D. S. Minenkov

The Maupertuis-Jacobi principle for Hamiltonians of the form F(x, |p|) in two-dimensional stationary semiclassical problems

We consider two-dimensional asymptotic formulas based on the Maslov canonical operator arising in stationary problems for differential and pseudodifferential equations. In the case of Lagrangian manifolds invariant with respect to Hamiltonian flow with Hamiltonians of the form F(x, |p|), we show how asymptotic formulas can be simplified by using the well-known (in classical mechanics) Maupertuis-Jacobi correspondence principle to replace the Hamiltonians F(x, |p|) by Hamiltonians of the form C(x)|p| arising, in particular, in geometric optics and related to the Finsler metric. As examples, we consider Hamiltonians corresponding to the Schrödinger equation, the two-dimensional Dirac equation, and the pseudodifferential equations for surface water waves.

On the Bose–Maslov statistics in the case of infinitely many degrees of freedom

We find the asymptotics of the counting function of elements of an additive arithmetical semigroup for the case of an exponential counting function of prime generators, which has a natural interpretation in terms of Bose statistics as well as in the problem of counting the number of Gaussian packets on decorated graphs.

Functions of Noncommuting Operators in an Asymptotic Problem for a 2D Wave Equation with Variable Velocity and Localized Right-hand Side

In the present paper, we use the theory of functions of noncommuting operators, also known as noncommutative analysis (which can be viewed as a far-reaching generalization of pseudodifferential operator calculus), to solve an asymptotic problem for a partial differential equation and show how, starting from general constructions and operator formulas that seem to be rather abstract from the viewpoint of differential equations, one can end upwith very specific, easy-to-evaluate expressions for the solution, useful, e.g., in the tsunami wave problem.

On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation

AbstractThe main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional anharmonic oscillator with potential V (x, τ) depending on the slow time τ = ɛt and with a small nonconservative term ɛg(

Asymptotics near the shore for 2D shallow water over sloping planar bottom

\dot x

Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems

, x, τ), ɛ ≪ 1. This problem was discussed in numerous papers, and in some sense the present paper looks like a “methodological” one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form

Об асимптотике считающей функции элементов в аддитивной арифметической полугруппе с экспоненциальной считающей функцией простых образующих@@@On the Asymptotics of the Element Counting Function in an Additive Arithmetic Semigroup with Exponential Counting Function of Prime Generators

X\left( {\frac{{S\left( \tau \right) + \varepsilon \varphi \left( \tau \right)}} {\varepsilon },I\left( \tau \right),\tau } \right)