Spectral Theory and Mathematical Physics : STMP 2018 Santiago Chile Commutator methods for N body Schrödinger operators Resolvent estimates and resonance free regions for Schrödinger operators with matrix valued potentials One dimensional discrete Anderson model in a decaying random potential: From a.c. spectrum to dynamical localization On non selfadjoint operators with finite discrete spectrum Pseudo differential perturbations of the Landau Hamiltonian Semiclassical surface wave tomography of isotropic media Persistence of point spectrum for perturbations of one dimensional operators with discrete spectra Resonances for a system of Schrödinger operators above an energy level crossing Nonexistence result for wave operators in massive relativistic system Quantized calculus for perturbed massive Dirac operator on noncommutative Euclidian space On the explicit semiclassical limiting eigenvalue (resonance) distribution for the Zeeman (Stark) hydrogen atom Hamiltonian Negative spectrum of the Robin Laplacian On some integral operators appearing in scattering theory and their resolutions The strong Scott conjecture: The density of heavy atoms close to the nucleous. This proceedings volume contains peer reviewed selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago Chile at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflect the intense cooperation between young researchers in spectral theory and mathematical physics and established specialists in this field. The list of topics covered includes: eigenvalues and resonances for quantum Hamiltonians; spectral shift function and quantum scattering; spectral properties of random operators; magnetic quantum Hamiltonians; microlocal analysis and its applications in mathematical physics. This volume can be of interest both to senior researchers and graduate students pursuing new research topics in Mathematical Physics.