This volume collects recent contributions on the contemporary trends in the mathematics of quantum mechanics, and more specifically in mathematical problems arising in quantum many-body dynamics, quantum graph theory, cold atoms, unitary gases, with particular emphasis on the developments of the specific mathematical tools needed, including: linear and non-linear Schrödinger equations, topological invariants, non-commutative geometry, resonances and operator extension theory, among others. Most of contributors are international leading experts or respected young researchers in mathematical physics, PDE, and operator theory. All their material is the fruit of recent studies that have already become a reference in the community. Offering a unified perspective of the mathematics of quantum mechanics, it is a valuable resource for researchers in the field.

Prof Gianfausto Dell'Antonios research focuses on axiomatic quantum field theory, local field theory, mathematics of quantum mechanics, critical point theory, stochastic processes, singular interactions, and many-body problems. He graduated in theoretical physics in Milan, was research associate in Copenhagen (Niels Bohr institute), Zurich (ETH), and Evanson (Northwestern), then professor of theoretical physics in Naples and professor of rational mechanics and mathematical physics at La Sapienza Rome. He held visiting professorships at the IHES Paris, Courant Insitute NY, The University of Marseille Luminy, Bielefeld University (as a recipient of a von Humboldt prize), CERN, SISSA Trieste, and the Interdisciplinary Laboratory of the Accademia dei Lincei. He held also visiting positions at the IAS Princeton, Ecole Politechnique Paris, Paris Dauphine, Harvard University, and the Max Planck Institute in Munich.

Dr Alessandro Michelangelis research is in the fiel

d at the interface between mathematical physics, functional analysis and non-linear dispersive PDE, and operator theory, with a special focus on the mathematical methods for quantum mechanical and condensed matter systems. He graduated in theoretical physics in Pisa and then mathematical physics at SISSA Trieste, held faculty positions at the LMU Munich and SISSA Trieste, and visiting positions at the University of Cambridge, SISSA Trieste, and Bilkent.

1 Shell interactions for Dirac operators.- 2 Correlation inequalities for classical and quantum XY models.- 3 Dissipatively generated entanglement.- 4 Abelian gauge potentials on cubic lattices.- 5 Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes.- 6 Analysis of fluctuations around non-linear effective dynamics.- 7 Logarithmic Sobolev inequalities for an ideal Bose gas.- 8 Spherical Schrödinger Hamiltonians: spectral analysis and time decay.- 9 On the Ground state for the NLS equation on a general graph.- 10 Self-adjoint extensions of the Dirac operator with Coulomb potential.- 11 Dispersive estimates for Schrödinger operators with point interactions in R3.- 12 Chern and FuKaneMele invariants as topological obstructions.- 13 Norm approximation for many-body quantum dynamics and Bogoliubov theory.- 14 Effective non-linear dynamics of binary condensates and open problems.- 15 Remarks on the derivation of the Gross-Pitaevskii equation withmagnetic Laplacian.- 16 On the inverse spectral problems for quantum graphs.- 17 Double-barrier resonances and time decay of the survival probability: a toy model