GRK 1838:  Spektraltheorie und Dynamik von Quantensystemen

• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2013 bis 2018
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 206649329

• Erstellungsjahr: 2020

Zusammenfassung der Projektergebnisse

Wissenschaft und Technik werden immer stärker durch Anwendungen quantenmechanischer Gesetze durchdrungen. Computerunterstützte Entwicklung neuer Medikamente, Kernspintomografie in der Medizin und Strahlentherapie, Mikroelektronik und Photovoltaik würden nicht existieren ohne unsere Kenntnisse der Quantenphysik. Obschon die grundlegenden quantenmechanischen Gesetze hinter diesen Anwendungen wohlbekannt sind als mathematische Gleichungen, ist es doch oft, auf Grund der Komplexität wechselwirkender Vielteilchenquantensysteme, unmöglich, wichtige Quantenphänomene mit den erwähnten grundlegenden Gesetzen in Verbindung zu bringen. Dieser Zustand ist nicht nur akademisch unbefriedigend, sondern auf lange Sicht auch hinderlich für den Fortschritt in Wissenschaft und Technik. Es erscheint daher unverzichtbar, dass der Fortschritt auf einem fundamentalen mathematischen Niveau versucht mit Entwicklungen in den angewandten Wissenschaften Schritt zu halten. Dieser mathematische Fortschritt und die Ausbildung der nächsten Generation von Experten in der Mathematik komplexer Quantensysteme sind die Kernanliegen dieses Graduiertenkollegs. Das Forschungsprogramm besteht aus einer Kollektion mathematischer Projekte betreffend wichtiger Modellsysteme aus der Festkörperphysik und der Quantenfeldtheorie. Die spezifischen Projekte betreffen mathematische Problemstellungen an der Grenze der gegenwärtig vorhandenen Möglichkeiten, und deren Lösung erfordert die Zusammenarbeit von Experten verschiedener Spezialgebiete der Mathematik. Die beteiligten Wissenschaftler aus Stuttgart und Tübingen sind führende Experten in mathematischer Quantenmechanik und in der Mathematik der Schrödingergleichung. Die Nachbarschaft von Stuttgart und Tübingen gestattet es ihnen, ihre Kräfte zu einem führenden Zentrum mathematischer Quantenmechanik in Deutschand zu bündeln. Auf Grund der Konzeption des Graduiertenkollegs werden Zusammenarbeiten iniziiert und intensiviert, die Einladung auswärtiger Spezialistinnen und Spezialisten wird vereinfacht und eine gemeinsame Plattform wird geschaffen für eine breite und gründliche Ausbildung junger Nachwuchswissenschaftlerinnen und Nachwuchswissenschaftler in der Mathematik der Vielteilchenquantensysteme. Der nachhaltige Einfluss dieses Graduiertenkollegs wird demonstriert durch die Schaffung eines neuen internationalen Masterstudiengangs “Mathematical Physics” in Tübingen. Dieser Studiengang wird unterstützt durch das Land Baden-Württemberg und die Universität Tübingen mit jeweils einer Dauerstelle der Klasse W3 bzw. E13.

Projektbezogene Publikationen (Auswahl)

• Dynamics and symmetries of a repulsively bound atom pair in an optical lattice, Phys. Rev. A 86, 013618 (2012)
  Deuchert, Andreas; Sakmann, Kaspar; Streltsov, Alexej I.; Alon, Ofir E. & Cederbaum, Lorenz S.
  
• Minimization of the energy of the nonrelativistic one-electron Pauli-Fierz model over quasifree states. Doc. Math., 18:1481–1519, 2013
  Bach, Volker & Breteaux, Sébastien
  
• 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
  
• 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)
  Schmid, J.
  
• Bose-Einstein condensation on quantum graphs Mathematical results in quantum mechanics, 221–226, World Sci. Publ., Hackensack, NJ, 2015
  Bolte, J. & Kerner, J.
  
• Corrections to Wigner type phase space methods, Nonlinearity 27 (2014), no. 12, 2951–2974
  Gaim, Wolfgang & Lasser, Caroline
  
• Ergodicity of the spin-boson model for arbitrary coupling strength. Comm. Math. Phys., 336(1):261–285, 2015
  Könenberg, M.; Merkli, M. & Song, H.
  
• Mathematical Aspects of the BCS Theory of Superconductivity and Related Theories
  Bräunlich, Gerhard
  
• 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
  Bräunlich, G.; Hainzl, C. & Seiringer, R.
  
• Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Acta App. Math. 137, 185–203 (2015)
  Monaco, Domenico & Panati, Gianluca
  
• Translation invariant quasi-free states for fermionic systems and the BCS approximation, Reviews in Mathematical Physics 26.7 (2014), p. 1450012
  Bräunlich, Gerhard; Hainzl, Christian & Seiringer, Robert
  
• Adiabatic theorems for general linear operators and wellposedness of linear evolution equations
  Schmid, Jochen
  
• Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17, 63–97 (2016)
  Fiorenza, Domenico; Monaco, Domenico & Panati, Gianluca
  
• 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
  
• Mean Field Limits in Strongly Confined Systems
  von Keler, Johannes
  
• Note on a family of monotone quantum relative entropies, Lett. Math. Phys. 105 (2015), no. 10, 1449–1466
  Deuchert, Andreas; Hainzl, Christian & Seiringer, Robert
  
• Stability of closed gaps for the alternating Kronig–Penney Hamiltonian. Anal. Math. Phys. 6, 67–83 (2016)
  Michelangeli, Alessandro & Monaco, Domenico
  
• A Fast Matrix-Free Algorithm for Spectral Approximations to High-Dimensional Partial Differential Equations
  Brumm, Bernd
  
• Bogolubov-Hartree-Fock Theory for Strongly Interacting Fermions in the Low Density Limit, Mathematical Physics, Analysis and Geometry 19.2 (2016), pp. 1–27
  Bräunlich, Gerhard; Hainzl, Christian & Seiringer, Robert
  
• Contributions to the mathematical study of BCS theory
  Deuchert, Andreas
  
• Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal. 54 (2016), 1020–1038
  Kieri, Emil; Lubich, Christian & Walach, Hanna
  
• Dynamics of a chlorophyll dimer in collective and local thermal environments. J. Math. Chem., 54(4):866–917, 2016
  Merkli, M.; Berman, G. P.; Sayre, R. T.; Gnanakaran, S.; Könenberg, M.; Nesterov, A. I. & Song, H.
  
• 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
  Bach, Volker; Breteaux, Sébastien; Petrat, Sören; Pickl, Peter & Tzaneteas, Tim
  
• Mixed parity pairing in a dipolar gas. J. Modern Optics 63, 1777-1782 (2016)
  Bruun, G. M.; Hainzl, C. & Laux, M.
  
• Numerical Integrators for Physical Applications
  Seyrich, Jonathan
  
• On the irreversible dynamics emerging from quantum resonances, J. Math. Phys., 57(3):033302, 26, 2016
  Könenberg, M. & Merkli, M.
  
• Self-adjointness and domain of the Fröhlich Hamiltonian, J. Math. Phys. 57 (2016), no. 2, 021902, 15 pp. 81Q12 (81V19)
  Griesemer, M. & Wünsch, A.
  
• Some new results on the well-posedness of hyperbolic evolution equations, Proc. Appl. Math. Mech. 16, 879–880 (2016)
  Schmid, Jochen
  
• Spectral Deformation for Two-Body Dispersive Systems with e.g. the Yukawa Potential, Math. Phys. Anal. Geom. 19 (2016), no. 4, 19:24
  Engelmann, Matthias & Rasmussen, Morten Grud
  
• The Adiabatic Limit of the Connection Laplacian with Applications to Quantum Waveguides
  Haag, Stefan
  
• The NLS Limit for Bosons in a Quantum Waveguide, Ann. Henri Poincaré 17 (2016), no. 12, 3321–3360
  von Keler, Johannes & Teufel, Stefan
  
• Validity of the NLS approximation for periodic quantum graphs. NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 6, Art. 63, 30 pp
  Gilg, Steffen; Pelinovsky, Dmitry & Schneider, Guido
  
• Z2 invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)
  Fiorenza, Domenico; Monaco, Domenico & Panati, Gianluca
  
• A lower bound for the BCS functional with boundary conditions at infinity. J. Math. Phys. 58 (2017), no. 8, 081901, 21 pp
  Deuchert, Andreas
  
• 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)
  Monaco, Domenico
  
• Energy estimates for the two-dimensional Fermi polaron
  Linden, Ulrich
  
• Effective equations in mathematical quantum mechanics
  Gilg, Steffen
  
• Effective Models for Many Particle Systems: BCS Theory and the Kac Model
  Geisinger, Alissa
  
• 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
  Jeblick, Maximilian; Mitrouskas, David; Petrat, Sören & Pickl, Peter
  
• 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
  Monaco, Domenico & Tauber, Clément
  
• 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
  Cornean, Horia D. & Monaco, Domenico
  
• On the dipole approximation with error estimates. Lett. Math. Phys. 108 (2018), no. 1, 185–193
  Boßmann, Lea; Grummt, Robert & Kolb, Martin
  
• 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
  Wirth, Jens
  
• Self-adjointness and domain of a class of generalized Nelson models
  Wünsch, Andreas
  
• Semiclassical Analysis in Infinite Dimensions: Wigner Measures. Bruno Pini Mathematical Analysis Seminar, 7(1), 18–35
  M. Falconi
  
• Spectral and Hardy inequalities for the Heisenberg Laplacian
  Ruszkowski, Bartosch
  
• Wannier functions and Z2invariants in time-reversal symmetric topological insulators, Reviews in Mathematical Physics, Vol. 29, No. 2 (2017) 1730001 (66 pages)
  Cornean, Horia D.; Monaco, Domenico & Teufel, Stefan
  
• 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
  Boßmann, Lea & Teufel, Stefan
  
• Diffusive stability for periodic metric graphs. Mathematische Nachrichten 292 (2019), no. 6, 1246-1259
  Chirilus‐Bruckner, Martina; Maier, Daniela & Schneider, Guido
  
• Entropy decay for the Kac evolution. Comm. Math. Phys. 363 (2018), no. 3, 847–875
  Bonetto, Federico; Geisinger, Alissa; Loss, Michael & Ried, Tobias
  
• Local spectral deformation. Ann. Inst. Fourier (Grenoble) 68 (2018), no. 2, 767–804
  Engelmann, Matthias; Møller, Jacob Schach & Rasmussen, Morten Grud
  
• On the domain of the Nelson Hamiltonian, J. Math. Phys., 59(4):042111, 21, 2018
  Griesemer, M. & Wünsch, A.
  
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators, Commun. Math. Phys. 359 (2018), no. 1, 61–100
  Monaco, Domenico; Panati, Gianluca; Pisante, Adriano & Teufel, Stefan
  
• Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21 (2018), no. 2, Art. 12, 37 pp
  Lampart, Jonas; Schmidt, Julian; Teufel, Stefan & Tumulka, Roderich
  
• Persistence of translational symmetry in the BCS model with radial pair interaction. Ann. Henri Poincare 19 (2018), no. 5, 1507–1527
  Deuchert, Andreas; Geisinger, Alissa; Hainzl, Christian & Loss, Michael
  
• Stability of the two-dimensional Fermi polaron, Lett. Math. Phys., 108(8):1837–1849, 2018
  Griesemer, Marcel & Linden, Ulrich
  
• Time integration of rank-constrained Tucker tensors, SIAM J. Numer. Anal. 56 (2018), 1273–1290
  Lubich, Christian; Vandereycken, Bart & Walach, Hanna
  
• Adiabatic currents for interacting electrons on a lattice, Rev. Math. Phys. 31 (2019), no. 3, 1950009
  Monaco, Domenico & Teufel, Stefan
  
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics. Rev. Math. Phys. 31 (2019), no. 8, 1950024, 36 pp
  Mitrouskas, David; Petrat, Sören & Pickl, Peter
  
• Complex charges, time reversal asymmetry, and interior-boundary conditions in quantum field theory, J. Physics A: Mathematical and Theoretical 52 (2019), 115301
  Schmidt, Julian & Tumulka, Roderich
  
• Convergence of a low-rank Lie-Trotter splitting for stiff matrix differential equations. SIAM J. Numer. Anal. 57 (2019), no. 4, 1947–1966
  Ostermann, Alexander; Piazzola, Chiara & Walach, Hanna
  
• 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
  Boßmann, Lea
  
• Effective dynamics of interacting bosons: Quasi-lowdimensional gases and higher order corrections to the meanfield description
  Boßmann, Lea
  
• High density limit of the Fermi polaron with infinite mass. Lett. Math. Phys. 109 (2019), no. 8, 1805–1825
  Linden, Ulrich & Mitrouskas, David
  
• Interior-Boundary Conditions as a Direct Description of QFT Hamiltonians
  Schmidt, Julian
  
• Interior-boundary conditions for many-body Dirac operators and codimension-1 boundaries. J. Phys. A 52 (2019), no. 29, 295202, 27 pp
  Schmidt, Julian; Teufel, Stefan & Tumulka, Roderich
  
• Nonlinear phenomena on metric and discrete necklace graphs
  Maier, Daniela
  
• On a direct description of pseudorelativistic Nelson Hamiltonians. J. Math. Phys. 60 (2019), no. 10, 102303, 21 pp
  Schmidt, Julian
  
• On Nelson-type Hamiltonians and abstract boundary conditions. Comm. Math. Phys. 367 (2019), no. 2, 629–663
  Lampart, Jonas & Schmidt, Julian
  
• Spectral asymptotics for Dirichlet Laplacians on random Cantor-like sets and on their complement
  Minorics, Lenon
  
• Spectral asymptotics for Krein-Feller operators with respect to V-variable Cantor measures. Forum Math. 32 (2020), no. 1, 121–138
  Minorics, Lenon Alexander
  
• Spectral theory of the Fermi polaron. Ann. Henri Poincaré, 20(6):1931– 1967, 2019
  Griesemer, M. & Linden, U.
  
• Time integration for the dynamical low-rank approximation of matrices and tensors
  Walach, Hannah
  
• Breather Solutions on Discrete Necklace Graphs, Special issue Differential Operators on Graphs and Waveguides, Journal Operators and Matrices, 2019
  Maier, Daniela
  
• Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs. J. Differ. Equations 268 (2020), no. 6, 2491–2509
  Maier, Daniela
  
• Eigenvalue Approximation for Krein-Feller-Operators, Analysis, Probability and Mathematical Physics on Fractals, World Scientific, Singapore, pp. 363-384 (2020)
  Freiberg, Uta & Minorics, Lenon
  
• Fractal Transformed Doubly Reflected Brownian Motions, Analysis, Probability and Mathematical Physics on Fractals, World Scientific, Singapore, pp. 131-161 (2020)
  Ehnes, Tim & Freiberg, Uta
  
• Higher order corrections to the mean-field description of the dynamics of interacting bosons, J. Stat. Phys. (2020)
  Boßmann, Lea; Pavlović, Nataša; Pickl, Peter & Soffer, Avy
  
• Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors, BIT Numer. Math. (2020)
  Ceruti, Gianluca & Lubich, Christian