Coalescing Saddle Points Research Articles

A Numerical Method for Oscillatory Integrals with Coalescing Saddle Points

The value of a highly oscillatory integral is typically determined asymptotically by the behavior of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points---roots of the derivative of the phase of the integrand---where the integrand is locally nonoscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points.

Semiclassical treatment of quantum propagation with nonlinear classical dynamics: A third-order thawed Gaussian approximation.

The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.

Uniform asymptotics of area-weighted Dyck paths

Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.

High-order harmonic generation in the presence of a static electric field

We consider high-order harmonic generation by a linearly polarized laser field and a parallel static electric field. We first develop a modified saddle-point method which enables a quantitative analysis of the harmonic spectra even in the presence of Coulomb singularities. We introduce a classification of the saddle-point solutions and show that, in the presence of a static electric field which breaks the inversion symmetry, an additional classification number has to be introduced and that the usual saddle-point approximation and the uniform approximation in the case of the coalescing saddle points have to be modified. The theory developed offers a simple and accurate explanation of the static-field-induced multiplateau structure of the harmonic spectra. The longer quantum orbits are responsible for a long extension of the harmonic plateau, while the larger initial electron velocities are the reason of lower harmonic emission rates.

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

There are two ways of deriving the asymptotic expansion of $J_\nu(\nu a)$ , as $\nu \to \infty$ , which holds uniformly for $a\geq 0$ . One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation $J_{\nu+1}(x)+J_{\nu-1}(x)=(2\nu /x)J_\nu(x)$ . Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations.

Uniform Asymptotic Expansions of Laguerre Polynomials

Two asymptotic expansions are obtained for the Laguerre polynomial $L_n^{(\alpha )} (x)$ for large n and fixed $\alpha > - 1$. These expansions are uniformly valid in two overlapping intervals covering the entire x-axis. The leading terms of both agree with the two asymptotic formulas given by Erdelyi who used the theory of differential equations. Our approach is based on two integral representations for the Laguerre polynomials. The phase function of one of these integrals has two coalescing saddle points, and to this one the cubic transformation introduced by Chester, Friedman, and Ursell is applied. The phase function of the other integral also has two coalescing saddle points, but in addition it has a simple pole. Moreover, the saddle points coalesce onto this pole. In this case a rational transformation is used, which mimics the singular behavior of the phase function. In both cases explicit expressions are given for the remainders associated with the asymptotic expansions.

Differential equations for the cuspoid canonical integrals

Differential equations satisfied by the cuspoid canonical integrals In(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(a1,a2,...,an−1). A set of linear coupled ordinary differential equations is derived for each step in the sequence In(0,0,...,0,0) →In(0,0,...,0,an−1) →In(0,0,...,an−2,an−1) →...→In(0,a2,...,an−2,an−1) →In(a1,a2,...,an−2,an−1). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives.

The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points

Methods that can be used in the numerical implementation of the uniform swallowtail approximation are described. An explicit expression for that approximation is presented to the lowest order, showing that there are three problems which must be overcome in practice before the approximation can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.

Uniform semiclassical evaluation of Franck–Condon factors and inelastic atom–atom scattering amplitudes

The problem of obtaining uniform semiclassical approximations for Franck–Condon factors and inelastic atom–atom scattering amplitudes is considered. Starting from Langer’s uniform Airy approximation for the wave function and an Airy function representation of the inelastic S matrix element, it is shown how both these problems can be expressed in terms of a multidimensional oscillatory integral, the exponent of which possesses two coalescing saddle points. Using a general formula for the uniform asymptotic evaluation of this multidimensional integral, simple derivations are presented for the uniform semiclassical approximation to the Franck–Condon factor and inelastic scattering amplitude. Previous derivations are shown to be special cases of the more general derivations given here. The use of catastrophe theory for choosing the simplest and most convenient canonical form for the transformation of the exponent in the uniform asymptotic evaluation of multidimensional integrals is discussed.

Uniformly Valid Expansions for Laplace Integrals

Uniformly valid asymptotic expansions for a class of parameter-dependent Laplace integrals involving coalescing saddle points are obtained by a direct method based on matched asymptotic expansion theory. Bessel functions of large order are treated as an example.

Multidimensional canonical integrals for the asymptotic evaluation of the S-matrix in semiclassical collision theory

The uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered. A concrete example of a non-separable two-dimensional integral with four coalescing saddle points is chosen since it exhibits many of the features of more general cases. It is shown how a uniform asymptotic approximation can be obtained in terms of a non-separable two-dimensional canonical integral and its derivatives. This non-separable two-dimensional canonical integral plays a similar role to the Airy integral in one-dimensional integrals with two coalescing saddle points. The uniform approximation is obtained by applying to the two-dimensional case the asymptotic techniques introduced by Chester et al. for one-dimensional integrals. An exact series representation is obtained for the canonical integral by means of complex variable techniques. The series representation can be used to evaluate the canonical integral for small to moderate values of its arguments, whilst for large values of its arguments existing asymptotic techniques may be used.