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Electron–positron pair cascades developed in the extreme electromagnetic fields of neutron star polar caps are considered a key source of magnetospheric plasma in these objects. We use a simplified model that maps the quantum electrodynamics processes governing pair cascades to analytically and numerically model these cascades, and show that large-amplitude oscillations of the electric field are inductively driven by the resulting plasma. A plasma instability arises in these oscillations, and particles accelerated in growing electric field perturbations can drive secondary pair bursts that damp the large-amplitude oscillations. An analytical model is proposed to describe this interplay between the pair production and kinetic collective plasma processes. All analytical results are shown to be in excellent agreement with particle-in-cell simulations. [ABSTRACT FROM AUTHOR]

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1, 1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M G Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical selfadjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical self-adjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces. [ABSTRACT FROM AUTHOR]