D. Minenkov
prosp.Vernadskogo 101/1, 119526, Moscow, Russian Federation
Institute for Problems in Mechanics, Russian Academy of Sciences
Publications:
Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.
Classical and Quantum Dynamics of a Particle in a Narrow Angle
2019, vol. 24, no. 6, pp. 704716
Abstract
We consider the 2D Schrödinger equation with variable potential in the narrow
domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding
classical problem is the billiard in this domain. In general, the corresponding dynamical system is
not integrable. The small angle is a small parameter which allows one to make the averaging and
reduce the classical dynamical system to an integrable one modulo exponential small correction.
We use the quantum adiabatic approximation (operator separation of variables) to construct the
asymptotic eigenfunctions (quasimodes) of the Schr¨odinger operator. We discuss the relation
between classical averaging and constructed quasimodes. The behavior of quasimodes in the
neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy
functions that follows from different representations of asymptotics near the cusp.

Dobrokhotov S. Y., Minenkov D. S.
On various averaging methods for a nonlinear oscillator with slow timedependent potential and a nonconservative perturbation
2010, vol. 15, nos. 23, pp. 285299
Abstract
The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the onedimensional anharmonic oscillator with potential $V(x, \tau)$ depending on the slow time $\tau=\varepsilon t$ and with a small nonconservative term $\varepsilon g(\dot x, x, \tau)$, $\varepsilon \ll 1$. 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 $X(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau), \tau)$, where the phase $S$, the "slow" parameter $I$, and the socalled phase shift $\phi$ are found from the system of "averaged" equations. The pragmatic result is that one can take into account the phase shift $\phi$ by considering it as a part of $S$ and by simultaneously changing the initial data for the equation for $I$ in an appropriate way.
