported by the Vienna Science and Technology Fund (WWTF) is to understand non-equilibrium quantum many-body systems, which is one of the grand challenges in current physics. The ability to describe time-dependent extended systems on a quantitative level promises to open up new technological developments such as laser control of chemical reactions, femtosecond lightfield electronics, and material design for sustainable energy. Most experimental probes of quantum many-body systems rely on so-called pump-probe schemes. The system is driven out of equilibrium and the ensuing dynamics is studied. While for weak pump and probe fields the dynamics can be understood within linear response theory where essentially equilibrium (or near ground state) properties enter, stronger pumping requires a full time-dependent propagation beyond the perturbative limit. While the high-dimensional many-particle Schrödinger equation is too complex to be solved for more than a few particles in three dimensions, practically all approximations to it give rise to coupled nonlinear partial differential equations in space and time in reduced dimensions. Still, propagation of these equations constitutes a major challenge. In this project we aim for the development and implementation of a highly efficient and accurate method, the adaptive time-splitting method, for the propagation of quantum many-body systems presently out of reach. We will investigate the efficiency, error, and speed up of this method and apply it to both fermionic and bosonic systems of current interest. So far, the project yielded the following articles and preprints and conference presentations. A collection of splitting methods constructed in the course of the project or taken from the literature and assessed as to their merits for computational purposes can be found at http://www.othmar-koch.org/splitting/.

Refereed Papers


• F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and J. Burgdörfer, The time-dependent two-particle reduced density matrix method: LiH in strong laser fields, J. Phys.: Conf. Ser. 635(2015), 112084.
  
  You can download this article here.
  
  
  
• W. Auzinger, Th. Kassebacher, O. Koch, M. Thalhammer, Adaptive Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Regime, Numer. Algorithms 72(2016), pp. 1-35.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1605.00429. Archived at https://uscholar.univie.ac.at.
  
  
  
• W. Auzinger, H. Hofstätter, D. Ketcheson, and O. Koch, Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations. Part I: Construction of Optimized Schemes and Pairs of Schemes, BIT 57(2017), pp. 55-74, DOI=dx.doi.org/10.1007/s10543-016-0626-9.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1604.01179
  
  
  
• W. Auzinger, W. Herfort, O. Koch, and M. Thalhammer, The BCH-Formula and Order Conditions for Splitting Methods, in: Lie Groups, Differential Equations, and Geometry, G. Falcone (ed.), UNIPA Springer Series, Vol. XIV(2017), pp. 71-84.
  
  
  
• W. Auzinger, Th. Kassebacher, O. Koch, M. Thalhammer, Convergence of a Strang Splitting Finite Element Discretization for the Schrödinger-Poisson Equation, M2AN - Math. Model. Numer. Anal. 51(2017), pp. 1245-1278. DOI=dx.doi.org/10.1051/m2an/2016059.
  
  You can download a reprint of this paper as a .pdf file. The original publication is available at www.esaim-m2an.org, the copyright is owned by EDP Sciences. Preprint also available from http://arxiv.org/abs/1605.00437
  
  
  
• T. Sato, K. L. Ishikawa, I. Brezinova, F. Lackner, S. Nagele, and J. Burgdörfer, Time-dependent complete-active-space self-consistent-field method for atoms: Application to high-harmonic generation, Phys. Rev. A 94, 023405 (2016).
  
  You can download a preprint of this paper from https://arxiv.org/abs/1605.00703.
  
  
  
• W. Auzinger, H. Hofstätter, and O. Koch, Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators, Springer LNCS 9890(2016), pp. 43-57.
  
  You can download a preprint of this paper from http://arxiv.org/abs/1605.00453 Archived at https://uscholar.univie.ac.at.
  
  
  
• W. Auzinger, W. Herfort, H. Hofstätter, and O. Koch, Setup of Order Conditions for Splitting Methods, Springer LNCS 9890(2016), pp. 30-42.
  
  You can download a preprint of this paper from http://arxiv.org/abs/1605.00445 Archived at https://uscholar.univie.ac.at.
  
  
  
• W. Auzinger, O. Koch, and M. Quell, Adaptive High-Order Splitting Methods for Systems of Nonlinear Evolution Equations with Periodic Boundary Conditions, Numer. Algorithms 75(2017), pp. 261-283. DOI=dx.doi.org/10.1007/s11075-016-0206-8.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1604.01201 .
  
  
  
• W. Auzinger, H. Hofstätter, O. Koch, and M. Thalhammer, Reduced Order of the Local Error Of Splitting For Parabolic Problems with Dirichlet Boundary Conditions, in `Numerical Analysis and Applied Mathematics' AIP Conference Proceedings1863(2017), 160008. DOI=dx.doi.org/10.1063/1.4992342
  
  
  
• F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and J. Burgdörfer, Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation, J. Phys.: Conf. Ser. 875(2017), 012011.
  
  You can download this article here.
  
  
  
• F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and J. Burgdörfer, Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation, J. Phys.: Conf. Ser. 875(2017), 022032.
  
  You can download this article here.
  
  
  
• W. Auzinger, I. Brezinova, H. Hofstätter, O. Koch, and M. Quell, Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations. Part II: Comparisons of Local Error Estimation and Step-Selection Strategies, Comput. Phys. Commun. 234(2019), pp. 55-71.
  
  You can download an extended preprint of this paper as a .pdf file. Archived at https://uscholar.univie.ac.at.
  
  
  
• X.-M. Tong, S. Borbely, I. Brezinova, F. Lackner, S. Nagele, J. Feist, L. Nagy, K. Tokesi, N. Toshima, and J. Burgdörfer, Energy straggling cross section for antiproton-atom collisions, J. Phys.: Conf. Ser. 875(2017), 092026.
  
  You can download this article here.
  
  
  
• F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and J. Burgdörfer, High-harmonic spectra from time-dependent two-particle reduced density matrix theory, Phys. Rev. A 95(2017), 033414.
  
  You can download a preprint from https://arxiv.org/abs/1611.00888.
  
  
  
• S. Donsa, H. Hofstätter, O. Koch, J. Burgdörfer, and I. Brezinova, Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization, Phys. Rev. A 96(2017), 043630.
  
  You can download a preprint from https://arxiv.org/abs/1707.02106.
  
  
  
• A. Blättermann, A. Kaldun, V. Stooß, S. Donsa, H. Wei, R. Pazourek, S. Nagele, C. Ott, C.-D. Lin, J. Burgdörfer, T. Pfeifer, Watching the emergence of a Fano resonance in doubly excited helium, J. Phys. Conf. Ser., 875 (2017), 012010.
  
  Available online from https://doi.org/10.1088/1742-6596/875/2/012010.
  
  
  
• V. Stooß, S. M. Cavaletto, S. Donsa, A. Blättermann, P. Birk, C. H. Keitel, I. Brezinova, J. Burgdörfer, C. Ott, and T.Pfeifer, Real-Time Reconstruction of the Strong-Field-Driven Dipole Response, Phys. Rev. Lett. 121(2018), 173005.
  
  You can download this article here.
  
  
  
• W. Auzinger and O. Koch, An improved local error estimator for symmetric time-stepping schemes, Appl. Math. Lett. 82(2018), pp.106-110.
  
  Archived at https://uscholar.univie.ac.at/.
  
  
  
• W. Auzinger, I. Brezinova, H. Hofstätter, O. Koch, and T. Sato, Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential, Journal of Computational Mathematics and Data Science 1(2021), 100014.
  
  You can download this paper here.
  
  
  
• O. Koch, Convergence of exponential Lawson-multistep methods for the MCTDHF equations, M2AN - Math. Model. Numer. Anal. 53(2019), pp. 2109-2119. DOI=10.1051/m2an/2019033.
  
  You can download a reprint of this paper as a .pdf file. The original publication is available at www.esaim-m2an.org, the copyright is owned by EDP Sciences.
  
  
  
• W. Auzinger, H. Hofstätter, and O. Koch, Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations, J. Comput. Appl. Math. 234(2019), pp. 195-201. Available online from https://doi.org/10.1016/j.cam.2019.02.011.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1806.07771
  
  
  
• S. Borbely, X.-M. Tong, S. Nagele, J. Feist, I. Brezinova, F. Lackner, L. Nagy, K. Tökesi, and J. Burgdörfer, Electron correlations in the antiproton energy-loss distribution in He, Phys. Rev. A 98, 012707 (2018).
  
  You can download a preprint of this paper from https://arxiv.org/abs/1901.10212
  
  
  
• S. Donsa, I. Brezinova, H. Ni, J. Feist, and J. Burgdörfer, Polarization tagging of two-photon double ionization by elliptically polarized XUV pulses, Phys. Rev. A 99, 023413.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1811.09110
  
  
  
• H. Overweg, A. Knothe, T. Fabian, L. Linhart, P. Rickhaus, L. Wernli, K. Watanabe, T. Taniguchi, D. Sanchez, J. Burgdörfer, F. Libisch, V.I. Falko, K. Ensslin, and T. Ihn, Topologically nontrivial valley states in bilayer graphene quantum point contacts, Phys. Rev. Lett. 121, 257702 (2018).
  
  You can download a preprint of this paper from https://arxiv.org/abs/1809.01920
  
  
  
• W. Auzinger, H. Hofstätter, K. Kropielnicka, O. Koch, and P. Singh, Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime, Appl. Math. Comput. 362 (2019), 124550.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1902.04324
  
  
  
• W. Auzinger, H. Hofstätter, and O. Koch Non-existence of generalized splitting methods with positive coefficients of order higher than four, Appl. Math. Lett. 97(2019), pp. 48-52. Available online from https://doi.org/10.1016/j.aml.2019.05.017.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1905.05492.
  
  
  
• W. Auzinger, H. Hofstätter, and O. Koch, An algorithm for computing coefficients of words in expressions involving exponentials and its application to the construction of exponential integrators, Springer LNCS 11661 (2019), pp. 197-214.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1912.01399v1.
  
  
  
• W. Auzinger, A. Grosz, H. Hofstätter, and O. Koch, Adaptive exponential integrators for MCTDHF, Springer LNCS 11958 (2020), pp. 557-565.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1905.05590
  
  
  
• S. Donsa, N. Douguet, J. Burgdörfer, I. Brezinova, L. Argenti, Circular holographic ionization-phase meter, Phys. Rev. Lett. 123, 133203 (2019). Available online from https://dx.doi.org/10.1103/PhysRevLett.123.133203.
  
  You can download a preprint of this paper from https://arxiv.org/abs/1904.04380.
  
  
  
• R. Wanzenböck, S. Donsa, H. Hofstätter, O. Koch, P. Schlagheck, I. Brezinova, Chaos-induced loss of coherence of a Bose-Einstein condensate, Phys. Rev. A 103, 023336.
  
  You can download a preprint of this paper from https://arxiv.org/abs/2007.07662.
  
  
  Oral Presentations
  
  
• W. Auzinger, Splitting Methods for Evolution Equations, Local Error Estimation, Adaptivity, International V. Skorobohatko Mathematical Conference, Drohobych, Ukraine, August 2015. Joint with H. Hofstätter, D. Ketcheson, and O. Koch.
  
  
• W. Auzinger, Splitting Schemes, Order Conditions, and Optimized Pairs, SciCADE - International Conference on Scientific Computing and Differential Equations Conference 2015, Potsdam, Germany, September 2015. Joint with H. Hofstätter, D. Ketcheson, and O. Koch.
  
  
• T. Kassebacher, Convergence of Adaptive Splitting and Finite Element Methods for the Schrödinger-Poisson Equation, SciCADE - International Conference on Scientific Computing and Differential Equations Conference 2015, Potsdam, Germany, September 2015. Joint with W. Auzinger, O. Koch, and M. Thalhammer.
  
  
• O. Koch, Adaptive High-order Time-Splitting Methods for Systems of Evolution Equations. Applications in Quantum Dynamics and Pattern Formation, SciCADE - International Conference on Scientific Computing and Differential Equations Conference 2015, Potsdam, Germany, September 2015. Joint with W. Auzinger, H. Hofstätter, Th. Kassebacher, and M. Thalhammer.
  
  
• O. Koch, Time-splitting FEM discretization of the Schrödinger-Poisson equation, 12th Austrian Numerical Analysis Day, Innsbruck, April 2016. Joint with W. Auzinger, Th. Kassebacher, and M. Thalhammer.
  
  
• O. Koch, Adaptive Full Discretization for Nonlinear Evolution Equations, University of Bath, U.K., April 2016.
  
  
• O. Koch (invited talk), Reduced Order of the Local Error of Splitting for Parabolic Problems, ICNAAM 2016, Rhodes, Greece, September 2016. Joint with W. Auzinger, H. Hofstätter, and M. Thalhammer.
  
  
• I. Brezinova (invited talk), A new theoretical approach for accurate multi-electron dynamics in strong fields: the time-dependent two-particle reduced density matrix method, International Workshop on Atomic Physics, Max-Plank-Institute for Physics of Complex Systems, Dresden, Germany, November 2016. Joint with F. Lackner, T. Sato, K. L. Ishikawa, and J. Burgdörfer.
  
  
• I. Brezinova (invited talk), Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation, 12th European Conference on Atoms Molecules and Photons (ECAMP12), Frankfurt/Main, Germany, September 2016. Joint with F. Lackner, T. Sato, K. L. Ishikawa, and J. Burgdörfer.
  
  
• O. Koch, Error Analysis of Splitting Methods for Parabolic Problems under Dirichlet Boundary Conditions, 9th Workshop Numerical Analysis of Evolution Equations, Innsbruck, November 2016. Joint with W. Auzinger, H. Hofstätter, and M. Thalhammer.
  
  
• O. Koch (invited talk), Adaptive Time-splitting FEM discretization of the Schrödinger-Poisson equation, Modern Numerical Methods for Quantum Mechanics, Warsaw, March 2017. Joint with W. Auzinger, Th. Kassebacher, and M. Thalhammer.
  
  
• H. Hofstätter, Time propagation for Schrödinger-type equations with expensive nonlinearity, Austrian Numerical Analysis Day 2017, Salzburg, May 2017. Joint with O. Koch.
  
  
• H. Hofstätter, Time propagation for Schrödinger-type equations with expensive nonlinearity, University of Innsbruck, Innsbruck, July 2017. Joint with O. Koch.
  
  
• O. Koch, Adaptivity for Splitting Schemes for Nonlinear Schrödinger Equations, University of Innsbruck, Innsbruck, July 2017. Joint with W. Auzinger, H. Hofstätter, and M. Thalhammer.
  
  
• O. Koch (invited talk), Adaptive exponential splitting and Lawson methods for Schrödinger equations, SciCADE 2017, Bath, U.K., September 2017. Joint with W. Auzinger, I. Brezinova, H. Hofstätter, and M. Thalhammer.
  
  
• O. Koch, Adaptive Exponential Integrators for Schrödinger Equations, University of Gdansk, Gdansk, November 2017.
  
  
• S. Donsa, Anderson localization of a one-dimensional Bose-Einstein condensate after long-time expansion, Vienna young Scientists Symposium, Vienna, June 2016. Joint with H. Hofstätter, O. Koch, J. Burgdörfer, and I. Brezinova.
  
  
• I. Brezinova (invited talk), Expansion of Bose-Einstein condensates in disorder: Anderson localization and wave chaos, International Workshop on Quantum-Classical Transition in Many-Body Systems, Dresden, Germany, February 2017. Joint with S. Donsa, H. Hofstätter, O. Koch, and J. Burgdörfer.
  
  
• S. Donsa, Ultrafast buildup of a Fano resonance in the time domain, 6th International Conference on Attosecond Physics, Xian, China, July 2017. Joint with A. Kaldun, A. Blättermann, V. Stooß, H. Wei, R. Pazourek, S. Nagele, C. Ott, C.-D. Lin, J. Burgdörfer, and T. Pfeifer.
  
  
• I. Brezinova (invited talk), A new theoretical approach for accurate multi-electron dynamics in strong fields: the time-dependent two-particle reduced density matrix method, 14th International Conference on Multiphoton Processes (ICOMP 2017), Budapest, Hungary, September 2017. Joint with F. Lackner, S. Donsa, T. Sato, K. L. Ishikawa, and J. Burgdörfer.
  
  
• F. Lackner, The time-dependent two-particle reduced density matrix method"; Vortrag: International Workshop on 'New challenges in Reduced Density Matrix Functional Theory: Symmetries, time-evolution and entanglement', CECAM-HQ-EPFL, Lausanne, Switzerland; September 2017. Joint with S. Donsa, T. Sato, K. L. Ishikawa, J. Burgdörfer, and I. Brezinova.
  
  
• S. Donsa (invited talk), The time-dependent two-particle reduced density matrix method, Informal seminar of the Departement of Theoretical Condensed Matter at the University Autonoma Madrid, Madrid, Spain, November 2017. Joint with F. Lackner, T. Sato, K. L. Ishikawa, J. Burgdörfer, and I. Brezinova.
  
  
• O. Koch, Adaptive exponential Lawson methods for nonlinear Schrödinger equations, 14th Austrian Numerical Analysis Day 2018, Klagenfurt, May 2018. Joint with W. Auzinger, I. Brezinova, and H. Hofstätter.
  
  
• H. Hofstätter, Order conditions for exponential integrators, 14th Austrian Numerical Analysis Day 2018, Klagenfurt, May 2018. Joint with W. Auzinger and O. Koch.
  
  
• W. Auzinger (invited talk), Sharp a posteriori error estimation for symmetric one-step integrators, International Conference on Modern Problems of Mechanics and Mathematics, Lviv, Ukraine, May 2018. Joint with H. Hofstätter and O. Koch.
  
  
• W. Auzinger (invited talk), Sharp local error estimation for time-reversible one-step schemes, Workshop `Modern Numerical Methods for Quantum Mechanics II', Gdansk, Poland, June 2018. Joint with H. Hofstätter and O. Koch.
  
  
• H. Hofstätter, Order conditions for exponential integrators, poster presentation at the Workshop `Modern Numerical Methods for Quantum Mechanics II', Gdansk, Poland, June 2018. Joint with W. Auzinger and O. Koch.
  
  
• S. Donsa, Accurate reconstruction of the non-linear response of matter from spectroscopic absorption data, 49th Annual Meeting of the APS Divison of Atomic, Molecular & Optical Physics, Fort Lauderdale, FL, USA. Published in Bulletin of the American Physical Society (2018), p. 218. Joint with I. Brezinova, J. Burgdörfer, V. Stooß, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer.
  
  
• S. Donsa, Long-time expansion of a Bose-Einstein condensate: Does the interacting many-particle system show Anderson localization? 49th Annual Meeting of the APS Divison of Atomic, Molecular & Optical Physicss, Fort Lauderdale, FL, USA. Published in Bulletin of the American Physical Society (2018), p. 173. Joint with H. Hofstätter, O. Koch, J. Burgdörfer, I. Brezinova,
  
  
• S. Donsa, Accurate reconstruction of the ultrafast non-linear response in the time domain from spectroscopy, poster presentation at Gordon Research Conference on Multiphoton Processes 2018, Smithfield, RI, USA, 2018. Joint with I. Brezinova, J. Burgdörfer, V. Stooß, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer,
  
  
• S. Donsa, Real-Time Reconstruction of the Strong-Field-Driven Dipole Response, poster presentation at Label Free Spectro-Microscopy, Garching 2018. Joint with I. Brezinova, J. Burgdörfer, V. Stooß, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer,
  
  
• S. Donsa, Expanding Bose-Einstein condensate in disorder: Role of many-body interactions for Anderson localization, poster presentation at Research Frontiers in Ultracold Quantum Gases. 685th WE-Heraus Seminar, Bad Honnef, Germany. Joint with R. Wanzenböck, H. Hofstätter, O. Koch, J. Burgdörfer, P. Schlagheck, I. Brezinova,
  
  
• S. Donsa, Accurate reconstruction of the non-linear response of matter from spectroscopic absorption data, 49th Annual Meeting of the APS Divison of Atomic, Molecular & Optical Physics, Fort Lauderdale, FL, USA. In Bulletin of the American Physical Society (2018), S. 218. Joint with I. Brezinova, J. Burgdörfer, V. Stooß, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer.
  
  
• S. Donsa, Long-time expansion of a Bose-Einstein condensate: Does the interacting many-particle system show Anderson localization? 49th Annual Meeting of the APS Divison of Atomic, Molecular & Optical Physics, Fort Lauderdale, FL, USA. In: "Bulletin of the American Physical Society", (2018), S. 173. Joint with H. Hofstätter, O. Koch, J. Burgdörfer, I. Brezinova.
  
  
• S. Donsa, Accurate reconstruction of the ultrafast non-linear response in the time domain from spectroscopy, Poster at Gordon Research Conference on Multiphoton Processes 2018, Smithfield, RI, USA. Joint with I. Brezinova, J. Burgdörfer, V. Stooß>, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer.
  
  
• S. Donsa, Real-Time Reconstruction of the Strong-Field-Driven Dipole Response, Poster at Label Free Spectro-Microscopy, Garching, Germany 2018. Joint with I. Brezinova, J. Burgdörfer, V. Stooß, S. Cavaletto, A. Blättermann, P. Birk, C. Keitel, C. Ott, T. Pfeifer.
  
  
• S. Donsa, Expanding Bose-Einstein condensate in disorder: Role of many-body interactions for Anderson localization, Poster at Research Frontiers in Ultracold Quantum Gases. 685. WE-Heraus Seminar, Bad Honnef, Germany 2018. Joint with R. Wanzenböck, H. Hofstätter, O. Koch, J. Burgdörfer, P. Schlagheck, I. Brezinova.
  
  
• O. Koch, Efficient adaptive time integrators for high-dimensional Schrödinger equations, NSIDE - Workshop on Numerical Solution of Integral and Differential Equations, Gdansk, Poland, July 2019. Joint with W. Auzinger, A. Grosz, and H. Hofstätter.
  
  You can view the abstract of this talk here.
  
  
• H. Hofstätter, An algorithm for computing coefficients of words in expressions envolving exponentials and its application to the construction of exponential integrators, 21st International Workshop on Computer Algebra in Scientific Computing, Moskow, Russia, August 2019. Joint with W. Auzinger, O. Koch.
  
  
• W. Auzinger, Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime, ParNum 2019, Dubrovnik, October 2019. Joint with H. Hofstätter, O. Koch, K. Kropielnicka, and P. Singh.
• W. Auzinger, Adaptive Time Propagation of the MCTDHF Equations, 6th Workshop on Stability and Discretization Issues in Differential Equations, Budapest, Hungary, June 2022. Joint with A. Grosz, H. Hofstätter, and O. Koch.
  
  
  

page written by Othmar Koch. last modification: Tue Nov 30 12:00 MET 2021