‘Killing Mie Softly’: Analytic Integrals for Complex Resonant States

Summary We consider integrals of products of Bessel functions and of spherical Bessel functions, combined with a Gaussian factor guaranteeing convergence at infinity. These integrals arise in wave and quantum mechanical scattering problems of open systems containing cylindrical or spherical scatterers, particularly when those problems are considered in the framework of complex resonant modes. Explicit representations are obtained for the integrals, building on those in the 1992 paper by McPhedran, Dawes and Scott. Attention is paid to those sums with a distributive part arising as the Gaussian tends towards the unit function. In this limit, orthogonality and normalisability of complex modes are investigated.

Covariant quantum-mechanical scattering via Stueckelberg–Horwitz–Piron theory

Based on the Stueckelberg–Horwitz–Piron theory of covariant quantum mechanics on curved spacetime, we solved the wave equations for a charged covariant harmonic oscillator in the background of a charged static spherically symmetric black hole. Using Green’s functions, we found an asymptotic form for the wave function in the lowest mode ([Formula: see text]-mode) and in higher moments. It has been proven that for [Formula: see text]-wave, in a definite range of solid angles, the differential cross-section depends effectively to on the magnetic and electric charges of the black hole.

Some useful classes with applications

In this learn-by-doing chapter, some widely available classes are introduced and employed to solve problems in the physical sciences. By using linear algebra classes from the Eigen package, collection classes from the C++ standard library, and function classes from the authors’ own Qat libraries, readers become familiar some classes that can be effectively applied in physics computation and which are illustrative of good class design. Practice obtained in using these classes is intended to be helpful in understanding the more formal treatment of C++ concepts in subsequent chapters. Physics applications include familiar problems such as coupled oscillations and quantum mechanical scattering in one dimension from piecewise-constant potentials, for which computational methods are necessary in all but the simplest cases. Visualization of functions with the Qt-based Qat library is also introduced.