arXiv subjects
Publications and source records attributed to Richard Chapling.
We consider the Schr\"odinger--Poisson system on the complete, simply-connected Riemannian manifolds of constant sectional curvature. We obtain closed-form stationary spherically-symmetric solutions for the homogeneous equations for certain dimensions, and give some basic examples of solutions with a nonzero background.
Using an unusual type of symmetric average, we show that for several common equations involving quite general potentials possessing symmetry, the ground state, if it exists, must also be symmetric.
We construct the meromorphic functions invariant under the action of the sense-preserving wallpaper groups on the complex plane. We discuss possible generalisa-tions of this to the general wallpaper groups. This provides the answer to a question posed on StackExchange regarding the possible symmetries of elliptic functions.
We provide a new analysis of the irreducible loop integrals first considered in a 2003 paper of Wu. Using convergence ideas from probability, we produce conditions on the regulator masses so that the integrals have well-defined limits in the limit required by the regularisation technique; we then derive closed-form expressions for the regularised forms of these integrals in terms of incomplete Gamma-functions.
We prove the existence of a ground state of the Maxwell--Schr\"odinger equations in one spatial dimension, describing a specified amount of free charge under the influence of a fixed charge. For one case (equal free and fixed charge, i.e., a neutral atom), we introduce a new type of quartic Banach space, in which the Hamiltonian is naturally coercive. We also show that for a point charge, for any ratio of charge such that the ground state exists, it is symmetric and decreasing.
We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop regularisation. We show all three are true, extend them to more general powers and produce their full asymptotic series. We also extend a classical result to produce an exact formula for the sum in the last.
We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form of the fundamental solutions for any number of dimensions in terms of generalised hypergeometric functions, with different closed forms for even and odd-dimensional cases.