Method Of Comparison Equations Research Articles

Semiclassical theory of potential scattering for massless Dirac fermions

In this paper we study scattering of two-dimensional massless Dirac fermions by a potential that depends on a single Cartesian variable. Depending on the energy of the incoming particle and its angle of incidence, there are three different regimes of scattering. To find the reflection and transmission coefficients in these regimes, we apply the Wentzel–Kramers–Brillouin (WKB), also called semiclassical, approximation. We use the method of comparison equations to extend our prediction to nearly normal incidence, where the conventional WKB method should be modified due to the degeneracy of turning points. We compare our results to numerical calculations and find good agreement.

Method of comparison equations and generalized Ermakov’s equation

The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to the construction of perturbative series for second-order differential equations with and without turning points.

Method of comparison equations for Schwarzschild black holes

We employ the method of comparison equations to study the propagation of a massless minimally coupled scalar field on the Schwarzschild background. In particular, we show that this method allows us to obtain explicit approximate expressions for the radial modes with energy below the peak of the effective potential which are fairly accurate over the whole region outside the horizon. This case can be of particular interest, for example, for the problem of black hole evaporation.

The method of comparison equations for cosmological perturbations

We apply the method of comparison equations to study cosmological perturbations duringinflation, obtaining the full power spectra of scalar and tensor perturbations to first and tosecond order in the slow-roll parameters. We compare our results with those derived bymeans of other methods, in particular the Green’s function method and the improvedWentzel–Kramers–Brillouin (WKB) approximation, and find agreement for the slow-rollstructure. The method of comparison equations, just like the improved WKBapproximation, can however be applied to more general situations where the slow-rollapproximation fails.

Analysis of structures in the cross sections for elastic scattering and spin exchange in low-energyH++Hcollisions

Elastic scattering and spin exchange cross sections in ${\mathrm{H}}^{+}+\mathrm{H}$ collisions are computed using accurate adiabatic potential curves for the center-of-mass energy range ${10}^{\ensuremath{-}4}lEl100\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. Both cross sections show considerable structure which necessitates computation on a fine energy grid to resolve them. We analyze the structures using a comparison equation method to find the poles of the scattering matrix in the complex energy plane. We show that many features of the cross sections can be correlated with these poles and tabulate the positions of the most important poles. Finding stationary phases in the partial wave cross sections, we also explain in all details the glory oscillations of the elastic cross section that extends below $100\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ down to the lowest energies.

Quantum transitions at a level crossing of decaying states

Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.

Semiclassical approximation for bound states of the Schrödinger equation with a Coulomb-like potential

The eigenvalue problem for the radial Schrödinger equation with an ‘‘almost’’ Coulomb potential is considered. This problem provides the simplest example of a system whose classical trajectories have singularities. Thus the standard semiclassical quantization procedure (in the sense of Einstein–Brillouin–Keller–Maslov) cannot be applied straightforwardly to this situation. The equation under consideration has two transition points on the half axis 0<x<∞: a regular singular point at the origin and a turning point for some x0≳0; these points coalesce at low energy levels. The well-known comparison equation method provides uniform asymptotics for equations of this kind, but has the following shortcomings: it does not appeal to the corresponding classical mechanical problem and the formulas for the eigenvalues and eigenfunctions contain phase integrals which are not analytic with respect to the parameter characterizing the closeness of the transition points. In the present article we derive new formulas for the eigenfunctions in the form of inverse Fourier transforms of rapidly oscillating exponentials multiplied by some powers of x, together with simple formulas for the eigenvalues. These formulas are directly connected with the classical trajectories and do not have the shortcomings of the comparison equation method, i.e., all the functions entering the asymptotic expansions are analytic in x and energy. As a by-product, an asymptotic expansion of integrals having two stationary phase points which coalesce for some value of the parameter and tend to infinity when the parameter tends to zero is obtained.

Asymptotic solutions of a Schrödinger equation with a second order pole

Asymptotic solutions of a Schrödinger equation in the vicinity of a second order pole singularity are constructed by means of the so-called comparison equation method. As an example the problem of the rotating oscillator with large length of the rotation axis is studied. An expansion for eigenenergies of this quantum system is deduced. Besides conventional series in ascending powers of the large parameter it includes the exponentially small term arising from tunneling phenomena.

Periodic orbit theory of broad resonances in two-dimensional hydrogenic Stark effect

The modified version of periodic orbit theory, developed by Robbins (1989) for systems which possess discrete symmetries is used to calculate parameters of broad resonances in the case of the two-dimensional hydrogenic Stark effect at energies above the zero field ionization threshold. It is shown that the contribution of single periodic orbits of the system give results that coincide with those of the comparison equation method also presented.

Periodic orbit theory of broad resonances in the hydrogenic Stark effect and the two-Coulomb-center problem

The periodic orbit theory is used to calculate the poles of the S -matrix which are related to broad resonances in two important realistic systems: the hydrogen atom in a uniform electric field and the two-Coulomb-center problem. It is shown that the contributions of certain complex periodic orbits in the periodic orbit sum give results that coincide with the comparison equation method calculations.

Local instabilities of weakly non-parallel large-scale flows: Uniform asymptotic expansion

Abstract A uniform asymptotic representation of local instabilities developing in weakly nonparallel large-scale flows that possess vertical and horizontal shear is presented. The analysis extends the comparison equation method of Langer to determine the local eigenfunctions and eigenvalues of the local instabilities. The advantages of the present method over the corresponding WKB analysis (Bar-Sever and Merkine, 1988) are discussed.

An improvement to the comparison equation method for solving the Schrödinger equation

A systematic improved comparison equation method to solve the Schrödinger equation is described. The method is useful in quantum mechanical calculations involving two or more transition or turning points and is applicable to real potentials with continuous derivatives. As a computational example of the method, a study of the bound-state problem using the Morse potential is given.

The quadratic Zeeman effect for highly excited hydrogen atoms in weak magnetic fields

The hydrogen Rydberg states in the presence of weak magnetic fields are analytically investigated by using quantum mechanical first-order perturbation theory. The unperturbed hydrogenic wavefunctions, which diagonalise the quadratic Zeeman interaction within the subspace of states with fixed principal quantum number n, are obtained by separation of variables on the Fock hypersphere in momentum space. By considering n as a large parameter, the comparison equation method is employed to find the uniform asymptotics of eigenfunctions and asymptotic expansions of quadratic Zeeman energies corresponding to the outermost components of the Zeeman n manifold. The results obtained are compared with other theoretical predictions.

An investigation of the exponential atomic collision model: analytical results

A simple, but versatile, atomic collision model for treating non-adiabatic phenomena associated with curve-crossings and non-crossings is the exponential model of Nikitin. This two-state model leads, within a semiclassical framework, to a collisionS-matrix made up of classical trajectory transition probabilities and semiclassical phases. It can be used to discuss physical processes such as excitation and charge transfer. In applying the model expressions to real situations, phase-integral expressions for theS-matrix elements are needed. Such expressions have been obtained earlier by Crothers through a process of interpretation and abstraction. To shed new light on this interpretation we give a phase-integral derivation of the two-state semiclassical collisionS-matrix within a general exponential model. Only analytical results are presented here; numerical ones will be reported in future publications. Representing theS-matrix as a product of two ‘half-way house’ matrices and assuming only one principal transition zone in each half we may parametrize the elements ofSin a semiclassically consistent way. The parameters are computed in terms of certain Stokes constants and complex Coulomb phases arising from the classical trajectory equations describing the evolution of the diabatic (l. c. a. o.) electronic states. The Stokes constants and complex Coulomb phases associated with the canonical (pure exponential model) one-pole, two-transition point problem are parametrized via the comparison equation method, supplemented by strong-coupling asymptotics. The corresponding quantities for a perturbed canonical form (general exponential model) are suitably abstracted in terms of simple known functions and physically significant phase integrals. In this way the complete semiclassical collisionS-matrix is derived. A short discussion of its applicability is given.

Reduction of quasipotential equations to Sturm-Liouville problems and the comparison equation method

Reduction of quasipotential equations to Sturm-Liouville problems and the comparison equation method

Asymptotic expressions for the widths of low-lying energy bands in one-dimensional periodical potentials

The motion of a quantum particle in a one-dimensional, slowly varying periodical potential is considered. By using the comparison equation method, the asymptotic expressions for exponentially small bandwidths of the low-lying energy bands is obtained. In the special case of a sinusoidal potential (corresponding to the Mathieu differential equation), the general formula reduces to an already existing result. Potentials, symmetrical and non-symmetrical, within the elementary period are discussed.

Asymptotic theory of discrete normal modes in inhomogeneous planar waveguides

Arnold [J. Acoust. Soc. Am. 69, 17 (1981)] has applied the method of comparison equations to obtain complete asymptotic solutions to the two turning point eigenvalue problem for the Schroedinger equation with boundary conditions imposed at finite boundaries. We present a different analysis of this problem employing the method of matched asymptotic expansions to connect higher order asymptotic solutions at each simple turning point obtained by means of Cherry's technique. The advantage of this approach is that it extends to the treatment of multiple turning point problems for which the canonical global comparison equations have no well‐studied solutions. As a further application of these ideas, we present a higher order asymptotic analysis of a three turning point eigenvalue problem for the Schroedinger equation with boundary conditions imposed at finite boundaries. Such a problem arises from a normal mode analysis of underwater acoustic propagation when a surface duct lies over a deep sound channel. This analysis includes an extension of Cherry's technique to the transition problem associated with the barrier between the surface duct and the deep sound channel.

Electromagnetic waves in two-dimensionally inhomogeneous media*

The investigation of electromagnetic waves in two-dimensionally inhomogeneous media leads to an approximation that decouples Maxwell’s equations. The two equations obtained for the longitudinal field components are free from the inconsistencies of the scalar wave approximation. For cylindrically symmetric media the resulting radial equations are basically equivalent to those already derived by Kurtz and Streifer, but a more refined WKB-type approximation of the solutions, using the “method of comparison equations,” leads to the result that all guided modes are split with respect to scalar propagation constants. Calculations carried out for quadratic media indicate that this splitting does not decrease with frequency in the way predicted by previous treatments, using first-order perturbation. The present theory is equally valid for scattered modes, for which an approximate phase shift formula is provided.

Perturbed symmetric resonance: the parabolic model

An exact parabolic model for perturbed symmetric resonance is developed and solved. To supplement the Zwaan-Stueckelberg theory (1932) a comparison-equation method is adopted. It reduces the problem to parabolic cylinder function factors for which the corres ponding diabatic crossing is classically inaccessible. Appeal to the author's previous strong-coupling expansions for parabolic cylinder functions of large order (1972) and argument is made to reinforce the view that the adiabatic treatment gives the more precise picture. Both semiclassical and impact-parameter derivations are given. Previous formulae as reported by the author are proven (1971).

Semiclassical approximation of the radial equation with two-dimensional potentials

The radial equation for scattering from a cylindrically symmetrical potential is examined, because two-dimensional scattering arises in high-energy electron diffraction from crystals. Particular attention is paid to the case of s waves, where there is a centripetal attractive potential for free particles. After showing that the Langer transformation, which leads to correct semiclassical wavefunctions for all other cases in two and three dimensions, fails for s waves, the method of comparison equations is applied, which enables the phase shifts and bound state conditions to be expressed in a simple form valid for all angular momenta. The theory is tested for s waves by comparison with exactly-calculated energy levels.