Maslov Canonical Operator Research Articles

Asymptotics of the Localized Bessel Beams and Lagrangian Manifolds

The Bessel beam-type asymptotic solutions of the three-dimensional Helmholtz equation, i.e., the solutions that have maxima in the vicinity of the -axis and are described by Bessel functions in the planes normal to it, are discussed. Since the Bessel functions slowly decrease at infinity, the energy of such solutions appears unlimited. Approaches to localizing such solutions by representing them in the form of the Maslov canonical operator on proper Lagrangian manifolds with simple caustics in the form of degenerate and nondegenerate folds are described. Efficient formulas for these solutions in the form of Bessel and Airy functions of a complex argument are obtained.

On uniform asymptotic approximations of whispering gallery modes propagating along curved penetrable interfaces

While in the classical theory of whispering gallery waves it is assumed that they propagate along a cylindrical perfectly reflecting wall, in many real situations it is important to consider similar waves propagating along penetrable interfaces or curvilinear level sets of the refractive index. An example of waves of the latter kind are whispering gallery modes formed in a shallow-water waveguide due to the horizontal refraction. Such modes are counterpart of quasistationary states in quantum mechanics, as they do not exponentially decay at infinity, and their respective eigenvalues are complex. Recently it was shown that Maslov canonical operator (MCO) theory can be used to compute uniform asymptotic expansions of the solutions to various wave propagation problems. Such approximations admit convenient analytical representation in terms of special functions that can be used for investigating their properties and for other purposes. The MCO theory is applicable only in cases where the solution exponentially decays at infinity, e.g., for computing bound states of a given quantum system. Nevertheless, the technique outlined here allows to obtain analytical expressions for the radial profiles of whispering gallery modes propagating along a penetrable wall or along a curvilinear level set of the refractive index.

New Integral Representations for the Maslov Canonical Operator on an Isotropic Manifold with a Complex Germ

New Integral Representations for the Maslov Canonical Operator on an Isotropic Manifold with a Complex Germ

Plancherel–Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations

We study the asymptotic properties of multiple orthogonal Hermite polynomials which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptotic expansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel–Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of bases of homogeneous difference equations, and the second on reducing difference equations to pseudodifferential ones and using the theory of the Maslov canonical operator. The results of these approaches agree.

Representation of Bessel functions by the Maslov canonical operator

Representation of Bessel functions by the Maslov canonical operator

Representations of Bessel functions via the Maslov canonical operator

Конструкция и применение канонического оператора Маслова иллюстрируются на примере функций Бесселя. Для них при вещественных значениях аргумента выводятся представления через канонический оператор и, как следствие, получаются хорошо известные асимптотики.

Quasi-classical approximation for magnetic monopoles

A quasi-classical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is given by a non-exact 2-form. For this, the multidimensional WKB method in the form of the Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a non-trivial line bundle. The constructed approximation is demonstrated for the example of the Dirac magnetic monopole on the two-dimensional sphere. Bibliography: 18 titles.

Localized Asymptotic Solution of a Variable-Velocity Wave Equation on the Simplest Decorated Graph with Initial Conditions on a Surface

A variable-velocity wave equation is studied on the simplest decorated graph, i.e., the topological space obtained by attaching a ray to $$\mathbb R^3$$ . The Cauchy problem with initial conditions localized on Euclidean space is considered. The leading term of an asymptotic solution of the problem under consideration as the parameter characterizing the size of the source tends to zero is described by using the construction of the Maslov canonical operator. It is assumed that the point on $$\mathbb R^3$$ at which the ray is attached is not a singular point of the wavefront.

Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp

We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type $$A_3$$ (standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.

Nonstandard caustics for localized solutions of the 2d shallow water equations with applications to wave propagation and run-up on a shallow beach

The usual semiclassical approximation is not directly applicable to linearized two-dimensional shallow water equations with localized initial data describing long waves (for example, tsunami waves) in a bounded basin of variable depth, because the momentum variables on the geometric optics rays become infinite at the boundary of the region. To obtain asymptotic solutions, one needs to extend the phase space and introduce a modified Maslov canonical operator. The paper gives a brief survey of some results obtained in this direction by the authors and their colleagues.

Квазиклассическое приближение для магнитных монополей

В работе строится квазиклассическое приближение для описания собственных значений магнитного лапласиана на компактном римановом многообразии в случае, когда магнитное поле не задается точной $2$-формой. Для этого применяется многомерный метод ВКБ в форме канонического оператора Маслова. В этом случае канонический оператор принимает значения в сечениях нетривиального линейного расслоения. Построенное приближение продемонстрировано на примере магнитного монополя Дирака на двумерной сфере. Библиография: 18 названий.

Localized Asymptotic Solution of a Variable-Velocity Wave Equation on the Simplest Decorated Graph

We consider a variable-velocity wave equation on the simplest decorated graph obtained by gluing a ray to the three-dimensional Euclidean space, with localized initial conditions on the ray. The wave operator should be self-adjoint, which implies some boundary conditions at the gluing point. We describe the leading part of the asymptotic solution of the problem using the construction of the Maslov canonical operator. The result is obtained for all possible boundary conditions at the gluing point.

Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems

We consider stationary scalar and vector problems for differential and pseudodifferential operators leading to the appearance of asymptotic solutions of one-dimensional problems localized in a neighborhood of intervals (“bound states”). Based on the semiclassical approximation and the Maslov canonical operator, we develop a constructive algorithm that allows writing an asymptotic solution globally under certain conditions using an Airy function of complex argument.

On the Application of Asymptotic Formulae Based on the Modified Maslov Canonical Operator to the Modeling of Acoustic Pulses Propagation in Three-Dimensional Shallow-Water Waveguides

In this study a technique for the modeling of propagation of acoustic pulses in shallow-water waveguides with three-dimensional bottom inhomogeneities is described. The described approach is based on the ray theory of sound propagation and the method of modified Maslov canonical operator. Representation of acoustical field in terms of the canonical operator gives several important advantages in practical computations. In particular, it is possible to compute the time series of a pulse at a reception point located on the caustics of a family of rays. Besides, a significant part of calculations within the proposed approach can be performed analytically; therefore, overall computational costs are substantially reduced. As an example, sound propagation in a wedge-shaped waveguide representing a shelf area near the coast line is considered. The ray geometry in such a waveguide is discussed both in the isovelocity case and in the presence of the thermocline in the water column. For both cases, the time series of an acoustical pulse propagating along the track aligned along the isobaths (parallel to the apex edge of the wedge) is calculated.

Maslov's Canonical Operator in Problems on Localized Asymptotic Solutions of Hyperbolic Equations and Systems

An analog of Maslov's canonical operator is defined for functions localized in a neighborhood of subsets of positive codimension.

Nonstandard Lagrangian Singularities and Asymptotic Eigenfunctions of the Degenerating Operator $$ - {d \over {dx}}D\left( x \right){d \over {dx}}$$

We express the asymptotic eigenfunctions of the operator $$ - {d \over {dx}}D\left( x \right){d \over {dx}}$$ that degenerates at the endpoints of an interval in terms of the modified Maslov canonical operator introduced in our previous studies.

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a spectral problem for the harmonic oscillator Schrodinger equation. The second approach is based on a reduction of the finite-difference equation for the Hermite polynomials to a pseudodifferential equation. Associated with each of the approaches are Lagrangian manifolds that give the asymptotics of Hermite polynomials via the Maslov canonical operator.

Asymptotics of Wave Functions of the Stationary Schrödinger Equation in the Weyl Chamber

We study stationary solutions of the Schrodinger equation with a monotonic potential U in a polyhedral angle (Weyl chamber) with the Dirichlet boundary condition. The potential has the form \(U\left( x \right) = \sum _{j = 1}^nV\left( {{x_j}} \right),x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}\) , with a monotonically increasing function V (y). We construct semiclassical asymptotic formulas for eigenvalues and eigenfunctions in the form of the Slater determinant composed of Airy functions with arguments depending nonlinearly on xj. We propose a method for implementing the Maslov canonical operator in the form of the Airy function based on canonical transformations.

Efficient Formulas for the Maslov Canonical Operator near a Simple Caustic

We revive well-known old formulas for the asymptotics of rapidly oscillating integrals with two nearly coincident saddle points and present an efficient approach to the computation of functions represented by Maslov’s canonical operator near a simple caustic in an algorithmic fashion convenient for implementation on software like Wolfram Mathematica. The exposition is elementary and assumes that no preliminary information about the canonical operator is be known to the reader.

Electronic optics in graphene in the semiclassical approximation

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameterh≪1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to aneffective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.