论文标题
用于扰动的谐波振荡器的准例浓度
Concentration of Quasimodes for Perturbed Harmonic Oscillators
论文作者
论文摘要
在这项工作中，研究了用于扰动的半经典谐波振荡器的准型浓度特性。这项研究的起点源于以下事实：在谐波振荡器的频率与通用界面扰动之间存在谐振的情况下，相当小的宽度的准膜的半经典测量值严格小于相应的宽度宽度的半经典测量，而不是针对未受扰动的系统的半经典测量值。在这项工作中，我们确切地考虑了频率矢量的毒液性特性，这些频率的diophantine特性，在存在扰动的存在中，特征值群体之间的分离以及经典汉密顿流动的动力学特性是由扰动的平均扰动符号所产生的。特别是，对于受扰动的二维周期性谐波振荡器，我们表征了针对该问题至关重要的宽度的半经典测量值。这些结果证明的两种主要成分具有独立感兴趣，是：（i）基于连贯状态传播的新型准植物通过几个通勤流的传播，以及（ii）和谐振荡器的一般量子Birkhoff正常形式。
In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the harmonic oscillator and for a generic bounded perturbation, the set of semiclassical measures for quasimodes of sufficiently small width is strictly smaller than the corresponding set of semiclassical measures for the unperturbed system. In this work we precise the description of this set taking into account the Diophantine properties of the vector of frequencies, the separation between clusters of eigenvalues in the spectrum produced by the presence of a perturbation, and the dynamical properties of the classical Hamiltonian flow generated by the average of the symbol of the perturbation by the harmonic oscillator flow. In particular, for the perturbed two-dimensional periodic harmonic oscillator, we characterize the set of semiclassical measures of quasimodes with a width that is critical for this problem. Two of the main ingredients in the proof of these results, which are of independent interest, are: (i) a novel construction of quasimodes based on propagation of coherent states by several commuting flows and (ii) a general quantum Birkhoff normal form for harmonic oscillators.
