D. Minenkov

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.

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.