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 initial-value problem of the one-particle Dirac equation on space-like 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 Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of square-integrable wave functions of two respective Cauchy surfaces. With a Picard-Lindelöf 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 well-known existence and uniqueness of classical solutions and their causal structure.
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We show that in graphene, modelled by the two-dimensional Dirac operator, charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power.
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At the interface of two two-dimensional 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 time-reversal invariant systems with half-integer spin, one of which is trivial and the other non-trivial, there are dissipationless spin-polarized interface currents.
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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.
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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 two-dimensional atom (with the three-dimensional Coulomb potential-one 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 two-dimensional atoms with the three-dimensional Coulomb potential.
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Weyl-von Neumann-Berg theorem for quaternionic operators
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We prove the Weyl-von Neumann-Berg theorem for right linear operators (not necessarily bounded) in a quaternionic Hilbert space: Let N be a right linear normal (need not be bounded) operator in a quaternionic separable infinite dimensional Hilbert spaceH. Then for a given ϵ > 0, there exists a compact operator K with and a diagonal operator D on H such that N = D + K.
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On the polar decomposition of right linear operators in quaternionic Hilbert spaces
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In this article, we prove the existence of the polar decomposition of densely defined closed right linear operators in quaternionic Hilbert spaces: If T is a densely defined closed right linear operator in a quaternionic Hilbert space H, then there exists a partial isometry U0 such that . In fact U0 is unique if N(U0) = N(T). In particular, if H is separable and U is a partial isometry with , then we prove that U = U0 if and only if either N(T) = {0} or R(T)⊥ = {0}.
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Continuity of the sequential product of sequential quantum effect algebras
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In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effectsA, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.
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On solutions of Maxwell’s equations with dipole sources over a thin conducting film
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We derive and interpret solutions of time-harmonic Maxwell’sequations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g., graphene sheet, lying between two unbounded isotropic and non-magnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmon-polariton of transverse-magnetic polarization of plane waves.
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On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components
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Using a Lagrangian framework, we study overdamping phenomena in gyroscopic systems composed of two components, one of which is highly lossy and the other is lossless. The losses are accounted by a Rayleigh dissipation function. As we have shown previously, for such a composite system, the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such system, a rather universal phenomenon of selective overdamping occurs. Namely, first of all, the high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes. Second of all, the rest of the low-loss modes remain oscillatory (i.e., the underdamped modes), each with an extremely high quality factor (Q-factor) that actually increases as the loss of the lossy component increases. We prove that selective overdamping is a generic phenomenon in Lagrangian systems with gyroscopic forces and gives an analysis of the overdamping phenomena in such systems. Moreover, using perturbation theory, we derive explicit formulas for upper bound estimates on the amount of loss required in the lossy component of the composite system for the selective overdamping to occur in the generic case and give Q-factor estimates for the underdamped modes. Central to the analysis is the introduction of the notion of a “dual” Lagrangian system and this yields significant improvements on some results on modal dichotomy and overdamping. The effectiveness of the theory developed here is demonstrated by applying it to an electric circuit with a gyrator element and a high-loss resistor.
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Dynamics of restricted three and four vortices problem on the plane
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The dynamics of a test particle (a particle with zero vorticity) advected by the velocity field of N point-vortices with vorticities Γj, j = 1, …N, is considered. Making an analogy with similar studies in celestial mechanics, we call such a study a “restricted N-vortex problem” or (N + 1)-vortex problem. In particular, we study and characterize the global planar dynamics of some restricted 3 and 4-vortex problems, as a function of the vorticities Γj of the vortices.
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