Methods for Cancellation of Apparent Cerenkov Radiation Arising From SME Models and Separability of Schrödinger’s Equation Using Exotic Potentials in Parabolic Coordinates

Abstract

In an attempt to merge the two prominent areas of physics: The Standard Model and General Relativity, there have been many theories for the underlying physics that may lead to Lorentz- and CPT-symmetry violations. At the present moment, technology allows numerous types of Planck-sensitive tests of these symmetries in a range of physical systems. We address a curiosity in isotropic CPT- and Lorentz-violating electrodynamics where there is a kinematic allowance for Cerenkov radiation of a charged particle in a vacuum moving with uniform motion. This however, should not be the case as it is known that constant motion in a vacuum should not cause the particle to lose any energy. Taking Fourier transforms of the modified magnetic field confirms the cancellation of the apparent radiation. The Fourier transform can be used to show that modes for short and long wavelengths cancel. In the second area of research we focus on solutions of the Schrödinger equation which may be found by separation of variables in more than one coordinate system. This class of potentials includes a number of important examples, including the isotropic harmonic oscillator and the Coulomb potential. There are multiple separable Hamiltonians that exhibit a number of interesting features, including “accidental” degeneracies in their bound state spectra and often classical bound state orbits that always close. We examine another potential, for which the Schrödinger equation is separable in both cylindrical and parabolic coordinates: a z-independent V ∝ 1/p2 = 1/ (x2 + y2) in three dimensions. All the persistent, bound classical orbits in this potential close, because all other orbits with negative energies fall to the center at ρ = 0. When separated in parabolic coordinates, the Schrödinger equation splits into three individual equations, two of which are equivalent to the radial equation in a Coulomb potential—one equation with an attractive potential, the other with an equally strong repulsive potential