On the theory of complex rays

The article surveys the application of complex-ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Sp{} in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterizations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex Wentzel--Kramers--Brilbuin expansion of these wavefields

The variational theory of complex rays for the calculation of medium-frequency vibrations

A new approach called the ``variational theory of complex rays’’ (VTCR) is developed for calculating the vibrations of weakly damped elastic structures in the medium-frequency range. Here, the emphasis is put on the most fundamental aspects. The effective quantities (elastic energy, vibration intensity, etc.) are evaluated after solving a small system of equations which does not derive from a finite element discretization of the structure. Numerical examples related to plates show the appeal and the possibilities of the VTCR

Semiclassical Description of Tunneling in Mixed Systems: The Case of the Annular Billiard

We study quantum-mechanical tunneling between symmetry-related pairs of regular phase space regions that are separated by a chaotic layer. We consider the annular billiard, and use scattering theory to relate the splitting of quasi-degenerate states quantized on the two regular regions to specific paths connecting them. The tunneling amplitudes involved are given a semiclassical interpretation by extending the billiard boundaries to complex space and generalizing specular reflection to complex rays. We give analytical expressions for the splittings, and show that the dominant contributions come from {\em chaos-assisted}\/ paths that tunnel into and out of the chaotic layer.Comment: 4 pages, uuencoded postscript file, replaces a corrupted versio

GPU accelerated cone based shooting bouncing ray tracing

2019 Summer.Includes bibliographical references.Ray tracing can be used as an alternative method to solve complex Computational Electromagnetics (CEM) problems that would require significant time using traditional full-wave CEM solvers. Ray tracing is considered a high-frequency asymptotic solver, sacrificing accuracy for speed via approximation. Two prominent categories for ray tracing exist today: image theory techniques and ray launching techniques. Image theory involves the calculation of image points for each continuous plane within a structure. Ray launching ray tracing is comprised of spawning rays in numerous directions and tracking the intersections these rays have with the environment. While image theory ray tracing typically provides more accurate solutions compared to ray launching techniques, due to more exact computations, image theory is much slower than ray launching techniques due to exponential time complexity of the algorithm. This paper discusses a ray launching technique called shooting bouncing rays (SBR) ray tracing that applies NVIDIA graphics processing units (GPU) to achieve significant performance benefits for solving CEM problems. The GPUs are used as a tool to parallelize the core ray tracing algorithm and also to provide access to the NVIDIA OptiX ray tracing application programming interface (API) that efficiently traces rays within complex structures. The algorithm presented enables quick and efficient simulations to optimize the placement of communication nodes within complex structures. The processes and techniques used in the development of the solver and demonstrations of the validation and the application of the solver on various structures and its comparison to commercially available ray tracing software are presented

Theory of x-ray absorption by laser-dressed atoms

An ab initio theory is devised for the x-ray photoabsorption cross section of atoms in the field of a moderately intense optical laser (800nm, 10^13 W/cm^2). The laser dresses the core-excited atomic states, which introduces a dependence of the cross section on the angle between the polarization vectors of the two linearly polarized radiation sources. We use the Hartree-Fock-Slater approximation to describe the atomic many-particle problem in conjunction with a nonrelativistic quantum-electrodynamic approach to treat the photon-electron interaction. The continuum wave functions of ejected electrons are treated with a complex absorbing potential that is derived from smooth exterior complex scaling. The solution to the two-color (x-ray plus laser) problem is discussed in terms of a direct diagonalization of the complex symmetric matrix representation of the Hamiltonian. Alternative treatments with time-independent and time-dependent non-Hermitian perturbation theories are presented that exploit the weak interaction strength between x rays and atoms. We apply the theory to study the photoabsorption cross section of krypton atoms near the K edge. A pronounced modification of the cross section is found in the presence of the optical laser.Comment: 13 pages, 3 figures, 1 table, RevTeX4, corrected typoe

The asymptotics of the generalised Bessel function

We demonstrate how the asymptotics for large |z| of the generalised Bessel function 0Ψ1(z) = X∞ n=0 z n Γ(an + b)n! , where a > −1 and b is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function pΨq(z). A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case a = − 1 2 , where the expansion for z → ±∞ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a function of arg z when −1 < a < 0, taking into account the Stokes phenomenon that occurs on the rays arg z = 0 and arg z = ±π(1 + a) for the associated function 1Ψ0(z). These regions are more precise than those given by Wright in his 1940 paper. Numerical computations are carried out to verify several of the expansions developed in the paper

Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict attention to primal & dual algorithms only, larger and more detailed experiments, more new crosscap number