S. Dobrokhotov
prosp. Vernadskogo 101, Moscow, 119526 Russia
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMech RAS)
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, no. 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.

Bruening J., Dobrokhotov S. Y., Semenov E. S.
Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem
2006, vol. 11, no. 2, pp. 167180
Abstract
We discuss the structure of asymptotic splitting formula for the lowest eigenvalues of multidimensional quantum double well problem. We show that the change of instanton by closed unstable trajectory of appropriate Hamiltonian system gives more natural and simpler preexponential factor (amplitude) in splitting formula. The projection of this trajectories onto configuration space are well know librations in classical mechanics.
