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Classical and Quantum Dynamics of a Particle in a Narrow Angle
Regular and Chaotic Dynamics ( IF 1.4 ) Pub Date : 2019-12-10 , DOI: 10.1134/s156035471906008x Sergei Yu. Dobrokhotov , Dmitrii S. Minenkov , Anatoly I. Neishtadt , Semen B. Shlosman
Regular and Chaotic Dynamics ( IF 1.4 ) Pub Date : 2019-12-10 , DOI: 10.1134/s156035471906008x Sergei Yu. Dobrokhotov , Dmitrii S. Minenkov , Anatoly I. Neishtadt , Semen B. Shlosman
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 Schrödinger 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.
中文翻译:
窄角粒子的经典动力学和量子动力学
我们考虑了具有Dirichlet边界条件的,在窄域中微分到楔形的具有可变电势的二维Schrödinger方程。相应的经典问题是该领域的台球。通常,相应的动力学系统是不可积分的。小角度是一个小参数,允许人们进行平均并将经典动力学系统简化为可积分的一个模指数小校正。我们使用量子绝热近似(变量的算子分离)来构造Schrödinger算子的渐近本征函数(准模)。我们讨论经典平均和构造的准模式之间的关系。研究了尖峰附近的准模态的行为。
更新日期:2019-12-10
中文翻译:
窄角粒子的经典动力学和量子动力学
我们考虑了具有Dirichlet边界条件的,在窄域中微分到楔形的具有可变电势的二维Schrödinger方程。相应的经典问题是该领域的台球。通常,相应的动力学系统是不可积分的。小角度是一个小参数,允许人们进行平均并将经典动力学系统简化为可积分的一个模指数小校正。我们使用量子绝热近似(变量的算子分离)来构造Schrödinger算子的渐近本征函数(准模)。我们讨论经典平均和构造的准模式之间的关系。研究了尖峰附近的准模态的行为。