Semiklassische Analysis mit Singularitäten

在这个项目中，我们将研究伪微分算子的半经典谱渐近的构造，利用相应的经典哈密顿量的奇异不变变分。对于一类广泛的伪微分算子，我们计划研究这些渐近的存在性，逼近的准确性，半经典渐近与临界流形拓扑分类的关系，以及各种相关问题。我们期望这些结果将适用于周期和规范周期问题的谱带的渐近描述，以及不可积哈密顿问题的谱渐近性的发现。

In this project we are going to study the construction of semiclassical spectral asymptotics for pseudodifferential operators using singular invariant varieties of the corresponding classical Hamiltonian. For a wide class of pseudodifferential operators, we plan to study the existence of such asymptotics, the accuracy of the approximation, the relationship between semiclassical asymptotics and the topological classification of critical manifolds, and various related questions. We expect that the results will be applicable to the asymptotic description of the spectral bands in periodic and gauge-periodic problems and to finding spectral asymptotics in problems with non-integrable Hamiltonians.