Quantum Mathematics I

This book is the first volume that provides an unique overview of the most recent and relevant contributions in the field of mathematical physics with a focus on the mathematical features of quantum mechanics. It is a collection of review papers together with brand new works related to the activities of the INdAM Intensive Period "INdAM Quantum Meetings (IQM22)", which took place at the Politecnico di Milano in Spring 2022 at Politecnico di Milano. The range of topics covered by the book is wide, going ranging from many-body quantum mechanics to semiclassical analysis, quantum field theory, Schrödinger and Dirac operators and open quantum systems

Review of recent breakthroughs on the mathematics of quantum mechanics Overview of the most recent contributions in several areas of mathematical physics Updated presentation of open problems in mathematical physics

Auteur
M. Correggi: Organizer of the INdAM Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March-May 2022;

Full Professor in Mathematical Physics since 2020 at Politecnico di Milano;

Coordinator of the Ph.D. program on Mathematical Models and Methods in Engineering at Politecnico di Milano (2022 - 2024);

Ph.D. in Mathematical Physics (SISSA, Trieste, 2004);

18 years of research experience after Ph.D.;

Principal Investigator of a MIUR FIR grant of 1 million EUR;

46 publications in international peer-reviewed journals;

Participant to more than 50 international events;

More than 60 invited talks;

Invited speaker at the XVIII International Congress on Mathematical Physics, Santiago de Chile, Chile;

Invited speaker at the XXI Congresso dell'Unione Matematica Italiana, Pavia, Italia;

Organizer of 14 international events (1 Intensive Period INdAM, 1 Nordita, 1 CMO/BIRS, 3 Schools MCQM & 1 EMS-IAMP School with more than 100 participants).

M. Falconi: Organizer of the INdAM Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March-May 2022;

Associate Professor in Mathematical Physics since 2023 at Politecnico di Milano;

Ph.D. in Analysis (Alma Mater Studiorum - Università di Bologna, 2012);

11 years of research experience after Ph.D.;

21 publications in international peer-reviewed journals; Participant to more than 30 international events;

More than 20 invited talks;

Organizer of 2 international events (1 Intensive Period INdAM, 1 Hausdorff Summer School).

Contenu
Chapter 1. Introduction.- Chapter 2. Many-Body Quantum Mechanics.- Chapter 3. Two Comments on the Derivation of the Time-Dependent Hartree-Fock Equation.- Chapter 4. Bogoliubov Theory for Ultra Dilute Bose Gases.- Chapter 5. Derivation of the Ginzburg-Landau Theory for Interacting Fermions in a Trap.- Chapter 6. Energy expansions for dilute Bose gases from local condensation results: a review of known results.- Chapter 7. Bogoliubov theory for the dilute Fermi gas in three dimensions.- Chapter 8. Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with External Potentials.- Chapter 9. Trial states for Bose gases: singular scalings and non-integrable potentials.- Chapter 10. Bogoliubov Transformations Beyond Shale-Stinespring: Generic v* v for bosons.- Chapter 11. Thermodynamic Game and The Kac Limit in Quantum Lattices.- Chapter 12. Topological Polarization in disordered systems.- Chapter 13. Open Quantum Systems.- Chapter 14. On the AsymptoticsDynamics of Open Quantum Systems.- Chapter 15. Boson quadratic GKLS generators.- Chapter 16. Semiclassical Analysis.- Chapter 17. Some Remarks on Semiclassical Analysis on Two-Steps Nilmanifolds.- Chapter 18. Waves in a Random Medium: Endpoint Strichartz Estimates and Number Estimates.- Chapter 19. Quasi-Classical Spin Boson Models.- Chapter 20. On the Semiclassical Regularity of Thermal Equilibria.- Chapter 21. Invariant measures as probabilistic tools in the analysis of nonlinear ODEs & PDEs.- Chapter 22. Quantum Field Theory.- Chapter 23. An Evolution Equation Approach to Linear Quantum Field Theory.- Chapter 24. Renormalization of spinboson interactions mediated by singular form factors:The Casimir-Polder effect for an approximate Pauli-Fierz model: the atom plus wall case.- Chapter 25. Dynamical Systems Involving Pseudo-Fermionic Operators and Generalized Quaternion Groups.- Chapter 26. Schroedinger and Dirac Operators.- Chapter 27. Spectral Asymptotics for Two-dimensional Dirac Operators in Thin Waveguides.- Chapter 28. Quadratic forms for Aharonov-Bohm Hamiltonians.- Chapter 29. On the magnetic Laplacian with a piecewise constant magnetic field in R3+.- Chapter 30. Quantum Systems at The Brink. - Chapter 31. Lowest Eigenvalue Asymptotics in Strong Magnetic Fields with Interior Singularities.- Chapter 32. Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions.