WKB Analysis Research Articles

Perturbative approach in the WKB analysis: Application to the Stark effect

We have considered a perturbative approach in the WKB framework. Based on an expression for $\ensuremath{\partial}E/\ensuremath{\partial}\ensuremath{\lambda},$ we have calculated perturbatively the Stark energy shifts for two important systems, a particle in a one-dimensional power-law potential and the hydrogen atom. The results provide important insight into the properties of these systems.

Singular-perturbative Reduction to BIrkhoff Normal Form and Instanton-Type Formal Solutions of Hamiltonian Systems

As is illustrated by the computation of monodromy groups of second-order Fuchsian equations (cf. [AKT1]), the exact WKB analysis provides us with a powerful tool for studying global behavior of solutions of linear ordinary differential equations. To generalize such an analysis to nonlinear equations, T. Kawai (RIMS, Kyoto Univ.), T. Aoki (Kinki Univ.) and the author have developed the WKB theory for Painleve equations with a large parameter in our series of articles ([KT1], [AKT2], [KT2]). (See [Tl], [T2] also.) In our treatment 2-parameter formal solutions called instanton-type solutions, which were constructed through the multiple-scale analysis in [AKT2], are playing a central role. Although we have succeeded in analyzing their local structure near simple turning points in [KT2], some of their important properties such as the behavior near fixed singular points have not been clarified yet. In this paper, to investigate their behavior near fixed regular-type singular points, we propose a new construction of 2-parameter formal solutions of Painleve equations with a large parameter. The new construction of formal solutions we propose here is based on Takano's work [Tkal] (see [Tka2] also), where he constructed a 2-parameter family of analytic solutions at each regular-type singular point of (ordinary) Painleve equations. He made use of the well-known fact that Painleve equations can be written in the form of Hamiltonian systems (which we call Painleve Hamiltonian systems here) and established some reduction theorem for

Short-wavelength instabilities of waves in rotating stratified fluids

Standing and propagating plane waves in a rigidly rotating stably stratified fluid are considered. Based on a WKB analysis, it is shown numerically that almost all of them are unstable with respect to three-dimensional short-wavelength perturbations. The waves whose wavenumber vector is parallel to the axis of rotation and stratification are stabilized as the Brunt-Väisälä frequency N (normalized by the angular velocity of rigid rotation) increases, whereas the inclined waves are unstable at any N. These instabilities are caused by parametric resonances of low order (first and second) and the instability growth rate is determined theoretically, when the wave amplitude is small.

Excess noise caused by transverse inhomogeneity of conductive channels

A type of excess noise which emerges in transversely inhomogeneous conductive channels is described. Its origin is traced back to the diffusion of the electric-field and carrier-density fluctuations across the channel. By making use of a WKB analysis, explicit analytical formulas for the current-noise spectrum are obtained. It is argued, that this type of noise should be manifested in a variety of currently used semiconductor devices.

WKB solution of the wave equation for a plane angular sector

The two angular Lamedifferential equations that satisfy boundary conditions on a plane angular sector ~PAS! are solved by the Wentzel-Kramers-Brillouin ~WKB! method. The WKB phase constants are derived by matching the WKB solution with the asymptotic solution of the Weber equation. The WKB eigenvalues and eigenfunctions show excellent agreement with the exact eigenvalues and eigenfunctions. It is shown that those WKB eigenvalues and eigenfunctions that contribute substantially to the scattering amplitude from a PAS can be computed in a rather simple way. An approximate formula for the WKB normalization constant, which is consistent with the WKB assumptions, is derived and compared with the exact normalization constant. @S1063-651X~98!00507-8# PACS number~s!: 42.25.2p I. INTRODUCTION In a previous paper @1#, formulas for the Wentzel- Kramers-Brillouin ~WKB! eigenvalues satisfying Dirichlet or Neumann boundary condition on a plane angular sector ~PAS! were reported and the WKB eigenvalues, $n,m%, were compared with the exact eigenvalues for PAS's of different corner angles ~60°, 90°, and 120° !. A historical review of the solution of the wave equation for a PAS was also given in the above paper. It suffices to say that, to our knowledge, no approximate solution of this problem has been reported in the literature. In this paper, the two coupled Lameequations are solved by the WKB method. The WKB analysis in this paper is valid for large values of n. Depending on the sign of m, one of the two Lameequations has turning points. When umu is small the turning points occur where the angles are small. In this region it proves more accurate to obtain the WKB phase constants by matching it with the asymptotic solution of the Weber equation. For large values of umu, it is shown that this phase constant reduces to the phase constant obtained when the solution is matched with the asymptotic solution of the Airy's equation as is commonly done in quantum mechanics. This paper is organized in the following way: In the sec- ond section the WKB solution is formulated and the WKB phase constants are derived. In the third section formulas for the WKB eigenvalues for Dirichlet and Neumann boundary conditions are derived and a comparison between the WKB and exact eigenfunctions is presented. The fourth section contains a derivation of approximate WKB solutions which are valid for small values of m/n. Finally, an approximate formula for the WKB normalization constant consistent with WKB assumptions is derived in section five.

Three-Dimensional Instabilities of Standing Waves in Rotating Fluids

Standing plane waves in a rigidly rotating ideal fluid are considered. It is shown that all of them are unstable with respect to three-dimensional perturbations. When the amplitude of standing waves is small, the growth rate of the instabilities increases as their wavenumber increases. It approaches the value predicted by a WKB analysis (Lifschitz & Fabijonas 1996) from below. The most dangerous modes are concentrated near the node-planes of the basic flow field. For long wavelength disturbances of small amplitude inertial waves resonant triad instabilities are observed.

Quantum-mechanical model of the Reissner-Nordström black hole

We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"{o}m black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"{o}m black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.

Simple spectral width estimation formula extracted from real energy shape resonance wavefunctions

A formula for estimating spectral widths of narrow or moderately broad isolated shape resonances is established, by simple arguments based on the three-turning point WKB theory. This formula enables one to handle all resonances of the potential at one go, by only using as input real-energy wavefunctions obtained from a standard Hamiltonian-matrix diagonalization. For its derivation, the ratio between the probability amplitude computed near the equilibrium position and that defined asymptotically, both referring to maxima of the (un-normalized) resonance wavefunction, is of major importance. The performance of the formula is demonstrated on the unimolecular dissociation of the ground electronic state of the exotic . Comparison is made with results from a complex-energy propagation and a previous WKB analysis.

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials { Q n } associated with the inner product 〈p,q〉= ∫ −1 1 p(x)q(x)p(x) dx + A 1p(1)q(1) + B 1p(−1)q(−1) + A 2p′(1)q′(1) + B 2p′(−1)q′(−1) , where p( x) = (1 − x) α (1 + x) β is the Jacobi weight function, α, β> − 1, A 1, B 1, A 2, B 2⩾0 and p, q ∈ P, the linear space of polynomials with real coefficients. The hypergeometric representation ( 6 F 5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q n ( x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.

Use of negative complex potential as absorbing potential

Exact numerical calculation shows that the use of negative complex potential (NCP) significantly improves the efficiency of wavefunction absorption over that of negative imaginary potential (NIP) in scattering applications. The improvement in absorption is especially significant in the case of low energy scattering with de Broglie wavelength larger than the length of absorbing potential. The addition of a negative real potential to the pure imaginary potential speeds up the absorption of wavefunction by effectively shortening its de Broglie wavelength. Explicit TD numerical calculation for a one-dimensional model demonstrates the effectiveness of the NCP and shows that the reflection from the absorbing potential can effectively be eliminated by using optimized absorbing parameters for the energy in question. In addition, comparison of the exact numerical calculation with semiclassical WKB analysis casts a serious doubt on the quantitative value of using WKB analysis at low energies with de Broglie wavelengths much larger than the absorbing length and/or for polynomial absorbing potentials higher than linear.

The polarizability of a particle in power-law potentials: a WKB analysis

We consider the problem of a particle confined to a one-dimensional power-law potential of the form , but also subject to a constant force F. This might represent, for example, the application of an external electric field, , via the perturbing potential . The electric polarizability of the system in a given state is defined via the energy shift due to the external field. Exact expressions for are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well . For the harmonic oscillator (infinite well) the polarizability scales as and is positive (negative) for large values of n indicating a qualitatively different response of the system to an external field for large quantum numbers. In order to examine this problem for a more general power-law potential, we apply WKB techniques to evaluate the energy shifts for large n and we find that (i) the n-dependence of the polarizability scales as as a function of k and (ii) the approximate value of k at which the crossover from to (for large n) occurs is . This study provides a useful example of numerical WKB techniques, the use of scaling ideas in simple one-dimensional quantum mechanical systems, and a better `feel' for the meaning of the polarizability which is an important physical quantity in more realistic atomic and molecular systems.

On the exact WKB analysis for the third order ordinary differential equations with a large parameter

n-th order with n 2 has not yet been well established; its local aspect is clari ed under the assumption that the discriminant of the characteristic equation is of simple zeros ([AKT, Theorem 1.4 and Theorem 1.6]), but its global aspect is far from complete understanding ([BNR], [AKT, Section 2]). The purpose of this paper is to examine the validity of our Ansatz concerning the Stokes geometry ([AKT, Section 2]) through the study of particular di erential equations whose solutions can be explicitly given in the form of integrals.

Effects of Low-Frequency Tropical Forcing on Intraseasonal Tropical–Extratropical Interactions

Abstract A spherical nondivergent barotropic model, linearized about a 300-mb climatological January flow, is used to examine the extratropical response to low-frequency tropical forcing. A two-dimensional WKB analysis shows that the energy propagation depends on the sum of three vectors: the basic state wind vector, a vector that is parallel to the absolute vorticity contours, and the local wave vector. The latter two vectors are functions of the slowly varying background flow and forcing frequency ω. As ω decreases, the ray paths approach that of the local wave vector, so that the energy propagates in a direction perpendicular to the wave fronts. The extratropical jet streams have a stronger influence on the long period (>30 day) ray paths than on those of intermediate period (∼10–30 day). Global and local energetics calculations show that the energy conversion from the zonally varying basic flow increases as ω decreases. The local energetics show that for the long period disturbances, both the energy c...

A WKB analysis of radical growth in the hydrogen-air mixing layer

The chain-branching process leading to ignition in the hydrogen-air mixing layer is studied by application of a novel WKB-like method with a four-step reduced scheme adopted for the chemistry description. Attention is restricted to initial free-stream temperatures above the crossover temperature corresponding to the second explosion limit of H2-O2 mixtures, thereby causing three-body recombination reactions to be negligible in the ignition process. It is shown that the initiation reactions, responsible for the early radical buildup, cease being important when the radical mass fractions reach values of the order of the ratio of the characteristic branching time to the characteristic initiation time, a very small quantity at temperatures of practical interest. The autocatalytic character of the chain-branching reactions causes the radical concentrations to grow exponentially with downstream distance in the process that follows. It is shown that, because of the effect of radical diffusion, the radical growth rate is uniform across the mixing layer in the first approximation, with an exponent given by that of a premixed branching explosion evaluated at the location where the effective Damkohler number based on the flow velocity is maximum. This exponent, as well as the leading-order representation of the radical profiles, are easily obtained by the imposition of a bounded, nonoscillatory behavior on the solution.

Multiple-scale analysis for Painlevé transcendents with a large parameter

1. Introduction. The purpose of this note is to give a survey of a part of the article[AKT3] which is concerned with the exact WKB analysis of Painlev´e transcendents witha large parameter. The exact WKB analysis is an analysis based on the systematic useof WKB solutions and Borel resummed WKB solutions of differential equations. A WKBsolution is a kind of formal solution that is expanded in the (negative) power series ofa large parameter. Such a series is divergent in general, but is easily constructed. Bytaking Borel resummation of a WKB solution, we get a holomorphic solution to theoriginal equation. The correspondence between WKB solutions to holomorphic solutionsobtained by Borel resummation is, however, not so simple (connection problems). Ifone knows the correspondence completely, then one can obtain large amount of globalinformation about the solutions. In fact, we know the correspondence, at least generically,in the case of second order linear ordinary differential equations of Fuchsian type (witha large parameter) and we can calculate the monodromy groups of the equations (cf.[AKT2]). In [AKT3], we investigate the Painlev´e equations from such a point of view (cf.[KT] also). Thus we are interested in(i) constructing formal solutions of Painlev´e equations,(ii) solving the connection problems for these formal solutions.In this note, we focus on the former problem and we give an outline of the constructionof formal solutions of the Painlev´e equations.2. Formal solutions of the first Painlev´e equation.2.1 Formal solution without free parameter.Let us consider the first Painlev´e equation with a large parameter η:

Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory.

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold. @S1063651X~96!00506-5#

Dynamics of MHD perturbations in the presence of singular lines

Typical effects relevant to space and laboratory plasmas, induced by the presence of singular lines in a magnetic configuration, are presented. They refer to modifications of flute perturbations in the scrape–off layer of a divertor, the onset of stochasticity around the magnetic separatrix, and the trapping of fast MHD waves in the vicinity of singular lines. In particular, a WKB analysis performed for X-point and current sheet configurations points out the occurence of global modes characterized by regular and irregular trajectories.

Asymptotics of radial wave equations

The Langer modification is an improvement in the WKB analysis of the radial Schrödinger equation. We derive a generalization of the Langer modification to any radial operator. For differential operators we write the modified classical symbols explicitly and show that the WKB wavefunctions with the modification have the exact limiting behavior for small radius. Unlike in the Schrödinger case, generally the modified radial analysis is not equivalent to the WKB analysis of the full problem before reduction by the spherical symmetry.

Fine structure in the spectra of low latitude field line resonances

The power spectra of low latitude (L<3), ground‐based recordings of dayside geomagnetic pulsations in the 10‐100 mHz band indicate that these pulsations come from discrete, driven field line resonances which are coupled to compressional, magnetohydrodynamic, cavity or waveguide modes. The spectral bands of individual harmonics of the field line resonances seen by the magnetometers have envelopes which are typically 10's of mHz wide but also show a fine scale structure of almost equispaced peaks with separations of about 3 to 5 mHz. A WKB analysis of a cavity or waveguide model shows that the fine structure is likely due to field line resonances driven by waveguide modes with turning points within the plasmasphere and reflection at the dayside magnetopause.

WKB Analysis to Global Solvability and Hypoellipticity

WKB Analysis to Global Solvability and Hypoellipticity