Journal of Mathematical Physics, since 1960, publishes some of the best papers from outstanding mathematicians and physicists. It was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods suitable for such applications and for the formulation of physical theories.
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With this paper, we provide a mathematical review on the initialvalue problem of the oneparticle Dirac equation on spacelike Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincaré group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient PaleyWiener and Sobolevtype estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of squareintegrable wave functions of two respective Cauchy surfaces. With a PicardLindelöf argument, this evolution map is generalized to the Dirac evolution including the external potential. For the latter, we introduce a convenient interaction picture on Cauchy surfaces. These tools immediately provide another proof of the wellknown existence and uniqueness of classical solutions and their causal structure.

We show that in graphene, modelled by the twodimensional Dirac operator, charge distributions with nonvanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power.

At the interface of two twodimensional quantum systems, there may exist interface currents similar to edge currents in quantum Hall systems. It is proved that these interface currents are macroscopically quantized by an integer that is given by the difference of the Chern numbers of the two systems. It is also argued that at the interface between two timereversal invariant systems with halfinteger spin, one of which is trivial and the other nontrivial, there are dissipationless spinpolarized interface currents.

In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.

The Hamiltonian of an atom with N electrons and a fixed nucleus of infinite mass between two parallel planes is considered in the limit when the distance a between the planes tends to zero. We show that this Hamiltonian converges in the norm resolvent sense to a Schrödinger operator acting effectively in whose potential part depends on a. Moreover, we prove that after an appropriate regularization this Schrödinger operator tends, again in the norm resolvent sense, to the Hamiltonian of a twodimensional atom (with the threedimensional Coulomb potentialone over distance) as a → 0. This makes possible to locate the discrete spectrum of the full Hamiltonian once we know the spectrum of the latter one. Our results also provide a mathematical justification for the interest in the twodimensional atoms with the threedimensional Coulomb potential.