C.J. Howls

The asymptotic quasinormal mode spectrum of non-rotating black holes

A conjectured connection to quantum gravity has led to a renewed interest in highly damped black-hole quasinormal modes (QNMs). In this paper we present simple derivations (based on the WKB approximation) of conditions that determine the asymptotic QNMs for both Schwarzschild and Reissner–Nordstrom black holes. This confirms recent results obtained by Motl and Neitzke, but our analysis fills several gaps left by their discussion. We study the Reissner–Nordstrom results in some detail, and show that, in contrast to the asymptotic QNMs of a Schwarzschild black hole, the Reissner–Nordstrom QNMs are typically not periodic in the imaginary part of the frequency. This leads to the charged black hole having peculiar properties which complicate an interpretation of the results in the context of quantum gravity.

High Orders of the Weyl Expansion for Quantum Billiards: Resurgence of Periodic Orbits, and the Stokes Phenomenon

A formalism is developed for calculating high coefficients cr of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂BThe cr are used to test the following conjectures. (a) The sequence of cr diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. (b) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B. (c) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series.

GLOBAL ASYMPTOTICS FOR MULTIPLE INTEGRALS WITH BOUNDARIES

Under convenient geometric assumptions, the saddle-point method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized Picard-Lefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided.

On the higher–order Stokes phenomenon

During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a‘higher–order Stokes phenomeno’, at which a Stokes multiplier itself can change value. We show that the higher–order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher–order Stokes phenomenon can have important effects on the large–time behaviour of partial differential equations.

Dispersive hyperasymptotics and the anharmonic oscillator

Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh–Schrodinger perturbation series.

Anharmonic oscillator discontinuity formulae up to second-exponentially-small order

The eigenvalues of the quartic anharmonic oscillator as functions of the anharmonicity constant satisfy a once-subtracted dispersion relation. In turn, this dispersion relation is driven by the purely imaginary discontinuity of the eigenvalues across the negative real axis. In this paper we calculate explicitly the asymptotic expansion of this discontinuity up to second-exponentially-small order.

Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

We introduce templates for exponential-asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential-asymptotic approach also reveals how boundary-value problems force the surprising presence of transseries in the linear case and negative powers of ε terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiple-scales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally, we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion.

Hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one

In this paper we discuss the special properties of hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one. We show that the re–expansions are independent of the inhomogeneity. We illustrate how this leads to a symmetry breaking in the Stokes constants within a pair of formal solutions of a differential equation. A consequence is that Stokes constants may exactly vanish in higher–order equations, leading to dramatic simplifications in the hyperasymptotic structures. Two examples are included.

Overlapping Stokes Smoothings: Survival of the Error Function and Canonical Catastrophe Integrals

We derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp ( –36).

Valuation of soccer spread bets

Simple statistical and probabilistic arguments are used to value the most commonly traded online soccer spread bets. Such markets typically operate dynamically during the course of a match and accurate valuations must, therefore, reflect the changing state of the match. Both goals and corners are assumed to evolve as Poisson processes with constant means. Although many of the bets that are typically traded are relatively easy to value, some (including the ‘four flags’ market) require more detailed analysis. Examples are given of the evolution of the spread during typical matches and theoretical predictions are shown to compare closely to spreads quoted by online bookmakers during some of the important matches of the EURO2004 tournament.