WKB Expansions Research Articles

Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence

We study the ODE/IM correspondence between two-dimensional WAr/WDr-type conformal field theories and the higher-order ordinary differential equations (ODEs) obtained from the affine Toda field theories associated with Ar(1)/Dr(1)-type affine Lie algebras. We calculate the period integrals of the WKB solution to the ODE along the Pochhammer contour, where the WKB expansions correspond to the classical conserved currents of the Drinfeld-Sokolov integrable hierarchies. We also compute the integrals of motion for WAr(WDr) algebras on a cylinder. Their eigenvalues on the vacuum state are confirmed to agree with the period integrals up to the sixth order. These results generalize the ODE/IM correspondence to higher-order ODEs and can be used to predict higher-order integrals of motion.

WKB analysis of the bifurcation for a three-dimensional neo-Hookean cylindrical tube under restricted compression

We study the bifurcation of a three-dimensional neo-Hookean elastic cylindrical tube under axial compression, where the movement of the outer surface is restricted. For the first time, the WKB method is applied to the three-dimensional eigenvalue problem and the bifurcation conditions are calculated for thick and thin-walled cylinders, separately. In this paper, two WKB expansions are considered for the two cases (i.e. for large axial or circumferential mode number) and the asymptotic expansion of the eigenvalue is obtained for each case separately. It is found that the changes of the axial stretch relative to the thickness of the cylinder have a boundary layer structure. The dependency of the axial stretch on the mode numbers, length and the wall thickness is assessed. The critical stretches occur for the finite critical mode numbers and are an decreasing function of the thickness of the tube, so thick cylinders are easier to buckle. Also, transitions between axial and circumferential modes occur in three points for mode numbers of 10,20 and at a point for mode number of zero. Using the compound matrix method to verify the analytical results, the comparison of outcomes of the numerical and analytical data shows a good agreement.

Eigenvalue spacing for 1D singular Schrödinger operators

The aim of this paper is to provide uniform estimates for the eigenvalue spacing of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.

Vacuum energy density and pressure inside a soft wall

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form [Formula: see text]. This “soft wall” model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, [Formula: see text], which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this paper, we apply the scheme to the wall with [Formula: see text] and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.

WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

Classical and quantum duality of quasi-exactly solvable problems

The concept of duality reflects a link between two seemingly different physical objects. An example from quantum mechanics is a situation where the spectra (or their parts) of two different Hamiltonians go into one another under a certain transformation.In this paper we resolve a long-noticed puzzle of matching between the perturbative and WKB expansions of dual energy levels in such potentials. Our approach to this class of problems is based on studying the global properties of the Riemann surface of the quantum momentum function, a natural quantum-mechanical analogue to the classical momentum. Being based on the position representation of the Schrödinger equation and the analytic properties of the wave functions, it explains the matching of the WKB and perturbative series to all orders. Our approach also reveals the classical origins of duality.

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

The WKB method for the quantum mechanical two-Coulomb-center problem

Using a modified perturbation theory, we obtain asymptotic expressions for the two-center quasiradial and quasiangular wave functions for large internuclear distances R. We show that in each order of 1/R, corrections to the wave functions are expressed in terms of a finite number of Coulomb functions with a modified charge. We derive simple analytic expressions for the first, second, and third corrections. We develop a consistent scheme for obtaining WKB expansions for solutions of the quasiangular equation in the quantum mechanical two-Coulomb-center problem. In the framework of this scheme, we construct semiclassical two-center wave functions for large distances between fixed positively charged particles (nuclei) for the entire space of motion of a negatively charged particle (electron). The method ensures simple uniform estimates for eigenfunctions at arbitrary large internuclear distances R, including R ≥ 1. In contrast to perturbation theory, the semiclassical approximation is not related to the smallness of the interaction and hence has a wider applicability domain, which permits investigating qualitative laws for the behavior and properties of quantum mechanical systems.

Anomalous transport

This article is concerned with the relativistic Vlasov equation, for collisionless axisymmetric plasmas immersed in a strong magnetic field, like in tokamaks. It provides a consistent kinetic treatment of the microscopic particle phase–space dynamics. It shows that the turbulent transport can be completely described through WKB expansions.

Magnetic WKB Constructions

This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in the presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman–Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born–Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions.We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal Agmon estimates and upper bound of the tunelling effect (in symmetric cases).We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades.

Curvature induced magnetic bound states: towards the tunneling effect for the ellipse

This article is devoted to the semiclassical analysis of the magnetic Laplacian on a smooth domain of the plane carrying Neumann boundary conditions. We provide WKB expansions of the eigenfunctions in generic situations. We also explain and illustrate a conjecture of magnetic tunneling when the domain is an ellipse.

Quasiclassical Two-Coulomb-Centre Wave Functions in the Spheroidal Coordinate System

The consistent scheme for obtaining WKB expansions for solutions of the quasiangular equation in quantum mechanical problem of two Coulomb centers Z1eZ2 is elaborated. It is shown that in each order of 1/R the corrections to the wave function can be expressed by a finite number of the discrete spectrum Coulomb wave functions with modified charge. The simple analytical expressions for the first and second corrections to the wave functions are obtained.

Stability and instability of Navier boundary layers

We study the inviscid limit problem for the incompressible Navier-Stokes equation on a half-plane with a Navier boundary condition depending on the viscosity. On one hand, we prove the $L^{2}$ convergence of Leray solutions to the solution of the Euler equation. On the other hand, we show the nonlinear instability of some WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^{s}$ ($s>1$) norms. These results are not contradictory, and in the periodic setting, we provide an example for which both phenomena occur simultaneously.

Axisymmetric and non-axisymmetric magnetostrophic MRI modes

The magnetorotational-instability is central to the understanding of many astrophysical magnetohydrodynamic flows (in particular accretion discs). We have recently shown that a modified version of this instability, the magnetostrophic MRI (MS-MRI), is relevant to the dynamics of the Earth liquid core (Petitdemange et al., 2008). Our previous study used a purely axial imposed magnetic field and considered only axisymmetric instabilities. We investigate here the effects of a large scale toroidal magnetic field on the development and saturation of the MS-MRI both in an axisymmetric setup and in fully three-dimensional configurations. We use direct numerical modeling of the full MHD equations (both in the linear and non-linear regimes) in a spherical geometry. We interpret our results using WKB expansions and shearing coordinates. We find that three-dimensional MS-MRI modes exhibit a strong helical structure for parameters relevant to planetary interiors. Three-dimensional MS-MRI modes share some similarities with their axisymmetric counterparts. They are amplified on the same short timescale. When non-linear effects become significant, they act to decrease the shear. During saturation, the magnetic structure expands spatially, while drifting at a constant rate along the direction of the rotation axis. A striking result is that an m=1 mode can dominate in the non-linear regime when a sufficiently large toroidal field is present. Three-dimensional MS-MRI modes could play an important role in planetary interior dynamics. They can cause rapid magnetic field variations and also act to limit the shear in the conducting liquid core.

Thin-Layer Solutions of the Helmholtz and Related Equations

This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.

On the number of reachable configurations for the chessboard pebbling problem

Recently, Chung, Graham, Morrison and Odlyzko [F. Chung, R. Graham, J. Morrison, A. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995) 113–123] studied various combinatorial and asymptotic enumeration aspects of chessboard pebbling. Here, a pebble is placed at the origin ( 0 , 0 ) of an infinite chessboard. At each step, a pebble is removed from ( i , j ) and replaced by two pebbles at positions ( i , j + 1 ) and ( i + 1 , j ) (provided these are unoccupied). After k steps, the board has k + 1 pebbles in various arrangements. Here we study the number of possible arrangements asymptotically, as k → ∞ . We analyze the recurrence derived in [F. Chung, R. Graham, J. Morrison, A. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995) 113–123] by methods of applied mathematics, such as WKB expansions and matched asymptotics. In particular, we obtain an analytic expression for the growth rate of the number of possible arrangements.

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with nr ∼ 1.

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to “cure” the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary, which include instanton configurations, characterized by non-analytic factors exp(− a/ g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a non-analytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det( H− E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function.

Two Asymptotic Forms of the Rotational Solution for Wave Propagation Inside Viscous Channels with Transpiring Walls

Summary In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel’s head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves with oscillations matching the natural frequencies of the enclosure are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyse. The mean flow is of the Berman type and can be obtained from the Navier–Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ordinary differential equations. Due to the oscillatory behaviour of the problem, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem’s characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem’s solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.

Exact WKB expansions for some potentials

We find WKB expansions for the spectra of the three-dimensional isotropic harmonic oscillator and Eckart and Morse potentials. It is shown that for all these potentials the series are convergent and, when summed, yield the exact energy spectra. These results support our working hypothesis that the WKB series for the energy spectra are convergent for all one-dimensional solvable potentials and their sums yield the exact energies.