Advances in Quantum Mechanics : Contemporary Trends and Open Problems 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 Fu Kane Mele 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 with magnetic Laplacian 16 On the inverse spectral problems for quantum graphs 17 Double barrier resonances and time decay of the survival probability: a toy model 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