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2013
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# 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.  704-716 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 (quasi-modes) of the Schr¨odinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes 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. Keywords: potential well, stationary Schrödinger equation, KAM theory, operator separation of variables, semiclassical asymptotics, Airy function, Bessel function Citation: Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.,  Classical and Quantum Dynamics of a Particle in a Narrow Angle, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 704-716 DOI:10.1134/S156035471906008X
 Dobrokhotov S. Y., Minenkov D. S. On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation 2010, vol. 15, no. 2-3, pp.  285-299 Abstract The 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 $V(x, \tau)$ 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 so-called 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. Keywords: nonlinear oscillator, averaging, asymptotics, phase shift Citation: Dobrokhotov S. Y., Minenkov D. S.,  On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 285-299 DOI:10.1134/S1560354710020152
 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.  167-180 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. Keywords: quantum double well problem, librations, splitting value, tunneling effect, normal forms, complex phase space Citation: Bruening J., Dobrokhotov S. Y., Semenov E. S.,  Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 167-180 DOI: 10.1070/RD2006v011n02ABEH000343