WKB Asymptotics Research Articles

Volume Conjecture and WKB Asymptotics

Volume Conjecture and WKB Asymptotics

WKB Asymptotics and Spectral Deformation in Semi-classical Limit

Phase-integral method is applied to study semi-classical limit for the spectral problem related to dynamical system describing quantum mechanical evolution with space-time reflection symmetry. The main goal of the paper is to investigate the relationship between analytic properties of the corresponding potential and the structure of limiting spectral graph. Quantization conditions of Bohr-Sommerfeld type are derived specifying localization of the spectrum for the model problem with a one-parameter family of polynomial potentials treated as deformation of a linear one.

WKB asymptotics of meromorphic solutions to difference equations

We consider the difference Schrödinger equation where z is a complex variable, E is a spectral parameter, and h is a small positive parameter. If the potential v is an analytic function, then, for h sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for differential equations. We assume that v has a simple pole and, in its neighborhood, we study the asymptotics of meromorphic solutions to the difference Schrödinger equation.

Quantization of band tilting in modulated phononic crystals

A general theory of the tilting of dispersion bands in phononic crystals whose properties are being slowly and periodically modulated in space and time is established. The ratio of tilt to modulation speed is calculated, for the first time, in terms of Berry's phase and curvature and is proven to be a robust integer-valued Chern number. Derivations are based on a version of the adiabatic theorem for elastic waves demonstrated thanks to WKB asymptotics. Findings are exemplified in the case of a 3-periodic discrete spring-mass lattice. Tilted dispersion diagrams plotted using fully numerical simulations and semianalytical calculations based on a numerically gauge invariant expression of Berry's phase show perfect agreement. One-way blocking of waves due to the tilt, and ultimately to the breaking of reciprocity, is illustrated numerically and shown to be highly significant across a limited number of unit cells, suggesting the feasibility of experimental demonstrations. Finally, a version of the bulk-edge correspondence principle relating the tilt of bulk bands to the number of one-way gapless edge states is demonstrated.

Splitting problem for WKB asymptotics in a nonresonant case and the reduction method for linear systems

As applied to the problem of asymptotic integration of linear systems of ordinary differential equations, we propose a reduction of order method that allows one to effectively construct solutions indistinguishable in the growth/decrease rate at infinity. In the case of a third-order equation, we use the developed approach to answer Bellman’s problem on splitting WKB asymptotics of subdominant solutions that decrease at the same rate. For a family of Wigner–von Neumann type potentials, the method allows one to formulate a selection rule for nonresonance values of the parameters (for which the corresponding second-order equation has a Jost solution).

The acoustic Green's function for swirling flow in a lined duct

This paper considers the acoustic field inside an annular duct carrying mean axial and swirling flow and with either acoustically hard or lined walls. The particular aim is to compute the Green's function, which is required for predicting the noise generated by known acoustic sources, both approximately and numerically. Asymptotic approximations for first the eigenmodes and then the Green's function are derived in the realistic limit of high reduced frequency, which are found to agree very favourably with results determined numerically, even for relatively modest frequencies. Using a blend of uniform WKB asymptotics and numerics, singular cases in which multiple turning points are present are treated, allowing computation of accurate results across a very wide range of parameter values and flows.

Large deviations for two-scale chemical kinetic processes

We formulate the large deviations for a class of two scale chemical kinetic processes motivated from biological applications. The result is successfully applied to treat a genetic switching model with positive feedbacks. The corresponding Hamiltonian is convex with respect to the momentum variable as a by-product of the large deviation theory. This property ensures its superiority in the rare event simulations compared with the result obtained by formal WKB asymptotics. The result is of general interest to understand the large deviations for multiscale problems.

Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation.

In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.

Superadiabatic evolution of acoustic and vorticity perturbations in Couette flow.

Nonadiabatic transitions between the acoustic and the vorticity modes perturbing a plane Couette flow are examined in the context of higher-order WKB asymptotics. In the case of the Schrödinger equation, it is known that looking at the solution expressed in the superadiabatic base, composed of higher-order asymptotic solutions, smoothes quantum state transitions. Then, increasing the order of the superadiabatic base causes these transitions to tend to the Gauss error function, and, once an optimal order is reached, the asymptotic process starts to diverge. We show that for perturbations in Couette flow, similar results can be applied on the amplitudes of the vorticity and acoustic modes. This allows us to more closely track the emergence of the acoustic modes in the presence of the vorticity mode.

Resurgent Analysis of the Witten Laplacian in One Dimension—II

The Witten Laplacian in one dimension is studied further by methods of resurgent analysis in order to approach Fukaya's conjectures relating WKB asymptotics and disc instantons. In this paper more precise connection formulae are presented, which allows the calculation of a subdominant exponential term in the hyperasymptotic expansion of a low-lying eigenvalue. Calculation of eigenfunctions corresponding to low-lying eigenvalues is presented in two examples.

Mathematical and computational methods for semiclassical Schrödinger equations

We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

Remarks on Non-Linear Schrödinger Equation with Magnetic Fields

We study the nonlinear Schrödinger equation with time-depending magnetic field without smallness assumption at infinity. We obtain some results on the Cauchy problem, WKB asymptotics and instability.

Wave Propagation through sparse potential barriers

AbstractWe prove that the three‐dimensional Schrödinger operator with slowly decaying sparse potential has an a.c. spectrum that fills ℝ+. A new kind of WKB asymptotics for Green's function is obtained. The absence of positive eigenvalues is established as well. © 2007 Wiley Periodicals, Inc.

Impact of localized inhomogeneity on the surface-wave velocity and bulk-wave reflection in solids

The effect of a weak surface, near-surface and interfacial inhomogeneity on the frequency dependence of the surface wave velocity and of the SH (shear horizontal) wave reflectivity in isotropic elastic media is studied analytically and numerically. The inhomogeneity is modeled as an infinite planar layer with continuously varying properties. Weak inhomogeneity may markedly affect the dispersion of the Rayleigh velocity and especially of the reflectivity. It is demonstrated how this effect, particularly pronounced at high frequency, depends on the extent of inhomogeneity. The material data for damaged and ideal concrete and several simple examples of inhomogeneity profiles are utilized for the numerical calculations based on the Peano expansion. The use of explicit low- and high-frequency approximations is also exemplified. Among these, simple WKB asymptotics are shown to be particularly helpful for the Rayleigh velocity in the case of a prominent inhomogeneity attached to the surface and for the reflection on weak interfaces.

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

The well-known Rayleigh criterion is a necessary and sufficient condition for inviscid centrifugal instability of axisymmetric perturbations. We have generalized this criterion to disturbances of any azimuthal wavenumber m by means of large-axial-wavenumber WKB asymptotics. A sufficient condition for a free axisymmetric vortex with angular velocity is real and equal to the inner cylinder radius. In sharp contrast, the maximum growth rate of non-axisymmetric disturbances is then independent of m.

Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation

A numerical scheme for the one-dimensional stationary Schrödinger–Poisson model is described. The scheme is used to simulate a resonant tunneling diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discretization of the energy variable, which is a parameter for the Schrödinger equation, is improved by approaching the wavefunctions in the double barrier region by its projection on the resonant states (following the work of Presilla–Sjöstrand and Jona-Lasinio [On Schrödinger equations with concentrated non-linearities, Ann. Phys. 240 (1995) 1–21]).

Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane K ( z ) = y 1 / 2 K i r ( 2 π m y ) e 2 π i m x \mathcal {K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x} , z = x + i y z=x+iy , λ = 1 4 + r 2 \lambda =\frac 14+r^2 the eigenvalue, s = 2 π m λ − 1 / 2 s=2\pi m\lambda ^{-1/2} and K i r K_{ir} the Macdonald-Bessel function. The phase velocity of K \mathcal {K} on { | s | I m z ≤ 1 } \{|s|Im z\le 1\} is a double-valued vector field, the tangent field to the pencil of geodesics G \mathcal {G} tangent to the horocycle { | s | I m z = 1 } \{|s|Im z =1 \} . For A ∈ S L ( 2 ; R ) A\in SL(2;\mathbb {R}) a multi-term stationary phase expansion is presented in λ \lambda for K ( A z ) e 2 π i n R e z \mathcal {K}(Az)e^{2\pi in\,Re z} uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for ψ \psi automorphic with coefficients { a n } \{a_n\} and eigenvalue λ \lambda it is shown for the special range n ∼ λ 1 / 2 n\sim \lambda ^{1/2} that a n a_n is O ( λ 1 / 4 e π λ 1 / 2 / 2 ) O(\lambda ^{1/4}\,e^{\pi \lambda ^{1/2}/2}) for λ \lambda large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound O ( | n | 1 / 2 λ 1 / 4 e π λ 1 / 2 / 2 ) O(|n|^{1/2}\lambda ^{1/4}\,e^{\pi \lambda ^{1/2}/2}) . An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.

Periodic multiphase nonlinear diffractive optics with curved phases

We describe diffraction for rapidly oscillating, periodically modulated nonlinear waves. This phenomenon arises for example when considering long-time propagation, or through perturbation of initial oscillations. We show existence and stability of solutions to variable coefficient, nonlinear hyperbolic systems, together with 3-scale multiphase infinite-order WKB asymptotics: the fast scale is that of oscillations, the slow one describes the modulation of the envelope, which is along rays for the oscillatory components, and the intermediate one corresponds to transverse diffraction. It gives rise to nonlinear Schrodinger equations on a torus for the profiles. The main difficulty resides in the fact that the coefficients in the original equations are variable: thus, phases are nonlinear, and rays are not parallel lines. This induces variable coefficients in the integro-differential system of profile equations, which in general is not solvable. We give sufficient (and, in general, necessary) geometrical coherence conditions on the phases for the formal asymptotics to be rigorously justified. Small divisors assumptions are also needed, which are generically satisfied.

Finite Gap Potentials and WKB Asymptotics¶for One-Dimensional Schrödinger Operators

Consider the Schrodinger operator H=−d2/dx2+V(x) with power-decaying potential V(x)=O(x−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi inversion problem on hyperelliptic Riemann surfaces.

Optique diffractive périodique à phases courbes

We give 3-scales WKB asymptotics for nonlinear hyperbolic systems with variable coefficients. Our profiles are transported, at the slower scale, along the rays of geometric optics, and are diffracted, at the intermediate scale, according to some (variable coefficients) Schrödinger equation.