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Periodic Spectral Problems

周期性频谱问题

基本信息

批准号：
EP/F029721/1
负责人：
Leonid Parnovski
金额：
$ 54.22万
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：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF029721%2F1
关键词：
Periodic Spectral Problems

项目摘要

Periodic differential operators arise in many areas of physics and mathematics, and studying their spectral properties is very important. 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 in a much more general setting. The solution is expected to require the use of the pseudo-differential calculus, geometry of lattices and geometrical combinatorics. An important quantitative characteristic of differential operators acting on a non-compact manifold in the so-called integrated density of states. This function is a natural analogue of the spectral counting function. We plan to study the behaviour of this function for large values of energy and, in particular, to prove that the density of states has a complete asymptotic expansion in the (negative as well as positive) powers of energy.We also plan to study more basic properties of the nature of the spectrum, namely whether the spectrum is absolutely continuous. We plan to give establish a wide range of sufficient conditions which guarantee the absolute continuity of the spectrum. Finally, we plan to study limit-periodic problems. These problems are natural generalisation of the periodic ones. While the class of limit-periodic operators is not as wide as the class of quasi-periodic or almost-periodic operators, some of the methods of the periodic theory are applicable to the limit-periodic case. We intend to prove the Bethe-Sommerfeld conjecture in the limit-periodic setting.

周期微分算子出现在物理和数学的许多领域，研究其谱性质是非常重要的。周期性算子的谱具有带隙结构，也就是说，它们由可能被带隙隔开的闭合区间的集合组成。有一个著名的假设，称为贝特-索末菲猜想，它声称间隙的数量是有限的。这已经证明了薛定谔算子与电场。本项目的一个目的是在一个更一般的设置证明猜想。该解决方案预计需要使用伪微分学，几何格和几何组合学。微分算子作用在非紧流形上的积分态密度的一个重要的定量特征。该函数是光谱计数函数的自然模拟。我们计划研究该函数在大能量下的行为，特别是证明态密度在能量的（负的和正的）幂次方上有一个完全的渐近展开，我们还计划研究谱的性质的更多基本性质，即谱是否绝对连续。我们计划建立一个广泛的充分条件，保证绝对连续的频谱。最后，我们计划研究极限周期问题。这些问题是周期问题的自然推广。虽然极限周期算子的类不像拟周期算子或概周期算子的类那样宽，但周期理论的一些方法适用于极限周期的情况。我们打算在极限周期背景下证明Bethe-Sommerfeld猜想。