Quantum field theory, statistical mechanics: Perspectives in statistical mechanics by M. Aizenman Coherent infrared representations in non-relativistic QED by T. Chen and J. Frohlich Quantum spin systems after DLS 1978 by B. Nachtergaele Barry Simon's contributions to quantum field theory by L. Rosen Nonrelativistic two-body and
N-body quantum systems, resonances: Isoperimetric inequalities for eigenvalues of the Laplacian by M. S. Ashbaugh and R. D. Benguria Non-self-adjoint operators and pseudospectra by E. B. Davies Barry Simon's contribution to non-relativistic quantum mechanics: Two-body and
N-body Schrodinger operators and resonances by R. Froese
N-body quantum scattering and quantum resonances: An overview by C. Gerard Mathematical analysis of Born-Oppenheimer approximations by G. A. Hagedorn and A. Joye Perturbation theory and atomic resonances since Schrodinger's time by E. M. Harrell II On a theorem for quantum mechanical scattering theory by A. Lenard Analytic criteria in the qualitative spectral analysis of the Schrodinger operator by V. Maz'ya The spectral geometry of geometrically finite hyperbolic manifolds by P. Perry Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations by Y. Pinchover Complex scaling in atomic physics: A staging ground for experimental mathematics and for extracting physics from otherwise impossible computations by W. P. Reinhardt Recent results on the Bethe-Sommerfeld conjecture by A. V. Sobolev Electric and magnetic fields, semiclassical limit: Recent developments in quantum mechanics with magnetic fields by L. Erdos Aspects of the integer quantum Hall effect by G. M. Graf Barry Simon's work on electric and magnetic fields and the semi-classical limit by I. W. Herbst Some bound state problems in quantum mechanics by D. Hundertmark.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.
Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density-functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric-space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density-functional theory, which relates the many-body wave function, particle density, and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers
m. Additionally, varying the magnetic field uncovers a richer complexity for the ``band structure'' present in ground-state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric-space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground-state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.