Abstract

We study an \(h\)-pseudodifferential operator (in particular, differential) acting on the \(n\)-D physical space with a principal symbol determining an integrable Hamiltonian system and with nontrivial subprincipal symbol which destroys the integrability. We propose a pragmatic method for calculating asymptotic eigenvalues and eigenfunctions (quasimodes) of such an operator. Unlike the approach of Lazutkin, where the correction to the principal symbol is included into the Hamiltonian, we deal with an integrable Hamiltonian system and nontrivial \(n\)-D transport equation. We propose an algorithm for constructing series of quasimodes in a neighborhood of a single Diophantine torus. We discuss the possibility to use this algorithm in practice using software like Wolfram Mathematica. As an example, we explicitly construct quasimodes for a Schrödinger operator describing a weakly coupled dimer, i.e., two weakly coupled particles in a one-dimensional potential field.

中文翻译：

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具有可积分主符号的算符的Diophantine花托和拟模的语用计算

摘要

我们研究一个作用于\（n \）- D物理空间的\（h \）-伪微分算子（特别是微分），其主符号确定可积的Hamilton系统，非平凡的次主符号破坏可积性。我们提出了一种实用的方法来计算这种算子的渐近特征值和特征函数（准模）。与Lazutkin的方法不同，在该方法中，对主要符号的校正已包含在哈密顿量中，我们处理可积的哈密顿量系统和非平凡的\（n \）-D运输方程式。我们提出了一种在单个丢番图圆环附近构造一系列准模式的算法。我们讨论了使用Wolfram Mathematica之类的软件在实践中使用此算法的可能性。例如，我们为Schrödinger算子显式构造拟模，描述了弱耦合二聚体，即在一维势场中的两个弱耦合粒子。