Periodic operators and related spectral distribution problems

周期算子和相关的谱分布问题

• 批准号: EP/D00022X/2
• 负责人: Alexander Sobolev
• 金额: --
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FD00022X%2F2
• 关键词: Periodic; operators; related; spectral; distribution

Spectra of periodic operators have a band-gap structure, that is, they consist of a collection of closed intervals possibly separated by gaps. There is a famous hypothesis, called the Bethe-Sommerfeld conjecture, which claims that the number of gaps is finite. It has been justified for the Schroedinger operator with an electric field. One aim of the present project is to prove the conjecture for the magnetic fields. This is a much more challenging problem, since the magnetic field induces a stronger perturbation than the electric one. The solution is expected to require the use of the pseudo-differential calculus and the geometry of lattices. The second objective is to investigate the distribution of the eigenvalues for some differential operators on manifolds of ``simple'' structure, for example, on the torus. Here there is a number of hypothesis and partial results describing subtle properties of the eigenvalue distribution function for the Laplace operator. The aim of the project is to find out how far these results can be extended to the case of the perturbed Laplace operator, for instance, to the Schroedinger operator with electric and magnetic fields.

周期算符的谱具有带隙结构，即它们由可能由间隙分隔的闭合区间的集合组成。有一个著名的假设，称为Bethe-Sommerfeld猜想，它声称空隙的数量是有限的。这对于有电场的薛定谔算符是合理的。本项目的一个目的是证明关于磁场的猜想。这是一个更具挑战性的问题，因为磁场引起的扰动比电场强得多。这一解决方案预计需要使用伪微积分和格子几何。第二个目标是研究一些微分算子在具有“简单”结构的流形上的特征值的分布，例如在环面上。这里有一些假设和部分结果描述了拉普拉斯算子的本征值分布函数的微妙性质。该项目的目的是找出这些结果可以在多大程度上扩展到微扰拉普拉斯算符的情况，例如，扩展到具有电场和磁场的薛定谔算符。

项目成果

DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

以丢番图点为中心的球中整数晶格点的分布

• DOI: 10.1112/s0025579309000527
• 发表时间: 2009
• 期刊: Mathematika
• 影响因子: 0.8
• 作者: Kang H

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

强扰动周期算子的 Bethe-Sommerfeld 猜想

• DOI: 10.48550/arxiv.0907.0887
• 发表时间: 2009
• 作者: Parnovski L