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Diophantine Tori and Pragmatic Calculation of Quasimodes for Operators with Integrable Principal Symbol
Russian Journal of Mathematical Physics ( IF 1.7 ) Pub Date : 2020-09-03 , DOI: 10.1134/s1061920820030024 A. Yu. Anikin , S. Yu. Dobrokhotov
中文翻译:
具有可积分主符号的算符的Diophantine花托和拟模的语用计算
更新日期:2020-09-03
Russian Journal of Mathematical Physics ( IF 1.7 ) Pub Date : 2020-09-03 , DOI: 10.1134/s1061920820030024 A. Yu. Anikin , S. Yu. Dobrokhotov
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:
具有可积分主符号的算符的Diophantine花托和拟模的语用计算