Comparison theorems for the position-dependent mass Schroedinger equation

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schroedinger equation are established. (i) If a constant mass $m_0$ and a PDM $m(x)$ are ordered everywhere, that is either $m_0\leq m(x)$ or $m_0\geq m(x)$, then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if $\nabla^2 (1/m(x))$ has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.Comment: 11 pages, 2 figure

Cosmic Rays from Gamma Ray Bursts in the Galaxy

The rate of terrestrial irradiation events by galactic gamma-ray bursts (GRBs) is estimated using recent standard-energy results. We assume that GRBs accelerate high-energy cosmic rays, and present results of three-dimensional simulations of cosmic rays moving in the Galactic magnetic field and diffusing through pitch-angle scattering. An on-axis GRB extinction event begins with a powerful prompt gamma-ray and neutron pulse, followed by a longer-lived phase from cosmic-ray protons and neutron-decay protons that diffuse towards Earth. Our results force a reinterpretation of reported ~ 10^{18} eV cosmic-ray anisotropies and offer a rigorous test of the model where high-energy cosmic rays originate from GRBs, which will soon be tested with the Auger Observatory.Comment: 9 pages, 4 figures, ApJ Letters, in press. Clarified limit of test-particle approximation, prediction that Auger will not confirm SUGAR source. (Data may not appear onscreen at low magnification.) Simulations at http://heseweb.nrl.navy.mil/gamma/~dermer/invest/sim/index.ht

Models and phenomenology

It is evident that models of the knee should match the observational phenomenology. In this talk I discuss a few aspects of phenomenology, which are important not only for the understanding of the knee origin, but also for the general problem of the origin of cosmic rays. Among them are the shape of the energy spectrum, its irregularity, the sharpness of the knee and its fine structure. The classification of models is given and some examples of the most recent models are discussed. The most probable conclusion deduced from this examination is that the knee has an astrophysical origin and the so called 'source' models of the knee are most likely among them.Comment: 13 pages, 4 figures, overview talk at 19th European Cosmic Ray Symposium, Firenze, Italy, 30.08-4.09.2004 (to be published in the Int.J.Mod.Phys.A