GRK 1838:  Spectral Theory and Dynamics of Quantum Systems

• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 206649329

• Final Report Year: 2020

Final Report Abstract

Science and technology are being pervaded by applications of quantum physical laws at an ever increasing rate. Computer-aided modern drug design, medical imaging (NMR) and (radiation-) treatment, micro-electronics and photovoltaics would all not exist without our knowledge of quantum physics. While the fundamental quantum mechanical laws behind these applications are well understood in terms of mathematical equations, the mathematical complexity of interacting manyparticle quantum systems prevents one from connecting important quantum phenomena with the aforementioned basic laws. This is not only unsatisfactory from an academic point of view, but in the long run it will likely hamper progress in science and technology. It therefore seems indispensable that progress on a fundamental mathematical level concerning many-body quantum systems attempts to keep pace with the developments in the applied sciences. This mathematical progress as well as the education of the next generation of experts in the mathematics of complex quantum systems is at the heart of this research training group. The research programme consists of a set of mathematical projects concerning important model systems from solid state physics and quantum field theory. The specific projects address mathematical problems at the forefront of what can be done with presently available mathematical technology and their solution requires the collaboration of experts in various fields of mathematical analysis. The participating scientists are foremost experts in mathematical quantum mechanics and the mathematics of the Schrödinger equation. Taking advantage of the vicinity of Stuttgart and Tübingen they combine their forces into a leading centre of expertise in mathematical quantum mechanics in Germany. The research training group is designed to initiate and intensify collaborations among them, to facilitate the invitation of leading scientists from abroad, and to offer a common platform for an efficient, broad and thorough education of young researchers in the mathematical analysis ofmany-body quantum systems. The lasting impact of this research training group is demonstrated by a newly established international master programme “Mathematical Physics” in Tübingen. This program is supported by the country of Baden-Württemberg and by the University of Tübingen with two new permanent positions on the levels W3-position and E13, respectively.

Publications

• Dynamics and symmetries of a repulsively bound atom pair in an optical lattice, Phys. Rev. A 86, 013618 (2012)
  A. Deuchert, K. Sakmann, A. I. Streltsov, O. E. Alon, L. S. Cederbaum
  
• Minimization of the energy of the nonrelativistic one-electron Pauli-Fierz model over quasifree states. Doc. Math., 18:1481–1519, 2013
  V. Bach, S. Breteaux, T. Tzaneteas
  
• On contact interactions as limits of short-range potentials, Methods Funct. Anal. Topology 19.4 (2013), pp. 364–375
  G. Bräunlich, C. Hainzl, R. Seiringer
  
• 2014 Mathematical Aspects of the BCS Theory of Superconductivity and Related Theories
  Bräunlich, Gerhard
  
• Corrections to Wigner type phase space methods, Nonlinearity 27 (2014), no. 12, 2951–2974
  W. Gaim, C. Lasser
  
• On the BCS gap equation for superfluid fermionic gases, Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference, World Scientific, Singapore, 2014, pp. 127–137
  G. Bräunlich, C. Hainzl, R. Seiringer
  
• Translation invariant quasi-free states for fermionic systems and the BCS approximation, Reviews in Mathematical Physics 26.7 (2014), p. 1450012
  G. Bräunlich, C. Hainzl, R. Seiringer
  
• 2015 Adiabatic theorems for general linear operators and wellposedness of linear evolution equations
  Schmid, Jochen
  
• 2015 Mean Field Limits in Strongly Confined Systems
  von Keler, Johannes
  
• Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values, Proceedings of the QMath 12 Conference, World Scientific, pp. 355–362 (2015)
  J. Schmid
  
• Bose-Einstein condensation on quantum graphs Mathematical results in quantum mechanics, 221–226, World Sci. Publ., Hackensack, NJ, 2015
  J. Bolte, J. Kerner
  
• Diffusion phenomena for partially dissipative hyperbolic systems, in Nonlinear Dynamics in Partial Differential Equations (Shin-Ichiro Ei, ed.), Advanced Studies in Pure Mathematics, 64 (2015) 303–310
  J. Wirth
  
• Ergodicity of the spin-boson model for arbitrary coupling strength. Comm. Math. Phys., 336(1):261–285, 2015
  M. Könenberg, M. Merkli, H. Song
  
• Note on a family of monotone quantum relative entropies, Lett. Math. Phys. 105 (2015), no. 10, 1449–1466
  A. Deuchert, C. Hainzl, R. Seiringer
  
• Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Acta App. Math. 137, 185–203 (2015)
  G. Panati and D. Monaco
  
• 2016 A Fast Matrix-Free Algorithm for Spectral Approximations to High-Dimensional Partial Differential Equations
  Brumm, Bernd
  
• 2016 Contributions to the mathematical study of BCS theory
  Deuchert, Andreas
  
• 2016 Numerical Integrators for Physical Applications
  Seyrich, Jonathan
  
• 2016 The Adiabatic Limit of the Connection Laplacian with Applications to Quantum Waveguides
  Haag, Stefan
  
• Bogolubov-Hartree-Fock Theory for Strongly Interacting Fermions in the Low Density Limit, Mathematical Physics, Analysis and Geometry 19.2 (2016), pp. 1–27
  G. Bräunlich, G. Hainzl, R. Seiringer
  
• Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17, 63–97 (2016)
  D. Fiorenza, D. Monaco, and G. Panati
  
• Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal. 54 (2016), 1020–1038
  E. Kieri, C. Lubich, H. Walach
  
• Dynamics of a chlorophyll dimer in collective and local thermal environments. J. Math. Chem., 54(4):866–917, 2016
  M. Merkli, G. P. Berman, R. T. Sayre, S. Gnanakaran, M. Könenberg, A. I. Nesterov, H. Song
  
• Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction. J. Math. Pures Appl. (9), 105(1):1–30, 2016
  V. Bach, S. Breteaux, S. Petrat, P. Pickl, T. Tzaneteas
  
• Mixed parity pairing in a dipolar gas. J. Modern Optics 63, 1777-1782 (2016)
  G. Bruun, C. Hainzl, M. Laux
  
• On the irreversible dynamics emerging from quantum resonances, J. Math. Phys., 57(3):033302, 26, 2016
  M. Könenberg, M. Merkli
  
• Self-adjointness and domain of the Fröhlich Hamiltonian, J. Math. Phys. 57 (2016), no. 2, 021902, 15 pp. 81Q12 (81V19)
  M.Griesemer, A. Wünsch
  
• Some new results on the well-posedness of hyperbolic evolution equations, Proc. Appl. Math. Mech. 16, 879–880 (2016)
  J. Schmid
  
• Spectral Deformation for Two-Body Dispersive Systems with e.g. the Yukawa Potential, Math. Phys. Anal. Geom. 19 (2016), no. 4, 19:24
  M. Engelmann, M.G. Rasmussen
  
• Stability of closed gaps for the alternating Kronig–Penney Hamiltonian. Anal. Math. Phys. 6, 67–83 (2016)
  A. Michelangeli and D. Monaco
  
• The NLS Limit for Bosons in a Quantum Waveguide, Ann. Henri Poincaré 17 (2016), no. 12, 3321–3360
  S. Teufel, J. von Keler
  
• Validity of the NLS approximation for periodic quantum graphs. NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 6, Art. 63, 30 pp
  S. Gilg, D. Pelinovsky, G. Schneider
  
• Z2 invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)
  D. Fiorenza, D. Monaco, and G. Panati
  
• 2017 Energy estimates for the two-dimensional Fermi polaron
  Linden, Ulrich
  
• 2017 Effective equations in mathematical quantum mechanics
  Gilg, Steffen
  
• 2017 Self-adjointness and domain of a class of generalized Nelson models
  Wünsch, Andreas
  
• 2017 Spectral and Hardy inequalities for the Heisenberg Laplacian
  Ruszkowski, Bartosch
  
• 2018 Effective Models for Many Particle Systems: BCS Theory and the Kac Model
  Geisinger, Alissa
  
• A lower bound for the BCS functional with boundary conditions at infinity. J. Math. Phys. 58 (2017), no. 8, 081901, 21 pp
  A. Deuchert
  
• Chern and Fu-Kane-Mele invariants as topological obstructions, Chapter 12 in: G. Dell’Antonio and A. Michelangeli (eds.), Advances in Quantum Mechanics: Contemporary Trends and Open Problems, vol. 18 in Springer INdAM Series (Spinger, Cham, 2017), pages 201–222. Proceedings volume for the INdAM meeting "Contemporary Trends in the Mathematics of Quantum Mechanics", July 4-8, 2016, Rome (Italy)
  D. Monaco
  
• Free time evolution of a tracer particle coupled to a Fermi gas in the high-density limit. Comm. Math. Phys. 356 (2017), no. 1, 143–187
  M. Jeblick, D. Mitrouskas, S. Petrat, P. Pickl
  
• Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys. 107 (2017), no. 7, 1315–1343
  D. Monaco, C. Tauber
  
• KK-theory, gauge theory and topological phases, Nieuw Archief voor Wiskunde 5/18 (2017), no. 4, 257–262
  F. Arici, D. Monaco
  
• On the construction of Wannier functions in topological insulators: the 3D case, Ann. Henri Poincaré 18 (2017), no. 12, 3863–3902
  H. Cornean, D. Monaco
  
• Regular singular problems for hyperbolic systems and their asymptotic integration, Proceedings of the 10th International ISAAC Congress. Pages 539–547 in P. Dang et al. (eds.), New Trends in Analysis and Interdisciplinary Applications, Trends in Mathematics, Springer 2017
  J. Wirth
  
• Semiclassical Analysis in Infinite Dimensions: Wigner Measures. Bruno Pini Mathematical Analysis Seminar, 7(1), 18–35
  M. Falconi
  
• Wannier functions and Z2invariants in time-reversal symmetric topological insulators, Reviews in Mathematical Physics, Vol. 29, No. 2 (2017) 1730001 (66 pages)
  H.D. Cornean, D. Monaco, S. Teufel
  
• Entropy decay for the Kac evolution. Comm. Math. Phys. 363 (2018), no. 3, 847–875
  F. Bonetto, A. Geisinger, M. Loss, T. Ried
  
• Local spectral deformation. Ann. Inst. Fourier (Grenoble) 68 (2018), no. 2, 767–804
  M. Engelmann, J. S. Møller, M. G. Rasmussen
  
• On the dipole approximation with error estimates. Lett. Math. Phys. 108 (2018), no. 1, 185–193
  L. Bossmann, R. Grummt, M. Kolb
  
• On the domain of the Nelson Hamiltonian, J. Math. Phys., 59(4):042111, 21, 2018
  M. Griesemer and A. Wünsch
  
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators, Commun. Math. Phys. 359 (2018), no. 1, 61–100
  D. Monaco, G. Panati, A. Pisante, S. Teufel
  
• Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21 (2018), no. 2, Art. 12, 37 pp
  J. Lampart, J. Schmidt, S. Teufel, R. Tumulka
  
• Persistence of translational symmetry in the BCS model with radial pair interaction. Ann. Henri Poincare 19 (2018), no. 5, 1507–1527
  A. Deuchert, A. Geisinger, C. Hainzl, M. Loss
  
• Stability of the two-dimensional Fermi polaron, Lett. Math. Phys., 108(8):1837–1849, 2018
  M. Griesemer, U. Linden
  
• Time integration of rank-constrained Tucker tensors, SIAM J. Numer. Anal. 56 (2018), 1273–1290
  C. Lubich, B. Vandereycken, H. Walach
  
• 2019 Effective dynamics of interacting bosons: Quasi-lowdimensional gases and higher order corrections to the meanfield description
  Boßmann, Lea
  
• 2019 Interior-Boundary Conditions as a Direct Description of QFT Hamiltonians
  Schmidt, Julian
  
• 2019 Nonlinear phenomena on metric and discrete necklace graphs
  Maier, Daniela
  
• 2019 Spectral asymptotics for Dirichlet Laplacians on random Cantor-like sets and on their complement
  Minorics, Lenon
  
• 2019 Time integration for the dynamical low-rank approximation of matrices and tensors
  Walach, Hannah
  
• Adiabatic currents for interacting electrons on a lattice, Rev. Math. Phys. 31 (2019), no. 3, 1950009
  D. Monaco, S. Teufel
  
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics. Rev. Math. Phys. 31 (2019), no. 8, 1950024, 36 pp
  D. Mitrouskas, S. Petrat, P. Pickl
  
• Breather Solutions on Discrete Necklace Graphs, Special issue Differential Operators on Graphs and Waveguides, Journal Operators and Matrices, 2019
  D. Maier
  
• Complex charges, time reversal asymmetry, and interior-boundary conditions in quantum field theory, J. Physics A: Mathematical and Theoretical 52 (2019), 115301
  J. Schmidt, R. Tumulka
  
• Convergence of a low-rank Lie-Trotter splitting for stiff matrix differential equations. SIAM J. Numer. Anal. 57 (2019), no. 4, 1947–1966
  A. Ostermann, C. Piazzola, H. Walach
  
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons. Ann. Henri Poincaré 20 (2019), no. 3, 1003–1049
  L. Bossmann, S. Teufel
  
• Derivation of the 1d nonlinear Schrödinger equation from the 3d quantum manybody dynamics of strongly confined bosons. J. Math. Phys. 60 (2019), no. 3, 031902, 30 pp
  L. Bossmann
  
• Diffusive stability for periodic metric graphs. Mathematische Nachrichten 292 (2019), no. 6, 1246-1259
  M. Chirilus-Bruckner, D. Maier, G. Schneider
  
• High density limit of the Fermi polaron with infinite mass. Lett. Math. Phys. 109 (2019), no. 8, 1805–1825
  U. Linden, D. Mitrouskas
  
• Interior-boundary conditions for many-body Dirac operators and codimension-1 boundaries. J. Phys. A 52 (2019), no. 29, 295202, 27 pp
  J. Schmidt, S. Teufel, R. Tumulka
  
• On a direct description of pseudorelativistic Nelson Hamiltonians. J. Math. Phys. 60 (2019), no. 10, 102303, 21 pp
  J. Schmidt
  
• On Nelson-type Hamiltonians and abstract boundary conditions. Comm. Math. Phys. 367 (2019), no. 2, 629–663
  J. Lampart, J. Schmidt
  
• Spectral theory of the Fermi polaron. Ann. Henri Poincaré, 20(6):1931– 1967, 2019
  M. Griesemer, U. Linden
  
• Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs. J. Differ. Equations 268 (2020), no. 6, 2491–2509
  D. Maier
  
• Eigenvalue Approximation for Krein-Feller-Operators, Analysis, Probability and Mathematical Physics on Fractals, World Scientific, Singapore, pp. 363-384 (2020)
  U.Freiberg, L. Minorics
  
• Fractal Transformed Doubly Reflected Brownian Motions, Analysis, Probability and Mathematical Physics on Fractals, World Scientific, Singapore, pp. 131-161 (2020)
  T. Ehnes, U. Freiberg
  
• Higher order corrections to the mean-field description of the dynamics of interacting bosons, J. Stat. Phys. (2020)
  L. Boßmann, N. Pavlović, P. Pickl, A. Soffer
  
• Spectral asymptotics for Krein-Feller operators with respect to V-variable Cantor measures. Forum Math. 32 (2020), no. 1, 121–138
  L.A. Minorics
  
• Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors, BIT Numer. Math. (2020)
  G. Ceruti, C. Lubich