Maslov Operator Research Articles

Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides

Abstract The article describes a method for constructing semiclassical asymptotic solutions of multidimensional (pseudo)differential partial differential equations with localized right-hand sides. The method is based on the asymptotic reduction of vector problems (under certain additional conditions) to scalar (pseudo)differential problems, for which efficient asymptotic formulas have recently been constructed. The method is based on the Feynman–Maslov operator calculus. Even if the original problems are stated for differential equations, the operators in reduced problems, as a rule, turn out to be pseudodifferential. Further, nontrivial problems of calculating operator symbols in the reduced scalar problems arise in the reduction process. We discuss approaches to the transformation of these symbols so as to enable their effective use when solving real problems. The method is illustrated by several physical examples.

Peierls substitution and the Maslov operator method

We consider a periodic Schrodinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: Open image in new window , where Open image in new window is the corresponding energy level of some auxiliary Schrodinger operator, assumed to be nondegenerate, and µ is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation Open image in new window with the operator Open image in new window represented as a function depending only on the operators of kinetic momenta \( \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) \).

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients.

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.

Book Review: Lagrangian manifolds and the Maslov operator

Book Review: Lagrangian manifolds and the Maslov operator

Canonical Maslov operator on Lagrangian manifolds with Hilbert model

Canonical Maslov operator on Lagrangian manifolds with Hilbert model

Quasiclassical scattering of wave packets on a narrow band in which the potential rapidly changes

Here ~0 (x), S o (x), dl) (x), V0 (x), V (T) are smooth real functions, the carrier ~0 is compact, the function V(T) decreases sufficiently rapidly as [TI + ~ (for definiteness, let us assume that V belongs to a Schwartz space). If V = 0, the asymptotics of the solutions of such a problem for times t~ [0, T] (T does not depend on h) is well known. It is given by the canonical Maslov operator [i, 2] on a Lagrangian manifold, invariant with respect to translations along the trajectories of the Hamiltonian system

A SCATTERING PROBLEM IN LAMINAR MEDIA

with a radiation condition at infinity is considered. The potential is periodic in the variable . Here is a large parameter, and is a small parameter with , . In this paper a formal asymptotic expansion of the solution of this problem is found. To construct it an operator analogous to the canonical Maslov operator is used which acts on a certain Lagrangian manifold not depending on . An analogous problem for the Schrödinger equation in a laminar medium is solved. Bibliography: 10 titles.

Geometry of Lagrangian manifolds and the canonical Maslov operator in complex phase space

A number of geometric questions related to the complex theory of the canonical Maslov operator are considered. The investigations center around the following themes: 1) quantization conditions and the problem of finding the asymptotics of eigenvalues; 2) universal characteristics of the theory of the complex canonical operator; 3) complex Hamiltonian fields and their trajectories.

Asymptotic solution of the Cauchy problem for equations with complex characteristics

Differential equations with a small parameter in the derivative are considered. A method is developed for constructing formal asymptotic solutions for the case of complex characteristics. For this a new class of manifolds is introduced which is a natural generalization of real Lagrangian manifolds to the complex case. The theory of the canonical Maslov operator is constructed in this class of manifolds. Asymptotic solutions are expressed in terms of the canonical Maslov operator.

Exact Green's function of the equations of quantum mechanics in an electromagnetic field. I

An exact Green's function of the Cauchy problem in arbitrary (nonparallel) stationary homogeneous electrical and magnetic fields is constructed for the Klein-Gordon-Fock and Dirac equations and for the Pauli equation in arbitrary nonstationary fields, on the basis of a single approach using the method of the canonical Maslov operator (extension of the WKB method to the multidimensional case) and a Fock idea about the proper time in relativitic mechanics.

The generating integral and the canonical maslov operator in the WKB method

The generating integral and the canonical maslov operator in the WKB method