Uniformly Valid Asymptotic Approximation for the Quantized Anharmonic Oscillator at High Energies

A uniformly valid asymptotic expansion is obtained for the solution of the one-dimensional time-independent Schrödinger equation which describes an anharmonic oscillator. The computational method is suggested by McKelvey's treatment of the two-turning-point problem, but has some practical advantages over his procedure. For turning points which lie close together, the energy levels are expressed as an asymptotic power series in ℏ/ (2m)1/2, with coefficients which depend only on the quantum number n and the derivatives of the potential at its minimum. The leading terms of this series are calculated and found to agree with the JWKB and perturbation results of earlier authors. For turning points which do not lie close together, the application of this method is indicated briefly and pursued up to the recovery of the Bohr—Sommerfeld quantum condition.

This paper provides an exact asymptotic analysis regarding the low-frequency dispersion characteristics of a multilayered coaxial cable. A layer-recursive description of the dispersion function is derived that is well suited for asymptotic analysis. The recursion is based on two well-behaved (meromorphic) subdeterminants defined by a perfectly electrically conducting (PEC) and a perfectly magnetically conducting termination, respectively. For an open waveguide structure, the dispersion function is a combination of two such functions, and there is only one branch point that is related to the exterior domain. It is shown that if there is one isolating layer and a PEC outer shield, then the classical Weierstrass preparation theorem can be used to prove that the low-frequency behavior of the propagation constant is governed by the square root of the complex frequency, and an exact analytical expression for the dominating term of the asymptotic expansion is derived. It is furthermore shown that the same asymptotic expansion is valid to its lowest order even if the outer shield has finite conductivity and there is an infinite exterior region with finite nonzero conductivity. As a practical application of the theory, a high-voltage direct current (HVDC) power cable is analyzed and a numerical solution to the dispersion relation is validated by comparisons with the asymptotic analysis. The comparison reveals that the low-frequency dispersion characteristics of the power cable is very complicated and a first-order asymptotic approximation is valid only at extremely low frequencies (below 1 Hz). It is noted that the only way to come to this conclusion is to actually perform the asymptotic analysis. Hence, for practical modeling purposes, such as with fault localization, an accurate numerical solution to the dispersion relation is necessary and the asymptotic analysis is useful as a validation tool.

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

The recommendations for electrical therapies described in this section are designed to improve survival from SCA and life-threatening arrhythmias. Whenever defibrillation is attempted, rescuers must coordinate high-quality CPR with defibrillation to minimize interruptions in chest compressions and to ensure immediate resumption of chest compressions after shock delivery. The high first-shock efficacy of newer biphasic defibrillators led to the recommendation of single shocks plus immediate CPR instead of 3-shock sequences that were recommended prior to 2005 to treat VF. Further data are needed to refine recommendations for energy levels for defibrillation and cardioversion using biphasic waveforms.

We show explicitly the relation between the uniform asymptotic and the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) wavefunctions, and between the matching of uniform asymptotic expansions and the complete JWKB connection formulae written in terms of Stokes multipliers and loop integrals. As an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and, via dispersion relations, the corresponding asymptotic behaviour of the Rayleigh-Schrodinger perturbation theory coefficients.

The book Turbulent Combustion by Norbert Peters is a concise monograph on single-phase gaseous low Mach number turbulent combustion. It is compiled from the author's review papers on this topic plus some additional material. Norbert Peters characterizes turbulent combustion both by the way fuel and air are mixed and by the ratio of turbulent and chemical time scales. This approach leads naturally to detailed models, which are based on results of turbulence modelling and asymptotic flame theory. In both areas Norbert Peters has contributed significantly over the last two decades. The book has four sections. In chapter 1 he discusses briefly the state of the art of combustion models as they are used by different authors. Important turbulent and chemical scales are introduced, which are then used to introduce and explain the different combustion models. He distinguishes between premixed and non-premixed combustion and also between infinitely fast and finite rate chemistry. The current turbulent combustion models are described in order of their complexity and physical accuracy. He explains Eddy BreakUp and Eddy Dissipation Models, the fundamentals of the PDF transport equation, and the laminar flamelet concept applied to non-premixed and premixed turbulent combustion. Then, the Conditional Moment Closure, the Linear Eddy Model, and combustion models used in Large Eddy Simulations are described very briefly. Chapter 2 is devoted to premixed turbulent combustion. After introducing some characteristic dimensionless numbers Peters uses the level set approach and the flamelet concept to formulate a combustion model valid in the thin zone and corrugated reaction zone regimes. He shows parallels between this more fundamental model and standard models like the Bray-Moss-Libby model. He also presents models for the turbulent burning velocity, the Flame Surface Area Ratio, and discusses the effects of gas expansion. Very helpful for the reader's understanding is the presentation of three worked examples of a slot burner, a propagating spherical flame and an oscillating counter flow. Peters' model for premixed turbulent combustion is based on the equations for the mean and the variance of the $G$-equation, some closure relations as well as the flamelet equation for premixed combustion. A numerical example is used to discuss its accuracy. Non-premixed turbulent combustion is the subject matter of chapter 3. Peters uses the mixture fraction variable and asymptotic flame theory to explain the regimes of non-premixed turbulent combustion. Two worked examples of a counterflow diffusion flame and the one-dimensional unsteady laminar mixing layer help the reader to understand the theory. After discussing turbulent jet diffusion flames and introducing the flamelet equation he develops steady and unsteady flamelet models for non-premixed turbulent combustion. In particular, the Eulerian Particle Flamelet Model and the RIF (Representative Interactive Flamelet) Model are discussed. These models have been used to predict pollutant formation in a gas turbine and a direct injection Diesel engine, respectively. Finally, partially premixed combustion is discussed in chapter 4. Lifted turbulent diffusion flames are reviewed and the prediction of the lift-off height is identified as a key problem. This leads directly to the introduction of the concept of a triple flame. Different models for partially premixed combustion are then presented and the numerical simulation and the scaling of lift-off heights in turbulent jet flames are studied. There is no doubt that this book is well written and is an important contribution to combustion literature. Peters uses asymptotic theory and scale separation to develop combustion models from first principles. Also, the book contains a comprehensive review of the current literature on turbulent combustion. It is clearly a `must have' for experienced combustion modellers and experimentalists. The book has not been written for non-experts and beginners; these readers would probably have liked more worked examples and exercises, a list of symbols, some material on the mathematical techniques of asymptotic analysis, and a more detailed discussion of the standard combustion models that are often used in the literature. Also, the numerical aspects of turbulent combustion modelling are not discussed in the book. Nevertheless, the book will be a useful source for advanced courses on combustion. Markus Kraft

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.

Let {\check{X}_0} be a semi-flat Calabi–Yau manifold equipped with a Lagrangian torus fibration {\check{p}:\check{X}_0 \rightarrow B_0} . We investigate the asymptotic behavior of Maurer–Cartan solutions of the Kodaira–Spencer deformation theory on {\check{X}_0} by expanding them into Fourier series along fibres of {\check{p}} over a contractible open subset {U\subset B_0} , following a program set forth by Fukaya [Graphs and Patterns in Mathematics and Theoretical Physics (2005)] in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer–Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich and Soibelman [The Unity of Mathematics (2006)] and Gross and Siebert [Ann. of Math. (2) 174 (2011)] on the reconstruction problem in mirror symmetry.

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in $\mathbf{R}^N$ for $N\geq2$, studies by Schatz in [Math. Comp., 67 (1998), pp. 877–899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269–1293].

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling Λ = λ M 3 is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for (E PT − E exact) E exact of at least 10 −7 (using 20 orders of perturbation theory) and 10 −3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10 −7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.

23 - Unified Treatment of Ideal Fermi, Bose, and Classical Gases

The effect of pressure and rock type on optimal dual energy levels and minimalisation of saturation uncertainties for three-phase core experiments was investigated. X-ray scanning of a composite core comprising of two chalk plugs and one Berea sandstone core was done. Reference scans using CO2, natural seawater, n-Ci10 and a CO2/n-Ci10 mixture at four pressure stages and five different pre-defined dual energy levels were acquired. Two unique typical given saturations were input together with data from the reference scans in a routine for back-calculating three-phase intensity data. A detailed uncertainty analysis using these synthetic three-phase data and combining low and high energies was performed. The resulting combination of low and high energy level giving the lowest uncertainty was selected as a given setting for all dynamic scans. Saturated with live oil, the core was waterflooded while dynamic X-ray scans were acquired. After the waterflood, the core was depleted below the bubble point to atmospheric pressure while scans were acquired continuously. The results show that the rock type largely effects the selection of energy levels for both two- and three-phase X-ray scans. To minimise the uncertainty, the chalks generally required a higher low energy level independent of pressure than the sandstone. Also, at high pressures and for all rocks the optimal high energy level was found in the same range. At lower pressures, a lower high energy level was favourable.

Large degree asymptotics of generalized Bessel polynomials

In this paper, we give an asymptotic analysis of the phase fieldAllen-Cahn and Cahn-Hilliard models of free surfaces with surfacetension. Unlike the traditional approach that approximates the solutionby the so-called matched asymptotic expansion involvingouter expansion, inner expansion and matching, our new approach utilizesa uniform double asymptotic expansion to expand the wholephase field function directly. Although the main result is not new, we would like to emphasize that we derive the result under a uniform double asymptotic expansion. Thus, in this paper the detailed structure of the phasefield functions in the equilibrium state is obtained, andthe consistency of the phase field models with the correspondingsharp interface models is discussed, including the free surfaceAllen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model withvolume constraint. The explicit asymptotic expansion of the phasefield function reveals rich details of itsstructures. Moreover, it nicely explains some unusual phenomena weobserved in numerical experiments. The theory introduced in thispaper can be applied to guide the future modeling and simulation ofother moving boundary problems by phase field models.